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201 Coherence in Three-Dimensional Category Theory

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Coherence in Three-DimensionalCategory Theory

NICK GURSKIUniversity of Sheffield

C A M B R I D G E U N I V E R S I T Y P R E S S

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Contents

Introduction page 11 Tricategories 12 Gray-monads 33 An outline 5

Acknowledgements 12

Part I Background 13

1 Bicategorical background 151.1 Bicategorical conventions 151.2 Mates in bicategories 17

2 Coherence for bicategories 212.1 The Yoneda embedding 212.2 Coherence for bicategories 222.3 Coherence for functors 29

3 Gray-categories 353.1 The Gray tensor product 363.2 Cubical functors 383.3 The monoidal category Gray 453.4 A factorization 48

Part II Tricategories 57

4 The algebraic definition of tricategory 594.1 Basic definition 594.2 Adjoint equivalences and tricategory axioms 654.3 Trihomomorphisms and other higher cells 664.4 Unpacked versions 78

v

vi Contents

4.5 Calculations in tricategories 834.6 Comparing definitions 85

5 Examples 865.1 Primary example: Bicat 865.2 Fundamental 3-groupoids 89

6 Free constructions 976.1 Graphs 976.2 The category of tricategories 996.3 Free Gray-categories 103

7 Basic structure 1067.1 Structure of functors 1077.2 Structure of transformations 1107.3 Pseudo-icons 1147.4 Change of structure 1237.5 Triequivalences 127

8 Gray-categories and tricategories 1298.1 Cubical tricategories 1298.2 Gray-categories 133

9 Coherence via Yoneda 1389.1 Local structure 1399.2 Global results 1419.3 The cubical Yoneda lemma 1449.4 Coherence for tricategories 154

10 Coherence via free constructions 15610.1 Coherence for tricategories 15710.2 Coherence and diagrams of constraints 16010.3 A non-commuting diagram 16110.4 Strictifying tricategories 16210.5 Coherence for functors 17110.6 Strictifying functors 177

Part III Gray-monads 181

11 Codescent in Gray-categories 18311.1 Lax codescent diagrams 18411.2 Codescent diagrams 18811.3 Codescent objects 190

Contents vii

12 Codescent as a weighted colimit 19612.1 Weighted colimits in Gray-categories 19712.2 Examples: coinserters and coequifiers 20012.3 Codescent 207

13 Gray-monads and their algebras 20913.1 Enriched monads and algebras 21013.2 Lax algebras and their higher cells 21313.3 Total structures 219

14 The reflection of lax algebras into strict algebras 22714.1 The canonical codescent diagram of a lax algebra 22814.2 The left adjoint, lax case 23014.3 The left adjoint, pseudo case 242

15 A general coherence result 24415.1 Weak codescent objects 24515.2 Coherence for pseudo-algebras 266

Bibliography 273Index 277

Introduction

In the study of higher categories, dimension three occupies an interestingposition on the landscape of higher dimensional category theory. From the per-spective of a “hands-on” approach to defining weak n-categories, tricategoriesrepresent the most complicated kind of higher category that the community atlarge seems comfortable working with. On the other hand, dimension three isthe lowest dimension in which strict n-categories are genuinely more restric-tive than fully weak ones, so tricategories should be a sort of jumping offpoint for understanding general higher dimensional phenomena. This workis intended to provide an accessible introduction to coherence problems inthree-dimensional category.

1 Tricategories

Tricategories were first studied by Gordon, Power, and Street in their 1995AMS Memoir. They were aware that strict 3-groupoids do not model homotopy3-types, and thus the aim of their work was to create an explicit defini-tion of a weak 3-category which would not be equivalent (in the appropriatethree-dimensional sense) to that of a strict 3-category. The main theorem ofGordon et al. (1995) is often stated: every tricategory is triequivalent to aGray-category. Triequivalence is a straightforward generalization of the usualnotion that two categories are equivalent when there is a functor between themwhich is essentially surjective, full, and faithful. The new and interesting fea-ture of this result is the appearance of Gray-categories. These are categorieswhich are enriched over the monoidal category Gray; this monoidal categoryhas the category of 2-categories and strict 2-functors as its underlying cat-egory, but its monoidal structure is not the Cartesian one. Gray-categoriescan thus be viewed as a maximally strict yet still completely general formof weak 3-category, and it is known, for instance, that Gray-groupoids modelall homotopy 3-types.

1

2 Introduction

My interest in tricategories began while carrying out joint work withEugenia Cheng on the Stabilization Hypothesis of Baez and Dolan. TheStabilization Hypothesis roughly states that k-degenerate weak (n + k)-categories correspond to what they called k-tuply monoidal n-categories. Here,k-degenerate means that the (n + k)-categories only have a single 0-cell, sin-gle 1-cell, and so on, up to having only a single (k − 1)-cell: thus the bottomk dimensions are degenerate. A k-tuply monoidal n-category is one which ismonoidal, and as k increases that monoidal structure becomes more and morecommutative until it stabilizes when k = n + 2. Some relevant examples tokeep in mind are

• the case k = 1, n = 0 gives 1-degenerate categories (categories with asingle object) on the one hand or 1-tuply monoidal 0-categories (sets withan associative and unital multiplication) on the other hand; and

• fixing n = 1 we get weak 2-categories with a single object, weak3-categories with a single object and single 1-cell, and weak 4-categorieswith a single object, 1-cell, and 2-cell on the one hand and monoidal cate-gories, braided monoidal categories, and symmetric monoidal categories onthe other hand.

The Stabilization Hypothesis is a guiding principle of higher category the-ory, yet we found that no systematic study of low dimensional cases had beencarried out.

As had already been discovered by Tom Leinster, k-degenerate (n + k)-categories and k-tuply monoidal n-categories were not precisely the samestructures, at least when using the explicit, algebraic notions of weakn-category. As an example, a bicategory with a single object and single 1-cellis not only a commutative monoid given by the set of 2-cells I ⇒ I undercomposition (where I is the single 1-cell), but is in fact a commutative monoidequipped with a distinguished invertible element. This element corresponds tothe left (or right, they are equal) unit isomorphism, and satisfies no axioms.So in fact it is the algebraic nature of the definition of bicategory that createsthis extra piece of data. To carry out the same analysis in dimension three, weneeded a fully algebraic definition of tricategory, and the definition of Gordon,Power, and Street was only partially algebraic.

The original definition was partially algebraic because it included data hav-ing certain properties but not the data necessary to check those properties. Inparticular, the associativity equivalence for 1-cell composition is a 2-cell

ah,g, f : (h ⊗ g)⊗ f ⇒ h ⊗ (g ⊗ f ),

2 Gray-monads 3

but the original definition did not include a 2-cell a�h,g, f nor invertible 3-cells

1 ∼= a�h,g, f ◦ ah,g, f , ah,g, f ◦ a�

h,g, f∼= 1 verifying that the 2-cell ah,g, f was

an equivalence. While this seems like a minor technical point, it does have animpact on how one goes about manipulating tricategories and the cells betweenthem. Making an algebraic definition was necessary for an examination of thestructures in the Stabilization Hypothesis, but one also requires a choice of thecells a�

h,g, f in order to define a composition law on transformations betweenfunctors of tricategories.

These concerns led me to consider a fully algebraic definition of tricate-gory in my 2006 University of Chicago Ph.D. thesis. While the changes tothe definition are minor, they do allow the definition of more constructionson tricategories such as functor tricategories and an explicit strictification. Themost important difference from the partially algebraic case is how coherence isapproached. While both proofs of coherence for tricategories involve embed-ding a tricategory in a Gray-category, the fully algebraic definition makesmore direct use of a Yoneda embedding, much like how coherence for bicat-egories is usually proved. Continuing to employ techniques similar to thoseused in the case of bicategories, it is also possible to use the fully algebraicdefinition to prove a coherence theorem for functors.

Tricategories have appeared in more applications recently, particularlyin topological applications. Carrasco, Cegarra, and Garzón (2011) study aGrothendieck construction for diagrams of bicategories (of which tricategoriesare an example) in order to understand the classifying spaces of braidedmonoidal categories. Lack (2011) has constructed a model category struc-ture on the category of Gray-categories and Gray-functors that restricts to amodel structure on Gray-groupoids. With these model structures in hand, Lackgoes on to prove that Gray-groupoids model homotopy 3-types. My paper(Gurski 2011) proves a coherence theorem for braided monoidal bicategoriesthat uses tricategorical techniques in a number of ways.

2 Gray-monads

The study of Gray-monads and their algebras has two distinct sides, reminis-cent of the study of 2-monads. First, Gray-monads are just monads enrichedover the monoidal category Gray, and thus carry with them the usual struc-ture associated to enriched monads. The category Gray of 2-categories and2-functors, but equipped with the Gray-tensor product, has many pleasantproperties so we can reproduce many of the usual constructions from monadtheory such as Eilenberg–Moore objects for a Gray-monad. The second halfof the story for Gray-monads is the three-dimensional picture, consisting of

4 Introduction

many different kinds of algebras and maps that all take advantage of the higherdimensional nature of a Gray-category. This side of the picture is much morecomplicated in terms of data and axioms, but the objects that arise from it aremuch more interesting from the perspective of applications in other parts ofhigher dimensional category theory. Comparing these two aspects of the theoryof Gray-monads is the study of a very general kind of coherence question.

This form of coherence goes back to the seminal work Two-dimensionalMonad Theory by Blackwell, Kelly, and Power (1989). That paper was con-cerned with 2-monads, and studied the two-dimensional aspects using the morewidely understood Cat-enriched theory for comparison. The basic situationwas as follows. Let A be a 2-category, and T a 2-monad (i.e., Cat-enrichedmonad) on it; a simple example to keep in mind is when A = Cat and T isthe 2-monad for strict monoidal categories. We can now form (at least) threedifferent 2-categories: the 2-category AT which is the Eilenberg–Moore objectin the enriched sense, the 2-category T -Alg of algebras with pseudo-algebramorphisms, and the 2-category T -Algl of algebras with lax algebra morphisms.Each of these 2-categories has the same objects, and there are inclusions

AT ↪→ T -Alg ↪→ T -Algl

which are locally full on 2-cells. The first main result of Blackwell et al. (1989)is that, under some conditions on A and T , the inclusions

AT ↪→ T -Alg, AT ↪→ T -Algl

each have a left 2-adjoint. The image of an object X under this left adjoint isoften denoted X ′, and the one-dimensional aspect of this 2-adjunction statesthat “weak” algebra maps (either pseudo-algebra morphisms or lax algebramorphisms, depending on the particular example) X → Y are in bijectionwith algebra morphisms X ′ → Y in the usual sense of monad theory.

What I have described so far is in fact the most basic situation, and wecan consider more complicated scenarios in which not only are the morphismsallowed to be weakened, but so is the notion of algebra as well. Once againthere will be an inclusion of AT into whatever 2-category of algebras wechoose to study, and it is possible to give conditions under which this inclusionhas a left 2-adjoint X → X ′. The unit of this adjunction will be a morphismX → X ′, and it is also possible to give conditions under which these com-ponents are equivalences. In other words, this very abstract form of coherencecan often be used to derive the usual kinds of coherence theorems such ascoherence for monoidal categories.

The conditions on the 2-category A and the 2-monad T to ensure that theseinclusions have a left adjoint, and then perhaps to show that the unit of the

3 An outline 5

adjunction has components which are equivalences, are conditions about theexistence of certain kinds of two-dimensional limits and colimits in A togetherwith the requirement that T preserve some of these. The most complete treat-ment of this perspective can be found in Codescent Objects and Coherence bySteve Lack (2002a). In this paper, Lack shows how the most important col-imit to consider is that of the codescent object which plays the role of a kindof two-dimensional coequalizer. Understanding codescent object turns out tobe essential in studying coherence through this kind of strategy (i.e., by con-structing a left adjoint to the inclusion of the “strict algebra case” into somelarger 2-category with weaker objects and/or morphisms), and leads to theo-rems about the existence of the left adjoint as well as showing the componentsof the unit are equivalences.

Far less has been studied in the three-dimensional world. The only workthusfar in this direction is a paper of John Power’s (2007) in which he beginsthe study of Gray-monads and their algebras. Here, the basic objects of studyare Gray-categories equipped with a Gray-monad; examples are much harderto come by, but one to keep in mind is that of 2-categories equipped with achoice of flexible limits or colimits. The work of Power should be seen asthe analogue of many parts of the original paper of Blackwell–Kelly–Power,and he proves many of the same basic theorems. He establishes the notions ofweak or lax algebra maps, together with the higher cells between them, andproves that these form a Gray-category containing the usual enriched cate-gory of algebras. Under some cocompleteness conditions, he proves that theinclusion of algebras with strict maps into algebras with weak maps has a leftadjoint, and using pseudo-limits of arrows he gives a sufficient condition forthe unit of this adjunction to have components which are internal biequiva-lences. He does not, however, pursue these using codescent techniques, butdoes remark that such a strategy might be useful for a complete understandingof coherence problems in dimension three.

3 An outline

This book is aimed at being a basic guide to coherence problems in three-dimensional category theory. From the above discussion, it should not besurprising that this is split into two parts. In the first part, we will discuss thecoherence theorem for tricategories and the related result for functors; muchof this material has been adapted from my 2006 Ph.D. thesis. The second partfocuses on the general coherence problem for algebras over a Gray-monadusing codescent methods. Just as Lack’s paper can be seen as a refinement ofthe basic results in Blackwell–Kelly–Power, the results in the second half of

6 Introduction

this work can be seen as a refinement of Power’s (2007) results. It is the inten-tion that this book can be read without any prior experience with tricategoriesor Gray-categories, and I have included background material in an attempt tokeep this book self-contained. The only exception is the inclusion of some cal-culational results that were proved by Gordon et al. (1995) and are of generaluse in the proofs leading up to the coherence theorem for tricategories. Most ofthese calculations are omitted because of the size of the diagrams involved soit might not be clear how these results might be used, but they are quite usefulfor performing many of these computations. Here is a detailed outline of whatis to come.

First, I will give some background information and establish notation. Sincetricategories and Gray-categories have three different composition operationson 3-cells, it is important to establish clear notation early on. With this in mind,I will use some slightly non-standard notation even at the level of bicategorieswhich can then easily be augmented when moving to the three-dimensionalworld later on. It is also important to keep in mind that at each dimensionthere are choices to be made about the canonical direction of the data presentin many different definitions. With this in mind, I will follow Gordon–Power–Street in using the oplax direction for transformations as the default notionalthough in practice this has little bearing since we will be more interested inthe pseudo-natural rather than the lax case. I will also recall the concept ofan icon, and remind the reader of the necessary calculational results from thetheory of mates that will be useful later.

The second piece of background material I will discuss is coherence forbicategories. I will present a number of formulations of this theorem, andwill follow the strategy used by Joyal and Street (1993) to prove these dif-ferent incarnations of coherence. Their approach provides a solid frameworkfor proving coherence for functors as well, and it is this feature in particularthat will be important later as the original work of Gordon–Power–Street didnot have a proof of coherence for functors between tricategories.

The final section of background will be a discussion of the Gray-tensorproduct and Gray-categories. I will present many different ways of think-ing about the Gray-tensor product, but will give very few proofs. My goalis less to give a fully rigorous account of this monoidal structure on the cat-egory of 2-categories and 2-functors and more to provide the reader with abasic understanding coupled with some intuition on how to manipulate thesestructures. Gray-categories will feature prominently in the rest of this work,and while the rules for working in a Gray-category are not much more com-plicated than those for working in a strict 2- or strict 3-category, there are

3 An outline 7

some important differences to keep in mind while doing calculations inside anarbitrary Gray-category.

With the background completed, we are ready to move on to discussing tri-categories and their coherence theory. I will begin with the relevant definitionsof tricategories and the higher cells between them. It is at this point that wediverge slightly from the treatment in Gordon–Power–Street, as the definitionI will give has a bit more structure than the one they work with. The specificdifference between the two definitions is that they require certain transforma-tions to be equivalences, while I specify an entire adjoint equivalence as partof the data. This difference does not change the definition in a conceptual way,but does make more techniques available.

Next I give some basic examples of tricategories and functors between them.The most important examples are Bicat and Gray, and they occupy the firstpart of this chapter. These examples will be used later in the proof of coherence,and so are worth constructing in detail. Then I give a topological examplewhich, to my knowledge, has not been explored in the literature thusfar: thefundamental 3-groupoid of a space. This is a straightforward construction, butimportant in studying the relationship between three-dimensional groupoidsand homotopy 3-types.

The next chapter is devoted to a discussion of the many different kinds offree objects that arise in this theory. There are at least four different types ofgraphs from which we can generate free tricategories or Gray-categories, andthis section is devoted to cataloguing all of the free constructions on thesedifferent types of underlying data. It is actually at this point that the changein the definition of tricategory makes its (technical) appearance, as it is simpleto freely generate an adjoint equivalence while it is not clear what it wouldmean to freely generate an equivalence. This chapter also begins the discussionof the category of tricategories and strict functors; this requires some care,as the composition law in this category does not give the same result as thecomposition of strict functors qua weak functors.

I will then discuss some of the basic constructions that would go into makinga weak four-dimensional category Tricat. In particular, I will give construc-tions of some composites of higher cells. Since this is largely a matter ofbookkeeping, I will only define the composites that we need later; thus thefirst obstruction to finishing the definition of Tricat is to define a few morekinds of composition. The second obstruction is providing all the rest of thedata, including things like associators and unit constraints for each of thedifferent levels of composition. This has to all be packaged to give a composi-tion functor between tricategories, with associativity and unit transformations,

8 Introduction

and so on, and each piece of data here has components which are themselvestransformations, etc., at which point it becomes clear that constructing Tricatby hand, without any tools, is a huge task that will, most likely, not producefruit in proportion to the work required (at least at this stage in the devel-opment of the theory). I would also like to point out that the changes in thedefinitions that I have made affect this section as well. Defining some ofthese composites actually requires using the pseudo-inverses of the data inthe Gordon–Power–Street definition of a tricategory, so Gordon et al. (1995)defined these composites only up to some ambiguity. This is the benefit ofmaking the definitions fully algebraic: whenever you want to define a new con-struction, every piece of data you might want is already on hand. The downside,of course, is that the things you are defining become much more complicated.In this case, though, the complications are all of a computational rather thanconceptual nature, and I believe that the drawback of having longer defini-tions is offset by being able to follow a more satisfying strategy for provingcoherence.

The next chapter details how Gray-categories can be seen as examples oftricategories. Here I will also explore the intermediate notion of a cubicaltricategory. This notion is important because it provides a stepping stone inthe proof of coherence. The simplest proof that every bicategory is biequiv-alent to a 2-category employs the Yoneda embedding together with the factthat every functor bicategory of the form [X, K ], where K is a 2-category, isitself a 2-category. Since the Yoneda embedding lands in a functor bicategory[Bop,Cat], and Cat is a 2-category, the strictification result follows, albeitwithout a particularly explicit construction of how to strictify a given bicate-gory. If we tried to follow the same idea for tricategories, we would land in afunctor tricategory [T op,Bicat], but since Bicat is not a Gray-category, thiswould not produce the desired result. Thus we seem to need a bit more struc-ture on the tricategory T to use Yoneda for the proof of coherence, and thisextra structure is that of a cubical tricategory.

With this Yoneda-style proof in mind, Chapter 9 begins with the construc-tion of functor tricategories when the target is a Gray-category. Since thefunctor tricategory inherits the compositional structure of the target, it alsobecomes a Gray-category. We will see this directly, although Power (2007)also notes that something very close to this structure can be constructed usingpseudo-algebras for a particular Gray-monad. I will also note that this cor-rects a mistake of Crans (1999). Finally, it is time to construct an appropriateYoneda embedding. Here we restrict ourselves to the case that the tricate-gory in question is cubical, as this will produce a Yoneda embedding of theform T ↪→ [T op,Gray], which by previous results is a Gray-category itself.

3 An outline 9

The full coherence result is then obtained by composing the Yoneda embed-ding with a canonical triequivalence S → st S from a generic tricategoryS to a cubical one. This triequivalence, and the cubical nature of st S, botharise directly from an explicit strictification that is a consequence of coher-ence for bicategories. In the search for a good notion of semi-strict 4-category,I would argue that understanding the properties of explicit strictifications oftricategories is the key to extending the strategy here up one dimension.

The next chapter analyzes the question of coherence from the perspective offree constructions. Using the Yoneda-embedding version of coherence, I provethat the free strict 3-category, free Gray-category, and free tricategory on a3-globular set are all triequivalent. From this, we obtain the corollary that anydiagram of coherence 3-cells in a tricategory which arises from a diagramof coherence 3-cells in a free tricategory must commute. As a non-example,I explain how the “categorified Eckmann–Hilton argument” produces a dia-gram which is not required to commute, and in general does not: any braided,but not symmetric, monoidal category gives an example of a tricategory inwhich it fails to commute (see Cheng and Gurski (2011) for a rigorous dis-cussion of the relationship between braided monoidal categories and doublydegenerate tricategories). I then prove a coherence theorem for functors, onceagain following the strategy of Joyal and Street. This is a result which was notpossible using the techniques in Gordon–Power–Street. As a consequence ofthis theorem, it is possible to unambiguously interpret the diagrams in Trim-ble’s webpage definition of a tetracategory, much as how the definition of atricategory requires the use of coherence for functors between bicategories tobe interpretted in a rigorous fashion. Finally, I consider the problem of findingan explicit strictification functor. This strictification is constructed directly, andI will show that it strictifies functors as well. It has a computadic flavor, but thatangle is not pursued any further here. It seems likely that a cleaner approachcould be made, but that it might require using an unbiased notion of tricategoryfrom the beginning.

Part III of this work concerns the general coherence problem for lax andpseudo-algebras over a Gray-monad. The first chapter in this section pro-vides the basic definitions of lax codescent diagrams, codescent diagrams, andthe (lax or pseudo) codescent objects associated to each. Codescent diagramsshould be considered higher dimensional versions of coequalizers, and everyalgebra over a monad is, in a canonical fashion, a coequalizer of free alge-bras. This fact is a generalization of the simple result that every group is aquotient of a free group, i.e., every group can be given a presentation, andis central in the study of monads and their algebras. The domains of thesecodescent diagrams are constructed explicitly, and in the lax case are given

10 Introduction

by the Gray-category associated to a Gray-computad (Batanin 1998b); thepseudo case can be expressed as the Gray-category associated to a Gray-computad modulo an equivalence relation on parallel 3-cells exhibiting thatcertain generators are inverse to each other.

The second chapter is a further exploration of the notion of codescent, thistime as a weighted colimit. I have included a short reminder on weighted col-imits in the context of Gray-categories, as well as some examples which helpbuild to the notion of a codescent object. Just as in the two-dimensional case,it is not very difficult to show that codescent objects can be constructed frommore basic weighted colimits. This chapter is largely of theoretical impor-tance, as much is already known about weighted colimits in the general caseof V-enriched categories. Thus being able to express codescent objects as thecolimits for a certain weight allows them to be studied using the general,and well-developed, machinery of enriched categories. In fact, that statementencapsulates the general philosophy of Part 3: we use enriched category theoryas much as possible in order to relate weak algebras for a monad to the strictones. This was the philosophy championed by Max Kelly, and it is applicableat the three-dimensional level in many of the same ways that it was applicableto the study of algebras over 2-monads.

The next chapter deals with constructing Gray-categories of algebras fora given Gray-monad. I first follow the standard theory of enriched monadsto construct the enriched Eilenberg–Moore object; this is the Gray-categorywhose objects are strict algebras, and whose higher cells are completely strictversions of functors, transformations, and modifications. I also give the def-initions of the lax and pseudo versions of these, and construct appropriateGray-categories of each. It should be noted that, in both the lax and pseudocases, the 2-cells are the pseudo-strength version of transformations in each.This requirement follows from using the pseudo-strength version of the Gray-tensor product rather than the lax version in Gray’s (1974) original work.I believe all of the definitions can be modified for the situation of the lax ten-sor, but at the moment I know of no applications that would benefit from thatalteration. Many of these definitions can be found, albeit with some slightlydifferent conventions on direction, in the paper of Power (2007).

We have now laid the foundations for work on the general coherence prob-lem, and Chapter 14 attacks this by using codescent objects to constructleft adjoints from the Gray-categories of lax and pseudo-T -algebras into theGray-category of strict algebras. This proceeds, much as in Lack’s work inthe two-dimensional case, by first constructing a canonical codescent diagramfrom an algebra, and then showing that its codescent object gives a left adjoint.I prove this style of theorem in the lax case, as the pseudo-strength version

3 An outline 11

is a simple extension. I also prove that this theorem has a partial converse,namely that if the inclusion does have a left adjoint, then the Gray-category ofstrict algebras is required to have certain codescent objects. This solves the firstpart of the general coherence problem, namely to construct a left adjoint of theinclusion of strict algebras into weak or lax algebras. Such a construction giveswhat is often called a pseudo-morphism classifier, as such a left adjoint givesfor every algebra A a strict algebra often denoted A′ in the literature followingBlackwell et al. (1989) such that weak morphisms A → B are in bijection withstrict morphisms A′ → B. The algebra A can be lax, pseudo, or strict, and thenotion of weak morphism can be either lax or pseudo, although changing thesehypotheses does change the construction of A′.

The final chapter of this work is concerned with determining when the unitof the adjunction between pseudo and strict T -algebras has components whichare biequivalences. This chapter is much more involved than the correspondingsection of Lack’s (2002a) paper since less of the background work on tricat-egorical colimits has been studied. The main new definition in this chapteris that of a weak codescent object, which is a tricategorical, conical style ofcolimit. The first main theorem, then, is that if a codescent diagram has acodescent object, then that object is also a weak codescent object. The secondmain theorem is that for a pseudo-algebra X , the underlying object is a weakcodescent object of the canonical codescent diagram (in the underlying Gray-category). Finally, the main theorem solves the general coherence problem byshowing that the components of the unit of the adjunction are biequivalencesif and only if the forgetful functor preserves certain codescent objects as weakcodescent objects. This is a generalization of Lack’s result, whose formulationwas pointed out to me by John Bourke, as the original hypothesis that T pre-serve certain weighted colimits is stronger than the hypothesis given here. Thuswe have completed the general coherence program for algebras over Gray-monads by characterizing precisely those pseudo-algebras which are able tobe strictified up to biequivalence.

I should point out that Part III does not contain any examples. In the realmof two-dimensional algebra, nearly every algebraic-looking structure on cat-egories, say, can be expressed using a 2-monad. The same is not true in theGray-enriched context, as being an enriched functor is a much more strin-gent requirement. For example, the canonical strictification functor st can beextended to a functor st : Gray → Gray, but it is not Gray-enriched becauseof issues with composition of transformations. Similarly, the free monoid func-tor Gray → Gray is also not Gray-enriched, once again because of theinteraction with transformations. One can see from these examples that thereare still many interesting problems open in the study of three-dimensional

12 Introduction

coherence, as applying the general theory developed here to specific examplesis much less straightforward than in the two-dimensional case.

Acknowledgements

I have had many wonderful conversations with a variety of delightful mathe-maticians that have helped shape this work. For those conversations and manyother forms of support, I would like to thank Peter May, Ross Street, EugeniaCheng, John Power, Martin Hyland, Steve Lack, John Baez, André Joyal, TomLeinster, John Bourke, Mike Shulman, Richard Garner, Alex Hoffnung, DanielShäppi, and Ignacio Lopez Franco; I would also like to thank everyone in theSheffield Category Theory group.

Part I

Background

1

Bicategorical background

This short chapter establishes some bicategorical conventions and notations.We will also quickly review mates with a focus on mates involving functorsand transformations.

1.1 Bicategorical conventions

In any bicategory B, we shall use the letters a, l, and r to denote the associativ-ity, left unit, and right unit isomorphisms, respectively. Vertical composition of2-cells will be written as concatenation, and the symbol * will be used to denotehorizontal composition of either 1- or 2-cells. The terms pseudo-functor, weakfunctor, and homomorphism of bicategories are all used throughout the lit-erature to refer to the same concept. We will always write functor for thisnotion; any strict or lax functor will be labeled as such. Given a functor F , wewill generically denote its constraints by ϕ since the source and target of thisconstraint make it clear what kind of constraint cell it is.

We follow the convention of Gordon et al. (1995) and not of the other ref-erences (Bénabou (1967) and Street (1972) for instance) in what is meant by alax transformation. For our purposes, a lax transformation α : F ⇒ G consistsof 1-cells αa : Fa → Ga and 2-cells

Fa FbF f �� Fb

Gb

αb

��

Fa

Ga

αa

��Ga Gb

G f��

α f��

��

subject to two axioms. A transformation is a lax transformation such thatthe cells α f are invertible for every f : a → b. A transformation between

15

16 Bicategorical background

strict 2-functors is a 2-natural transformation if the cells α f are identitiesfor all f .

Since we have changed the orientation of the naturality isomorphism in thedefinition of transformation, it is necessary to alter the definition of modifi-cation by changing its axiom. These changes are not substantive, they merelyavoid excessive use of the prefix op-.

A numbered prefix, such as in 2-category or 2-functor, will always refer tothe strict notion.

Our naming conventions for the corresponding concepts for tricategorieswill be the same, as we reserve the terms functor, transformation, etc., to meanthe weak version. Any strict or lax version of these concepts will always becalled such.

Finally, we will use bold letters such as f to represent adjoint equivalencesin a bicategory. We will generally follow the notation of Gurski (2012), so thatf consists of a left adjoint f , a right adjoint f �, invertible unit η, and invertiblecounit ε. When η, ε are reserved for other purposes, such as when discussingGray-monads or codescent objects, we will sometimes use u, c for the unitand counit, respectively; it should be clear from context if this is the case.

Finally, we remind the reader of the definition of an icon (seeLack and Paoli (2008) and Lack (2010a, b)) between lax functors. Since iconsarise when studying lax, weak, and strict functors between bicategories, wegive this most general form of the definition and leave it to the reader to inter-pret the constraints in the following diagrams as being invertible or identitieswhen appropriate.

Definition 1.1 Let F,G : B → C be lax functors between bicategories withconstraints ϕ0, ϕ2 for F , and ψ0, ψ2 for G. Assume that F and G agree onobjects. An icon α : F ⇒ G consists of natural transformations

αab : Fab ⇒ Gab : B(a, b) → C(Fa, Fb)

(note here that we require Fa = Ga, Fb = Gb so that the functors Fab,Gab

have a common target) such that the following diagrams commute. (Note thatwe suppress the 0-cell source and target subscripts for the transformations αab

and instead only list the 1-cell for which a given 2-cell is the component.)

IFa F Iaϕ0 �� F Ia

G Ia

αI

��

IFa

G Ia

ψ0 ��

�� F f ∗ Fg F( f ∗ g)ϕ2 �� F( f ∗ g)

G( f ∗ g)

α f ∗g

��

F f ∗ Fg

G f ∗ Gg

α f ∗αg

��G f ∗ Gg G( f ∗ g)

ψ2

��

1.2 Mates in bicategories 17

Definition 1.2 The 2-category Icon is defined to have objects bicategories,1-cells functors, and 2-cells icons between them.

Remark 1.3 Note that this 2-category was called Hom in Lack and Paoli (2008).To actually prove these cells, with their obvious composition laws, give a2-category is a relatively basic calculation that we leave to the reader.

Definition 1.4 The 2-category Grayicon is defined to be the locally full sub-2-category of Icon consisting of 2-categories, 2-functors, and icons.

The following result is easy to prove, but will be quite useful.

Proposition 1.5 Let F,G : B → C be functors between bicategories. Thenevery invertible icon α : F ⇒ G gives rise to an adjoint equivalence betweenF and G in the functor bicategory Bicat(B,C). Conversely, given such anadjoint equivalence such that the transformations α : F ⇒ G, β : G ⇒ Fhave all their components at objects an identity 1-cell, there is an invertibleicon α : F ⇒ G whose component at a 1-cell f is the composite

F fl−1=⇒ IFb ∗ F f

α f=⇒ G f ∗ IFar=⇒ G f.

1.2 Mates in bicategories

In this section, we will quickly review the necessary results from the theoryof mates in a bicategory that are used in our definitions. The main refer-ence in the case that the bicategory involved is actually a strict 2-category isKelly and Street (1974).

Lemma 1.6 Let B be a bicategory, and let ( f, f �, ε f , η f ) and (g, g�, εg, ηg)

be a pair of adjunctions in B. Then there is a bijection between 2-cells α :t f � ⇒ g�s and 2-cells β : gt ⇒ s f .

Proof Define the isomorphism by sending the 2-cell α to the 2-cell α+ givenby the following pasting diagram.

a

b

f

��a a

1��

b

a

f �

��

η f

b cs �� c c1 ��b c

s

��l

a dt��a d

t

��r−1

α

��

��

d

c

g

��

c

d

g��

��εg


The inverse function β → β+ should be obvious, and this is an isomorphismby the triangle identities and coherence for bicategories.

We call α+ the mate of α under the pair of adjunctions f, g. It should benoted that the mate of an invertible 2-cell is invertible. The rest of this sec-tion will be devoted to stating a variety of propositions that will be needed indealing with tricategories; no proofs will be provided, as they generally followfrom large diagram chases involving only the triangle identities, coherence forbicategories, and the axioms for functors and transformations.

If f = ( f, f �, ε, η) is an adjoint equivalence in B, and (F, ϕ) : B → C is afunctor, then

(F f, F f �, ϕ−10 · Fε · ϕ2, ϕ

−12 · Fη · ϕ0)

is an adjoint equivalence in C .

Proposition 1.7 Assume that F,G : B → C are weak functors, and thatα : F ⇒ G is a transformation between them. If f is an adjoint equivalence inB, then

α+f = (α f � )−1.

It should be noted that here we are using the opposite adjoint equivalence ofthe one stated above.

Proposition 1.8 Assume that F,G : B → C are functors, and that(α, α�, ε, η) is an adjoint equivalence in Bicat(B,C) with α : F ⇒ G andα� : G ⇒ F. Then

α�f = (α−1

f )+.

There is a special case that will be important to us, and that is when thereare no additional 1-cells s, t . In that case, we obtain an isomorphism between2-cells α : f � ⇒ g� and 2-cells α : g ⇒ f . Here we define the mate α† byfirst defining α = r−1αl, and then

α† = lα+r−1.

Proposition 1.9 Let f, g,h be three adjunctions in a bicategory B. If α :f � ⇒ g� and β : g� ⇒ h� are composable 2-cells, then

(βα)† = (α)†(β)†,

where the mate of βα is taken via the pair of composite adjunctions.

1.2 Mates in bicategories 19

An important case is the following. Let B be a bicategory, and letf1, f2, g1, g2 be four adjoint equivalences such that the left adjoints form asquare as below.

f1��

g2��

g1 ��

f2��

If α : g2 f1 ⇒ f2g1 is a 2-cell, then we denote by α+ the mate of α withrespect to the opposite of the adjunctions f1, f2. Similarly, we denote by β−the mate of β : f2g1 ⇒ g2 f1 under the adjunctions g1, g2. Note the differentdirections of the 2-cells, and the necessary choices of which adjunction to use;thus α+− makes sense, while α−+ does not.

Proposition 1.10 Given the situation above, let α : g2 f1 ⇒ f2g1 be aninvertible 2-cell. Then1. (α+−)−1 = (α−1)+−,2. (α−+)−1 = (α−1)−+, and3. α+− = α†.

Corollary 1.11 Assume that F,G : B → C are weak functors, and that(α, α�, ε, η) is an adjoint equivalence in Bicat(B,C) with α : F ⇒ G andα� : G ⇒ F. If f is an adjoint equivalence in B, then

α�f � = α+−

f = α†f .

Proof Combining Proposition 1.7 and Proposition 1.8 gives the first equality,and the second is the third part of Proposition 1.10.

We next turn to the relationship between mates and the constraint 2-cellsϕ : f g : F f Fg ⇒ F( f g) of a weak functor (F, ϕ) : B → C .

Proposition 1.12 Let (F, ϕ) : B → C be a weak functor between bicate-gories. Then the following equation holds for any appropriate pair of adjointequivalence f, g in B.

(ϕ f g)† = ϕ−1

g� f �

Finally, we end this section with a discussion of the relationship between thebicategory constraint cells a f gh, l f , r f and their mates.


Proposition 1.13 Let B be a bicategory with constraints given above. Thenthe following equations hold for any appropriate triple of adjoint equivalencesf, g,h.

(r f )† = l−1

f �

(l f )† = r−1

f �

(a f gh)† = a−1

h�g� f �

Using these results, we can now take mates of diagrams of 2-cells inside abicategory B.

2

Coherence for bicategories

In this chapter, we will give a rapid treatment of the coherence theory forbicategories, including a full proof of the coherence theorem for functors. Thegoal of this chapter is to prepare the reader for the path we will take throughthe coherence theory for tricategories, as well as to recall some crucial factsthat will be used throughout. The overall strategy here is adapted from the oneused by Joyal and Street (1993) for monoidal categories.

We will give two proofs that every bicategory is biequivalent to a strict2-category, each having its own flavor. The first proof can be dispensed withquickly. The second proof requires some of the tools developed for the first,but also allows us to prove the coherence theorem for functors.

2.1 The Yoneda embedding

This section is devoted to proving a coherence theorem by first developing anappropriate Yoneda lemma for bicategories. We will not provide any proofs inthis section, we instead refer the reader to Street (1974, 1996).

Proposition 2.1 Let B,C be bicategories. There is a bicategory Bicat(B,C)whose 0-cells are the functors F : B → C, whose 1-cells are the transforma-tions α : F ⇒ G, and whose 2-cells are the modifications : α � β.

The proof of this proposition requires identifying the constraint cells andthen checking the bicategory axioms. These constraint cells are obtained fromthe constraint cells in the target, giving the following corollary.

Corollary 2.2 If C is a strict 2-category and B is any bicategory, then thefunctor bicategory Bicat(B,C) is a strict 2-category.

21

22 Coherence for bicategories

Definition 2.3 Let B be a bicategory. Then the bicategory Bop has the samecells as B, the 1-cell source and target maps are switched, rop = l, lop = r ,and aop

f gh = a−1hg f .

Now we are in a position to define the Yoneda map y : B → Bicat(Bop,Cat)and state the Yoneda lemma for bicategories.

Definition 2.4 Let B be a bicategory. Then the Yoneda map

y : B → Bicat(Bop,Cat)

is defined on the underlying 2-globular set as follows. The functor y acts bysending an object a to the functor B(−, a). The functor y acts on the 1-cellf : a → a′ by sending it to the transformation f ∗ − : B(−, a) ⇒ B(−, a′).The functor y acts on 2-cells by sending α : f ⇒ f ′ to the modification withcomponent α ∗ 1g . We leave it to the reader to construct the rest of the data,and verify the axioms.

Definition 2.5 Let P be a property of functors between categories. A func-tor F : B → C between bicategories is locally P if each functor Fab hasproperty P .

Theorem 2.6 (Bicategorical Yoneda lemma) The Yoneda functor y : B →Bicat(Bop,Cat) is locally an equivalence.

Corollary 2.7 Every bicategory is biequivalent to a strict 2-category.

Proof Let I be the sub-2-category of Bicat(Bop,Cat) consisting of those0-cells which are in the image of y; it is immediate that this is a 2-category.Then y : B → I is locally an equivalence by Theorem 2.6, and it is surjectiveon objects by definition.

2.2 Coherence for bicategories

This section is devoted to proving a coherence theorem of the form “everyfree bicategory is biequivalent to a strict free 2-category via a strict functor.”Using this, we obtain biequivalences e : stB → B, f : B → stB for everybicategory B, where stB is an explicitly constructed strict 2-category. Othernotions of coherence are mentioned.

The approach here closely follows that of Joyal and Street (1993). The maintechnical difference is that we work with bicategories instead of monoidal cat-egories. We can view monoidal categories as single-object bicategories, so thenew feature to keep track of is any data related to the objects of the bicategory.

2.2 Coherence for bicategories 23

In particular, it is more convenient to work with icons as our 2-cells instead oftransformations. In fact, the proof given here actually says that every bicate-gory is equivalent to a strict 2-category in the 2-category Icon; this theorem isnot new, see Lack and Paoli (2008) for a full treatment of this perspective oncoherence.

2.2.1 Graphs and free constructions

Definition 2.8 The category Gr(Cat) of category-enriched graphs (whichwe will also call Cat-graphs) has objects G consisting of a set G0 of objectsand for every pair of objects a, b, a category G(a, b). A map f : G → G ′ ofCat-graphs consists of functions f0 : G0 → G ′

0 and functors fab : G(a, b) →G ′( f0a, f0b).

Remark 2.9 Since Cat is a 2-category, we can give Gr(Cat) the structureof a 2-category as well. A 2-cell α : f ⇒ g in Gr(Cat) only exists whenf0 = g0, and then it consists of natural transformations αab : fab ⇒ gab foreach pair of objects in the source category-enriched graph. This data obviouslyunderlies that of an icon, and it is this 2-category Gr(Cat), together with the2-monad T for bicategories, that gives rise to icons as the algebra 2-cells for T .

The free bicategory on a Cat-graph G, denoted FG, has the followingunderlying 2-globular set. The set of 0-cells of FG is G0. The set of 1-cells is inductively defined to include new 1-cells Ia for each a ∈ G0, 1-cellsf : a → b for each object f ∈ G(a, b), and 1-cells f ◦g if f, g are both 1-cellsof FG. The source and target functions are defined in the obvious fashion.

The set of 2-cells of FG is defined in three steps. The first is to define a basic2-cell. These are built inductively from the arrows in all of the G(a, b) and newisomorphism 2-cells a f gh, l f , r f by binary horizontal composition. Secondly,we form composable strings of these basic 2-cells. Finally, we quotient out bythe equivalence relation generated by naturality of the 2-cells a f gh, l f , r f , themiddle-four interchange law, the rule that the composition α ◦ β in FG agreeswith that of G if α, β are arrows in some G(a, b), and the two bicategoryaxioms. Note that there is an obvious inclusion i : G → FG of category-enriched graphs.

Proposition 2.10 1. The data above satisfy the necessary axioms to consti-tute a bicategory.2. Let B be a bicategory. Then given a map f : G → B of category-enrichedgraphs, there is a unique strict functor of bicategories f : FG → B such thatf i = f in Gr(Cat).


Proof The first statement is obvious by the definition. The second statementfollows by defining f using induction and strictness.

Now we define the free 2-category on a Cat-graph G, denoted Fs G. The setof 0-cells is the set G0. The set of 1-cells is the set of composable strings oflength ≥ 0, where the unique string of length zero will be the identity 1-cell.The set of 2-cells from one string fn ∗ fn−1 ∗ · · · ∗ f1 to another gm ∗ · · · ∗ g1

is empty if n = m, and otherwise consists of the strings αn ∗ αn−1 ∗ · · · ∗ α1

where αi : fi → gi in some G(a, b).Composition of 1-cells is by concatenation, and composition of 2-cells is

given by

(αn ∗ · · · ∗ α1) ◦ (βn ∗ · · · ∗ β1) = (αnβn) ∗ · · · ∗ (α1β1).

It is a simple matter to verify the following proposition, where here j denotesthe inclusion of G into Fs G.

Proposition 2.11 1. The data above satisfy the necessary axioms to consti-tute a 2-category.2. Let X be a 2-category. Then given a map f : G → X of category-enrichedgraphs, there is a unique 2-functor f : Fs G → X such that f j = f inGr(Cat).

Remark 2.12 Both of the previous two propositions can be strengthened byshowing that the free construction is the object-part of a left 2-adjoint to theforgetful functor from an appropriate 2-category to Gr(Cat) viewed as a 2-category.

Thus the statement of the coherence theorem for bicategories becomes thefollowing.

Theorem 2.13 (Coherence for bicategories) The functor : FG → Fs Ginduced by j : G → Fs G is a strict biequivalence.

2.2.2 Proof of the coherence theorem

Definition 2.14 Let G,G ′ be category-enriched graphs, and let S, T :G → G ′ be maps between them. The category-enriched graph Eq(S, T ) isdefined to have objects those a ∈ G0 such that S0a = T0a. The categoryEq(S, T )(a, b) has objects pairs (h, α) where h : a → b in G and α : Sh →T h is an isomorphism in G ′(S0a, S0b). The morphisms β : (h, α) → (h′, α′)are those β : h → h′ in G such that

α′ ◦ S(β) = T (β) ◦ α.


Note that there is a map π : Eq(S, T ) → G defined by

π(a)= aπ(h, α)= hπ(β)=β.

Lemma 2.15 Let B,C be bicategories, and F,G : B → C be functorsbetween them. Then Eq(F,G) supports a bicategory structure such that πcan be extended to a strict functor Eq(F,G) → B. Furthermore, there isan invertible icon

σ : Fπ ⇒ Gπ.

Proof For the first claim, we must define composition, identity 1-cells, con-straint 2-cells, and check the bicategory axioms. To fix notation, the constraintcells for F will be ϕ f g and ϕ0, while those for G will be ψ f g and ψ0.Composition of 1-cells is then defined by the formula

(g, β) ∗ ( f, α) = (g ∗ f, ψ f g ◦ (β ∗ α) ◦ ϕ−1f g ).

The identity 1-cell for the object a is (ida, ψ0 ◦ ϕ−10 ). It is simple to check

that the associativity and unit constraints from B are 2-cells in Eq(F,G) withthe appropriate sources and targets; from this the bicategory axioms followimmediately.

It is trivial to check that π can be extended to a strict functor.Finally, we define the icon σ : Fπ ⇒ Gπ . The component at ( f, α) is just

α; this is a natural transformation by the definition of morphisms in Eq(F,G),and the icon axioms follow easily.

Remark 2.16 In the language of two-dimensional limits, the category-enriched graph Eq(S, T ) is the iso-inserter of S and T . Since Icon is the2-category of algebras, pseudo-algebra maps, and algebra 2-cells for the freebicategory 2-monad T (see Lack and Paoli 2008), it follows that Icon has allPIE-limits, of which iso-inserters are one variety; it also follows that the for-getful functor Icon → Gr(Cat) preserves these, so the content of the proof ofthe previous lemma is really just an identification of the bicategory structureon the iso-inserter.

Proposition 2.17 Let F : FX → B be a functor from a free bicategory intoany bicategory. Then there is a strict functor G : FX → B and an invertibleicon α : F ⇒ G.

Proof Since FX is free, there is a unique strict functor G : FX → B suchthat Fi = Gi as maps X → B. We also have a map ι : X → Eq(F,G) which


is the identity on objects, sends f to ( f, idF f ), and sends β to β. Note thatπι = i and the icon σ ∗ 1ι is the identity by construction. This produces, bythe universal property of FX , a unique strict functor ι : FX → Eq(F,G) suchthat ιi = ι. This gives the equality πιi = i , and since πι is strict, it must be theidentity functor on FX . Then the icon σ ∗ 1ι has source Fπι = F and targetGπι = G by construction.

It should be noted that we have used that functors of bicategories composein a strictly associative and unital fashion in this proof; this will not be the casein the tricategorical version of this lemma, introducing some extra steps in thatproof.

Let f : X → B be a map of category-enriched graphs into a bicategory B.Then we can extend f to a map of category-enriched graphs f : Fs X → Bwhich is defined as follows. The object function f0 agrees with f0. The identity1-cell on a gets mapped to the identity 1-cell on f0a, and f (h) = f (h) whereh : a → b is an object of X (a, b). If hn ∗ · · · ∗ h1 : a → b in Fs X , then

f (hn ∗ · · · ∗ h1) = (· · · ( f hn ∗ f hn−1) ∗ f hn−2) ∗ · · · ∗ f h2) ∗ f h1.

Similarly, f (αn ∗ · · · ∗ α1) is the 2-cell

(· · · ( f αn ∗ f αn−1) ∗ · · · ∗ f α2) ∗ f α1.

Lemma 2.18 Let G be a category-enriched graph, and let F : FG → X bea strict functor into a 2-category X. Then there exists a unique strict functorFs : Fs G → B such that F = Fs.

Proof This is an immediate consequence of the universal properties of F, Fs ,and the fact that i = j .

Lemma 2.19 Let F,G : B → C be functors between bicategories, and letα : F ⇒ G be an invertible icon between them. Then F is locally faithful(locally full) if and only if G is locally faithful (locally full).

Proof We need only show that F locally faithful implies G locally faithfulby symmetry. Using the naturality of the 2-cells α f , we get

Gα = α f ′ Fαα−1f ,

where α : f ⇒ f ′. Thus G is locally faithful since the the composite on theright is a locally faithful function of α. The same proof shows local fullness.

Proof of 2.13 It is clear that is surjective on objects, so we need onlyshow that it is locally an equivalence of categories. We have the map


i : Fs G → FG, and it is simple to check that the composite map of category-enriched graphs

Fs Gi−→ FG

−→ Fs G

is the identity, so is locally essentially surjective.Now the functor i is also locally essentially surjective. A 1-cell in FG from

a to b is a composite of the generating 1-cells in G together with new, formalidentity 1-cells. Each such 1-cell f is isomorphic, using unit isomorphisms,to one with no identity cells, denoted f ′. It is then clear that f ′ is isomor-phic to a 1-cell in the image of i using associativity isomorphisms. Restrictingattention to a single hom-category, we then have that (i)ab is full whileiab is essentially surjective; from this it follows that ab is also full, so islocally full.

To show that is locally faithful, first note that there is a locally faithfulfunctor T : FG → X into a strict 2-category X by the Yoneda lemma. There isa strict functor S : FG → X and an invertible icon α : S ⇒ T by Proposition2.17. By the universal property of the map , there is a unique strict functorR : Fs G → B such that R = S. Now S is locally faithful since T is, hence must be locally faithful as well.

2.2.3 Using coherence: strictification

Let B be a bicategory. We use the coherence theorem to construct a strictifica-tion stB of B, along with a biequivalence e : stB → B.

The 2-category stB will have the same objects as B. A 1-cell from a to b willbe a string of composable 1-cells of B, where there is a unique empty stringwhich will be the identity 1-cell. Before defining 2-cells, we define e on 0- and1-cells. On 0-cells, e is the identity. On 1-cells, we define

e( fn ∗ fn−1 ∗ · · · ∗ f1) = (· · · ( fn ∗ fn−1) ∗ fn−2) · · · ∗ f2) ∗ f1;for the empty string ∅ : a → a, we set e(∅) = Ia . The set of 2-cells betweenthe strings fn fn−1 · · · f1 and gm gm−1 · · · g1 is defined to be the set of 2-cellsbetween e( fn ∗ fn−1 ∗ · · · ∗ f1) and e(gm ∗ gm−1 ∗ · · · ∗ g1) in B. It is nowobvious how e acts on 2-cells.

The 2-category structure of stB is defined as follows. Composition of 1-cellsis given by concatenation of strings, with the empty string as the identity. Itis immediate that this is strictly associative and unital. Vertical compositionof 2-cells is as in B, and this is strictly associative and unital since verticalcomposition of 2-cells in a bicategory is always strict in this way.


Let A be the sub-category-enriched graph of B with all the same objects butwith A(a, b) the discrete category with obA(a, b) = obB(a, b). By coherence,the strict functor : FA → Fs A is a biequivalence, and it is easy to see that the2-category Fs A is locally discrete. Thus, in FA, the set of 2-cells between anytwo 1-cells is either empty or a singleton, depending on whether these 1-cellsare mapped to the same 1-cell by . (Note that this is one way to prove the “alldiagrams of constraint cells commute” form of coherence for bicategories.) Inparticular, we have a unique coherence isomorphism

e( fn ∗ · · · ∗ f1) ∗ e(gm ∗ · · · ∗ g1) ∼= e( fn ∗ · · · ∗ f1 ∗ gm ∗ · · · ∗ g1).

Thus we can now define the horizontal composition α ∗ β in stB as thecomposite

e( fn · · · f1gm · · · g1) ∼= e( fn · · · f1)e(gm · · · g1)α∗β−→ e( f ′

n · · · f ′1)e(g

′m · · · g′

1)∼= e( f ′n · · · f ′

1g′m · · · g′

1)

in B, where the unlabeled isomorphisms are induced by the strict mapFA → B. The uniqueness of these isomorphisms ensures that this definitionsatisfies the middle-four interchange laws as well as being strictly associativeand unital.

By definition, e is functorial on vertical composition of 2-cells. The con-straint cells for e are induced by the strict map FA → B in a similar fashionas above. The uniqueness of these cells immediately forces the functor axiomsto hold. Finally, it is trivial to see that e is a biequivalence as it is surjective onobjects, locally surjective on 1-cells, and a 2-local isomorphism on 2-cells bydefinition. Thus we have completed the task of producing, for each bicategoryB, a strict 2-category stB and a biequivalence e : stB → B.

It will be useful later to note that there exists a biequivalence f : B → stBdefined as follows. The map f is the identity on objects, includes each 1-cellas the string of length 1, and then is the identity on 2-cells as well. This isfunctorial on 2-cells, and we can take both constraint cells to be representedby identity 2-cells in B (although they are not identities in stB). The func-tor axioms are then easy to check. The only thing to check to show that fis a biequivalence is that it is locally essentially surjective, but this is easyas every 1-cell fn · · · f1 is clearly isomorphic to a 1-cell of length 1, namelye( fn · · · f1); the empty string is isomorphic to the identity map viewed as a1-cell of stB, so f is locally essentially surjective. It should be noted thate f = 1B , and f e is biequivalent to 1stB in Bicat(stB, stB) by a transforma-tion whose components on objects can all be taken to be identities and whose

2.3 Coherence for functors 29

components on 1-cells all come from coherence; this in fact shows that f e isisomorphic to 1stB in Icon, proving that every bicategory is equivalent to a2-category in Icon. This result was first noted by Lack and Paoli (2008).

2.3 Coherence for functors

In this section, we prove a coherence result for functors of bicategories. Thistheorem is analogous to Theorem 2.13 in that it states that “free functors arebiequivalent to free strict functors.” The statement is slightly more delicate asthe universal property of a free functor involves squares, but it produces similarresults to those in Section 2.2.3. As in the previous section, the treatment heremirrors that of Joyal and Street (1993).

2.3.1 Free functors

Let ϕ : G → G ′ be a map in Gr(Cat). Our goal is to produce the free functorgenerated by ϕ; the source of this functor will be the free bicategory generatedby G, but the target is a more complicated object. The idea is that the targetwill be the free bicategory generated by G ′ and new 2-cells that will play therole of constraint cells.

We define the bicategory F(G ′, ϕ) as follows. The 0-cells of F(G ′, ϕ) arethe same as the objects of G ′. The 1-cells are generated (using binary compos-ites) by new 1-cells Ia : a → a, the 1-cells of G ′, and new 1-cells ϕ(r) forevery 1-cell r in FG. These are subject to the requirement that ϕ(r) = s inF(G ′, ϕ) if r is an object G(a, b) and ϕ(r) = s in G ′, and we extend this overcomposition.

The 2-cells are defined in a sequence of steps analogous to how we definedthe 2-cells of FG. The first step is to form basic 2-cells from the 2-cells ofG ′, 2-cells ϕ(α) with α a 2-cell of FG (subject to the same kind of condi-tion that we imposed on the 1-cells ϕ(r)), and isomorphism constraint cellsa f gh, l f , r f , ϕa, ϕ f g by binary horizontal composition. Then we form stringsof vertically composable basic cells, and finally we quotient out by the equiv-alence relation formed by the necessary naturality conditions along with theaxioms for a bicategory and those required of the 2-cells ϕa, ϕ f g to force ϕ toextend to a weak functor FG → F(G ′, ϕ). The universal property of F(G ′, ϕ)is expressed by the following proposition.

Proposition 2.20 Let ϕ : G → G ′ be a map of category-enriched graphs.Then there is a commutative square


G G ′ϕ ��G

FG

i��

G ′

F(G ′, ϕ)k��

FG F(G ′, ϕ)ϕ

��

in Gr(Cat) such that for all commutative squares

G G ′ϕ ��G

X

R��

X YF

��

G ′

Y

S��

in Gr(Cat) with F : X → Y a functor between bicategories, there exists aunique commutative square of functors

FG F(G ′, ϕ)ϕ ��FG

X

U��

F(G ′, ϕ)

Y

V��

X YF

��

such that

(1) the functors U, V are strict and

(2) Ui = R and V k = S.

Proof There is an obvious inclusion k : G ′ → F(G ′, ϕ) and the definitionof F(G ′, ϕ) forces the first square to commute. Now assume we have a com-mutative square of the form Sϕ = F R. The functor U is already determinedby the universal property of FG. We define V as follows. On 0-cells, V agreeswith S. The action of V on 1-cells is determined inductively by strictness andthe relations Ui = R, V k = S; the same holds for 2-cells, with the additionalrequirement that the constraint cells in F(G ′, ϕ) required for ϕ to be a functorare mapped to the constraint cells in Y for the composite functor FU . Thisdemonstrates uniqueness and forces the required diagrams to commute.

Given any ϕ : G → G ′ as above, we can consider the following square.

G G ′ϕ ��G

Fs G

j��

Fs G Fs G ′Fsϕ

��

G ′

Fs G ′j��


By our universal property, we thus have the following commutative square.


Fs G

��

F(G ′, ϕ)

Fs G ′ ��

Fs G Fs G ′Fsϕ

��

The coherence theorem for functors now takes the following form.

Theorem 2.21 (Coherence for functors) The functor : F(G ′, ϕ) → Fs G ′is a strict biequivalence.

2.3.2 Proof of the coherence theorem

Lemma 2.22 Assume that the following squares commute in Gr(Cat) wherethe Fi are functors between bicategories.

G G ′ϕ �� G ′

B

Si

��

G

A

R

��A B

Fi

��

Let the following squares be those induced by the universal property.

FG F(G ′, ϕ)ϕ ��F(G ′, ϕ)

B

Si

��

FG

A

R

��A B

Fi

��

Assume that the Si have the same object-map, and that the Fi have the sameobject-map. Then for every pair consisting of an invertible icon α : F1 ⇒ F2

and an (obG ′ × obG ′)-indexed collection of isomorphisms β : S1(x, x ′) ∼=S2(x, x ′) between functors G ′(x, x ′) → B(S1x, S1x ′) such that

α ∗ 1R = β ∗ 1ϕ

as (obG × obG)-indexed collections of natural isomorphisms, there is aninvertible icon β : S1 ⇒ S2 such that

α ∗ 1R = β ∗ 1ϕ

as icons.

Proof First, we must construct a new bicategory B I . It has the same 0-cellsas B. A 1-cell a → b is a triple (h1, h2, γ ) which consists of a pair of 1-cells


h1, h2 : a → b and a 2-cell isomorphism γ : h1 ⇒ h2. A 2-cell (h1, h2, γ ) ⇒(k1, k2, δ) consists of a pair of 2-cells σi : hi ⇒ ki such that σ2γ = δσ1.The identity 1-cell for a is the triple (ida, ida, 1), and composition is givenby composition of each pair of 1-cells together with the horizontal compositefor the 2-cells. The constraints are given by those from B, and the axioms areimmediate.

Now α induces a functor F : A → B I by the formulas F(x) = F1x = F2x ,F( f ) = (F1 f, F2 f, α f ), and F(σ ) = (F1σ, F2σ). The constraint cells forF are given by the constraint cells of F1 and F2. We must now check thatthese constraint cells satisfy the necessary equation to be valid 2-cells, butthis follows immediately from the icon axioms. Using the obvious map G ′ →B I and the universal property of F(G ′, ϕ), we obtain the commutative squarepictured below.


A

R��

F(G ′, ϕ)

B I

S��

A B IF

��

There are strict functors πi : B I → B given by πi (a) = a, πi (h1, h2, γ ) =hi , πi (σ1, σ2) = σi , and an invertible icon : π1 ∼= π2 whose component at(h1, h2, γ ) is γ . It is immediate that πi F = Fi , so by the universal propertyof F(G ′, ϕ), we get that πi S = Si as well. Thus we define β to be ∗ 1S ; therequired properties are easy to verify.

Proof of 2.21 We have the inclusion j : G ′ → Fs G ′ and thus an inducedmap of category-enriched graphs j : Fs G ′ → F(G ′, ϕ). It is easy to checkthat the composite

Fs G ′ j−→ F(G ′, ϕ) −→ Fs G ′

is the identity in Gr(Cat), so is locally full and locally essentially surjective.We know that is surjective on objects, so we need only show that it is locallyfaithful.

By Proposition 2.17, there is a strict functor S : FG → F(G ′, ϕ) and aninvertible icon α : S ⇒ ϕ. Thus the universal property of F(G ′, ϕ) gives thefollowing commutative square.


FG

1��

F(G ′, ϕ)

F(G ′, ϕ)E��

FG F(G ′, ϕ)S��


We also have the identity square.


FG

1��

F(G ′, ϕ)

F(G ′, ϕ)1��

FG F(G ′, ϕ)ϕ

��

Using α together with the identity isomorphisms, we can apply Lemma 2.22;since the identity functor is locally full and faithful, we can use Lemma 2.19to conclude that E is locally full and faithful.

The universal property of F(G ′, ϕ) provides the following commutativesquare.


FG

1��

F(G ′, ϕ)

FG ′ 1��

FG FG ′Fϕ

��

The universal property also implies that 1 = ; since we already know that is locally faithful, we need only show that 1 is locally faithful to completethe proof. There is a unique strict functor T : FG ′ → F(G ′, ϕ) which extendsthe inclusion of G ′ into F(G ′, ϕ). It is a simple calculation to check thatS = T ◦ Fϕ. Then T 1 is a strict functor F(G ′, ϕ) → F(G ′, ϕ) and it iseasy to check that it makes the following square commute using the fact thatall of the functors are strict.


FG

1��

F(G ′, ϕ)

F(G ′, ϕ)

T 1��

FG F(G ′, ϕ)S��

Thus E = T 1, and hence 1 is locally faithful since E is.

2.3.3 Using coherence: strictification

In this section, we use Theorem 2.21 to produce for each functor F : B → B ′a strict 2-functor stF : stB → stB ′. Thus, up to biequivalence, we can replacefunctors by strict maps. Since this construction will commute with composi-tion, we can replace diagrams by biequivalent diagrams of strict 2-categoriesand strict 2-functors between them.


Let F : X → Y be a functor between bicategories. We define the strictfunctor stF : stX → stY as follows. On 0-cells, stF agrees with F . On 1-cells,we define

stF( fn · · · f1) = F fn · · · F f1,

and stF(∅a) = ∅Fa . We will define the action of stF on 2-cells using the sametechnique as in Section 2.2.3. Let α : e( fn · · · f1) ⇒ e(gm · · · g1) be a 2-cellin stX . Then we define stF(α) to be the 2-cell

e(F fn · · · F f1) ∼= F(

e( fn · · · f1))

Fα−→ F(

e(gm · · · g1)) ∼= e(Fgm · · · Fg1),

where the unlabeled isomorphisms are the unique isomorphism 2-cells pro-vided by our coherence theorem by considering the sub-Cat-graph of Y withno non-identity 2-cells.

The same proof as in Section 2.2.3 shows that this is a strict functor; thesame techniques also prove that st(F ◦ G) = stF ◦ stG. The commutativity ofthe square

X YF ��X

stX

f��

Y

stY

f��

stX stYstF

��

is immediate from the definitions. It is not the case that Fe = e ◦ stF , butthere is an invertible icon ω between these with each component given by theunique coherence 2-cell; in particular, this equation does hold when X,Y are2-categories and F is a 2-functor, as the unique coherence 2-cell is necessarilyan identity.

Theorem 2.23 The assignment B → stB can be extended to a 2-functor

st : Icon → Grayicon .

This 2-functor is the left 2-adjoint to the inclusion Grayicon ↪→ Icon.

Remark 2.24 We will not prove the previous theorem, as it will not beneeded here. For a proof, we refer the reader to the author’s upcoming paper(Gurski 2013). We will, however, need two consequences of this theoremwhich are relatively easy to prove independently. First, we will need thateX : stX → X is a 2-functor when X is a 2-category; this is a simple cal-culation. Second, we will need that, for any bicategory B, there is an invertibleicon estB ∼= st(eB); this is most easily seen as a consequence of one of thetriangle identities for the 2-adjunction, but can also be computed directly.

3

Gray-categories

This chapter will be a basic introduction to the theory of Gray-categories.There are a variety of natural ways to motivate the Gray-tensor product of2-categories, and I would like to mention a few of them briefly without worry-ing about proofs of the various technical results that make this theory work. Tobe clear, I do not believe any of the material in this chapter is new; I have onlycollected together material on the Gray tensor product and Gray-categoriesthat we will need later in studying either coherence for tricategories or the gen-eral coherence problem for algebras over Gray-monads. The main referencesare Gray’s (1974, 1976) work, although the handwritten notes of Street (1988)provide another perspective. I have also drawn heavily from the material inGordon–Power–and Street (1995). I do not know of a reference for the expla-nation of the Gray-tensor product in terms of a factorization, although it ismentioned in passing by Lack (2010b), and it was certainly from the lecturesupon which that article is based that I learned that the Gray-tensor productcould be expressed in this way.

This chapter proceeds as follows. First, I will give the generators-and-relations definition of the Gray-tensor product. While this definition willnot be particularly useful in the discussion of coherence for tricategories, itwill be used with regularity when we turn to discussing algebras for Gray-monads. I should also point out that the tensor product given here is theup-to-isomorphism version, not the lax version as in Gray’s original work.Next, we define the notion of cubical functor and relate it to the Gray-tensorproduct. Cubical functors of two variables are weak functors F : A1×A2 → Bthat are 2-functors in each variable separately (as well as satisfying someadditional strictness), and so they are very much like bilinear maps betweenabelian groups. This perspective explains in what way the Gray-tensor prod-uct of 2-categories is similar to the usual tensor product of abelian groups. Thisdescription in terms of cubical functors will be useful later, both in practice

35

36 Gray-categories

and as the motivation for the definition of a cubical tricategory. Next we showthat the Gray-tensor product is part of an adjunction, specifically that − ⊗ Bis left adjoint to the functor Hom(B,−) whose value at a 2-category C isthe 2-category of strict 2-functors B → C , pseudo-natural transformationsbetween them, and modifications. With this in hand, we complete the descrip-tion of Gray as a closed, symmetric monoidal category. Finally, we go onto see how the Gray tensor product fits into a factorization of the canonicalcomparison map from the “funny” tensor product to the Cartesian product.

3.1 The Gray tensor product

Our goal in describing the Gray tensor product of 2-categories will be touse the resulting monoidal structure as a category over which to enrich. Theresulting objects, categories enriched over 2Cat with the Gray tensor product,will be a semi-strict form of 3-category used in our coherence theorem. Itis possible to define this tensor using only a universal property, but we pre-fer to define it from the ground up and show later that it satisfies a universalproperty.

The Gray tensor product of X and Y , denoted X ⊗ Y , has objects orderedpairs (A, B), where A ∈ obX and B ∈ obY . The morphisms of X ⊗ Y aregenerated by two kinds of morphisms. The first type of generator is an orderedpair of the form ( f, 1) : (A, B) → (A′, B) with f : A → A′ a morphism ofX ; the second type is (1, g) : (A, B) → (A, B ′) with g : B → B ′ a morphismof Y . The morphisms of X ⊗ Y are equivalence classes of composable stringsof these two types of generators. The equivalence relation is the smallest onesuch that the following conditions hold, when they make sense.

• ( f, 1) ∗ ( f ′, 1) ∼ ( f ∗ f ′, 1)

• (1, g) ∗ (1, g′) ∼ (1, g ∗ g′)• If w,w′ are two equivalent strings, then w ∗ v ∼ w′ ∗ v and u ∗w ∼ u ∗w′.

Note that if w ∼ w′, then w and w′ have the same source and target. Note that(1, 1) is the identity 1-cell in this 2-category.

The 2-cells of X ⊗ Y are formed in a similar, but slightly more complicatedmanner. There are three basic types of generating 2-cells, and a 2-cell in thetensor product is an equivalence class of composites, vertical and horizontal, ofthese basic 2-cells. The first type of 2-cell is one of the form (α, 1) : ( f, 1) ⇒( f ′, 1) where α : f ⇒ f ′ is a 2-cell in X . The second type of 2-cell is oneof the form (1, β) : (1, g) ⇒ (1, g′) where β : g ⇒ g′ is a 2-cell in Y . Thethird kind of 2-cell is an isomorphism � f,g : ( f, 1)(1, g) ⇒ (1, g)( f, 1), with

3.1 The Gray tensor product 37

inverse�−1f,g : (1, g)( f, 1) ⇒ ( f, 1)(1, g), where both f and g are non-identity

morphisms in their respective 2-categories. If either f or g is the identity, then� f,g is the identity. We now form equivalence classes of formal compositesof such 2-cells in two steps. First, we compose them horizontally with thesame conditions we placed on composing 1-cells. Second, we compose themvertically and impose conditions like the ones above and additional ones toforce the resulting structure to be a 2-category.

First we deal with horizontal composition. Let w,w′ be strings of the threebasic types of generating 2-cells in X ⊗ Y . Then w ∼ w′ if they are made soby the smallest equivalence relation such that the following conditions hold,when they make sense.

• (α, 1) ∗ (α′, 1) ∼ (α ∗ α′, 1)• (1, β) ∗ (1, β ′) ∼ (1, β ∗ β ′)• If σ, σ ′ are two equivalent strings, then σ ∗ τ ∼ σ ′ ∗ τ and ρ ∗ σ ∼ ρ ∗ σ ′.

Note that if σ ∼ σ ′, then σ and σ ′ have the same source and target 0-cells. Weshall denote these equivalence classes by [σ ], [τ ], etc.

A 2-cell in X ⊗ Y is then an equivalence class of vertically composablestrings [α1][α2] · · · [αn], where the equivalence relation is the smallest onesuch that the following conditions hold, when they make sense.

• (α, 1)(α′, 1) ∼ (αα′, 1)• (1, β)(1, β ′) ∼ (1, ββ ′)•

(� f ′,g ∗ (1 f , 1)

)((1 f ′ , 1) ∗� f,g

)∼ � f ′ f,g

•((1, 1g′) ∗� f,g

)(� f,g′ ∗ (1, 1g)

) ∼ � f,g′g

•((1, 1g′) ∗ (1 f ′ , 1) ∗� f,g

)(� f ′,g′ ∗ (1 f , 1) ∗ (1, 1g)

)∼

(� f ′,g′ ∗ (1 f , 1) ∗ (1, 1g)

)((1, 1g′) ∗ (1 f ′ , 1) ∗� f,g

)• If α : f ⇒ f ′ and β : g ⇒ g′ in X and Y , respectively, then(

(1, β) ∗ (α, 1))� f,g ∼ � f ′,g′

((α, 1) ∗ (1, β)

).

• If [α] ∼ [α′], then [α][β] ∼ [α′][β] and [δ][α] ∼ [δ][α′]; the samecondition holds for horizontal composition as described below.

It is now easy to write down horizontal and vertical composition ofsuch equivalence classes. For vertical composition, we have concatenation ofstrings. For horizontal composition, let w be represented by [α1][α2] · · · [αn].Note that the 0-cell source and target ofw can be computed by taking the 0-cellsource and target of any of the αi . Thus ifw′ is represented by [α′

1][α′2] · · · [α′

m]and has the same 0-cell source asw’s 0-cell target, we make the following con-struction. If m < n, insert n − m vertical identity 2-cells in any way into w′;

38 Gray-categories

we write the resulting string as [α1][α2] · · · [αn] and define w ∗ w′ to be theequivalence class of ([α1] ∗ [α1])([α2] ∗ [α2]) · · · ([αn] ∗ [αn]). If m ≥ n, weperform a similar construction on w. It is easy to show that this equivalenceclass is independent of how the identities were inserted.

We omit the details that X ⊗ Y forms a 2-category; the only difficult axiomto check is the interchange law. We also omit the details that the above ten-sor product gives a monoidal structure on 2Cat. This monoidal structurehas a symmetry defined on generating objects and 1-cells by switching theorder, on generating 2-cells of the form (α, 1) or (1, β) by switching theorder, and on generating 2-cells of the form � f,g as �−1

g, f . Additionally, thismonoidal structure is closed, with an adjoint hom-functor to be determinedlater.

3.2 Cubical functors

In this section, we present a different perspective on the Gray tensor productusing cubical functors. This is in analogy with the definition of the usual tensorproduct of R-modules, in which the module A ⊗R B is the target of a universalbilinear map A × B → A ⊗R B. The Gray tensor product X ⊗ Y will receivea universal cubical functor X × Y → X ⊗ Y . We first define cubical functorsof n variables, describe them in elementary terms, and then prove the aboveuniversal property.

3.2.1 Defining cubical functors

Definition 3.1 A functor F : A1 × A2 × · · · × An → B is cubical if thefollowing condition holds:

If ( f1, f2, . . . , fn)∗ (g1, g2, . . . , gn) is a composable pair of morphisms in the2-category A1 × A2 × · · · × An such that for all i > j , either gi or f j is anidentity map, then the comparison 2-cell

φ : F( f1, f2, . . . , fn) ∗ F(g1, g2, . . . , gn)

⇒ F(( f1, f2, . . . , fn) ∗ (g1, g2, . . . , gn)

)is an identity.

First, note that every cubical functor strictly preserves identity 1-cells. Thisfollows from the unit axioms for a functor and the fact that the 2-cell

φ f I : F f ∗ F I ⇒ F( f ∗ I )

3.2 Cubical functors 39

is always an identity 2-cell (similarly for φI f ) since it satisfies the cubicalcondition. For the case n = 1, a cubical functor is trivially a strict 2-functor.

Proposition 3.2 A cubical functor F : A1 × A2 → B determines, and isuniquely determined by

(1) For each object a1 ∈ obA1, a strict 2-functor Fa1;(2) For each object a2 ∈ obA2, a strict 2-functor Fa2;(3) For each pair of objects a1, a2 in A1, A2, respectively, the equation

Fa1(a2) = Fa2(a1) := F(a1, a2)

holds;(4) For each pair of 1-cells f1 : a1 → a′

1, f2 : a2 → a′2 in A1, A2,

respectively, a 2-cell isomorphism

F(a1, a2) F(a1, a′2)

Fa1 ( f2) �� F(a1, a′2)

F(a′1, a′

2)

Fa′2( f1)

��

F(a1, a2)

F(a′1, a2)

Fa2 ( f1)

��F(a′

1, a2) F(a′1, a′

2)Fa′1( f2)

��

� f1, f2

∼=��

��

which is an identity 2-cell if either f1 or f2 is an identity 1-cell;

subject to the following three axioms for all diagrams of the form

(a1, a2) (a′1, a′

2)

( f1, f2)

��(a1, a2) (a′

1, a′2)

(g1,g2)

��(α1,α2)��(a′

1, a′2) (a′′

1 , a′′2 )

(h1,h2) ��

in A1 × A2.

F(a1,a2) F(a1,a′2)

Fa1 ( f2)

��F(a1,a2) F(a1,a′

2)Fa1 (g2)��F(a1,a2)

F(a′1,a2)

Fa2 (g1)

��

F(a1,a2)

F(a′1,a2)

Fa2 ( f1)

��

F(a1,a′2)

F(a′1,a

′2)

Fa′2( f1)

��F(a′

1,a2) F(a′1,a

′2)Fa′

1(g2)

��

F(a1,a2)

F(a′1,a2)

Fa2 (g1)

��

F(a1,a′2)

F(a′1,a

′2)

Fa′2( f1)

��

F(a1,a2) F(a1,a′2)

Fa1 ( f2) ��

F(a′1,a2) F(a′

1,a′2)

Fa′1(g2)

��

F(a1,a′2)

F(a′1,a

′2)

Fa′2(g1)

F(a′

1,a2) F(a′1,a

′2)

Fa′1( f2)

��

=

⇓Fa1α2

⇐Fa2α1 ⇓� ⇓�

⇓Fa′1α2

⇐Fa′

2α1

40 Gray-categories

F(a1,a2) F(a1,a′2)

Fa1 ( f2) �� F(a1,a′2)

F(a′1,a

′2)

Fa′2( f1)

��F(a′

1,a′2)

F(a′′1 ,a

′2)

Fa′2(h1)

��

F(a1,a2)

F(a′1,a2)

Fa2 ( f1)

��F(a′

1,a2)

F(a′′1 ,a2)

Fa2 (h1)

��F(a′′

1 ,a2) F(a′′1 ,a

′2)Fa′′

1( f2)

��

F(a′1,a2) F(a′

1,a′2)

Fa′1( f2)

�� =

F(a1,a2) F(a1,a′2)

Fa1 ( f2) ��

F(a′′1 ,a2) F(a′′

1 ,a′2)Fa′′

1( f2)

��

F(a1,a2)

F(a′′1 ,a2)

Fa2 (h1 f1)

��

F(a1,a′2)

F(a′′1 ,a

′2)

Fa′2(h1 f1)

��

⇓�

⇓�

⇓�

F(a1,a2) F(a1,a′2)

Fa1 ( f2) �� F(a1,a′2) F(a1,a′′

2 )Fa1 (h2) �� F(a1,a′′

2 )

F(a′1,a

′′2 )

Fa′′2( f1)

��

F(a1,a2)

F(a′1,a2)

Fa2 ( f1)

��F(a′

1,a2) F(a′1,a

′2)

Fa′1( f2)

�� F(a′1,a

′2) F(a′

1,a′′2 )

Fa′1(h2)

��

F(a1,a′2)

F(a′1,a

′2)

Fa′2( f1)

��

F(a1,a2)

F(a′1,a2)

Fa2 ( f1)

��

F(a1,a′′2 )

F(a′1,a

′′2 )

Fa′′2( f1)

��

F(a1,a2) F(a1,a′′2 )

Fa1 (h2 f2) ��

F(a′1,a2) F(a′

1,a′′2 )Fa′

1(h2 f2)

��

⇓� ⇓�

⇓�

Proof It is easy to see that Fa1 : A2 → B is a 2-functor, as

F(1, f ∗ g) = F(1, f ) ∗ F(1, g)

F(1, 1) ∗ F(1, g) = F(1, g) = F(1, g) ∗ F(1, 1),

where all displayed equalities are actually 2-cell constraints. The same argu-ment shows that Fa2 is a strict 2-functor.


Let f1 : a1 → a′1, f2 : a2 → a′

2 be a pair of 1-cells in A1, A2, respectively.Then the 2-cell� f1, f2 is the composite of the constraint 2-cell with the identity2-cell.

F(a1, a2) F(a1, a′2)

F(1, f2) ��F(a1, a2)

F(a′1, a2)

F( f1,1)

��F(a′

1, a2) F(a′1, a′

2)F(1, f2)��

F(a1, a′2)

F(a′1, a′

2)

F( f1,1)

��

F(a1, a2)

F(a′1, a′

2)

F( f1, f2)��

��∼=��

��

=��

��

Coherence for functors gives that each of the three axioms holds.Given the data above, we construct a cubical functor F . The functor F is

already defined on objects, so we define it on 1-cells by

F( f1, f2) = F(1, f2) ∗ F( f1, 1)

and on 2-cells by

F(α1, α2) = F(1, α2) ∗ F(α1, 1).

Here we have written F(1,−), F(−, 1), for Fa1(−), resp. Fa2(−). The con-straint cells are given by � or are identities as necessitated by the definition ofcubical functor, and it is simple to check that the axioms above give the axiomsfor a weak functor.

Proposition 3.3 A cubical functor F : A1 × A2 × A3 → B determines, andis uniquely determined by

(1) For each object a1 ∈ A1, a cubical functor of two variables Fa1 : A2 ×A3 → B, and similarly for objects a2 ∈ A2, a3 ∈ A3;

(2) For each pair of objects a1, a2 in A1, A2, respectively, the equation

Fa1(a2,−) = Fa2(a1,−)

holds, and similarly for pairs a1, a3 and a2, a3;

such that the following axiom holds:

Given a 1-cell ( f1, f2, f3) : (a1, a2, a3) → (a′1, a′

2, a′3) in A1 × A2 × A3, the

equation below holds.

42 Gray-categories

F(a1,a2,a3)

F(a1,a2,a′3)Fa1 (1, f3) ��

F(a1,a2,a3)

F(a1,a′2,a3)

Fa1 ( f2,1)

��

F(a1,a2,a′3)

F(a1,a′2,a

′3)

Fa1 ( f2,1)

��

F(a1,a′2,a3)

F(a1,a′2,a

′3)Fa1 (1, f3) ��

F(a1,a2,a3)

F(a′1,a2,a3)

Fa3 ( f1,1)

��F(a′

1,a2,a3)

F(a′1,a

′2,a3)

Fa3 (1, f2) ��

F(a1,a′2,a3)

F(a′1,a

′2,a3)

Fa3 ( f1,1)

��

F(a1,a′2,a

′3)

F(a′1,a

′2,a

′3)

Fa2 ( f1,1)

��

F(a′1,a

′2,a3)

F(a′1,a

′2,a

′3)

Fa2 (1, f3)

��

F(a1,a2,a3)

F(a1,a2,a′3)Fa1 (1, f3) ��

F(a1,a2,a3)

F(a′1,a2,a3)

Fa3 ( f1,1)

��

F(a1,a2,a′3)

F(a1,a′2,a

′3)

Fa1 ( f2,1)

��

F(a1,a′2,a

′3)

F(a′1,a

′2,a

′3)

Fa2 ( f1,1)

��F(a′

1,a2,a3)

F(a′1,a

′2,a3)

Fa3 (1, f2) ��

F(a′1,a

′2,a3)

F(a′1,a

′2,a

′3)

Fa2 (1, f3)

��

F(a′1,a2,a3)

F(a′1,a2,a′

3)

Fa2 (1, f3)

��

F(a1,a2,a′3)

F(a′1,a2,a′

3)

Fa2 ( f1,1)

��F(a′

1,a2,a′3)

F(a′1,a

′2,a

′3)

Fa′1( f2,1)

��

⇓�

⇓� ⇓�

⇓� ⇓�

⇓�

Proposition 3.4 A cubical functor F : A1 × A2 × · · · × An → B, n ≥ 3,determines, and is uniquely determined by

(1) For each (a1, a2, . . . , an) ∈ A1 × A2 × · · · An and each i < j < k, therestriction to

F(a1, a2, . . . , ai , . . . , a j , . . . , ak, . . . , an) : Ai × A j × Ak → B

is a cubical functor of three variables (where ai indicates that object hasbeen omitted and the variable is free), and

(2) These functors are compatible in the sense of Proposition 3.3.

Proposition 3.5 Let i1, . . . , ik be positive integers, and let

Fj : A j,1 × · · · × A j,i j → B j

be cubical functors. Then for any cubical functor

F : B1 × · · · × Bk → C,

the composite F ◦ (F1 × · · · × Fk) is a cubical functor.

Proof This is a functor, so we must check that certain constraints are iden-tities. Recall that the constraint 2-cell for a composite G ◦ F is given by theformula

φG F = φG ∗ G(φF ).


Since G preserves identity 2-cells, it is enough to establish that φF and φG areappropriate identities.

Let ( fm,n), ( f ′m,n) be composable arrows in the product 2-category

∏Ap,q

such that whenever (a, b) < (a′, b′), either f ′a′,b′ or fa,b is the identity. For

(a, b) to be less than (a′, b′) in the total order, either a < a′ or b < b′.In particular, for a fixed a, either f ′

a,b′ or fa,b is the identity. Thus the con-straint φa for Fa is the identity, so the contraint for F1 × · · · × Fk is theidentity.

Now we must show that the constraint for F is the identity. Thisamounts to proving that if a > a′, then either Fa( fa,1, fa,2, . . . , fa,ia ) orFa′( f ′

a′,1, . . . , f ′a′,ia′ ) is the identity. If, for a fixed a, all the fa,b are identi-

ties, then Fa( fa,1, . . . , fa,ia ) will be an identity as well since Fa preserves1-cell identities strictly because it is cubical. Now assume that a′ > a andthat Fa′( f ′

a′,1, . . . , f ′a′,ia′ ) is not the identity. Then some fa′,b is not the iden-

tity. Since (a′, b) > (a, c) for every c, every fa,c must be the identity by thecubical assumption. This shows that Fa( fa,1, . . . , fa,ia ) is the identity, and wemay now conclude that φF is the identity by the fact that F is cubical. Thiscompletes the proof that the composition constraint for F at ( fa,b) ∗ ( f ′

a,b) isthe identity, so the composite functor is cubical.

Recall that a multicategory M consists of

• a set of objects, M0,

• sets M(a1, . . . , an; b) for ai , b ∈ M0 (including the case of the empty sourcestring),

• identities 1 ∈ M(a, a), and

• composition laws: if ai are strings of objects as above, then composition isa function

M(b1, . . . , bn; c)× M(a1; b1)× · · · × M(an; bn) → M(a1, . . . , an; c);

this collection of data satisfies unit and associativity laws which we will notwrite down here. The interested reader should consult Leinster (2004) forfurther information.

Corollary 3.6 There is a multicategory 2Catc whose objects are 2-categoriesfor which the set

2Catc(A1, A2, . . . , An; B)

consists of the cubical functors A1 × · · · × An → B.

44 Gray-categories

3.2.2 The universal cubical functor

We are now in a position to prove that the Gray tensor product provides asolution to the problem of finding a universal cubical functor

A × B → C.

Theorem 3.7 Let A, B, and C be 2-categories. There is a cubical functor

c : A × B → A ⊗ B,

natural in A and B, such that composition with c induces an isomorphism

2Catc(A, B; C) ∼= 2Cat(A ⊗ B,C).

Proof We define c using Proposition 3.2. We define the 2-functor ca by

ca(b) = (a, b)

ca( f ) = (1a, f )

ca(α) = (11a , α);the 2-functor cb is defined similarly. The 2-cell isomorphism � f,g is the same� f,g that is part of the data for A⊗B. The three axioms for a cubical functor areexactly the axioms for the Gray tensor product, so we have defined a cubicalfunctor c : A × B → A ⊗ B. Naturality in both variables is clear.

To prove that this cubical functor has the claimed universal property, assumethat F : A × B → C is a cubical functor. We define a strict 2-functor F :A ⊗ B → C by the following formulas.

F(a, b) = F(a, b)

F( f, 1) = Fb( f )

F(1, g) = Fa(g)

F(α, 1) = Fb(α)

F(1, β) = Fa(β)

F(�A⊗Bf,g ) = �F

f,g.

This defines F on objects, generating 1-cells, and generating 2-cells. Weextend F to the whole of A⊗ B by making it a strict 2-functor, i.e., it preservesall types of compositions and identities. The axioms for cubical functors andthe Gray tensor product ensure that this is well-defined. It is clear that F is theunique strict 2-functor making the diagram

3.2 The monoidal category Gray 45

A × B

c��

F �� C

A ⊗ BF

��

commute, completing the proof.

Remark 3.8 One could go on and prove that 2Catc is a representable multi-category in the language of Hermida (2000). This also follows from the (stated,but unproven here) fact that 2-categories equipped with the Gray tensor prod-uct form a monoidal category, but at this point one could show representabilitydirectly.

3.3 The monoidal category Gray

In this section, we will establish the basic results necessary to introducethe monoidal category Gray. We will not prove that this monoidal structuresatisfies the necessary coherence laws (Gray 1974, 1976).

Notation 3.9 Let Hom(A, B) denote the full sub-bicategory of Bicat(A, B)with objects the strict functors.

Recall that if B is a 2-category, then Hom(A, B) is a 2-category for anybicategory A.

Proposition 3.10 Let A, B be 2-categories.

1 The evaluation map e : Hom(A, B)× A → B is a cubical functor.2 The function which sends a 2-functor F : A1 → Hom(A2, B) to the

composite

A1 × A2F×1→ Hom(A2, B)× A2

e→ B

is a natural isomorphism between 2-functors A1 → Hom(A2, B) andcubical functors A1 × A2 → B.

Proof For the first part, the evaluation map e is defined by the followingformulas.

e(F, a) = Fae(1F , f ) = F f

e(σ, 1a) = σa (the component of σ at a)e(11F , α) = Fαe(�, 11a ) = �a

�σ, f = σ f .

46 Gray-categories

It is easy to check that this is a 2-functor when each variable is held fixed, andsatisfies the necessary conditions to give a cubical functor.

For the second claim, first note that the composite displayed is actually acubical functor by Proposition 3.5. Now given a cubical functor F : A1 ×A2 → B, we must construct a strict 2-functor F : A1 → Hom(A2, B). Tofix notation, we have objects a1, a′

1 in A1, morphisms f1, f ′1 in A1 each with

source a1 and target a′1, and a 2-cell α1 : f1 ⇒ f ′

1 in A1; similarly for A2 withsubscript 2 instead of 1. The strict 2-functor is given by the formulas below.

F(a1) = Fa1

F( f1)a2 = Fa2( f1)

F( f1) f2 = � f1, f2

F(α)a2 = Fa2(α).

It is easy to check that F( f1) gives a weak transformation, that F(α) gives amodification, that this assignment is a 2-functor, and that it is inverse to theassignment F → e ◦ (F × 1).

Corollary 3.11 For any 2-category B, the functor −⊗ B is left adjoint to thefunctor Hom(B,−).

Proposition 3.12 Let A, B,C be 2-categories, and G : A × B → C be afunctor such that each G(a,−), G(−, b) is a strict 2-functor. Then there is acubical functor F : A × B → C such that

(1) F agrees with G on objects and

(2) there is an invertible icon ν : G ⇒ F.

Proof Define F to agree with G on objects. For a 1-cell ( f, g) : (a, b) →(a′, b′), define F to be the composite

F(a, b)G( f,1)−→ F(a′, b)

G(1,g)−→ F(a′, b′),

where we have already used that F(a, b) = G(a, b). For a 2-cell (α, β) :( f, g) → ( f ′, g′), define F(α, β) to be the horizontal composite

F(a, b) F(a′, b)

G( f,1)��

F(a, b) F(a′, b)

G( f ′,1)

�� F(a′, b) F(a′, b′).

G(1,g)��

F(a′, b) F(a′, b′).

G(1,g′)��

G(α,1)�� G(1,β)��

3.3 The monoidal category Gray 47

The structure constraint for this functor is given by the following formula,where the indicated isomorphisms are obtained from the structure constraintsfor the functor G.

F( f2, g2) ∗ F( f1, g1) := G(1, g2) ∗ G( f2, 1) ∗ G(1, g1) ∗ G( f1, 1)∼= G(1, g2) ∗ G( f2, g1) ∗ G( f1, 1)∼= G(1, g2) ∗ G(1, g1) ∗ G( f2, 1) ∗ G( f1, 1)= G(1, g2 ∗ g1) ∗ G( f2 ∗ f1, 1)=: F( f2 ∗ f1, g2 ∗ g1).

Since this is defined using only the structure constraints of the functor G,coherence immediately implies that this new constraint will also satisfy theaxioms necessary for F to be a cubical functor. Thus we have defined a cubicalfunctor F : A × B → C .

To define the invertible icon ν, we need a 2-cell isomorphism ν( f,g) :G( f, g) ⇒ F( f, g). This amounts to a 2-cell isomorphism G( f, g) ∼=G(1, g) ∗ G( f, 1), and for this we take the structure 2-cell for the functor G.Once again, coherence for functors immediately implies that this choice willsatisfy the axioms for being a transformation.

Remark 3.13 Gordon et al. (1995) call the procedure above nudging, andthe transformation ν nudges G into a cubical functor.

Definition 3.14 A functor F : A1 × A2 × · · · × An → B is opcubical if thefollowing condition holds:If ( f1, f2, . . . , fn)∗ (g1, g2, . . . , gn) is a composable pair of morphisms in the2-category A1 × A2 × · · · × An such that for all i < j , either gi or f j is anidentity map, then the comparison 2-cell

φ : F( f1, f2, . . . , fn) ∗ F(g1, g2, . . . , gn)

⇒ F(( f1, f2, . . . , fn) ∗ (g1, g2, . . . , gn)

)is an identity.

Remark 3.15 Note that the difference between the definitions of cubical andopcubical functors is the switching of i > j for cubical functors to i < jfor opcubical functors. It is easy to check that, given a cubical functor F :A × B → C , we can produce an opcubical functor F∗ : A × B → C bydefining

F∗( f, g) = F( f, 1)F(1, g)

and replacing the necessary structure 2-cells with their inverses. Nudging F∗gives back F , and in this case the icon ν has components at each morphism

48 Gray-categories

given by the structure constraints for F . Thus we obtain an isomorphismbetween cubical functors A × B → C and opcubical functors A × B → C .

On the other hand, it is clear that there is an isomorphism between cubicalfunctors A × B → C and opcubical functors B × A → C by the definitionof opcubical functor. Combining these gives an isomorphism between cubicalfunctors A × B → C and cubical functors B × A → C . This procedure is oneway of producing a symmetry isomorphism A ⊗ B ∼= B ⊗ A.

We have now established the basic results necessary to state the followingtheorem.

Theorem 3.16 The category 2Cat of 2-categories and 2-functors has thestructure of a closed symmetric monoidal category when equipped with

• the Gray tensor product, A ⊗ B,• unit object the terminal 2-category,• the internal hom-functor Hom(A, B), and• symmetry either given by the construction given in Section 1 or by the

procedure above.

Remark 3.17 Note that this is a different closed symmetric monoidal struc-ture than the one given by Cartesian product and the usual hom-2-categoryhaving 0-cells 2-functors, 1-cells 2-natural transformations, and 2-cells modi-fications. We shall refer to the monoidal structure using the Gray tensor productas Gray, and the Cartesian monoidal structure as 2Cat. While these twomonoidal structures have the same underlying category, they do have differentproperties. One example of this is that there is a model structure on the cate-gory 2Cat, and it is a monoidal model structure with respect to the Gray-tensorproduct, but not the Cartesian one. For more on this model structure, and modelstructures on 2-categories in general, see the papers of Lack (2002b, 2004).

3.4 A factorization

The final section of this chapter will construct the Gray tensor product byshowing that it arises naturally from a factorization system on the category2Cat. This factorization system will have as its left class of maps those2-functors which induce an isomorphism on the underlying category and as itsright class of maps those 2-functors which are locally full and faithful. Thus theGray tensor product A ⊗ B appears as the middle term by using this factoriza-tion on the canonical map A�B → A × B where A�B is what is often calledthe funny tensor product of 2-categories. Both A�B, A× B exist in the context

3.4 A factorization 49

of V-enriched categories for a wide class of enriching categories V, so we seethe Gray tensor product arise from a very general phenomenon. We begin thissection with a review of orthogonal factorization systems, particularly in thecase of categories of the form V-Cat, and then move on to show that the Graytensor product appears as a result of this factorization when V = Cat.

Definition 3.18 Let C be a category. An orthogonal factorization system onC consists of two classes of maps, L,R such that

• every morphism f : A → B factors as f = pi , i ∈ L, p ∈ R, and thisfactorization is unique up to unique isomorphism; and

• both L,R contain all the isomorphisms of C and are closed undercomposition.

An important consequence of this definition is the unique solution to certainlifting problems. Given a commutative square

A Bf �� B

Y

p

��

A

X

i

��X Yg

��

with i ∈ L, p ∈ R, there is a unique lift

A Bf �� B

Y

p

��

A

X

i

��X Yg

��X

B

l

��

making both triangles commute.The key result for our use is the following theorem.

Theorem 3.19 Let V be a monoidal category with an orthogonal factor-ization system (L,R). Assume that V has the property that if f, g ∈ L thenf ⊗ g ∈ L. Then the category V-Cat has an orthogonal factorization system(L∗,R∗) in which the class L∗ consists of those functors which are bijectiveon objects and locally in L and the class R∗ consists of those functors whichare locally in R.

Proof First, it is clear that both L∗,R∗ contain all isomorphisms and areclosed under composition since L,R both do. Thus we only need to provethe unique factorization.

50 Gray-categories

Let F : A → B be a V-functor. Factor this functor into

Ai−→ X

p−→ B

as follows. The set of objects of X will be the same as the set of objects ofA, and i will be the identity on objects while p will agree with F on objects.Then define X (a, b) by using the factorization system in V to factor the mapFa,b : A(a, b) → B(Fa, Fb) as below,

A(a, b)ia,b−→ X (a, b)

pa,b−→ B(Fa, Fb)

where ia,b ∈ L, pa,b ∈ R. It is clear that this factorization is unique up tounique isomorphism in the category of V-graphs, and now we must show thatX is a V-category, that i, p are V-functors, and that in fact this factorization isunique up to unique isomorphism in V-Cat.

First, we must provide X with unit maps I → X (a, a). We define this to bethe composite

I −→ A(a, a)ia,a−→ X (a, a)

of the unit for A and ia,a . The unit axiom for the putative V-functor i is thenimmediate, and then unit axiom for p then follows from the fact that F is aV-functor.

Second, we must define composition in X . Consider the following diagramin V.

A(b, c)⊗ A(a, b) A(a, c)�� A(a, c) X (a, c)ia,c �� X (a, c)

B(Fa, Fc)

pa,c

��

A(b, c)⊗ A(a, b)

X (b, c)⊗ X (a, b)

ib,c⊗ia,b

��X (b, c)⊗ X (a, b) B(Fb, Fc)⊗ B(Fa, Fb)

pb,c⊗pa,b

�� B(Fb, Fc)⊗ B(Fa, Fb) B(Fa, Fc)��

By the assumption on tensoring maps in L, the left-hand vertical map is in L;by definition, the right-hand vertical map is in R. Thus there exists a uniquemorphism in V X (b, c)⊗ X (a, b) → X (a, c)making both triangles commute.This defines composition in X , and the two triangles are the two compositionaxioms for checking that i, p are V-functors. The only things left to check arethe V-category axioms for X . The unit axioms follow from the unit axioms inA together with the naturality of the unit isomorphisms in V. The associativityaxiom follows by showing that the two possible triple composites both solvethe same lifting problem in V and then invoking uniqueness. Finally, it is easyto check that this factorization is unique in V-Cat and not just V-graphs byusing the uniqueness of the factorizations in V.


As a first example, every category has two trivial factorization systems on it.The right-maximal factorization system has L being the class of isomorphismsand R being the class of all maps. Any monoidal V satisfies the hypotheses ofthe theorem above for the right-maximal factorization system, and the result-ing factorization system (L∗,R∗) on V-Cat is once again the right-maximalfactorization system.

The left-maximal factorization has L being the class of all maps and R beingthe class of isomorphisms. Once again, any monoidal V satisfies the hypothe-ses of the theorem for the left-maximal factorization system, and the resultingfactorization (L∗,R∗) on V-Cat is the V-enriched version of the standard bo-fffactorization system on Cat in which the left class consists of functors whichare bijective on objects and the right class consists of functors which are fulland faithful. Applying our theorem again to this factorization system on V-Catgives the following immediate corollary.

Corollary 3.20 There is a factorization system on the category of V-enriched2-categories (i.e., 2-categories in which the sets of 2-cells are replaced by anobject of V, with suitably adjusted axioms) in which the left class consists ofthose morphisms which are isomorphisms on underlying categories and theright class consists of those morphisms F : A → B which are isomorphismson the V-objects of 2-cells A(a, b)( f, g) → B(Fa, Fb)(F f, Fg). In particu-lar, there is a factorization system on 2Cat with left class the 2-functors whichare isomorphisms on 0- and 1-cells and right class the 2-functors which arelocally full and faithful, i.e., isomorphisms between sets of 2-cells as above.

This factorization system is the one that we will be using on the category2Cat to exhibit the Gray tensor product as arising naturally. In order to do so,we will have to first produce a 2-functor to factor, and then we require someadditional analysis of the relationship between the Gray tensor product and theCartesian product to prove that a given factorization is the one induced by thisfactorization system. The first of these two steps is itself a completely generalphenomenon of enriched category theory, so we examine it now.

If V is a nice category over which to enrich, then V-Cat becomes a closedmonoidal category. The tensor product X ⊗Y is given by the Cartesian producton objects and the formula

(X ⊗ Y )((x, y), (x ′, y′)

)= X (x, x ′)⊗ Y (y, y′)

on morphism objects. The closed structure is obtained by setting [Y, Z ] to bethe V-category whose objects are V-functors Y → Z and whose hom-objects

52 Gray-categories

are the V-enriched objects of natural transformations; this can be computed asan end in V.

There is another closed structure on V-Cat. Its internal hom, which wewill denote Trans(Y, Z), is given by the V-category of not-necessarily-naturaltransformations; the notation Trans is meant to evoke that these are transfor-mations, but they might not be natural ones. Therefore Trans(Y, Z) has objectswhich are V-functors Y → Z just as before, but the hom-object between twosuch F,G : Y → Z is now given by

Trans(Y, Z)(F,G) =∏y∈Y

Z(Fy,Gy).

(In order to compute this product, we require that the V-categories in questionare small.) The functor Z → Trans(Y, Z) has a left adjoint which we denoteX → X�Y and is often called the funny tensor product of X and Y . It can begiven explicitly as the pushout, in V-Cat, of the diagram below, where we havewritten ob X, ob Y for the discrete V-categories whose sets of objects are thesame as those for X and Y ; the morphism ob X → X is given by unit maps,and similarly for Y .

ob X ⊗ ob Y X ⊗ ob Y��ob X ⊗ ob Y

ob X ⊗ Y��

We leave it to the reader to verify this is the left adjoint as we have claimed.Now it is clear that the following square of V-functors commutes.

ob X ⊗ ob Y X ⊗ ob Y��ob X ⊗ ob Y

ob X ⊗ Y��

X ⊗ ob Y

X ⊗ Y��

ob X ⊗ Y X ⊗ Y��

By the universal property of the pushout, we get an induced V-functor j :X�Y → X ⊗ Y . In the case when V = Cat, the object that we havedenoted thusfar as X ⊗ Y is actually the Cartesian product X × Y since themonoidal structure on Cat is given by products. The 2-category X�Y has thesame objects as X × Y . It has 1-cells which are generated by basic 1-cellsof two types, one type being ( f, 1) : (x, y) → (x ′, y) for f : x → x ′ inX and the other type being (1, g) : (x, y) → (x, y′) for g : y → y′ in Y ;


these generators are subject to the relations that ( f, 1)( f ′, 1) = ( f f ′, 1) and(1, g)(1, g′) = (1, gg′). The 2-cells are similarly generated by 2-cells of theform (α, 1) and (1, β) subject to the following relations.

(α, 1)(α′, 1) = (αα′, 1)(α, 1) ∗ (α′, 1) = (α ∗ α′, 1)(1, β)(1, β ′) = (1, ββ ′)

(1, β) ∗ (1, β ′) = (1, β ∗ β ′).

The induced 2-functor X�Y → X × Y sends every cell to the “same” cell inthe Cartesian product. It is this induced 2-functor that we will factor to producethe Gray tensor product.

In order to produce the desired factorization, we need to examine the rela-tionship between the Cartesian product and the Gray tensor product. We havealready seen that there is a universal cubical functor c : A × B → A ⊗ B; itsuniversal property is that given any cubical functor F : A × B → C , there isa unique 2-functor F : A ⊗ B → C such that F = Fc. The following propo-sition is crucial in constructing the desired factorization, and will also play arole in our discussion of coherence in Part II of this work.

Proposition 3.21 The 2-functor F given by the universal property of theGray tensor product is locally full (resp. locally faithful) if and only if F islocally full (resp. locally faithful).

Proof First, note that the cubical functor c : A×B → A⊗B is locally faithfulby construction. Thus if F is locally faithful, then F = Fc is the composite oftwo locally faithful functors, hence is also locally faithful.

Second, note that c is also locally full. Let ( f, g), ( f ′, g′) be a pair ofparallel 1-cells in A × B. Then c( f, g) = (1, g)( f, 1) in A ⊗ B. Now the2-cells in A ⊗ B are generated by 2-cells of the form (α, 1), (1, β), and� f,g : ( f, 1)(1, g) ⇒ (1, g)( f, 1) together with its inverse; additionally, wehave identity 2-cells ( f, 1)( f ′, 1) = ( f f ′, 1) and (1, g)(1, g′) = (1, gg′),which we will denote generically as i↓ for the identity which reduces the num-ber of generating 1-cells and i↑ for the identity which increases the number ofgenerating 1-cells. Therefore any 2-cell c( f, g) ⇒ c( f ′, g′) can be written as

a composite of these. It is immediate that(

1 ∗ (α, 1))� = �

((α, 1) ∗ 1

), and

similarly for cells with (1, β). By construction of the Gray tensor product, wealso have that the composite

( f, 1)( f ′, 1)(α,1)∗1−→ ( f , 1)( f ′, 1)

i−→ ( f f ′, 1)

54 Gray-categories

is equal to the composite

( f, 1)( f ′, 1)i−→ ( f f ′, 1)

(α∗1,1)−→ ( f f ′, 1).

Now, if a 2-cell δ : c( f, g) ⇒ c( f ′, g′) is written as a composite of generating2-cells of the form (α, 1), (1, β), and �, we are done since we can use thenaturality of � and the interchange law to show that all of the instances of �and so δ = c(α, β). If δ involves i↓ or i↑, note that it must be written as acomposite of 2-cells involving (α, 1), (1, β),�, and i↑ before any instances ofi↓. The equations above imply that we can write this initial segment of δ as

�i↑�i↑ · · ·�i↑c(α, β).

The portion of this segment excluding c(α, β) can actually be written as �′i ′where �′ is a (unique, by coherence for functors) composite of �’s and i ′ is a(unique) composite of i↑’s. Thus δ equals

δ′i↓�′i ′c(α, β).

Now i↓�′ = �′′i↓ by interchange and the Gray tensor product axioms.Therefore δ actually equals

δ′�′′i↓i ′c(α, β) = δ′�′′i ′′c(α, β)

since the single instance of i↓ cancels with a single (unique) instance of i↑ in i ′to give i ′′. This presentation reduces the total number of i↑’s, i↓’s by one each.Since any presentation of δ must have an equal number of instances of i↑ andi↓, repeating this procedure will eventually cancel all of them, showing thatδ can be written with only 2-cells of the form (α, 1), (1, β), and �, in whichcase δ = c(α, β) for some choice of α, β as shown previously. In particular,if F is locally full, then so is F since F = Fc would be a composite of twolocally full functors.

Now it is clear that c is an isomorphism on objects, and it is also imme-diate from the definition of c and the Gray tensor product that it is locallyessentially surjective, so c is a biequivalence. In particular, we can choose apseudo-inverse d : A ⊗ B → A × B, and then if F = Fc we would have thatFd � F in the functor bicategory. The functor d would then be locally full andlocally faithful since it is also a biequivalence, showing that F would be locallyfull (resp. locally faithful) if F is locally full (resp. locally faithful).

Corollary 3.22 The universal cubical functor c : A × B → A ⊗ B has apseudo-inverse i : A ⊗ B → A × B which is a 2-functor that is the identity onobjects. Thus A × B and A ⊗ B are equivalent in the 2-category Icon.


Proof The identity functor A × B → A × B is strict, so in particular it iscubical. It induces a unique 2-functor i = 1 in the diagram below.

A × B A ⊗ Bc �� A ⊗ B

A × B

i

��

A × B

A × B

1

��

This shows that ic = 1, and there is an invertible icon 1 ∼= ci whose componentat a 1-cell of A ⊗ B is a composite of 2-cells which are whiskerings of � f,g

together with identities of the form i↑ or i↓ (using the notation of the previousproof).

For any 2-categories A, B, it is clear that the following diagram commutes.

ob A × ob B A × ob B��ob A × ob B

ob A × B��

A × ob B

A ⊗ B��

ob A × B A ⊗ B��

Therefore by the universal property of the pushout, there is a unique 2-functorm : A�B → A ⊗ B such that the obvious triangles commute. Recall that wealso have the canonical 2-functor j : A�B → A × B that always exists in anycategory of enriched categories (provided V is suitably nice as a category overwhich to enrich). We now come to the main result of this section.

Theorem 3.23 The triangle

A�B A ⊗ Bm �� A ⊗ B

A × B

i

��

A�B

A × B

j

��

commutes, and moreover is the factorization of j into a 2-functor which is anisomorphism on underlying categories followed by a 2-functor which is locallyfull and faithful. The triangle

56 Gray-categories

A�B A × Bj �� A × B

A ⊗ B

c

��

A�B

A ⊗ B

m

��

commutes up to an invertible icon.

Proof The final statement follows from the commutativity of the first triangletogether with the final statement in Corollary 3.22.

To prove that the first triangle commutes, first note that each 2-functorinvolved is the identity on objects. Since the 1-cells of A�B are generatedby 1-cells of the form ( f, 1) and (1, g), we need only show that the diagramcommutes on those to show that it commutes for all 1-cells; checking this isentirely trivial, as both 2-functors send ( f, 1) to ( f, 1) and (1, g) to (1, g).Similarly, every 2-cell can be written as a horizontal composite of generating2-cells of the form (α, 1) and (1, β), and it is once again simple to check thatboth of the 2-functors involved agree on these cells. Thus im = j and we haveproved that the first triangle commutes.

Finally we must check that m induces an isomorphism on underlying cat-egories and that i is locally full and faithful. The first of these is true bydefinition, and the second follows from Corollary 3.22.

Remark 3.24 We have focused on producing the Gray tensor product byapplying a factorization system to the canonical map between a pair of leftadjoints, the functors A�− and A × −. By taking mates, this induces a factor-ization using the corresponding right adjoints, and this factorization turns outto be the composite of inclusions

2Cat(B,C) ↪→ Hom(B,C) ↪→ Trans(B,C)

where 2Cat(B,C) is the 2-category of 2-functors, 2-natural transformations,and modifications from B to C . While there do seem to be some factorizationsystems lurking nearby, the author is not aware of any explanation using fac-torizations and the right adjoints that is as complete as the one given aboveusing left adjoints.

Part II

Tricategories

4

The algebraic definition of tricategory

In this chapter, we give the definition of an algebraic tricategory. We shallmake note of when this differs from the definition of tricategory given byGordon et al. (1995). Next, we give the definitions of functor, transformation,modification, and perturbation. Following this, we provide a separate sectionwith an “unpacked” view of these definitions, with all of the data given explic-itly but without axioms. Finally, we make some quick comparisons betweenthe definitions here and those in Gordon–Power–Street.

4.1 Basic definition

Definition 4.1 A tricategory T consists of the following data subject to thefollowing axioms.

DATA:

• A set obT of objects of T ;

• For (a, b) ∈ obT × obT , a bicategory T (a, b), called the hom-bicategoryof T at a and b. The objects of T (a, b) will be referred to as the 1-cells ofT with source a and target b, the arrows of T (a, b) will be referred to as2-cells of T (with their same source and target), and the 2-cells of T (a, b)will be referred to as 3-cells of T (also with their same source and target);

• For objects a, b, c of T , a functor ⊗ : T (b, c) × T (a, b) → T (a, c) calledcomposition;

• For an object a of T , a functor Ia : 1 → T (a, a), where 1 denotes the unitbicategory;

59

60 The algebraic definition of tricategory

• For objects a, b, c, d of T , an adjoint equivalence a

T (c, d)× T (b, c)× T (a, b)⊗×1 ��

1×⊗��

T (b, d)× T (a, b)

⊗��

⇓a

T (c, d)× T (a, c) ⊗�� T (a, d)

in Bicat(T (c, d)× T (b, c)× T (a, b), T (a, d));

• For objects a, b of T , adjoint equivalences l and r

T (a, b)

T (b, b)× T (a, b)

Ib×1��

T (a, b) T (a, b)1

��

T (b, b)× T (a, b)

T (a, b)

⊗

��

⇓ l

T (a, b) T (a, b)1

��T (a, b)

T (a, b)× T (a, a)

1×Ia

��

T (a, b)× T (a, a)

T (a, b)

⊗

��

⇓ r

in Bicat(T (a, b), T (a, b));

• For objects a, b, c, d, e of T , an isomorphism 2-cell π (i.e., an invertiblemodification)

T 3

T 2

1×⊗ ��

T 2 T⊗��

T 4

T 3

1×1×⊗��

T 4

T 3

1×⊗×1��

��

T 3

T 2

1×⊗��

T 4 T 3⊗×1×1 ��

T 3 T 2⊗×1 ��

T 3

T 2

⊗×1

��

T 2

T

⊗��

π�� T 3

T 2

1×⊗ ��

T 2 T⊗��

T 3 T 2⊗×1 ��

T 4

T 3

1×1×⊗��

T 4 T 3⊗×1×1 �� T 3

T 2

1×⊗��

��

T 2

T

⊗��

T 3

T 2

⊗×1

��

T 2

T

⊗��

⇐a

⇓a

=⇓a×1

⇓a

1×a⇐

in the bicategory Bicat(T 4(a, b, c, d, e), T (a, e)), where T 4 =T 4(a, b, c, d, e) is an abbreviation for T (d, e)×T (c, d)×T (b, c)×T (a, b),for example;

• For objects a, b, c of T , invertible modifications

T 2

T 2

1

�T 2

T 31×I×1��

!��T 2

T 2

1

��

T 3 T 2⊗×1

��T 3

T 2

1×⊗��

T 2 T⊗��

T 2

T

⊗��

T 2

T 2

1

�T 2

T 2

1

��T 2 T⊗

��

T 2

T

⊗��

⇓r �×1

⇐1×l ⇓a

1

��

��μ ��

4.1 Basic definition 61

T 2

T 3I×1×1 ��

T 3

T 2

⊗×1��

T 2

T

⊗��

T T1

��

T 2

T

⊗��

T 2 T 21

�� T 2

T

⊗��

T T1

��

T 2

T 3I×1×1 ��

T 3

T 2

⊗×1��

T 2

T

⊗��

T

T 2

I×1 ��

T 2

T

⊗" ��

T 3

T 2

1×⊗��

=

⇓l×1

= ⇓a

⇓l

λ ��

T T1 ��

T 2

T

⊗#!

T 2

T 31×1×I ��

T 3

T 2

1×⊗��T 2

T

⊗#!

T 2 T 21 �� T 2

T

⊗#!T T

1 ��

T 2

T 31×1×I ��

T 3

T 2

1×⊗��T 2

T

⊗#!T

T 21×I " ��

T 2

T

⊗

��

T 3

T 2

⊗×1#!

⇓1×r �

= ⇓r �

= ⇒a

ρ ��

AXIOMS:

• The following equation of 2-cells holds in the bicategory T (a1, a5), wherewe have used parentheses instead of ⊗ for compactness and the unmarkedisomorphisms are naturality isomorphisms for a.

(((k j)h)g) f

((k( jh))g) f

(a1)1

#!((k( jh))g) f

(k(( jh)g)) fa1 �"��

(k(( jh)g)) f

(k( j (hg))) f(1a)1 ��(k( j (hg))) f

k(( j (hg)) f )a ��

k(( j (hg)) f )

k( j ((hg) f ))

1a!��

k( j ((hg) f ))

k( j (h(g f )))

1(1a)

��(((k j)h)g) f

((k j)h)(g f )

a !��

((k j)h)(g f ) (k j)(h(g f ))a

�� (k j)(h(g f ))

k( j (h(g f )))

a

�"��

(k(( jh)g)) f

k((( jh)g) f )

a��

k((( jh)g) f )

k(( j (hg)) f )1(a1) ��

k((( jh)g) f )

k(( jh)(g f ))1a ��

k(( jh)(g f ))

k( j (h(g f )))1a ��

((k( jh))g) f

(k( jh))(g f )

a !��

(k( jh))(g f ) k(( jh)(g f ))a

��

((k j)h)(g f )

(k( jh))(g f )

a(11)

#!

(((k j)h)g) f

((k( jh))g) f

(a1)1

#!((k( jh))g) f

(k(( jh)g)) fa1 �"��

(k(( jh)g)) f

(k( j (hg))) f(1a)1 ��(k( j (hg))) f

k(( j (hg)) f )a ��

k(( j (hg)) f )

k( j ((hg) f ))

1a!��

k( j ((hg) f ))

k( j (h(g f )))

1(1a)

��(((k j)h)g) f

((k j)h)(g f )

a !��

((k j)h)(g f ) (k j)(h(g f ))a

�� (k j)(h(g f ))

k( j (h(g f )))

a

�"��

(((k j)h)g) f

((k j)(hg)) f

a1

$#��

((k j)(hg)) f

(k j)((hg) f )a ��

(k j)((hg) f )

(k j)(h(g f ))

(11)a

%�

(k j)((hg) f )

k( j ((hg) f ))

a$#��

((k j)(hg)) f

(k( j (hg))) f

a1

&$!!!!!!!!

∼=

∼=

⇓π ⇓1π

⇓π

∼=

⇓π1 ⇓π

⇓π


• The following equation of 2-cells holds in the bicategory T (a1, a4), wherethe unmarked isomorphisms are either naturality isomorphisms for a orunique coherence isomorphisms from the hom-bicategory.

(hg) f

((hI )g) f(r �1)1 �"��

((hI )g) f

(h(Ig)) fa1

�"��

(h(Ig)) f

(hg) f

(1l)1

!��

(hg) f

h(g f )

a

��

(hg) f

h(g f )a !��

h(g f ) h(g f )

1

�"

((hI )g) f

(hI )(g f )

a��

h(g f )

(hI )(g f )

r �(11)#!

(h(Ig)) f

h((Ig) f )

a

��h((Ig) f )

h(I (g f ))

1a��

h((Ig) f )

h(g f )

1(l1)

'%""""""""""""

h(I (g f ))h(g f )1l��

(hI )(g f )

h(I (g f ))

a��

(hg) f

((hI )g) f(r �1)1 �"��

((hI )g) f

(h(Ig)) fa1

�"��

(h(Ig)) f

(hg) f

(1l)1

!��

(hg) f

h(g f )

a

��(hg) fh(g f )a ��h(g f ) h(g f )

1��

(hg) f

(hg) f

11

��

∼=

⇓π ∼=

⇓1λ

⇓μ

⇓μ1

∼=

• The following equation of 2-cells holds in the bicategory T (a1, a4).

(hg) f

h(g f )

a

#!h(g f )

h((gI ) f )1(r �1) �"��

h((gI ) f )

h(g(I f ))

1a

!��

h(g(I f ))

h(g f )

1(1l)

!��

(hg) f (hg) f

1

�" (hg) f

h(g f )

a

�"��(hg) f

(h(gI )) f

(1r �)1

��############(hg) f

((hg)I ) f

r �1��((hg)I ) f

(h(gI )) f

a1

#!

((hg)I ) f

(hg)(I f )a ��

(hg)(I f )

(hg) f

(11)l��

(hg)(I f )

h(g(I f ))

a#!

(h(gI )) f

h((gI ) f )

a

#!

(hg) f

h(g f )

a

#!h(g f )

h((gI ) f )1(r �1) �"��

h((gI ) f )

h(g(I f ))

1a

!��

h(g(I f ))

h(g f )

1(1l)

!��

(hg) f (hg) f1

�� (hg) fh(g f )

a��

h(g f )

h(g f )11 ��

∼=

⇓ρ1

⇓π

∼=

⇓μ

⇓1μ

∼=

4.1 Basic definition 63

Definition 4.2 A tricategory T is strict if each of the adjoint equivalencesa, l, r is the identity adjoint equivalence and the modifications π,μ, λ, ρ aregiven by unique coherence isomorphisms.

Remark 4.3 (Adjoint equivalences) The major difference between the def-inition given by Gordon et al. (1995) and the one given here is that all ofthe equivalences in the definition from Gordon et al. have been replaced withadjoint equivalences. For example, the associator a of Gordon et al. is anequivalence ⊗ ◦ ⊗ × 1 → ⊗ ◦ 1 × ⊗ in the appropriate bicategory; wehave replaced this with an adjoint equivalence which includes a distinguishedpseudo-inverse a� as well as unit and counit isomorphisms that satisfy thetriangle identities.

Remark 4.4 The definition of r has been changed from that of Gordon et al.;our r � here is the r of Gordon et al.. This has been arranged so that the unitisomorphisms always have an identity cell in the source and never in the target.This makes the definition look more symmetrical, but at the price of makingthe definition of lax tricategory less obvious. We will not be considering laxtricategories in this work, but the reader interested in them should keep thischange in mind.

Remark 4.5 (Suppression of constraints) Note that the diagrams abovenever have associations given for their sources and targets as they are merelyshorthand. By the coherence theorem for bicategories, a pasting diagram of2-cells in a bicategory has a unique value once a choice of association has beenmade for the source and target. Unless there is an obvious choice of associa-tion, we will always assume that 1-cells in a bicategory have been associatedby applying the function e used in the construction of the strictification stB inthe previous chapter.

Additionally, the diagrams do not all type-check in the following sense.Written down in equational form, the axioms would take the form of an equa-tion of 2-cells in some bicategory. This equation would not be well-formed,though, as the sources and targets would not always match up to allow adja-cent terms in this equation to be composed. These sources and targets can bemade to match up by appropriately inserting constraint 2-cells which arise aseither the constraint cells in a bicategory or as the constraint cells of a functor.By the coherence theorem for functors, such a pasting diagram has a uniquevalue regardless of how these constraint cells are inserted. It is in this sensethat we interpret the axioms given above.


It should be noted that we must use the full power of coherence for func-tors to interpret the definition as given above. In the same way, the coherencetheorem for functors between tricategories is necessary to interpret Trim-ble’s definition of tetracategory as presented on his website using pastingdiagrams.

Remark 4.6 It should be noted that λ and ρ seem to have a different statusthan μ. In particular, the reader will note that the cells are not categorified ver-sions of bicategory axioms, but instead categorified versions of useful resultsabout constraint cells in a bicategory. See Joyal and Street (1993) for a proofof the one-object versions of these bicategorical results and to see how theyassist in the proof of coherence for monoidal categories. Thus λ and ρ pro-vide an interesting example of how new data arises in the categorificationprocess.

It should be noted, however, that these cells are determined by the rest ofthe data for a tricategory and the requirement that the second and third axiomshold. This can be seen for λ by using the second axiom, setting h = I , andusing unit constraints. These axioms are not redundant, though, and do providenew information, as generating λ and ρ in this fashion does not guarantee thatthe second and third axioms hold, but only that they hold in the special casesused in this strategy for defining λ and ρ.

Remark 4.7 Most of the data for a tricategory can be seen as a direct cat-egorification of the axioms in the definition of bicategory. The datum π isplainly seen to be a categorified version of the Mac Lane pentagon, which iswritten

(1 ∗ a) ◦ a ◦ (a ∗ 1) = a ◦ a

in equational form. The final data consists of three parts, two of which havealready been discussed. The modification μ is a direct categorification of thesingle unit axiom for bicategories:

r ∗ 1 = 1 ∗ l ◦ a.

The axioms are less transparent. The first tricategory axiom is called thenon-abelian 4-cocycle condition. The picture should be familiar to topologistsas K5 and to category theorists as O5. See Stasheff (1963) and Street (1987b)for more discussion of these objects.

4.2 Adjoint equivalences and tricategory axioms 65

The other two axioms were introduced by Gordon et al. (1995), and are nor-malized versions of the cocycle condition. At this point, one might be readyto conjecture that the axioms for a “hands-on” definition of weak n-categorymight take the following form. First, there will always be one associativityaxiom. It should always be a pasting diagram of the shape of K (n + 2).Second, there should always be n − 1 unit axioms, one for each interior posi-tion to place a unit in a string of n + 1 1-cells. In the bicategory case, thisis a single unit axiom concerning the string f I g, while in the tricategorycase we have two unit axioms for f I gh and f g I h. Trimble’s definition alsofollows this convention, having three unit axioms for the definition of tetra-category. The two unit axioms for the exterior positions (I f g and f g I forbicategories, and I f gh, f gh I for tricategories, for example) should be deriv-able from the others; this is well-known for bicategories as mentioned above,and Gordon–Power–Street prove the analogous result for tricategories. As faras I am aware, the corresponding results for tetracategories have not beenproven.

4.2 Adjoint equivalences and tricategory axioms

It should be noted that we have only included axioms for the left adjoints ofthe adjoint equivalences that are the basic data for a tricategory, except in thecase of r where we have only used the right adjoint. Thus the major differ-ence between the definition given here and that of Gordon et al. (1995) is theaddition of specified pseudo-inverses and the necessary units and counits, butwe require these to satisfy no additional axioms. This is not necessary by thetheory of mates in a bicategory.

Mates allow us to define the opposite tricategory of T , T op, and by this wesee that the relevant axioms for the right adjoints (or left adjoint in the caseof r) are already satisfied.

Definition 4.8 Let T be a tricategory. Then the opposite tricategory, denotedT op, is given by the following data. The tricategory T op has the same objectset as T , and

T op(a, b) = T (b, a).

The composition functor ⊗op is given by ⊗ ◦ τ , where τ is the twist iso-morphism. We take the same unit homomorphism. The adjoint equivalencesaop, lop, rop are the opposite adjoint equivalences of a, r, l, in which case we


switch the left and right adjoints and take the new unit to be the inverse of theold counit and the new counit to be the inverse of the old unit. We take theisomorphisms πop, μop to be (π−1)†, (μ−1)†, similarly for λop, ρop.

As a corollary to the computations with mates given earlier, we have thefollowing.

Corollary 4.9 The data for T op given above satisfy the axioms necessary tobe a tricategory.

The general style of definition will then be as follows. All of the data involv-ing 2-cells in a tricategory (i.e., 1-cells in some hom-bicategory) will be given,when appropriate, as adjoint equivalences. The 3-cells isomorphisms betweencomposites of these will be given in terms of the left adjoints whenever pos-sible. Any required 3-cells isomorphisms between the dual data can then beobtained by taking the relevant mates. The axioms for these 3-cells will betreated similarly. It should be noted that, since we are dealing with adjointequivalences, whenever necessary we can take the opposite adjoint equiva-lences by switching the left and right adjoints and modifying the unit andcounit as required.

4.3 Trihomomorphisms and other higher cells

Definition 4.10 Let T and T ′ be tricategories. A trihomomorphism F : T →T ′ consists of the following data subject to the following axioms.

DATA:

• A function obT → obT ′;

• For objects a, b of T , a functor Fab : T (a, b) → T ′(Fa, Fb);

• For objects a, b, c of T , an adjoint equivalence χ : ⊗′ ◦ (F × F) ⇒ F ◦ ⊗with left adjoint shown below:

T (b, c)× T (a, b)

T (a, c)

⊗��

T (b, c)× T (a, b) T ′(Fb, Fc)× T ′(Fa, Fb)F×F �� T ′(Fb, Fc)× T ′(Fa, Fb)

T ′(Fa, Fc)

⊗′��

T (a, c) T ′(Fa, Fc)F

��

χ

��

4.3 Trihomomorphisms and other higher cells 67

• For each object a of T , an adjoint equivalence ι : I ′Fa ⇒ F ◦ Ia with left

adjoint shown below:

1 T ′(Fa, Fa)I ′

Fa ��1

T (a, a)Ia ��

T (a, a)

T ′(Fa, Fa)

F

��$$$$$ι��

• For objects a, b, c, d of T , an invertible modification as pictured below:

T 2

T 1

⊗ ��

T T ′F

��

T 3

T 2

1×⊗��

T 3

T 2

⊗×1��

��

T 2

T

⊗��

T 3 T ′3F×F×F ��

T 2 T ′2F×F ��

T ′3

T ′2

⊗′×1

��

T ′2

T ′⊗′��

ω�� T 2

T

⊗ ��

T T ′F

��

T 2 T ′2F×F ��

T 3

T 2

1×⊗��

T 3 T ′3F×F×F �� T ′3

T ′21×⊗′��

��T ′2

T ′⊗′ ��

T ′3

T ′2

⊗′×1

��

T ′2

T ′⊗′��

⇐a′

⇓χ

⇓1×χ⇓χ×1

⇓χa⇐

• For objects a, b of T , invertible modifications γ and δ as pictured below:

T ′

T ′2I ′×1

(&%%%%%%%%

T ′2

T ′

⊗′

)'&&&&&&&&

T

T ′

H

#!

T T1

�� T

T ′

H

#! T ′ ′1

��

T

T ′

H

#!

T T1

��

T ′

T ′2I ′×1

(&%%%%%%%%

T ′2

T ′

⊗′

)'&&&&&&&&

T

T ′

H

#!

T

T 2

I×1*(��

T 2

T

⊗

�'''''''''T 2

T ′2

H×H

#!

=

⇓l ′⇒ι×1

⇒χ

⇓l

γ ��

T ′ T ′1 ��

T

T ′

H

#!

T

T 21×I !��

T 2

T

⊗�"��

T

T ′

H

#!

T T1 �� T

T ′

H

#!T ′ T ′1 ��

T

T 21×I !��

T 2

T

⊗�"��

T

T ′

H

#!T ′

T ′21×I ′

'%(((((((((

T ′2

T ′

⊗′

��)))))))))

T 2

T ′2

H×H

#!

⇓r �

= ⇓r ′�

⇒1×ι ⇒

χ

δ ��


AXIOMS:

• For all 1-cells (x, y, z, w) ∈ T (d, e) × T (c, d) × T (b, c) × T (a, b), thefollowing equation of modifications holds:

((H f Hg)H j)Hk

(H( f g)H j)Hk

(χ1)1

#!(H( f g)H j)Hk

H(( f g) j)Hk

χ1��*******

H(( f g) j)Hk

H((( f g) j)k)χ ��

H((( f g) j)k)

H(( f (g j))k)

H(a1)��

H(( f (g j))k)

H( f ((g j)k))

Ha

��+++++++

H( f ((g j)k))

H( f (g( jk)))

H(1a)

��((H f Hg)H j)Hk

(H f Hg)(H j Hk)

a��+++++++

(H f Hg)(H j Hk)

H f (Hg(H j Hk))

a��+++++++

H f (Hg(H j Hk)) H f (HgH( jk))1(1χ)

�� H f (HgH( jk))

H f H(g( jk))

1χ

��*******

H f H(g( jk))

H( f (g( jk)))

χ

��*******

H(( f g) j)Hk H( f (gh))HkHa1 �� H( f (gh))Hk H(( f (g j))k)

χ ��

((H f Hg)H j)Hk

(H f (HgH j))Hka1 ��

(H f (HgH j))Hk

H f ((HgH j)Hk)

a

+),,,,,,,

H f ((HgH j)Hk)

H f (Hg(H j Hk))

1a

,*----

----

-

(H f (HgH j))Hk

(H f H(g j))Hk

(1χ)1��.....

(H f H(g j))Hk

H( f (gh))Hk

χ1

��.....(H f H(g j))Hk

H f (H(g j)Hk)

a

��

H f ((HgH j)Hk)

H f (H(g j)Hk)

1(χ1)

��///////

H f (H(g j)Hk)

H f H((g j)k)1χ $#��H f H((g j)k)

H( f ((g j)k))χ $#��H f H((g j)k)

H f H(g( jk))

1Ha

-+000

0000

0000

∼=

⇓ω1

⇓π∼=

⇓ω

∼=

⇓1ω

((H f Hg)H j)Hk

(H( f g)H j)Hk

(χ1)1

#!(H( f g)H j)Hk

H(( f g) j)Hk

χ1��*******

H(( f g) j)Hk

H((( f g) j)k)χ ��

H((( f g) j)k)

H(( f (g j))k)

H(a1)��

H(( f (g j))k)

H( f ((g j)k))

Ha

��+++++++

H( f ((g j)k))

H( f (g( jk)))

H(1a)

��((H f Hg)H j)Hk

(H f Hg)(H j Hk)

a��+++++++

(H f Hg)(H j Hk)

H f (Hg(H j Hk))

a��+++++++

H f (Hg(H j Hk)) H f (HgH( jk))1(1χ)

�� H f (HgH( jk))

H f H(g( jk))

1χ

��*******

H f H(g( jk))

H( f (g( jk)))

χ

��*******

(H( f g)H j)Hk

H( f g)(H j Hk)

a" 111111

(H f Hg)(H j Hk)

H( f g)(H j Hk)

χ1

.,222222222(H f Hg)(H j Hk)

(H f Hg)H( jk)1χ /-��

H( f g)(H j Hk)H( f g)H( jk)1χ0.333

(H f Hg)H( jk)

H( f g)H( jk)

χ1

#!

(H f Hg)H( jk)

H f (HgH( jk))

a

��

H( f g)H( jk)

H(( f g)( jk))

χ

��$$$$$

H(( f g)( jk))

H( f (g( jk)))

Ha" ��

H((( f g) j)k)

H(( f g)( jk))

Ha

1/4444444444444

∼=∼=

⇓ω

∼=

⇓Hπ

⇓ω


• For all 1-cells (x, y) ∈ T (b, c) × T (a, b), the following equation ofmodifications holds:

H f Hg

H( f g)

χ

#!H( f g)

H(( f I )g)

H(r �1)

��$$$$$$$$$$$$$$$$$

H(( f I )g) H( f (Ig))Ha �� H( f (Ig))

H( f g)

H(1l)

��55555555555555555

H f Hg H f Hg1

�� H f Hg

H( f g)

χ

#!

H f Hg

H( f I )Hg

Hr �1

&$666666666666

H( f I )Hg

H(( f I )g)

χ

20777777777777777

H f Hg

(H f I )Hg

r �1

$#��

(H f I )Hg H f (I Hg)a �� H f (I Hg)

H f Hg1l ��(H f I )Hg

(H f H I )Hg(1ι)1 *(��

(H f H I )HgH( f I )Hgχ131 (H f H I )Hg

H f (H I Hg)a ��

H f (H I Hg)

H f H(Ig)1χ

��

H f H(Ig)

H( f (Ig))

χ

&$!!!!

H f (I Hg)

H f (H I Hg)

1(ι1)

#!

H f H(Ig)

H f Hg

1Hl

��

∼=⇓ω

∼=

⇓μ

⇓1γδ1⇒

∼=

H f Hg

H( f g)

χ

#!H( f g)

H(( f I )g)

H(r �1)

��$$$$$$$$$$$$$$$$$

H(( f I )g) H( f (Ig))Ha �� H( f (Ig))

H( f g)

H(1l)

��55555555555555555

H f Hg H f Hg1

�� H f Hg

H( f g)

χ

#!H( f g) H( f g)1

��

⇓Hμ

∼=

From this point, we will refer to trihomomorphisms as merely functorsunless there is a need to draw attention. Since these are the primary kind of1-cells between tricategories, there should be no confusion.

Definition 4.11 A lax funtor F : T → T ′ consists of the same data as atrihomomorphism F : T → T ′ with the following changes:

• each Fab is only a lax functor,

• a lax transformation χ in place of χ ,


• a lax transformation ι in place of ι, and

• the modifications are no longer required to be invertible.

Remark 4.12 Note that there are interesting intermediate definitionsbetween lax functors and functors. One such is a lax homomorphism whichis a lax functor in which each Fab is a functor instead of merely a laxfunctor. Lax homomorphisms are the right notion of morphism between 1-object tricategories viewed as monoidal bicategories for many applications, seeDay and Street (1997), Garner and Gurski (2009), Cheng and Gurski (2011).

Definition 4.13 1. A functor F is locally strict if each Fab is a strict functorbetween bicategories.2. A strict functor is a trihomomorphism F : T → T ′ such that

• F is locally strict,

• χ and ι are the identity adjoint equivalences,

• F maps the adjoint equivalences a, l, r in the source tricategory to the sameadjoint equivalences in the target tricategory,

• and the modifications ω, γ , and δ are given by the diagrams below, whereall unmarked isomorphisms are unique coherence cells arising either fromthe functor ⊗ or the hom-bicategory.

(Fh ⊗′ Fg)⊗′ F f F(h ⊗ g)⊗′ F f

1⊗′1!�

(Fh ⊗′ Fg)⊗′ F f F(h ⊗ g)⊗′ F f1

�� F(h ⊗ g)⊗′ F f F((h ⊗ g)⊗ f )1 �� F((h ⊗ g)⊗ f )

F(h ⊗ (g ⊗ f ))

Fa

��

(Fh ⊗′ Fg)⊗′ F f

Fh ⊗′ (Fg ⊗′ F f )

a′

��Fh ⊗′ (Fg ⊗′ F f ) Fh ⊗′ F(g ⊗ f )

1⊗′1

�"Fh ⊗′ (Fg ⊗′ F f ) Fh ⊗′ F(g ⊗ f )1 �� Fh ⊗′ F(g ⊗ f ) F(h ⊗ (g ⊗ f ))

1��

(Fh ⊗′ Fg)⊗′ F f

F(h ⊗ (g ⊗ f ))

Fa=a′

��∼=∼=

∼=

∼=

I ′ ⊗′ F f F I ⊗′ F f

1⊗′1)'

I ′ ⊗′ F f F I ⊗′ F f1

�� F I ⊗′ F f F(I ⊗ f )1 �� F(I ⊗ f )

F f

Fl=l ′��

I ′ ⊗′ F f

F fl ′

��88888888888888888888888888888888∼=

∼=


F f F f ⊗′ I ′(r ′)� �� F f ⊗′ I ′ F f ⊗′ F I

1⊗′1)'

F f ⊗′ I ′ F f ⊗′ F I1

�� F f ⊗′ F I

F( f ⊗ I )

1��

F f

F( f ⊗ I )(r ′)�=Fr �

��88888888888888888888888888888888∼=

∼=

Remark 4.14 It is clear from the definition above that given a function onobjects F0 and strict functors of hom-bicategories Fab, there is at most onestructure of a strict functor with this underlying data.

Remark 4.15 This definition differs from the definition of strict functorgiven by Gordon et al. (1995) in two ways. First, we require local strict-ness while the original definition did not. Second, the definition given byGordon et al. (1995) requires that the modifications ω, γ , and δ are identities,when this is in fact impossible as their sources do not equal their targets in gen-eral; we have remedied this mistake by requiring these modifications to haveunique coherence isomorphisms as their components.

Definition 4.16 Let F,G : T → T ′ be functors with the same source andtarget. A tritransformation θ : F → G consists of

• a family of 1-cells θa : Fa → Ga of T ′, indexed by the objects of T ,

• adjoint equivalences

T (a, b)

T ′(Ga,Gb)

G��

T (a, b) T ′(Fa, Fb)F �� T ′(Fa, Fb)

T ′(Fa,Gb)

T ′(1,θb)

��T ′(Ga,Gb) T ′(Fa,Gb)

T ′(θa ,1)��

θ

��

in Bicat(T (a, b), T ′(Fa,Gb)) for all objects a, b of T , and

• invertible modifications as shown below. We have abbreviated T (a, b) by[a, b], T (b, c) × T (a, b) by [b, c; a, b], and similarly in T ′; no distinctionis made between T and T ′, as lower case letters such as a, b, c, etc., areobjects in T while Fa,Gb, etc., are objects in T ′.


[b,c;a,b]

[a,c]

⊗

42[a,c]

[Ga,Gc]

G

539999999999999999999999999999

[Ga,Gc]

[Fa,Gc]

T ′(θa ,1)

*(��

[b,c;a,b] [Fb,Fc;Fa,Fb]F×F �� [Fb,Fc;Fa,Fb]

[Fb,Gc;Fa,Fb]

T ′(1,θc)×1

��[Fb,Gc;Fa,Fb]

[Fa,Gc]

⊗��

[b,c;a,b]

[Gb,Gc;Fa,Fb]

G×F

��[Gb,Gc;Fa,Fb]

[Gb,Gc;Fa,Gb]

1×T ′(1,θb)

��[Gb,Gc;Fa,Gb] [Fa,Gc]⊗

��

[Gb,Gc;Fa,Fb] [Fb,Gc;Fa,Fb]��T ′(θb,1)×1

[b,c;a,b]

[Gb,Gc;Ga,Gb]

G×G

64:::::::::::::::::::::::

[Gb,Gc;Ga,Gb] [Gb,Gc;Fa,Gb]1×T ′(θa ,1) ��[Gb,Gc;Ga,Gb]

[Ga,Gc]

⊗

��

�;75

[b,c;a,b]

[a,c]

⊗

42[a,c]

[Ga,Gc]

G

539999999999999999999999999999

[Ga,Gc]

[Fa,Gc]

T ′(θa ,1)

*(��

[b,c;a,b] [Fb,Fc;Fa,Fb]F×F �� [Fb,Fc;Fa,Fb]

[Fb,Gc;Fa,Fb]

T ′(1,θc)×1

��[Fb,Gc;Fa,Fb]

[Fa,Gc]

⊗��

[b,c;a,b]

[Fb,Fc;Fa,Fb]

F×F

��[Fb,Fc;Fa,Fb]

[Fa,Fc]

⊗��

[Fa,Fc] [Fa,Gc]T ′(1,θc) ��[a,c] [Fa,Fc]F

��

⇓θ×1

⇓a1×θ⇐

⇓a�

χ⇐

⇓a

χ⇐

⇓θ

1

T (a,a)

Ia

86T (a,a)

T ′(Ga,Ga)

G

��T ′(Ga,Ga) T ′(Fa,Ga)

T ′(θa ,1)��

1

T ′(Fa,Ga)

θa

86

1

T ′(Fa,Fa)

IFa��

T ′(Fa,Fa)

T ′(Fa,Ga)

T ′(1,θa)

'%((((((((((T (a,a) T ′(Fa,Fa)F��

M � ��

1

T (a,a)

Ia

97��

T (a,a)

T ′(Ga,Ga)

G

��T ′(Ga,Ga) T ′(Fa,Ga)

T ′(θa ,1)��

1

T ′(Fa,Ga)

θa

:8

1

T ′(Ga,Ga)

IGa

��

ι⇐

⇓θ

r�⇐ι⇐

l�⇐

The functor θa is the functor whose value at the single object of 1 is the 1-cellθa and all of whose constraints are given by unique coherence isomorphisms.


This collection of data is subject to the following three axioms.


θF f

(θ I )F f

r �1��

(θ I )F f

(θF I )F f

(1ι)1��

(θF I )F f

(G Iθ)F fθ1

��))))))

(G Iθ)F f

G I (θF f )a *(<<<<<

G I (θF f )

G I (G f θ)

1θ �=====

G I (G f θ)

(G I G f )θ

a�

'%((((((

(G I G f )θ

G(I f )θ

χ1

��

G(I f )θ

G f θ

Gl1

��

θF f

(Iθ)F f

l�1��

(Iθ)F f

I (θF f )a ��

I (θF f )

I (G f θ)

1θ

��

I (G f θ)

G f θ

l

��$$$$$$$$$$$$$

(θF I )F f

θ(F I F f )

a��θ(F I F f )

θF(I f )

1χ��θF(I f )

G(I f )θ

θ��

(θ I )F f

θ(I F f )

a��8888888888

θ(I F f )

θF f

1l��88888888888

θF f G f θθ

��

θ(I F f )

θ(F I F f )

1(ι1)

;9-------

θF(I f )

θF f

1Fl

��θF f θF f

1��

I (θF f )

θF f

l

��

⇓�

∼=

⇓μ

⇓1γ

∼=

∼=⇓λ

θF f

(θ I )F f

r �1��

(θ I )F f

(θF I )F f

(1ι)1��

(θF I )F f

(G Iθ)F fθ1

��))))))

(G Iθ)F f

G I (θF f )a *(<<<<<

G I (θF f )

G I (G f θ)

1θ �=====

G I (G f θ)

(G I G f )θ

a�

'%((((((

(G I G f )θ

G(I f )θ

χ1

��

G(I f )θ

G f θ

Gl1

��

θF f

(Iθ)F f

l�1��

(Iθ)F f

I (θF f )a ��

I (θF f )

I (G f θ)

1θ

��

I (G f θ)

G f θ

l

��$$$$$$$$$$$$$(Iθ)F f

(G Iθ)F f

(ι1)1

:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

I (θF f )

G I (θF f )

ι1

#!

I (G f θ)

G I (G f θ)

ι1

<;

I (G f θ)

(I G f )θ

a�

#!(I G f )θ

(G I G f )θ

(ι1)1

#!

(I G f )θ

G f θ

l1��⇓M1 ∼= ∼=

⇓γ 1

⇓λ

∼=


θF f

G f θ

θ

=<??????

G f θ

G f (θ I )

1r ��

G f (θ I )

G f (θF I )

1(1ι)��$$$$$$$$$$

G f (θF I )

G f (G Iθ)

1θ

��

G f (G Iθ)

(G f G I )θ

a�

��

(G f G I )θ

G( f I )θ

χ1

��

θF f

θF f1 /- θF f

G f θ

θ

��

G f θ

G( f I )θ

Gr �1

��)))))))))

θF f(θF f )I

r �/-@@@@@@@@@@(θF f )I

(G f θ)I

θ1

>=AAAAA

G f θ(G f θ)I

r �� (G f θ)I

G f (θ I )

a

#!

(θF f )I

(θF f )F I

1ι

�"��

(θF f )F I

(G f θ)F I

θ1?>44444

(G f θ)F I

G f (θF I )

a

.,22222222

(G f θ)I

(G f θ)F I1ι �"��

(θF f )I

θ(F f I )

a ��

θ(F f I )

θ(F f F I )

1(1ι)@?��

(θF f )F I

θ(F f F I )

aA@BBBB

θ(F f I )

θF f

1r BACCCCCC

θF f

θF( f I )

1Fr ��

θ(F f F I )

θF( f I )

1χ!��

θF( f I )G( f I )θθ ��

∼=

⇓ρ

∼=

∼=

∼=

⇓1δ ∼=⇓ρ

⇓�

θF f

G f θ

θ

=<??????

G f θ

G f (θ I )

1r ��

G f (θ I )

G f (θF I )

1(1ι)��$$$$$$$$$$

G f (θF I )

G f (G Iθ)

1θ

��

G f (G Iθ)

(G f G I )θ

a�

��

(G f G I )θ

G( f I )θ

χ1

��

θF f

θF f1 /- θF f

G f θ

θ

��

G f θ

G( f I )θ

Gr �1

��)))))))))

G f θ

G f (Iθ)1l� �� G f (Iθ)

G f (G Iθ)

1(ι1)

�"��

G f θ

G f θ

1

!�DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD

G f (Iθ)

(G f I )θ

a�

�BEEEEEEE

(G f I )θ

G f θ

r1

��(G f I )θ

(G f G I )θ(1ι)1 ��

⇓1M

∼=

⇓μ∼=

⇓δ1

Once again, we will generally refer to these as transformations, omitting theprefix tri-.

Definition 4.17 A lax transformation θ : F → G between lax functors con-sists of the same data as a transformation between functors with the followingchanges:

• lax transformations θ in place of the adjoint equivalences θ and• the modifications are no longer required to be invertible.


Definition 4.18 Let θ and φ be transformations with the same source F andtarget G. A trimodification m : θ ⇒ φ consists of a family of 2-cells ma :θa ⇒ φa in the target tricategory T ′, indexed by the objects of the sourcetricategory T , and invertible modifications

T (a,b)

T ′(Ga,Gb)

G��

T (a,b) T ′(Fa,Fb)F �� T ′(Fa,Fb)

T ′(Fa,Gb)

T ′(1,θb)��

T ′(Ga,Gb) T ′(Fa,Gb)T ′(θa ,1) ��

θ

��

T ′(Ga,Gb) T ′(Fa,Gb)

T ′(φa ,1)

=<(ma)

∗��

m ��

T (a,b) T ′(Fa,Fb)F ��T (a,b)

T ′(Ga,Gb)

G

��

T ′(Fa,Fb)

T ′(Fa,Gb)

T ′(1,φb)

��T ′(Ga,Gb) T ′(Fa,Gb)

T ′(φa ,1)��

T ′(Fa,Fb)

T ′(Fa,Gb)

T ′(1,θb)

CC

(mb)∗DDφ

62 FFFFFFFFFF

FFFFFFFFFF

such that the following two axioms hold, where we have written tensor asconcatenation and all unmarked isomorphisms are naturality isomorphisms.This collection of data is subject to the following two axioms.

(θF f )Fg (G f θ)Fgθ1 �� (G f θ)Fg G f (θFg)

a �� G f (θFg) G f (Ggθ)1θ �� G f (Ggθ) (G f Gg)θ

a�� (G f Gg)θ G( f g)θ

χ1 �� G( f g)θ

G( f g)φ

G1m

��

(θF f )Fg

(φF f )Fg

(m F1)F1

��(φF f )Fg

φ(F f Fg)

a!�DDDDDDDDDDDDDD

φ(F f Fg) φF( f g)1χ

�� φF( f g)

G( f g)φ

φ

�"��

(φF f )Fg (G f φ)Fgφ1 �� (G f φ)Fg G f (φFg)

a��G f (φFg) G f (Ggφ)

1φ�� G f (Ggφ) (G f Gg)φ

a�� (G f Gg)φ G( f g)φ

χ1��

(G f θ)Fg

(G f φ)Fg

(G1m)F1

��

G f (θFg)

G f (φFg)

G1(m F1)

��

G f (Ggθ)

G f (Ggφ)

G1(G1m)

��

(G f Gg)θ

(G f Gg)φ

(G1G1)m

��

(θF f )Fg (G f θ)Fgθ1 �� (G f θ)Fg G f (θFg)

a �� G f (θFg) G f (Ggθ)1θ �� G f (Ggθ) (G f Gg)θ

a�� (G f Gg)θ G( f g)θ

χ1 �� G( f g)θ

G( f g)φ

G1m

��

(θF f )Fg

(φF f )Fg

(m F1)F1

��(φF f )Fg

φ(F f Fg)

a!�DDDDDDDDDDDDDD

φ(F f Fg) φF( f g)1χ

�� φF( f g)

G( f g)φ

φ

�"��

(θF f )Fg

θ(F f Fg)

a

!�DDDDDDDDDDDDDD

θ(F f Fg) θF( f g)1χ

�� θF( f g)

G( f g)θ

θ

�"��θ(F f Fg)

φ(F f Fg)

m(F1F1)

��

θF( f g)

φF( f g)

m F(11)

��

⇓m1 ∼= 1m⇓ ∼= ∼=

⇓�

⇓�

∼= ⇓m

∼=


θ θ Ir �

�� θ I θF I1ι �� θF I G Iθ

θ �� G Iθ

G Iφ

1m��

θ

φ

m

��φ

Iφ

l��GGGGGGGGGGGGGGGGGGG

Iφ

G Iφ

ι1

��HHHHHHHHHHHHHHHHHH

φ φ Ir �

�� φ I 1ι1ι

�� 1ι G Iφφ ��

θ I

φ I

m1��

θF I

1ι

m1

��

θ θ Ir �

�� θ I θF I1ι �� θF I G Iθ

θ �� G Iθ

G Iφ

1m��

θ

φ

m

��φ

Iφ

l��GGGGGGGGGGGGGGGGGGG

Iφ

G Iφ

ι1

��HHHHHHHHHHHHHHHHHH

θ

Iθ

l��

Iθ

G Iθ

ι1

��Iθ

Iφ

1m

��

∼= ∼= ⇓m

⇓M

⇓M

∼= ∼=

Remark 4.19 Note that in equational form, the modifications m above canbe written as

m : (ma)∗ ∗ 1G ◦ θ ⇒ φ ◦ (mb)∗ ∗ 1F

in the appropriate hom-bicategory. This is how the data is presented byGordon et al. (1995).

Definition 4.20 A lax modification consists of the same data as a modifica-tion, without the requirement that the modifications m be invertible.

Definition 4.21 A perturbation σ : m � n between trimodifications withthe same source and target consists of a family of 3-cells σa : ma � na in thetarget tricategory T ′, indexed by objects of the source tricategory T , such thatthe following axiom holds.

θ ⊗ F f G f ⊗ θθ �� G f ⊗ θ

G f ⊗ φ

1⊗m

��

θ ⊗ F f

φ ⊗ F f

m⊗1

��φ ⊗ F f G f ⊗ φ

φ��

θ ⊗ F f

φ ⊗ F f

n⊗1

53

θ ⊗ F f G f ⊗ θθ �� G f ⊗ θ

G f ⊗ φ

1⊗n

��

G f ⊗ θ

G f ⊗ φ

1⊗m

EE

θ ⊗ F f

φ ⊗ F f

n⊗1

��φ ⊗ F f G f ⊗ φ

φ��

σ⊗1⇐ 1⊗σ⇐m

��

��n

��

��


Remark 4.22 This equation is presented by Gordon et al. (1995) as theequality of modifications shown below.

(θb)∗◦F (θa)∗◦G

θ �� (θa)∗◦G

(φa)∗◦G

(ma)∗∗1

��

(θb)∗◦F

(φb)∗◦FFF

(φb)∗◦F (φa)∗◦G

φ��

(mb)∗∗1

⇐m

⇐(σb)∗∗1

(θb)∗◦F

(φb)∗◦F

(nb)∗∗1

'%

(θb)∗◦F (θa)∗◦G

θ ��

(φb)∗◦F (φa)∗◦G

φ��

(θb)∗◦F

(φb)∗◦F

(nb)∗∗1

��

(θa)∗◦G

(φa)∗◦G

(ma)∗∗1

FF

(θa)∗◦G

(φa)∗◦G

'%(na)

∗∗1

⇐n ⇐

(σa)∗∗1

=

Since two modifications are equal if and only if they have the same com-ponents, the equation given in the definition and the one here are equivalentaxioms.

Notation 4.23 As a reminder, we shall drop the prefixes bi- and tri- whenthe context is understood. Thus both homomorphisms of bicategories andtrihomomorphisms of tricategories will be called functors, with weak beingunderstood; a lax map will always be called such.

4.4 Unpacked versions

This section will give unpacked versions of the previous definitions. We haveincluded this section both as a reference and to display all of the constraintcells necessary for the construction of the free tricategory. Only data will beunpacked as the formulas for the axioms are already presented as the equal-ity of pasting diagrams using the cells from the unpacked definitions. Anunpacked version of the definition of perturbation is not given, as the origi-nal definition is already maximally unpacked.

Unpacked tricategoriesA tricategory T has the data of

• a set obT of objects,• for each pair of objects a, b, a bicategory T (a, b),• for each triple of objects a, b, c, a functor

⊗ : T (b, c)× T (a, b) → T (a, c)

which includes isomorphisms

(β ′ ⊗ α′) ∗ (β ⊗ α) ∼= (β ′ ∗ β)⊗ (α′ ∗ α),1g ⊗ 1 f ∼= 1g⊗ f ,

4.4 Unpacked versions 79

• for each object a, an object Ia ∈ T (a, a) and a morphism ia : Ia → Ia

along with an isomorphism ia ∼= 1Ia ,

• for each triple of composable 1-cells h, g, f , 2-cells

ahg f : (h ⊗ g)⊗ f → h ⊗ (g ⊗ f )a�

hg f : h ⊗ (g ⊗ f ) → (h ⊗ g)⊗ f

and invertible 3-cells

εahg f : ahg f ∗ a�

hg f∼= 1h⊗(g⊗ f )

ηahg f : 1(h⊗g)⊗ f ∼= a�

hg f ∗ ahg f ,

• for each pair of triples of composable 1-cells, h, g, f and h′, g′, f ′, and atriple of 2-cells between them γ, β, α, invertible 3-cells (natural in γ, β, α)

aγ,β,α : ah′g′ f ′ ∗ (γ ⊗ β)⊗ α ∼= γ ⊗ (β ⊗ α) ∗ ahg f

a�γ,β,α : a�

h′g′ f ′ ∗ γ ⊗ (β ⊗ α) ⇒ (γ ⊗ β)⊗ α ∗ a�hg f ,

• for each 1-cell f , 2-cells

l f : Ib ⊗ f → fl�f : f → Ib ⊗ f

r f : f ⊗ Ia → fr �

f : f → f ⊗ Ia


εlf : l f l�f ⇒ 1 f

ηlf : 1Ib⊗ f ⇒ l�f l f

εrf : r f r �

f ⇒ 1 f

ηrf : 1 f ⊗Ia ⇒ r �

f r f ,

• for each pair of 1-cells f, f ′ and 2-cell between them α, invertible 3-cells(natural in α)

lα : l f ′ ∗ (1 ⊗ α) ⇒ α ∗ l f

l�α : l�f ′ ∗ α ⇒ (1 ⊗ α) ∗ l�frα : r f ′ ∗ (α ⊗ 1) ⇒ α ∗ r f

r �α : r �

f ′ ∗ α ⇒ (α ⊗ 1) ∗ r �f ,


• for every quadruple of composable 1-cells j, h, g, f , an invertible 3-cell asdisplayed below,

(( j⊗h)⊗g)⊗ f

( j⊗(h⊗g))⊗ f

a⊗1*(��

( j⊗(h⊗g))⊗ f j⊗((h⊗g)⊗ f )a �� j⊗((h⊗g)⊗ f )

j⊗(h⊗(g⊗ f ))

1⊗a

��

(( j⊗h)⊗g)⊗ f

( j⊗h)⊗(g⊗ f )

a��GGGGGGGGGGGGGGG

( j⊗h)⊗(g⊗ f )

j⊗(h⊗(g⊗ f ))

a

$#��

π jhg f

��

• and for every pair of composable 1-cells f, g, invertible 3-cells as displayedbelow.

g⊗ f

(g⊗I )⊗ f

r �⊗1@?::::::::

(g⊗I )⊗ f g⊗(I⊗ f )a �� g⊗(I⊗ f )

g⊗ f

1⊗l

�BIIIIIIII

g⊗ f g⊗ f1

��

μg f

��

(I⊗g)⊗ f g⊗ fl⊗1 ��(I⊗g)⊗ f

I⊗(g⊗ f )

a!�JJJJJJJJJJJ

I⊗(g⊗ f )

g⊗ f

l

�"KKKKKKKKKKKλg f��

g⊗ f g⊗( f ⊗I )1⊗r �

��g⊗ f

(g⊗ f )⊗I

r �!�JJJJJJJJJJJ

(g⊗ f )⊗I

g⊗( f ⊗I )

a

�"KKKKKKKKKKKρg f

��

Unpacked functorsLet S, T be tricategories. A functor F : S → T has the data of

• a function obF : obS → obT ,

• for each pair of objects a, b in S, a functor

Fab : S(a, b) → T (Fa, Fb),

• for all pairs of composable 1-cells f, g in S, 2-cells in T

χg f : Fg ⊗′ F f → F(g ⊗ f )χ �

g f : F(g ⊗ f ) → Fg ⊗′ F f

4.4 Unpacked versions 81


εχg f : χg f ∗ χ �

g f ⇒ 1F(g⊗ f )

ηχg f : 1Fg⊗′ F f ⇒ χ �

g f ∗ χg f ,

• for all pairs of pairs of composable 1-cells, f, g and f ′, g′, and all pairs of2-cells between them β, α, invertible 3-cells (natural in β, α) as displayedbelow,

Fg ⊗′ F f F(g ⊗ f )χ ��Fg ⊗′ F f

Fg′ ⊗′ F f ′

Fβ⊗′ Fα��

Fg′ ⊗′ F f ′ F(g′ ⊗ f ′)χ

��

F(g ⊗ f )

F(g′ ⊗ f ′)

F(β⊗α)��

χβα�(��

��

F(g ⊗ f ) Fg ⊗′ F fχ �

�� Fg ⊗′ F f

Fg′ ⊗′ F f ′

Fβ⊗′ Fα��

F(g′ ⊗ f ′) Fg′ ⊗′ F f ′χ �

��

F(g ⊗ f )

F(g′ ⊗ f ′)

F(β⊗α)��

χ �βα

�(��

��

• for all objects a in S, 2-cells

ιa : IFa → F Ia

ι�a : F Ia → IFa


ειa : ιa ∗ ι�a ⇒ 1F Ia

ηιa : 1IFa ⇒ ι�a ∗ ιaι : ιa ∗ ia ⇒ Fia ◦ ιaι� : ι�a ∗ Fia ⇒ ia ◦ ι�a,

• for every triple of composable 1-cells h, g, f in S, an invertible 3-cell in Tas displayed below,

(Fh ⊗′ Fg)⊗′ F f F(h ⊗ g)⊗′ F fχ⊗′1 �� F(h ⊗ g)⊗′ F f F((h ⊗ g)⊗ f )

χ �� F((h ⊗ g)⊗ f )

F(h ⊗ (g ⊗ f ))

Fa

��

(Fh ⊗′ Fg)⊗′ F f

Fh ⊗′ (Fg ⊗′ F f )

a′

��Fh ⊗′ (Fg ⊗′ F f ) Fh ⊗′ F(g ⊗ f )

1⊗′χ�� Fh ⊗′ F(g ⊗ f ) F(h ⊗ (g ⊗ f ))

χ��

ωhg fG4 LLLLLLLLLLL

LLLLLLLLLLL


• for every 1-cell f in S, two invertible 3-cells as displayed below.

I ′ ⊗′ F f F I ⊗′ F fι⊗′1 �� F I ⊗′ F f F(I ⊗ f )

χ �� F(I ⊗ f )

F f

Fl

��

I ′ ⊗′ F f

F f

l ′��88888888888888888888888888888888888

γ fH7 ��

F f F f ⊗′ I ′(r ′)� �� F f ⊗′ I ′ F f ⊗′ F I1⊗′ι �� F f ⊗′ F I

F( f ⊗ I )

χ

��

F f

F( f ⊗ I )

Fr ��88888888888888888888888888888888888

δ f

/��

Unpacked transformationsLet F,G : S → T be functors. Then the data of a transformation α : F → Gconsists of

• for every object a of S, a 1-cell αa : Fa → Ga in T ,

• for every 1-cell f : a → b in S, 2-cells

α f : αb ⊗′ F f → G f ⊗′ αa

α�f : G f ⊗′ αa → αb ⊗′ F f

and invertible 3-cells as displayed below,

εαf : α f ∗ α�f ⇒ 1G f ⊗′αa

ηαf : 1αb⊗′ F f ⇒ α�f ∗ α f ,

• for every 2-cell θ : f → g in S, an invertible 3-cell (natural in θ ) in T asdisplayed below,

αb ⊗′ F f G f ⊗′ αaα f ��αb ⊗′ F f

αb ⊗′ Fg

1⊗′ Fθ��

αb ⊗′ Fg Gg ⊗′ αaαg��

G f ⊗′ αa

Gg ⊗′ αa

Gθ⊗′1��

αθ

$G///////

///////

4.5 Calculations in tricategories 83

• for every pair of composable 1-cells g, f in S, an invertible 3-cell in T asdisplayed below,

(αc⊗′ Fg)⊗′ F f (Gg⊗′αb)⊗′ F fαg⊗′1 �� (Gg⊗′αb)⊗′ F f Gg⊗′(αb⊗′ F f )

a′�� Gg⊗′(αb⊗′ F f ) Gg⊗′(G f ⊗′αa)

1⊗′α f �� Gg⊗′(G f ⊗′αa)

(Gg⊗′G f )⊗′αa

(a′)��

(Gg⊗′G f )⊗′αa

G(g⊗ f )⊗′αa

χG⊗′1��

(αc⊗′ Fg)⊗′ F f

αc⊗′(Fg⊗′ F f )

a′

��αc⊗′(Fg⊗′ F f ) αc⊗′ F(g⊗ f )

1⊗′χ F�� αc⊗′ F(g⊗ f ) G(g⊗ f )⊗′αaαg⊗ f

��

�g f

IF KKKKKKKKKKKKKKKKKKK

KKKKKKKKKKKKKKKKKKK

• and for every object a in S, an invertible 3-cell as displayed below.

αa αa ⊗′ I ′(r ′)� �� αa ⊗′ I ′ αa ⊗′ F I1⊗′ιF �� αa ⊗′ F I G I ⊗′ αa

αI ��αa

I ′ ⊗′ αa

(l ′)� ��

I ′ ⊗′ αa

G I ⊗′ αa

ιG⊗′1

��Ma��

Unpacked modificationsLet α, β : F → G be transformations. Then the data for a modification m :α ⇒ β consists of

• for every object a of S, a 2-cell ma : αa → βa of T and

• for every 1-cell f : a → b in S, an invertible 3-cell in T as displayed below.

αb ⊗′ F f

G f ⊗′ αaα f �"��

G f ⊗′ αa

G f ⊗′ βa

1⊗′ma

!��

αb ⊗′ F f

βb ⊗′ F fmb⊗′1 !��

βb ⊗′ F f

G f ⊗′ βa

β f

�"��

m f

��

4.5 Calculations in tricategories

The calculations involving three-dimensional pasting diagrams often lookquite daunting, but are generally not more complicated than similar argumentsfor two-dimensional pasting diagrams. In fact, we have represented all of the3-pasting diagrams in a two-dimensional fashion; in doing so, we have takenthe notation (⊗) for composition along 0-cell boundaries somewhat seriously,treating it as we would a monoidal structure. This fits perfectly with the ideathat a tricategory with a single object is a “suspended” monoidal bicategory.


The vast majority of the calculational proofs that we need can be obtained inthe standard way, using

• functoriality of ⊗,• the naturality of a, l, r ,• the modification axioms for π,μ, λ, ρ,• the calculus of mates, when necessary, and• the tricategory axioms.

In addition, some results might require composing an entire diagram with anidentity 1-cell on one side or the other. The most important calculations of thistype were already carried out by Gordon et al. (1995), and are summarized inthe following proposition.

Proposition 4.24 Let T be a tricategory. Then the following equations of3-cells hold.

f (gh) f (g(hI ))1(1r �) ��

( f g)h

f (gh)

a

#!

( f g)h

(( f g)h)Ir � ��

(( f g)h)I

( f g)(hI )

a

��

( f g)(hI )

f (g(hI ))

a

#!f (gh)

( f (gh))I

r ��

(( f g)h)I

( f (gh))I

a1

#!( f (gh))I

f ((gh)I )

a��

f (gh)

f ((gh)I )

1r ��88888888888

f ((gh)I )

f (g(hI ))

1a

��

( f g)h

f (gh)

a

#!f (gh) f (g(hI ))1(1r �) ��

( f g)h

(( f g)h)Ir � �B"""""

(( f g)h)I

( f g)(hI )

a

@?#####

( f g)(hI )

f (g(hI ))

a

#!

( f g)h ( f g)(hI )1r ��

∼=⇓ ρ

⇓ 1ρ

⇓ π

⇓ ρ

∼=

((I f )g)h

(I f )(gh)

a

��(I f )(gh)

I ( f (gh))a ��

I ( f (gh))

f (gh)

l

*(��

((I f )g)h f (gh)(l1)1 �� f (gh)

f (gh)

a

��

((I f )g)h

(I ( f g))ha1��

(I ( f g))h

I (( f g)h)a ��

I (( f g)h)

I ( f (gh))

1a��

(I ( f g))h

f (gh)l1 ��MMMMMMMMMMMM

I (( f g)h)

f (gh)

l

��

((I f )g)h

(I f )(gh)

a

��(I f )(gh)

I ( f (gh))a �B"""""

I ( f (gh))

f (gh)

l

@?#####

((I f )g)h ( f g)h(l1)1 �� ( f g)h

f (gh)

a

��(I f )(gh) f (gh)

l1 ��∼=

⇓ λ

⇓ λ1

⇓ π

⇓ λ

∼=

I f

f

l

#!f f Ir ��

I f

(I f )Ir � " ��

(I f )I

I ( f I )

a

��

I ( f I )

f I

l

#!

(I f )I

f I

l1

��

I f

f

l

#!f f Ir ��

I f

(I f )Ir � " ��

(I f )I

I ( f I )

a

��

I ( f I )

f I

l

#!

I f I ( f I )1r ��

∼= ⇓ λ

∼=

⇓ ρ

4.6 Comparing definitions 85

We will not give a proof of this proposition, as it is long, not particularlyenlightening, and the proof of Gordon et al. (1995) is likely as simple aspossible. Gordon et al. did these calculations under the assumption that the tri-category in question was cubical. This is not a necessary assumption, althoughit does make the calculations somewhat simpler (although still far from sim-ple). Adding these equations to the above list of calculational tools, we obtainall of the techniques necessary to perform the tricategorical calculations thatwill be mentioned but not explicitly worked in future proofs; whenever a moreinteresting technique is required, it will be explained.

4.6 Comparing definitions

This section will briefly compare the definitions given here with those givenby Gordon et al. (1995). The only way in which we have changed the defini-tions of Gordon et al. is by adding additional data, so that the “pseudo-naturalequivalences” of Gordon et al. have been replaced with ajoint equivalences inthe functor bicategory. Thus it is obvious that every tricategory in our sense(we shall refer to these as algebraic tricategories) gives rise to a tricategory inthe sense of Gordon, Power, and Street by neglect of structure. Given a tricat-egory in the sense of Gordon, Power, and Street, it is possible to construct analgebraic tricategory by choosing adjoint equivalences. There is no canonicalchoice, but any two such choices are related by a functor that is the identityon objects and hom-bicategories. It is similarly easy to see that every functorin our sense gives rise to a functor in the sense of Gordon, Power, and Streetby neglect of structure, and every functor in the sense of Gordon, Power, andStreet can be non-canonically given additional data to produce a functor in oursense.

Every transformation in our sense also gives a transformation in the senseof Gordon, Power, and Street, but the converse is more delicate. Since the def-inition of transformation involves the cells a�, the definition given by Gordon,Power, and Street is ambiguous; in fact, for any choice of adjoint equivalencea �eq b, one can define a notion of transformation. Since any two such choicesare uniquely isomorphic as adjoints of a, it is always possible to produce,from a transformation using an adjoint equivalence a �eq b, a transforma-tion using our chosen adjoint equivalence a �eq a�. Using this procedure, wecan once again produce from a transformation of Gordon, Power, and Street atransformation as defined here.

The definitions given here of modification and perturbation are exactly thesame as those given by Gordon et al.

5

Examples

In this chapter, we will explicitly construct two example tricategories. Oneof these examples will be useful later, and the other is just basic, but impor-tant. The main common feature of these examples is that they can, withoutmuch effort, be constructed directly, without any sophisticated understandingof tricategories.

5.1 Primary example: Bicat

This section will establish two key results. The basic result is that the collectionof bicategories, functors, transformations, and modifications forms a tricate-gory. This will be shown directly by calculation. Later, we will also be ableprove it by transporting the tricategory structure from the tricategory Gray.This will also prove that Bicat is triequivalent to an easily determined fullsub-Gray-category of Gray which we call Gray′.

It should also be noted that there are two natural tricategory structures onthe collection of bicategories, functor, transformations, and modification; thisbecomes clear when defining the horizontal composite of transformations,as there are two obvious choices and a canonical comparison map betweenthem. This bifurcation will be noted, but it will not be important to the theorydeveloped here. The line of proof followed here is largely calculational.

The first piece of data we must construct is the hom-bicategory Bicat(A, B)for bicategories A and B. It has objects the functors F : A → B, 1-cells thetransformations α : F ⇒ G, and 2-cells the modifications : α � β. Wewill not construct this bicategory explicitly, but only mention that the structureconstraints in it are obtained from the structure constraints of the target B. Theunit IA : 1 → Bicat(A, A) is given by a functor whose value on the uniqueobject of 1 is the identity functor idA : A → A.

86

5.1 Primary example: Bicat 87

Proposition 5.1 There is a functor

⊗ : Bicat(B,C)× Bicat(A, B) → Bicat(A,C)

whose function on objects is given by G ⊗ F = G F.

Proof We have defined ⊗ on objects, now we must define it on hom-categories. Let α : F ⇒ F ′ and β : G ⇒ G ′ be transformations. Thenwe define the transformation G ⊗ α : G F ⇒ G F ′ to have its component at aas Gαa and its component at f : a → a′ to be the 2-cell

Gαa′ ∗ G F fφ−→ G(αa′ ∗ F f )

Gα f−→ G(F ′ f ∗ αa)φ−1

−→ G F ′ f ∗ Gαa,

where φ is the structure constraint for G. It is easy to check that this is atransformation with the claimed source and target. We similarly define β ⊗ F .Now define

β ⊗ α := (G ′ ⊗ α) ∗ (β ⊗ F).

This transformation has as its component at a the 1-cell

G ′αa ∗ βFa .

Thus given modifications : α � α′ and : β � β ′, we define ⊗ to bethe modification with component

( ⊗ )a = G ′a ∗ Fa .

It is a simple matter to check that this does define a modification with sourceβ ⊗ α and target β ′ ⊗ α′.

These assignments preserve composition of modifications and preserveidentities by the interchange law. Thus we have defined a functor on hom-categories, so the next step is to give structure constraints.

Let α′ : F ′ ⇒ F ′′ and β ′ : G ′ ⇒ G ′′ be transformations. We must providean isomorphism modification between (β ′⊗α′)∗(β⊗α) and (β ′ ∗β)⊗(α′ ∗α).The first of these transformations has component

(G ′′α′a ∗ β ′

F ′a) ∗ (G ′αa ∗ βFa)

at a, while the second has component

G ′′(α′a ∗ αa) ∗ (β ′

Fa ∗ βFa)

88 Examples

at a. The structure constraint for composition is the modification ⊗2 with com-ponent at a given by the following composite, where coherence 2-cells areunmarked isomorphisms.

(G ′′α′a ∗ β ′

F ′a) ∗ (G ′αa ∗ βFa) ∼= G ′′α′a ∗ ((β ′

F ′a ∗ G ′αa) ∗ βFa)1∗(β ′

G′αa∗1)

−→ G ′′α′a ∗ ((G ′′αa ∗ β ′

Fa) ∗ βFa)∼= (G ′′α′a ∗ G ′′αa) ∗ (β ′

Fa ∗ βFa)

φ−1∗1−→ G ′′(α′a ∗ αa) ∗ (β ′

Fa ∗ βFa).

A lengthy calculation shows that this is a modification; it is clearly invertible.The constraint cell for the identity is constructed similarly.

Finally, we must check the functor axioms. These follow directly fromcoherence and the transformation axioms.

Remark 5.2 Note that we could have defined β ⊗ α by the formula

(β ⊗ F ′) ∗ (G ⊗ α).

This has the effect of giving Bicat an opcubical composition instead of thecubical one defined here. The rest of the results of this section can be reformu-lated in terms of this composition, giving a different tricategory structure onbicategories, functors, transformations, and modifications. We will refer to thistricategory structure as Bicat∗.

Proposition 5.3 There is an adjoint equivalence a : ⊗ (⊗ × 1) ⇒ ⊗(1 × ⊗)with the component of a at the object (H,G, F) being the identity transforma-tion and the component of a� at (H,G, F) being the identity transformation.

Proof We need only give each component at a triple (γ, β, α) of transforma-tions, check that this does give the claimed transformation, provide the unitand counit modifications, and check the triangle identities. The modificationaγβα has component at a given by the following composite.

id ◦ (H ′G ′αa ◦ (H ′βFa ◦ γG Fa)) ∼= (H ′G ′αa ◦ H ′βFa) ◦ γG FaφH ∗1−→

H ′(G ′αa ◦ βFa) ◦ γG Fa ∼= (H ′(G ′αa ◦ βFa) ◦ γG Fa) ◦ id.

This is easily shown to be a modification, and the component a�γβα is defined

similarly. The transformation axioms follow from coherence for functors as allthe 2-cells involved are constraint cells.

Now we must define the unit and counit of this adjoint equivalence. Theseare modifications 1 � a�a and aa� � 1; both are given by coherence cells,from which the triangle identities follow immediately.

5.2 Fundamental 3-groupoids 89

Remark 5.4 It is clear that we have actually constructed invertible iconsa, a�, and then realized them as actual transformations in the proof above. Infact, the previous proposition combined with the next three results essentiallyconstitute a proof that there is a functor of tricategories

Icon → Bicat

from the 2-category of bicategories, functors, and icons, seen as a discrete tri-category, to the full tricategory of bicategories which is the identity on objectsand 1-cells.

We state the next two propositions without proof, as they follow from similararguments as the previous propositions.

Proposition 5.5 There is an adjoint equivalence l : ⊗(IA × 1) ⇒ 1 withthe component of l at the object F being the identity transformation and thecomponent of l� at the object F being the identity.

There is an adjoint equivalence r : ⊗(1 × IA) ⇒ 1 with the component ofr at the object F being the identity transformation and the component of r � atthe object F being the identity.

Proposition 5.6 There are invertible modifications π,μ, λ, ρ as in the def-inition of a tricategory with the component of each at any object being themodification given by unique coherence isomorphisms.

Theorem 5.7 The data provided above gives a tricategory structure on thecollections of bicategories, functors, transformations, and modifications.

Proof All three axioms follow from the observation that for any of the modi-fications involved, the components are all given by constraint cells in the targetbicategory. Thus coherence implies that all necessary diagrams commute.

The last two results of this section are presented without proof. They willnot be used in the remainder of this work.

Lemma 5.8 There is a biequivalence Bicat(A, B) → Bicat∗(A, B) which isthe identity on objects.

Theorem 5.9 There is a triequivalence Bicat → Bicat∗ which is the identityon 0- and 1-cells.

5.2 Fundamental 3-groupoids

There are two fundamental examples of tricategories that arise directly fromspaces. The first is the tricategory whose objects are spaces, whose 1-cells are

90 Examples

continuous maps, and whose higher cells are appropriate kinds of mediatinghomotopies. This tricategory was constructed completely by Gurski (2011).The second such example is the fundamental 3-groupoid of a given space X ;its objects are the points in X , and the higher cells are paths, two-dimensionaldisks, and equivalence classes of three-dimensional disks in X . This is theexample that we construct here.

Fix a space X . The tricategory �3 X will have as its objects the points of X .A 1-cell f : x → y in �3 X is a path from x to y, hence a continuous mapf : I → X such that f (0) = x, f (1) = y. A 2-cell α : f ⇒ g in �3 Xis a homotopy from f to g, relative to x, y. This means that α consists of acontinuous function α : I × I → X such that

• α(0, t) = f (t),• α(1, t) = g(t),• α(s, 0) = x, and• α(s, 1) = y,

for all s, t ∈ I . Put another way, each α(s,−) is also a path from x to y.Finally, a 3-cell [] : α � β in �3 X is an equivalence class of homotopies,where

• : I × I × I → X is a continuous function such that (0,−,−) = α,(1,−,−) = β, and each (r,−,−) is a 2-cell f ⇒ g, and

• ∼ if there exists a continuous H : I 4 → X such that H(0,−,−,−) =, H(1,−,−,−) = , and each H(q,−,−,−) : I 3 → X is a continuousfunction satisfying the conditions placed upon above.

Proposition 5.10 The 1-cells, 2-cells, and 3-cells of �3 X with fixed 0-cellsource x and fixed 0-cell target y form a bicategory, �3 X (x, y).

Proof First, fix 1-cells f, g : x → y; we will show that the 2-,3-cells withf as 1-cell source and g as 1-cell target form a category. If α : f ⇒ g, thenthe identity 1α is given by 1α(r, s, t) = α(s, t); it is clear that this functionrepresents an equivalence class of 3-cells [1α] : α � α. Given [] : α �β, [ ] : β � γ , we define the composite

[ ] ◦ [] : α � γ

to be represented by the function

� ={

(2r,−,−) 0 ≤ r ≤ 12

(2r − 1,−,−) 12 ≤ r ≤ 1.

It is clear that if ∼ ′ and ∼ ′, then [ ] ◦ [] = [ ′] ◦ [′]; therequired function H : I 4 → X can be obtained by patching together the


functions H, H which show ∼ ′, ∼ ′ in the obvious way. Thestandard associativity and unit homotopies show that � satisfies the axioms forcomposition in a category up to homotopy, and these homotopies show that thecomposition ◦ on equivalence classes gives the set of 2- and 3-cells with fixedsource and target the structure of a category.

The identity 1 f : f ⇒ f is given by the function 1 f (s, t) = f (t); it is clearthat this is a valid 2-cell f ⇒ f . Given α : f ⇒ g, β : g ⇒ h, we defineβ ∗ α by the formula

β ∗ α ={

α(2s,−) 0 ≤ s ≤ 12

β(2s − 1,−) 12 ≤ s ≤ 1.

Given [] : α � α′, [ ] : β � β ′ where α, α′ : f ⇒ g and β, β ′ : g ⇒ h,we can define [ ] ∗ [] to be represented by the function

� (r, s, t) ={

(r, 2s, t) 0 ≤ s ≤ 12

(r, 2s − 1, t) 12 ≤ s ≤ 1.

It is simple to check that the equivalence class of this function gives a 3-cellwith the claimed source and target, and that if ∼ ′, ∼ ′, then � ∼ ′ � ′ so that [ ] ∗ [] = [ ′] ∗ [′]. Finally, it is immediate that[1β ] ∗ [1α] = [1β∗α], and it is a simple check to verify interchange since weactually have the equality

( � ) � ( ′ � ′) = ( � ′) � ( � ′)

without having to find a mediating function H : I 4 ⇒ X . Thus we have shownthat there is a composition functor

�3 X (x, y)(g, h)×�3 X (x, y)( f, g) → �3(x, y)( f, h)

which sends (β, α) to β ∗ α and ([ ], []) to [ ] ∗ [].It only remains to produce the associativity and unit constraints, and then

check the two bicategory axioms. We leave the axioms to the reader, as pro-ducing the required homotopies is fairly simple. Given a composable triple(γ, β, α), the associator

aγ,β,α : (γ ∗ β) ∗ α � γ ∗ (β ∗ α)is defined to be the equivalence class of the function below.

A(r, s, t) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

α

(4s

2 − r, t

)0 ≤ s ≤ 1

2 − r4

β(4s − 2 + r, t) 12 − r

4 ≤ s ≤ 34 − r

4

γ

(4s − 3 + r

1 + r, t

)34 − r

4 ≤ s ≤ 1.

92 Examples

The right unit constraint for α is defined to be the equivalence class of thefunction below.

R(r, s, t) =

⎧⎪⎨⎪⎩

f (t) 0 ≤ s ≤ 1 − r

2

α

(2s − 1 + r

1 + r, t

)1 − r

2≤ s ≤ 1.

The left unit constraint is defined similarly. We should note that these arerequired to be natural transformations, and we leave those computations tothe reader; they are messy, but unenlightening.

Proposition 5.11 There is a strict functor

⊗ : �3 X (y, z)×�3 X (x, y) → �3(x, z)

whose action on objects sends a pair of paths (g, f ) to the standard compositepath:

(g ⊗ f )(t) ={

f (2t) 0 ≤ t ≤ 12

g(2t − 1) 12 ≤ t ≤ 1.

Proof We have already given the action of this functor on objects it remainsto do so on higher cells and then to check that this assignment gives a strictfunctor. For α : f → f ′, β : g → g′ where the target of f is the source of g,β ⊗ α is given by the formula below.

(β ⊗ α)(s, t) ={

α(s, 2t) 0 ≤ t ≤ 12

β(s, 2t − 1) 12 ≤ t ≤ 1.

The formula for [ ] ⊗ [] is analogous, and is defined to be the equivalenceclass of the function below.

( ⊗ )(r, s, t) ={

(r, s, 2t) 0 ≤ t ≤ 12

(r, s, 2t − 1) 12 ≤ t ≤ 1.

It is easy to check that this gives a well-defined 3-cell of �3 X .Now we must check that constraint data for the functor ⊗ can all be defined

to be identities. The composition constraint is a 3-cell

(β ′ ⊗ α′) ∗ (β ⊗ α) � (β ′ ∗ β)⊗ (α′ ∗ α).Writing out the formulas for both sides shows that these 2-cells are the same;we give these explicitly below, leaving out the constraints on s, t as they canbe determined from the variables in the formulas as 2s (resp., 2t) means that s


(resp., t) is required to lie between 0 and 12 while 2s − 1 (resp., 2t − 1) means

that s (resp., t) is required to lie between 12 and 1.

(β ′ ⊗α′) ∗ (β⊗α)(s, t) = (β ′ ∗β)⊗ (α′ ∗α)(s, t) =

⎧⎪⎪⎨⎪⎪⎩

α(2s, 2t)β(2s, 2t − 1)α′(2s − 1, 2t)

β ′(2s − 1, 2t − 1).

The unit constraint is then a 3-cell

1g ⊗ 1 f � 1g⊗ f ,

and we leave it to the reader to calculate that the source 2-cell 1g ⊗ 1 f is alsoconstant in the s-direction, so this constraint can be taken to be the identityas well.

We must now verify the functor axioms; since we have bicategories, and not2-categories, these do not follow from being able to define the constraint cellsas identities. This amounts to checking that ⊗ sends the left and right unit andassociativity constraints in �3 X (y, z) × �3 X (x, y) to the same constraintsin �3 X (x, z). It is relatively easy to compute that this is the case, as writingdown the formulas in each case will in fact give equal, not just homotopic,functions.

Theorem 5.12 There is a tricategory�3 X with objects the points in X, hom-bicategories as in Proposition 5.10, and composition as in Proposition 5.11.

Proof As we have given the cells, hom-bicategories, and composition, all thatremains is to give the unit and associativity adjoint equivalences, the invertible3-cells π, λ, ρ, μ, and to check the tricategory axioms. Before doing so, wenote that all of the 1-, 2-, and 3-cells of�3 X are appropriately invertible. Givena 3-cell [] : α � β, there is a 3-cell [−1] : β � α which is represented bythe function

(r, s, t) → (1 − r, s, t);it is easy to check that this gives a well-defined 3-cell [−1] : β � α andthat this is an inverse to []. Similarly, every 2-cell α : f ⇒ g can be canoni-cally equipped with the rest of the data constituting an adjoint equivalence. Itspseudo-inverse is the 2-cell

α�(s, t) = α(1 − s, t),

with unit and counit the obvious contracting 3-cells. Finally, a 1-cell f :x → y can be equipped with the rest of the data constituting a biadjointbiequivalence by choosing f �(t) = f (1 − t), although we will not need this

94 Examples

construction to finish the proof of this theorem. We will, though, now freelyuse that every 3-cell is invertible and that every 2-cell is part of a canonicaladjoint equivalence, and omit those details in the remainder of this proof.

The unit Ix : x → x is given by Ix (t) = x . For a composable triple (h, g, f )of 1-cells, the associator ahg f : (h ⊗ g) ⊗ f ⇒ h ⊗ (g ⊗ f ) is given by thefunction below.

a(s, t) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

f

(4t

2 − s

)0 ≤ t ≤ 1

2 − s4

g(4t − 2 + s) 12 − s

4 ≤ t ≤ 34 − s

4

h

(4t − 3 + s

1 + s

)34 − s

4 ≤ t ≤ 1.

For a 1-cell f : x → y, the right unit constraint 2-cell r f : f ⊗ I ⇒ f is givenby the function below.

r(s, t) =

⎧⎪⎨⎪⎩

x 0 ≤ t ≤ 1 − s

2

f

(2t − 1 + s

1 + s

)1 − s

2≤ t ≤ 1.

The left unit is defined similarly.Next we define the constraint 3-cells. In each case, the method is cumber-

some but straightforward. First, we compute the source and target 2-cells, andthen produce a piecewise-linear homotopy between them. We will explicitlycompute the mate of ρ with source (1 f ⊗ rg) ∗ a f gI and target r f g .

r f g(s, t) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

x 0 ≤ t ≤ 1 − s

2

g

(4t − 2 + 2s

1 + s

)1 − s

2≤ t ≤ 3 − s

4

f

(4t − 3 + s

1 + s

)3 − s

4≤ t ≤ 1

(1 f ⊗ rg) ∗ a f gI (s, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x 0 ≤ t ≤ 1 − s

2, 0 ≤ s ≤ 1

2

g(4t − 2 + 2s)1 − s

2≤ t ≤ 3 − 2s

4, 0 ≤ s ≤ 1

2

f

(4t − 3 + 2s

1 + 2s

)3 − 2s

4≤ t ≤ 1, 0 ≤ s ≤ 1

2

x 0 ≤ t ≤ 1 − s

2, 1

2 ≤ s ≤ 1

g

(4t − 2 + 2s

2s

)1 − s

2≤ t ≤ 1

2 ,12 ≤ s ≤ 1

f (2t − 1) 12 ≤ t ≤ 1, 1

2 ≤ s ≤ 1.


We therefore define (the mate of) ρ to be given by the equivalence class of thefunction below.

ρ(r, s, t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x 0 ≤ t ≤ 1 − s

2

g

(4t − 2 + 2s

1 + rs

)1 − s

2≤ t ≤ 3 − (2 − r)s

4, 0 ≤ s ≤ 1

2

f

(4t − 3 + (2 − r)s

1 + (2 − r)s

)3 − (2 − r)s

4≤ t ≤ 1, 0 ≤ s ≤ 1

2

g

(4t − 2 + 2s

2s + r − rs

)1 − s

2≤ t ≤ 2 + r − rs

4, 1

2 ≤ s ≤ 1

f

(4t − 2 − r + rs

2 − r + rs

)≤ 2 + r − rs

4≤ t ≤ 1, 1

2 ≤ s ≤ 1.

The source and target 2-cells can be represented graphically as the domainsof functions from I 2. In each case, we label the axes and state what the functionis in the horizontal direction. It is then possible to compute the entire function,as everything is done in a piecewise linear fashion. The pictures correspondingto the source and target of the 3-cell ρ are shown below.

Ix g f

g f

I g f��

��

��

Ix g f

��

NNNN

NNNN

NNNN

NNNN

NNN

g f

t→s↓t→s↓

(1 f ⊗ rg) ∗ a f gI r f g

The formula for ρ(r, s, t) given above is then just the obvious piecewise linearhomotopy between these two functions.

We leave the matter of giving the formulas for the other constraint 3-cells tothe reader. In each case, we define these to be the piecewise linear homotopiesobtained in the same method that we used to define ρ above. Additionally, wealso leave verification of the axioms to the reader. To verify one of the tricate-gory axioms, we must produce a homotopy I × I 3 → X between the sourceand target composite 3-cells satisfying certain properties. The same methodusing piecewise linear homotopies works, although one has to rely on the for-mulas involved as it is quite difficult to draw useful pictures for the domains aswe did in defining ρ.

96 Examples

Remark 5.13 A more satisfactory proof would involve showing that the stan-dard disks Dn for 0 ≤ n ≤ 3 give a co-tricategory object D3 in the categoryof spaces, and then defining �3 X as Top(D3, X). Unfortunately, this strategydoes not work, as the co-tricategory axioms only hold up to higher homotopies.It should be possible to show that the collection of disks of all dimensionsforms a co-ω-category object (of the weak variety) in Top; if we call this objectD, then Top(D, X) should give the fundamental ω-groupoid of X , from whichwe should be able to obtain �3 X by a quotienting process that identifies twoparallel 3-cells when there is a 4-cell between them. This is the strategy (with-out the quotienting) used to define fundamental ω-groupoids for an operadicdefinition like that of Batanin (1998a) or Leinster (2004).

Remark 5.14 At this point, it is clear that the assignment X → �3 X shouldbe the action on objects of some kind of a functor. Since �3 X is a tricategory(actually, a trigroupoid since all of the cells are invertible in the appropriatesense), and tricategories should form a weak 4-category, it would be reasonableto postulate that �3 should be a functor between weak 4-categories. Makingthis precise would require a substantial amount of work as one would first needto construct weak 4-categories of both tricategories and spaces. This strategyis carried out one dimension down by Gurski (2011), and a construction of afunctor �2 : Top → Bicat between a tricategory Top of spaces and the tri-category of bicategories is given. More is actually accomplished in that paper,as certain operad actions on spaces are shown to give rise to monoidal andbraided monoidal structure on their fundamental 2-groupoids.

6

Free constructions

This chapter will develop the basic tools necessary to construct freetricategories and free Gray-categories. First we must decide on the under-lying data from which a tricategory is to be generated freely. Second, wemust construct both the free tricategory and the free Gray-category on thisdata. This requires a bit of care as one must pay careful attention to how theuniversal property is stated; the issue here is that, as we will see, the cate-gory of tricategories and strict maps has to be constructed directly, and notas a sub-object of some structure involving more general 1-cells. Finally, weprove some results analogous to those leading up to the proof of the coherencetheorem for bicategories.

6.1 Graphs

The first step in producing a free tricategory is to decide from what data we willgenerate such a tricategory. The natural choice is that of a bicategory-enrichedgraph, but we wish to construct free Gray-categories as well and so we mustalso work with category-enriched 2-graphs.

Definition 6.1 1. A category-enriched 2-graph X consists of a directed graphX1 ⇒ X0 along with, for each pair of parallel arrows f, g in X1, a categoryX ( f, g). The category of category-enriched 2-graphs, written 2Gr(Cat), hasfor morphisms X → X ′ the pairs (P, F), where P is a map of the under-lying directed graphs and F is a collection of functors F f,g : X ( f, g) →X ′(P f, Pg).2. A bicategory-enriched graph Y consists of a set Y0 along with, for eachpair of elements a, b ∈ Y0, a bicategory Y (a, b). The category of bicategory-enriched graphs has for morphisms Y → Y ′ the pairs (Q,G), where

97

98 Free constructions

Q : Y0 → Y ′0 is a function and G is a collection of functors Ga,b : Y (a, b) →

Y ′(Qa, Qb). We shall write this category as Gr(Bicat).

Notation 6.2 We shall denote by Gr(2Cat) the subcategory of the categoryof bicategory-enriched graphs for which each Y (a, b) is a strict 2-category andeach Ga,b is a strict 2-functor. We also write Gr(Bicats) for the subcategoryof Gr(Bicat) consisting of all the objects and only the maps for which eachfunctor Ga,b is a strict functor. There are obvious inclusions

Gr(2Cat) ⊂ Gr(Bicats) ⊂ Gr(Bicat).

Remark 6.3 1. There is an obvious forgetful functor Gr(Bicat) →2Gr(Cat). This functor, when restricted to Gr(Bicats), has an obvious leftadjoint induced by the free bicategory functor.2. There is also an obvious underlying 3-globular set functor

U : 2Gr(Cat) → 3GlobSet

which assigns to the category-enriched 2-graph X the 3-globular set U Xhaving

• U X0 = X0,

• U X1 = X1,

• U X2 = ∐f,g obX ( f, g),

• U X3 = ∐f,g arX ( f, g),

and having the obvious source and target functions. The functor U has aleft adjoint FCat, which when applied to a 3-globular set G, produces thecategory-enriched 2-graph FCatG having FCatG0 = G0, FCatG1 = G1, andFCatG( f, g) = F(G3 ⇒ G2), where F is the free category functor from thecategory of directed graphs to Cat and this free category functor is appliedto the 2- and 3-cells of G whose 1-cell source is f and whose 1-cell targetis g. This is the prototype for how we will construct free tricategories on acategory-enriched 2-graph.

Recall that the free bicategory on a category-enriched graph, FG for G acategory-enriched graph, has the following universal property: given any bicat-egory B and a map of Cat-graphs G → B, there is a unique strict functorFG → B making the following triangle commute.

G FG��G

B�� FG

B��

6.2 The category of tricategories 99

Let FB : 2Gr(Cat) → Gr(Bicat) be the functor that is defined by lettingFB X be the bicategory-enriched graph with FB X0 = X0, and FB X (x, y) =F(tr Xx,y), where tr Xx,y is the category-enriched graph with

(tr Xx,y)0 = { f ∈ X1 : s( f ) = x, t ( f ) = y)}and (tr Xx,y)( f, g) = X ( f, g); the functor is defined on morphisms in theobvious fashion. Similarly, we can define a functor

F2C : 2Gr(Cat) → Gr(2Cat).

The following result is now an obvious consequence of the coherence theoremfor bicategories.

Theorem 6.4 Let X be a category-enriched 2-graph. Then the mapFB(X) → F2C (X) which is the identity on objects and is given on hom-bicategories by the universal property of free bicategories is, for every pairof objects x, y, a strict biequivalence FB X (x, y) → F2C X (x, y).

This theorem motivates the following definition.

Definition 6.5 A map (Q,G) of bicategory-enriched graphs is a localbiequivalence if each functor Ga,b is a biequivalence. We will say that themap (Q,G) is a locally strict local biequivalence if it is a local biequivalenceand a map in Gr(Bicats).

6.2 The category of tricategories

As we will explicitly compute in the next chapter, tricategories and functorsbetween them do not form a category: composition of functors is neither asso-ciative nor unital. One might expect this to be remedied by restricting to strictfunctors, but this is not the case unless the composition law is changed. Thecomposition of strict functors qua weak functors is still neither associativenor unital, but there is a composition law which is, namely that inherited byviewing tricategories as algebras for a monad.

Definition 6.6 Let F : S → T be a map of 3-globular sets, where S, Tare both tricategories. We say that F is a virtually strict functor if it can beequipped with constraint data making it a strict functor.

Definition 6.7 The category Tricat1 is defined to be the category of tricate-gories and virtually strict functors, with composition defined to be compositionof the maps of underlying 3-globular sets.


For this to be a category, we must check that it is closed under com-position and contains identities for every object. Both of these are entirelystraightforward to check, so we leave the details to the reader.

Proposition 6.8 The forgetful functors

Tricat1 → Gr(Bicats) → 2Gr(Cat) → GSet3,

together with the three possible composite functors, are all monadic.

Proof Each of these statements is easy to check with the explicit left adjointsthat we construct below. The claim that the functor Tricat1 → Gr(Bicats)

is monadic also follows from the work of Batanin–Cisinski–Weber, specif-ically Theorem 7.3 of their E-print. Theorem 7.3 and Lemma 2.4 of Ibid.together also show that Gr(Bicats) → 2Gr(Cat), 2Gr(Cat) → GSet3, andGr(Bicats) → GSet3 are all monadic as well.

We will now explicitly construct left adjoints to the first two forgetful func-tors above (and hence their composite), as these constructions will be neededlater for coherence.

We will first define the free tricategory, FA, generated by a bicategory-enriched graph A. This tricategory will have a universal property with respectto locally strict maps A → T of Bicat-graphs.

Let A be a bicategory-enriched graph. Then the free tricategory on A,denoted FA, has object set

obFA = A0.

Let a, b ∈ A0. Then FA(a, b) is the bicategory whose objects are builtinductively from the basic building blocks

(1) objects f of A(c, d) and(2) new objects Ia : a → a

by tensoring when source matches target. Thus a generic object of FA(a, b)might look like

(( f ⊗ g)⊗ Ia2)⊗ (h ⊗ j),

where j ∈ A(a, a1), h ∈ A(a1, a2), g ∈ A(a2, a3), and f ∈ A(a3, b); wewrite these as f .

The 1-cells of FA(a, b) are built from the basic building blocks

(1) 1-cells α : f → g in A(c, d),(2) new 1-cells ia : Ia → Ia ,(3) the constraint cells l f : I ⊗ f → f , l�f : f → I ⊗ f ,

6.2 The category of tricategories 101

(4) the constraint cells r f : f ⊗ I → f , r �f : f → f ⊗ I , and

(5) the constraint cells a f gh : ( f ⊗g)⊗h → f ⊗(g⊗h), a�f gh : f ⊗(g⊗h) →

( f ⊗ g)⊗ h

by tensoring along object boundaries and composing along 0-cell boundaries(the 0-cells in each FA(a, b), not the objects of the new tricategory), subjectto the equivalence relation generated by setting

(α) ∗ (β) = (α ∗ β),where the left-hand side is a composite in the free tricategory while the rightis a composite in A.

The 2-cells are similarly built up inductively from the 2-cells in each A(a, b)and from constraint 2-cells. These constraint 2-cells are units and counitsfor the various adjoint equivalences, the constraint isomorphisms forcing theunit and composition to be functors, π , μ, λ, ρ, and new hom-bicategoryconstraint cells involving the new 1-cells. These 2-cells are subject to therequired relations relating composition along 0- and 1-cell boundaries inthe new hom-bicategories with those of the old hom-bicategories, for thefunctoriality and naturality conditions of the functors, transformations, andmodifications involved, for the conditions forcing certain pairs of cells to beadjoint equivalences, and for the three axioms for a tricategory.

Proposition 6.9 Let A be a bicategory-enriched graph. Then there is alocally strict map i : A → FA which is the identity on objects and sendseach cell to the cell of the same name in FA.

The following theorem follows immediately from the previous propositionand the definition of strict functor.

Theorem 6.10 Let A be a bicategory-enriched graph, and let T be a tricate-gory. If F : A → T is a locally strict map of bicategory-enriched graphs, thenthere is a unique strict functor F : FA → T making the following trianglecommute in Gr(Bicats).

A FA��A

T�� FA

T��

Hence F provides a left adjoint to the forgetful functor Tricat1 → Gr(Bicats).

Remark 6.11 We have not yet defined the action of F on morphisms, butthis is determined by the universal property. Let F : X → Y be a map of


bicategory-enriched graphs. Then the universal property of F gives a uniquestrict functor making the diagram below commute in Gr(Bicats).

X FXi �� FX

FY

!��

X

Y

F

��Y FY

i��

We shall call this functor FF .

Definition 6.12 Let X be a category-enriched 2-graph. The free tricategoryon X , also denoted FX , is F(FB X).

Before moving on to the construction of free Gray-categories, we prove amuch-needed result about the free tricategory construction.

Theorem 6.13 Let A, B be Bicat-graphs, and let f : A → B be a mapbetween them. If f is a locally strict local biequivalence and an isomorphismon objects, then the strict functor F f : FA → FB is a local biequivalence,hence a triequivalence.

Proof We must show that each functor F fa,b is locally full, locally faithful,locally essentially surjective, and biessentially surjective. First note that F fa,b

sends constraint cells to constraint cells, tensors to tensors, and compositionsto compositions since it is both strict and locally strict.

We prove the first two claims by induction over tensor length. If α : g ⇒ hand β : g ⇒ h are parallel 2-cells in FA(a, b)which are represented by 2-cellsin A(a, b), then if is clear that

F f (α) = F f (β) =⇒ α = β

since f is locally faithful; the same holds if α or β is a constraint cells. Thissuffices, by induction, to show that F fa,b is locally faithful as it strictly pre-serves tensors, all compositions, and the equivalence relation imposed by thetricategory axioms.

To show that F fa,b is locally full, first let β : F f (g) ⇒ F f (h) be a 2-cellwhich is represented by a 2-cell in B( f a, f b). Then we can find an α : g ⇒ hsuch that F f (α) = β since f is locally full; the same holds if β is a constraintcell. Since F fa,b is strict, tensors of 2-cells and constraint cells are also in theimage.

To show that F fa,b is locally essentially surjective, first note that tensorsof isomorphism 2-cells are again isomorphism 2-cells; similarly with horizon-tal compositions of isomorphism 2-cells. Since every 1-cell of FB( f a, f b)

6.2 Free Gray-categories 103

is built from the 1-cells of the B(c, d)’s and new 1-cells, it suffices to showthat all of these are isomorphic to the images of 1-cells in FA(a, b). Thisfollows immediately from the strictness of F f , the fact that each fa,b is abiequivalence, and the fact that f is an isomorphism on objects.

Now all that remains is to show that F fa,b is biessentially surjective. Theproof is analogous to the one given in the previous paragraph.

6.3 Free Gray-categories

In this section, we construct the free Gray-category on a 2Cat-graph Y .This is less messy than constructing the free tricategory as there are fewer“interesting” pieces of new data to generate.

Let Y be a 2-category-enriched graph, so Y consists of a set Y0 and for eacha, b ∈ Y0, a 2-category Y (a, b). The free Gray-category on Y , denoted FGY ,has object set

obFGY = Y0.

The 2-category FGY (a, b) is constructed as follows. The objects of FGY (a, b)are composable strings in

(1) the objects f ∈ Y (c, d) and(2) a new object Ia : a → a for each a ∈ Y0,

subject to the condition that s I t = st for all strings s and t . Thus a typicalobject of FGY (a, b) is

f g Ia2 hj = f gh j,

where j ∈ Y (a, a1), h ∈ Y (a1, a2), g ∈ Y (a2, a3), and f ∈ Y (a3, b); we writethese as f , just as we did in the free tricategory.

The set of 1-cells of FGY (a, b) between strings fn ⊗· · ·⊗ f1 and gm ⊗ · · ·⊗g1 is empty if n = m. If n = m then it consists of composites of strings of1-cells αn ⊗ · · · ⊗ α1 where at most one αi is a non-identity 1-cell in someY (c, d). These strings shall be written 1 jα1i−1, indicating that α is the 1-cellin the i th position and that the total string has length i + j . We subject theseto the relation that if α : fk → f ′

k and α′ : f ′k → f ′′

k , then the compositionof strings (1mα′1k−1) ∗ (1mα1k−1) is equal to the string 1m(α′

k ∗ αk)1k−1. Inparticular, the identity is when every αi is an identity.

The basic 2-cells between 1 jα1i−1 and 1 jβ1i−1 are of the form 1 j1i−1

with a 2-cell α ⇒ β in some Y (a, b), and we impose the same conditionon vertical composition that we did on composition of 1-cells. Each 2-cell is


a composable string built from formal horizontal composites of basic 2-cellsand invertible 2-cells of the form

1kα1l−1,1 jβ1i−1 : (1kα1l−1+i+ j ) ∗ (1k+l+ jβ1i−1)∼=⇒

(1k+l+ jβ1i−1) ∗ (1kα1l−1+i+ j ).

We impose on these 2-cells the axioms required for the Gray tensor product.Composition of 1-cells is given by concatenation, as is vertical composition of2-cells. Horizontal composites of 2-cells are obtained in the same fashion thatwe obtained them for the 2-category X ⊗ Y in Chapter 4. Once again, we omitthe details for showing that FGY (a, b) is a 2-category.

The last thing to define is a composition map

FGY (b, c)⊗ FGY (a, b) → FGY (a, b),

where we must use the Gray tensor product on the left. On the 0-cellsof these 2-categories, composition is just concatenation. If 1kα1l−1 and1 jβ1i−1 are generating 1-cells in FGY (b, c),FGY (a, b), respectively, thentheir composition is the 1-cell given by the string

(1k+l+ jβ1i−1) ∗ (1kα1l−1+i+ j );this can be extended over composites using that composition is required to be2-functorial in each variable separately. The tensor product of a pair of 2-cellsfrom these 2-categories is defined by a similar formula on basic 2-cells andextended in the obvious manner. The following proposition is now routine tocheck.

Proposition 6.14 Let Y be a 2-category-enriched graph. Then the data givenabove for FGY satisfy the axioms for being a Gray-category.

If Y is a category-enriched 2-graph, then we call FG(F2C Y ) the free Gray-category on Y . This is justified by the following theorem.

Theorem 6.15 1. Let Y be a 2-category enriched graph. Then for everyGray-category T and every map of 2-category-enriched graphs F : Y → T ,there is a unique Gray-functor F : FGY → T such that the following diagramcommutes in Gr(2Cat).

Y FGY��Y

T�� FGY

T��

6.3 Free Gray-categories 105

2. Let X be a category-enriched 2-graph. Then for every Gray-category Tand every map of category-enriched 2-graphs F : X → T , there is a uniqueGray-functor F : FG(F2C X) → T such that the following diagram commutesin 2Gr(Cat).

X FG(F2C X)��X

T�� FG(F2C X)

T��

Proof The second statement follows from the first since it expresses the factthat FGF2C is a left adjoint because it is a composite of two left adjoints.The first statement follows immediately by noting that a Gray-functor strictlypreserves all types of units and compositions, and sends the isomorphism �

in FG X to the corresponding isomorphism in T . Since the entire structure ofFG X is built from these cells using the Gray-category axioms, the functor Fis uniquely determined.

7

Basic structure

This chapter will be devoted to studying some aspects of the total algebraicstructure consisting of tricategories, functors, transformations, modifications,and perturbations. This chapter will only establish some basic properties thatwill be used later. There should be a weak 4-category Tricat, but constructingthe entire structure would involve a substantial investment, much of which wewill not need for the purposes of proving coherence. Since we will be prov-ing a version of the Yoneda lemma for cubical tricategories, we will need toconstruct functor tricategories of the form [T op,Gray]. This functor tricate-gory would be the hom-tricategory in the putative construction of Tricat, butwe only construct this in the special case when the target is a Gray-category,and this restriction greatly simplifies many of the calculations. The Yonedaembedding for cubical tricategories will be constructed in Chapter 9; fornow we will focus on some basic composition formulas that will be requiredlater.

The third section of this chapter proves some basic results using the notionof pseudo-icons as originally defined by Garner and Gurski (2009). It isoften useful to employ the simpler pseudo-icons than transformations in theproofs leading up to coherence. The reason is that many of these trans-formations have identity components, and much of the data defining thosetransformations is made more complicated by the insertion of unit con-straints because of those identity components. Using equivalence pseudo-iconsinstead makes for cleaner calculations, and just as in the two-dimensionalcase equivalence pseudo-icons and transformations with identity componentsboth provide the same information. We have made some constructions usingidentity-component transformations in order to show what is required, butmost of the proofs later will use equivalence pseudo-icons. It is impor-tant to remember, however, that using equivalence pseudo-icons makes forsimpler calculations at the cost of having to prove that they provide the

106

7.1 Structure of functors 107

same information as transformations with identity components, a conceptuallysimple but calculationally intense result.

7.1 Structure of functors

This section will give an explicit formula for composing functors between tri-categories. The formula here is more interesting than the corresponding one forfunctors between bicategories in the following way. Functors between bicate-gories compose strictly so that bicategories with lax, weak, or strict functorsform a category. This gives rise to the fact that the tricategory Bicat has strictcomposition of 1-cells. We will see that this is not the case with tricategories,and this will justify the definitions made in the previous chapter about themorphisms in the category of tricategories.

Let R, S, and T be tricategories, and let H : R → S and J : S → T befunctors. We now define the composite functor J H : R → T .

• The function on objects obR → obT is given by the composite of the objectfunctions for H and J .

• The functors on hom-bicategories J Hab : R(a, b) → T (J Ha, J Hb) aregiven by

R(a, b)Hab−→ S(Ha, Hb)

JHaHb−→ T (J Ha, J Hb).

• The adjoint equivalence χ J H is defined as follows. The transformation χ J H

is the composite

⊗T ◦ J × J ◦ H × Hχ J ∗1−→ J ◦ ⊗S ◦ H × H

1∗χH

−→ J ◦ H ◦ ⊗R .

Similarly, the transformation (χ J H )� is the composite

J ◦ H ◦ ⊗R 1∗χ �−→ J ◦ ⊗S ◦ H × H

χ �∗1−→ ⊗T ◦ J × J ◦ H × H.

The counit of this adjunction is the composite displayed below.

(χχ �)g f = (J (χg f ) ∗ χHgH f

) ∗ (χ �

Hg,H f ∗ J (χ �g f )

) ∼=−→J (χg f ) ∗

((χHgH f ∗ χ �

HgH f

) ∗ J (χ �g f )

)1∗(ε∗1)−→

J (χg f ) ∗(

1 ∗ J (χ �g f )

)1∗l−→ J (χg f ) ∗ J (χ �

g f )φ J

2−→J (χg f χ

�g f )

Jε−→ J (1H(g⊗ f ))φ J

0−→ 1J H(g⊗ f ).

The unit is defined similarly, and a check shows that this gives an adjointequivalence in the appropriate bicategory.

108 Basic structure

• If we denote the units by I R : 1 → R(a, a), etc., then the adjointequivalence ιJ H is defined as follows. The transformation ιJ H is thecomposite

I T ιJ−→ J ◦ I S 1∗ιH−→ J ◦ H ◦ I R .

The transformation (ιJ H )� is the composite

J ◦ H ◦ I R 1∗ι�−→ J ◦ I S ι�−→ I T .

The unit and counit of this adjunction are determined in a manner similar tothat used for χ .

• The component at (h, g, f ) of the invertible modification ωJ H is definedby the pasting diagram below. The unmarked isomorphisms are given byunique coherence cells by the coherence for functors theorem or natural-ity isomorphisms, and the unlabeled 2-cells are uniquely determined as thesource and target of JωH .

(J Hh⊗′′ J Hg)⊗′′ J H f J Hh⊗′′(J Hg⊗′′ J H f )a ��(J Hh⊗′′ J Hg)⊗′′ J H f

J H(h⊗g)⊗′′ J H f

χ⊗′′1

��

(J Hh⊗′′ J Hg)⊗′′ J H f

J (Hh⊗′ Hg)⊗′′ J H f

Jχ⊗1

��


J H((h⊗g)⊗ f )

χ

��



Jχ⊗′′1

��OOOOOOOOOOO


J (H(h⊗g)⊗′ H f )

χ

��

J (H(h⊗g)⊗′ H f )

J H((h⊗g)⊗ f )

Jχ

��OOOOOOOOOOO


J ((Hh⊗′ Hg)⊗′ H f )

χ

+)PPPPPPPP

J ((Hh⊗′ Hg)⊗′ H f )

J (H(h⊗g)⊗′ H f )

J (χ⊗′1)86QQQQQQQQ

J ((Hh⊗′ Hg)⊗′ H f ) J (Hh⊗′(Hg⊗′ H f ))Ja ��

J Hh⊗′′(J Hg⊗′′ J H f )

J Hh⊗′′ J (Hg⊗′ H f )

1⊗′′χ

J��

J Hh⊗′′ J (Hg⊗′ H f )

J (Hh⊗′(Hg⊗′ H f ))

χ

86RRRRRRRRJ Hh⊗′′ J (Hg⊗′ H f )

J Hh⊗′′ J H(g⊗ f )

1⊗′′ Jχ��SSSSSSSSSSS

J Hh⊗′′(J Hg⊗′′ J H f )

J Hh⊗′′ J H(g⊗ f )

1⊗′′χ

��J Hh⊗′′ J H(g⊗ f )

J (Hh⊗′ H(g⊗ f ))

χ

J��J (Hh⊗′(Hg⊗′ H f ))

J (Hh⊗′ H(g⊗ f ))

J (1⊗′χ)+),

,,,,,,, J Hh⊗′′ J H(g⊗ f )

J H(h⊗(g⊗ f ))

χ

��

J (Hh⊗′ H(g⊗ f ))

J H(h⊗(g⊗ f ))

Jχ

��SSSSSSSSSSS

J H((h⊗g)⊗ f ) J H(h⊗(g⊗ f ))J Ha

��

J ((Hh⊗′ Hg)⊗′ H f )

J H(h⊗(g⊗ f )))'��J ((Hh⊗′ Hg)⊗′ H f )

J H(h⊗(g⊗ f ))0.

∼=

:=

∼=

⇓ ωJ

∼=

=:

∼=∼=

∼=⇓ JωH

• The component at f of the invertible modification γ J H is defined by thepasting diagram below, where the unmarked isomorphisms are once againunique coherence cells or naturality isomorphisms; δ J H is defined similarly.

7.1 Structure of functors 109

I ′′⊗′′ J H f

J H I⊗′′ J H f

ι⊗′′1

��J H I⊗′′ J H f

J H(I⊗ f )

χ

��J H(I⊗ f ) J H f

J Hl��

I ′′⊗′′ J H f

J I ′⊗′′ J H f

ι⊗′′153��

J I ′⊗′′ J H f

J H I⊗′′ J H fJ ι⊗′′1EE��

J H I⊗′′ J H f

J (H I⊗′ H f )

χ 53��

J (H I⊗′ H f )

J H(I⊗ f )

χEE��

J I ′⊗′′ J H f

J (I ′⊗′ H f )

χ

��555555

J (I ′⊗′ H f )

J (H I⊗′ H f )

J (ι⊗′1)��//////

J (I ′⊗′ H f )

J H f

Jl ′

+)PPPPPPPPPPP

I ′′⊗′′ J H f

J H f

l ′′

KH

∼=

∼=

∼=

⇑Jγ H

⇑γ J

Note that in the definitions above no associations were given. This is becausefunctors between bicategories compose in a strictly associative manner. Calcu-lation then yields the following result.

Proposition 7.1 The data above satisfies the axioms for a functor betweentricategories.

Proposition 7.2 Tricategories and strict functors do not form a categorywhen equipped with the composition law above.

Proof We show that the composite functor id ◦ id does not have the sameunderlying data as the functor id, so that composition of functors is not strictlyunital.

The identity functor idT on a tricategory T has each component functor theidentity, and χ is the identity transformation ⊗ ⇒ ⊗. For an object (g, f ), thecomponent of this transformation is id : g ⊗ f → g ⊗ f . The transformationχ for the composite idT ◦ idT has component id ◦ id : g ⊗ f → g ⊗ f . Ingeneral, this is not equal to the identity map on g ⊗ f .

The following corollary is the reason for the definition of Tricat1 given inChapter 6.

Corollary 7.3 Tricategories and functors do not form a category using theabove composition law.

110 Basic structure

7.2 Structure of transformations

It will be necessary in later sections to understand some of the basic structureof transformations, so we collect in this section the relevant results. Most of theproofs are simple diagram chases, so we omit these details whenever possible.

Proposition 7.4 Let α : F → G and β : G → H be transformations. Thenthere is a transformation βα : F → H with (βα)a = βa ⊗ αa.

Sketch of proof The adjoint equivalence βα is given by setting (βα) f equalto the composite

(βb ⊗ αb)⊗ F fa−→ βb ⊗ (αb ⊗ F f )

1⊗α f−→ βb ⊗ (G f ⊗ αa)a�−→

(βb ⊗ G f )⊗ αaβ f ⊗1−→ (H f ⊗ βa)⊗ αa

a−→ H f ⊗ (βa ⊗ αa)

and (βα)�f is the obvious adjoint, with the unit and counit given by theobvious composition of constraint cells with units and counits for all of theadjoint equivalences involved. The definitions of � and M are given by dia-grams similar to those in Theorem 9.3 with additional coherence cells insertedwhere necessary. Checking the necessary axioms requires using the tricat-egory axioms in the target as well as the axioms for each transformationseparately.

Proposition 7.5 Let F, F ′ : R → S and G,G ′ : S → T be functors, and letα : F → F ′, β : G → G ′ be transformations. Then there are transformationsβ ∗ 1F : G F → G ′F and 1G ∗ α : G F → G F ′ whose components are givenby βFa and Gαa, respectively.

Proof We will only prove the statement for β ∗ 1F as the other proof isanalogous. The adjoint equivalences β ∗ 1F are defined by the followingformulas.

(β ∗ 1F ) f = βF f

(β ∗ 1F )�f = β�

F f

εβ∗1Ff = ε

βF f

ηβ∗1Ff = η

βF f

(β ∗ 1F )θ = βFθ .

These define appropriate transformations and modifications since these cellsare just components of β. The component at f, g of the invertible modification� is given by the diagram below.

7.2 Structure of transformations 111

(βFc⊗G Fg)⊗G F f (G′ Fg⊗βFb)⊗G F fβFg⊗1

�� (G′ Fg⊗βFb)⊗G F f G′ Fg⊗(βFb⊗G F f )a �� G′ Fg⊗(βFb⊗G F f ) G′ Fg⊗(G′ F f ⊗βFa )

1⊗βF f �� G′ Fg⊗(G′ F f ⊗βFa )

(G′ Fg⊗G′ F f )⊗βFa

a��


G′(Fg⊗F f )⊗βFa

χG′ ⊗1

LI!!!!!!!

G′(Fg⊗F f )⊗βFa

G′ F(g⊗ f )⊗βFa

G′(χF )⊗1

42��

(βFc⊗G Fg)⊗G F f

βFc⊗(G Fg⊗G F f )

a

��βFc⊗(G Fg⊗G F f )

βFc⊗G(Fg⊗F f )

1⊗χG

��

βFc⊗G(Fg⊗F f )

βFc⊗G F(g⊗ f )

1⊗G(χF ) '%""""""""""

βFc⊗G F(g⊗ f ) G′ F(g⊗ f )⊗βFaβF(g⊗ f )

��

βFc⊗G(Fg⊗F f ) G′(Fg⊗F f )⊗βFa

βFg⊗F f ��

βFc⊗(G Fg⊗G F f )

βFc⊗G F(g⊗ f )1⊗χG F

��


G′ F(g⊗ f )⊗βFa

1⊗χG′ F

�J

H7 HHHHHHHHHHHHHHHHH

HHHHHHHHHHHHHHHHH

β−1χF

3K TTTTTTTTTTTTTTTTTTTTTT

�Fg,F f

∼=∼=

The two isomorphisms are the composites of unit isomorphisms in the hom-bicategory with the functoriality isomorphism for ⊗.

For each object a, the single component of the invertible modification M isgiven by the diagram below.

βFa βFa⊗IG Far �

�� βFa⊗IG Fa βFa⊗G F Ia1⊗ιG F

�� βFa⊗G F Ia G ′ F Ia⊗βFa

βF Ia ��βFa

IG′ Fa⊗βFa

l�

0. IG′ Fa⊗βFa

G ′ F Ia⊗βFa

ιG′ F ⊗1

<;βFa⊗IG Fa

βFa⊗G IFa

1⊗ιG '%(((((((

βFa⊗G IFa

βFa⊗G F Ia

1⊗GιF

@?:::::::βFa⊗G IFa

G ′ IFa⊗βFa

βIFa �BIIIIIII

IG′ Fa⊗βFa

G ′ IFa⊗βFa

ιG′⊗1

��

G ′ IFa⊗βFa

G ′ F Ia⊗βFa

G ′ιF ⊗1

��)))))))))))))))))

∼=

∼=MaC� )))))))

)))))))

βιF

A��

��

The transformation axioms are now easy to check using that β is a transforma-tion and the fact that all the coherence cells used in the definitions above areeither those of the hom-bicategory or of the functor ⊗.

It will be necessary later to use associativity and unit transformations.If we were to construct the tetracategory Tricat from first principles, thesetransformations would be a necessary part of that structure.

Proposition 7.6 1. Let F : Q → R,G : R → S, H : S → T be functors.Then there are transformations

α : (H ◦ G) ◦ F → H ◦ (G ◦ F)α� : H ◦ (G ◦ F) → (H ◦ G) ◦ F

which have as their components at the object a the identity 1-cell IH G Fa.

112 Basic structure

2. Let idT denote the identity functor on the tricategory T . Then there aretransformations

ρ : F ◦ id → Fρ� : F → F ◦ id

which have as their components at the object a the identity 1-cell IFa.

Proof We will only prove the first claim, as the second follows by analo-gous arguments. First note that (H ◦ G) ◦ F and H ◦ (G ◦ F) have the sameunderlying map on cells. The components are the identity cells IH G Fa , and thetransformations α are defined by the formulas below.

α f = r � ◦ lα�

f = l� ◦ r.

The unit and counit of this adjoint equivalence are given by the obvious com-posites of constraint cells in the target hom-bicategory and units and counitsof the adjoint equivalences l, r. The naturality isomorphism αθ is given by thecomposite of the naturality constraints for the transformations involved. Thusthis adjoint equivalence is just the adjoint equivalence r � ◦ l � l� ◦ r .

The component at f, g of the invertible modification � is given by thediagram below.

(I⊗H G Fg)⊗H G F f (H G Fg⊗I )⊗H G F f(r� l)⊗1�� (H G Fg⊗I )⊗H G F f H G Fg⊗(I⊗H G F f )

a �� H G Fg⊗(I⊗H G F f ) H G F f ⊗(H G F f ⊗I )1⊗(r� l)�� H G F f ⊗(H G F f ⊗I )

(H G Fg⊗H G F f )⊗I

a�

��(H G Fg⊗H G F f )⊗I

H G F(g⊗ f )⊗I

χH(G F)⊗1

��

(I⊗H G Fg)⊗H G F f

I⊗(H G Fg⊗H G F f )

a

��I⊗(H G Fg⊗H G F f )

I⊗H G F(g⊗ f )

1⊗χ(H G)F

" I⊗H G F(g⊗ f ) H G F(g⊗ f )

l�� H G F(g⊗ f ) H G F(g⊗ f )⊗I

r��

(I⊗H G Fg)⊗H G F f

H G Fg⊗H G F f

l⊗1'%'''''''''

H G Fg⊗H G F f

(H G Fg⊗I )⊗H G F f

r�⊗1

��.......

H G Fg⊗(I⊗H G F f )

H G Fg⊗H G F f

1⊗l

1/UUUUUUU

H G Fg⊗H G F f H G Fg⊗H G F f1

�� H G Fg⊗H G F f

H G F f ⊗(H G F f ⊗I )

1⊗r�

*(��H G Fg⊗H G F f

(H G Fg⊗H G F f )⊗Ir� 53999999

I⊗(H G Fg⊗H G F f ) I⊗H(G Fg⊗G F f )1⊗χH

��I⊗(H G Fg⊗H G F f )

H G Fg⊗H G F f

l

��I⊗H(G Fg⊗G F f ) H(G Fg⊗G F f )

l ��

H G Fg⊗H G F f

H(G Fg⊗G F f )

χH

�LVVVVVVVVV

I⊗(H G Fg⊗H G F f )

I⊗H G(Fg⊗F f )

1⊗χH G

MMWWWWWWWWWWWWW

I⊗H G(Fg⊗F f )

I⊗H G F(g⊗ f )

1⊗H GχF

��

I⊗H G(Fg⊗F f ) H G(Fg⊗F f )l

��

I⊗H(G Fg⊗G F f )

I⊗H G(Fg⊗F f )

1⊗HχG

,*-----------

(H G Fg⊗H G F f )⊗I

H(G Fg⊗G F f )⊗I

χH ⊗1

86QQQQQQQQQQ

H(G Fg⊗G F f )


r�" 11111H(G Fg⊗G F f )

H G(Fg⊗F f )

HχG

LIXXXXXXXXXXXXX


H G F(g⊗ f )⊗I

Hχ⊗1

%�YYY

YYYY

YYYY

YYYY

YYYY


H G(Fg⊗F f )⊗I

HχG ⊗1

,*ZZZZ

ZZZZ

ZZZ

H G(Fg⊗F f )

H G F(g⊗ f )

H Gχ

NN[[[[[[[[[[[[H G(Fg⊗F f )

H G(Fg⊗F f )⊗Ir� ��

H G(Fg⊗F f )⊗I

H G F(g⊗ f )⊗I

H GχF'%((((((((((

∼= ⇓ μ ∼=

⇓ λ ∼=

⇓ ρ

∼=

∼=

∼=

∼=

∼=

∼= ∼=∼=

∼=

The unmarked isomorphisms are either given by the composite of a unitconstraint from the hom-bicategory with the functoriality constraint of ⊗ in

7.2 Structure of transformations 113

the case of the triangular regions, or by a naturality isomorphism in the case ofthe square regions. Note that we actually require a mate of ρ and not ρ itselfin the upper right corner.

For each object a, the single component of the invertible modification M isgiven by the diagram below.

I I⊗Ir �

�� I⊗I I⊗H G F I1⊗ι(H G)F

�� I⊗H G F I H G F Il �� H G F I H G F I⊗I

r ��I

I⊗I

l�

0. I⊗I

H G F I⊗I

ιH(G F)⊗1

<;I⊗I

I⊗H I

1⊗ι��

I⊗H I I⊗H G I1⊗H ι �� I⊗H G I

I⊗H G F I

1⊗H Gι

#!

I⊗H G I H G Il �� H G I

H G F I

H Gι

#!

H G I H G I⊗Ir �� H G I⊗I

H G F I⊗I

H Gι⊗1

2077777777I⊗H I

H I

l

)'&&&&&&&&&&&&&&&&

H I

H G I

H ι

��############H I H I⊗I

r �� H I⊗I

H G I⊗I

H ι⊗1

20\\\\\\\\\I H I

ι ��I

I⊗I

r �

)'��

I⊗I

H I⊗I

ι⊗1

(&��

I⊗I

I

l

%�

I

I

1

1/UUUUUUUUUUUUUUUUUUUU

∼= ∼= ∼= ∼=

∼= ∼= ∼=

∼=

CC

The two regions marked C have isomorphisms given by composites of unitisomorphisms with the functoriality constraint for ⊗, the isomorphisms lr � ∼= 1and r �1 ∼= l� are mates of the isomorphism lI ∼= rI in Lemma 7.7 below(composed with a unit in the latter case), and all the other isomorphisms arenaturality isomorphisms.

The three transformation axioms can now be checked by lengthy calculation.

Lemma 7.7 Let T be a tricategory. Then the 1-cells lI and rI are isomorphicin the bicategory T (a, a).

Proof An isomorphism is given by the following composite; isomorphismscoming from the constraint cells in the bicategory T (a, a) are unmarked.

l ∼= l11∗ηr∼= l(rr �)

∼= (lr)r � Nat.∼= (r ∗ l ⊗ 1)r �(1∗λ)∗1∼= (r(la))r � ∼= (r(l(1 ⊗ l� ∗ r ⊗ 1)))r �

∼= (r((l ∗ 1 ⊗ l�)r ⊗ 1))r � Nat.∼= (r((l�l)r ⊗ 1))r �(1∗(εl∗1))∗1∼= (r(1 ∗ r ⊗ 1))r � ∼= r(r ⊗ 1 ∗ r �)

Nat.∼= r(r �r)1∗εr∼= r1 ∼= r.

114 Basic structure

Remark 7.8 We could have stated and proved the proposition above usingthe terminology of pseudo-icons as developed in the next section. We havechosen not to do so, as it does not substantially reduce the work involved,and it would not help were one to attempt to construct the tetracate-goy Tricat directly. On the other hand, that is the strategy employed byGarner and Gurski (2009).

Remark 7.9 It should be noted that, once we have proven our coherencetheorem, the isomorphism in the proof of Lemma 7.7 will be the unique iso-morphism constructed from the tricategory coherence cells from rI to lI . Inany definition below, an isomorphism between lI and rI is assumed to be theone constructed in the lemma, or the mate thereof.

7.3 Pseudo-icons

This section is devoted to proving the tricategorical versions of the results inChapter 2 concerning icons and free bicategories. In it, we will recall the defi-nition of a pseudo-icon between functors from Garner and Gurski (2009), andthen prove some basic results.

Definition 7.10 Let S, T be tricategories, and F,G : S → T be functorsbetween them. Assume that F and G agree on objects. Then a pseudo-iconα : F ⇒ G consists of

• a transformation αa,b : Fa,b ⇒ Ga,b between the functors Fa,b : S(a, b) →T (Fa, Fb) and Ga,b : S(a, b) → T (Ga,Gb) (noting that Fa = Ga, Fb =Gb by assumption),

• an invertible 3-cell Ma as shown below, and

IFa F IaιFa �� F Ia

G Ia

αIa

��

IFa

IGaIGa G IaιGa

��

⇓ Ma

7.3 Pseudo-icons 115

• invertible 3-cells � f,g as shown below which are the components of amodification �.

F f ⊗ Fg F( f ⊗ g)χ F

�� F( f ⊗ g)

G( f ⊗ g)

α f ⊗g

��

F f ⊗ Fg

G f ⊗ Gg

α f ⊗αg

��G f ⊗ Gg G( f ⊗ g)

χG��

⇓ � f,g

These are subject to three axioms, which we do not produce here, that aremodified versions of the transformation axioms. The interested reader shouldconsult Garner and Gurski (2009) for the precise axioms.

Definition 7.11 An equivalence pseudo-icon is a pseudo-icon α whosecomponents α f for 1-cells are all equivalences.

Remark 7.12 Garner and Gurski (2009) construct a tricategory of tricate-gories in which the pseudo-icons are the 2-cells. The equivalence pseudo-iconsare precisely the 2-cells in that tricategory which are equivalences as calculatedin the hom-bicategories.

Lemma 7.13 Let F,G : S → T be functors between tricategories, andassume that F,G agree on objects. Given a transformation α : F ⇒ G suchthat αa = IFa for all objects a, there is a pseudo-icon β : F ⇒ G suchthat every component of β at a 1-cell is an equivalence. Conversely, given anequivalence pseudo-icon β : F ⇒ G, there is a transformation α : F ⇒ Gsuch that αa = IFa for all objects a.

Proof Let α be such a transformation. Then β f is the composite rG f ∗α f ∗l�F f ,with naturality 3-cells given by the obvious pastings of the naturality cells forr, α, l�; the transformation axioms are easy to check, and it is immediate thatthis is an equivalence. The invertible 3-cell Mβ

a is given by the pasting diagrambelow in which M denotes a mate of the appropriate 3-cell M for α and all ofthe isomorphisms between left and right units I I → I or their pseudo-inversesare obtained from the standard one lI ∼= rI by taking mates if necessary.

116 Basic structure

IFa IGa1�� IGa G Ia

ιG��IFa

F Ia

ιF

20\\\\\\\\\\\\\\

F Ia

I F Ia

l�

(&%%%%%%%%%%%%%%%%%%

I F Ia

G Ia I

αIa

)'&&&&&&&&&&&&&&&&&&

G Ia I

G Ia

r

1/44444444444444

IFa

I Il�

�"��

I I

IGa

r

!��I I

I F Ia

1ιF

&$XXXXXXXXXXXXXXXXXXXXX

IGa

I I

r �

��<<<<<<<<IGa

I Il� OO I I

G Ia IιG 1

��<<<<<<<<

∼=

∼=⇓ M

∼= ∼=

We now construct the invertible 3-cell�βf g . The pasting diagram that defines

�β is much more complicated than the one for Mβ , so we simplify it by break-ing it into two steps. We do this by giving the pasting for�, but with one of thecells being undefined for the moment. This interior cell will itself be a pastingof five different 3-cells, and we define it afterwards.

F f Fg

F( f g)

χ

:]]]]]]]]]]]]]]]]

F( f g)

I F( f g)l�

��

I F( f g)

G( f g)I

α f g

��

G( f g)I

G( f g)

r

%�YYY

YYYY

YYYY

YYYY

Y

F f Fg

(I F f )(I Fg)

l�l�

%�YYY

YYYY

YYYY

YYYY

Y

(I F f )(I Fg)

(G f I )(GgI )α f αg ��

(G f I )(GgI )

G f Gg

rr

��

G f Gg

G( f g)

χ

:]]]]]]]]]]]]]]]]

F f Fg

I (F f Fg)

l�

=<???????????

I (F f Fg)

I F( f g)

1χ

�"��I (F f Fg)

(I F f )Fg

a�)'JJJJ

(I F f )Fg

(G f I )Fg

α1)'JJJJ

(G f I )Fg

G f (I Fg)

a)'JJJJ

G f (I Fg)

G f (GgI )1α

*(��

G f (GgI )

(G f Gg)Ia� *(��

(G f Gg)I

G( f g)Iχ1 ��$$$$$$

F f Fg

(I F f )Fg

l�1

(&%%%%%%%%%%%%

(I F f )(I Fg)

(I F f )Fg

1l

PP^^^^^^^^^^^^^^^^^^^^^^^^

(I F f )(I Fg)

(I G f )(I Fg)

α11

QG��

(I G f )(I Fg)

(G f I )(GgI )

11α

��(I G f )(I Fg)

(G f I )Fg

11l

RQRRRRRRRR(I G f )(I Fg)

G f (I Fg)

r11

�"��

(G f I )(GgI )

G f (GgI )

r11

:]]]]]]]]]]]]]]]]]]]]]]]]]]]

G f (GgI )

G f Gg

1r

_________________________

(G f Gg)I

G f Gg

r

S5``

````

````

````

````

````

````

`

�

∼= ⇓(�α)−1

⇓λ

∼=

∼=

∼=

∼=∼=

⇓ρ∼=


The cell labeled � is then given by the composite below, where we have writtencomposition along 0-cell boundaries as concatenation to save space.

a ∗((11)l

) Nat.∼=(

1(1l))

∗ a1μ∼=

(1(r1)

)∗ (1a�) ∗ a

∼=(

1(l1))

∗ (1a�) ∗ a1λ∼= (1l) ∗ aμ∼= r(11).

The unmarked isomorphism is lI ∼= rI . The proof that these cells satisfy thepseudo-icon axioms is a very long calculation using both the transformationaxiom and the tricategory axioms; as the pasting diagrams which we mustshow are equal take up several pages to write down, we omit these calculationshere.

For the converse, let β : F ⇒ G be an equivalence pseudo-icon. We mustnow produce a transformation α whose components on objects are all identi-ties. For a 1-cell f : a → b, the 2-cell α f is the composite r �

G f β f lF f , and theadjoint equivalence is then the composite of the three adjoint equivalences forr, l, β.

Next we must give invertible 3-cells�α,Mα , and check the transformationsaxioms. The 3-cell � is given by the pasting below.

(I F f )Fg

F f Fg

l1

#!F f Fg

G f Fgβ1 ��))))

G f Fg

(G f I )Fgr �1 ��))))

(G f I )Fg G f (I Fg)a �� G f (I Fg) G f Fg

1l �� G f Fg G f Gg1β �� G f Gg

G f (GgI )

1r �'%((((

G f (GgI )

(G f Gg)I

a�'%((((

(G f Gg)I

G( f g)I

χ1��

(I F f )Fg

I (F f Fg)a ��I (F f Fg)

1F( f g)1χ��1F( f g) F( f g)

l�� F( f g)

G( f g)

χ

��G( f g)

G( f g)I

r ��

I (F f Fg)

F f Fg

lTRF f Fg

F( f g)

χ

��

G f Fg

G f Fg

1

$# G f Gg

G( f g)

χ

US

G f Gg

(G f Gg)Ir � VT

⇐λ ∼=

⇓(�β)−1

⇓μ

⇓ρ

∼=

The 3-cell M is given by the pasting below.

I I Ir �

�� I I I F I1ιF �� I F I F I

l �� F I G Iβ �� G I G I I

r ��I I

Il 53DDDDDDD

I

F I

ιF

$#��

I

I1

�� I

G I

ιG

*(I

I Il�

��

I

I Ir � ��

I I

G I I

ιG 1

20∼=∼= ⇓Mβ

∼= ∼=

118 Basic structure

Each of the unlabelled isomorphisms is either naturality or obtained as themate of the isomorphism lI ∼= rI . Now there are the transformation axioms tocheck, and as before we omit the calculations.

Remark 7.14 The previous lemma does not establish a bijection betweenequivalence pseudo-icons α : F ⇒ G and identity-component transforma-tions β : F ⇒ G. The former are objects in a category, while the latterare objects in a bicategory. The correspondence given above does show thatthere is an embedding of equivalence pseudo-icons into transformations whichis biessentially surjective on the collection of transformations with identitycomponents.

The following lemma has a simple proof that we omit. Recall that a functorF : S → T is 2-locally P if each of the induced maps on the categories of2- and 3-cells F : S(a, b)( f, g) → T (Fa, Fb)(F f, Fg) have property P .

Lemma 7.15 Let F,G : S → T be functors between tricategories, andα : F → G be a pseudo-icon between them whose components at every 1-cellare equivalences. Then F is 2-locally faithful (2-locally full) if and only if G is2-locally faithful (2-locally full).

Definition 7.16 Let X,Y be bicategory-enriched graphs, and let F,G :X → Y be maps between them. The category-enriched 2-graph Eq(F,G) isdefined to have objects those a ∈ X0 such that F0a = G0a. The category-enriched graph Eq(F,G)(a, b) has objects pairs (h,α) with h : a → bin X and α : F(h) → G(h) an adjoint equivalence in Y . The category

Eq(F,G)(a, b)((h,α), (h′,α′)

)has objects the pairs (β, ) with β : h → h′

in X and an invertible 2-cell in Y (Fa,Ga) of the form

: G(β) ∗ α ⇒ α′ ∗ F(β).

The category Eq(F,G)(a, b)((h,α), (h′,α′)

)has 1-cells with source (β, )

and target (β ′, ′) those 2-cells : β ⇒ β ′ such that

(1α′ ∗ F ) ◦ = ◦ (G ∗ 1α).

Lemma 7.17 1. The category-enriched 2-graph Eq(F,G) can be equippedwith the structure of a bicategory-enriched graph admitting a locally strictmap

π : Eq(F,G) → X.

2. If X, Y are tricategories and F,G are functors between them, then Eq(F,G)admits the structure of a tricategory such that


(1) π can be given the structure of a strict functor and(2) there is an equivalence pseudo-icon σ : Fπ → Gπ .

Proof For the first claim, we need to define horizontal compositions, 1-cellidentities, and the requisite constraint isomorphisms to provide each category-enriched graph Eq(F,G)(a, b) with the structure of a bicategory. The 1-cellidentity for (h, α) is(

1h, (1α ∗ (φF0 )

−1) ◦ r−1α ◦ lα ◦ (φG

0 ∗ 1α)).

Composition of 1-cells is given by setting the first component of (β ′, ′) ∗(β, ) equal to β ′ ∗ β and the second component equal to the pasting diagrambelow.

Fh Ghα �� Gh

Gh′

Gβ��

Gh′

Gh′′

Gβ ′��

Fh

Fh′

Fβ

��Fh′

Fh′′

Fβ ′��

Fh′ Gh′α′

��

Fh′′ Gh′′α′′

��

Gh

Gh′′

G(β ′β)

�K

Fh

Fh′′

F(β ′β)

!�

��

′�� (φG )−1

DDφFDD

Horizontal composition of 2-cells is given by horizontal composition in Y ; it issimple to check that this gives a composition functor. The constraint 2-cells areall given by the constraint 2-cells in the hom-bicategories of Y , and coherenceimplies that these satisfy the two bicategory axioms. We now define π by

π(h, α) = hπ(β, ) = β

π( ) = .

It is trivial to check that we can equip π with the structure of a map inGr(Bicats).

For the second claim, we must first give a composition functor

Eq(F,G)(b, c)× Eq(F,G)(a, b) → Eq(F,G)(a, c).

On objects, we define (h′,α′)⊗(h,α) to have its first component be h′⊗h andits second component have left adjoint be given by the following composite.

F(h′ ⊗ h)χ �

−→ F(h′)⊗ F(h)α′⊗α−→ G(h′)⊗ G(h)

χ−→ G(h′ ⊗ h).

The remainder of the adjoint equivalence is then defined in the obvious way.On 1-cells, we define

(δ′, ′)⊗ (δ, ) : (h′,α′)⊗ (h,α) → ( j ′,β ′)⊗ ( j,β)

120 Basic structure

to have its first component be δ′ ⊗ δ. The second component is defined by thepasting diagram below. (Note that we have used u = (Gδ′ ∗α′)⊗ (Gδ ∗α) andv = (β ′ ∗ Fδ′)⊗ (β ∗ Fδ) for space reasons.)

F(h′⊗h) Fh′⊗Fhχ �

�� Fh′⊗Fh Gh′⊗Ghα′⊗α �� Gh′⊗Gh G(h′⊗h)

χ �� G(h′⊗h)

G( j ′⊗ j)

G(δ′⊗δ)

��

F(h′⊗h)

F( j ′⊗ j)

F(δ′⊗δ)

��F( j ′⊗ j) F j ′⊗F j

χ �� F j ′⊗F j G j ′⊗G j

β ′⊗β�� G j ′⊗G j G( j ′⊗ j)

χ��

Fh′⊗Fh

F j ′⊗F j

Fδ′⊗Fδ

��

Gh′⊗Gh

G j ′⊗G j

Gδ′⊗Gδ

��

Fh′⊗Fh

G j ′⊗G j

u

WU

Fh′⊗Fh

G j ′⊗G j

v

)'

∼= ∼=

∼=

∼=

′⊗XI ��

The isomorphisms in the square regions are naturality isomorphisms and theisomorphisms in the triangular regions are the functoriality isomorphisms of⊗. It is immediate that this is an invertible 2-cell.

On 2-cells, we define the composition ′ ⊗ by the same formula in X .Naturality of the isomorphisms Fβ ′ ∗ Fβ ⇒ F(β ′ ∗β), Gβ ′ ∗Gβ ⇒ G(β ′ ∗β)ensures that this cell satisfies the required axiom. The unit constraint cell isgiven by the isomorphism 1 ⊗ 1 ∼= 1 for the functor ⊗, and the constraint cellfor composition is given by the isomorphism

(β ′ ⊗ β) ∗ (α′ ⊗ α) ∼= (β ′ ∗ α′)⊗ (β ∗ α)obtained from the functor ⊗. Coherence for functors implies that the requisitediagrams commute.

The associativity transformation a is defined to have its component at thetriple (h′′,α′′), (h′,α′), (h,α) be given by the 1-cell with first componentah′′h′h and second component the composite below.

F((h′′⊗h′)⊗h

)F(h′′⊗h′)⊗Fh

χ �� F(h′′⊗h′)⊗Fh G(h′′⊗h′)⊗Gh

((χ◦α′′⊗α′)◦χ �

)⊗α�� G(h′′⊗h′)⊗Gh G

((h′′⊗h′)⊗h

)χ �� G((h′′⊗h′)⊗h

)

G(

h′′⊗(h′⊗h))

Ga

��

F((h′′⊗h′)⊗h

)

F(

h′′⊗(h′⊗h))

Fa

��F(

h′′⊗(h′⊗h))

Fh′′⊗F(h′⊗h)χ �

�� Fh′′⊗F(h′⊗h) Gh′′⊗G(h′⊗h)α′′⊗

((χ◦α′⊗α)◦χ �

)�� Gh′′⊗G(h′⊗h) G(

h′′⊗(h′⊗h))

χ��

F(h′′⊗h′)⊗Fh

(Fh′′⊗Fh′)⊗Fh

χ �⊗1��

(Fh′′⊗Fh′)⊗Fh (Gh′′⊗Gh′)⊗Gh(α′′⊗α′)⊗α

�� (Gh′′⊗Gh′)⊗Gh

G(h′′⊗h′)⊗Gh

χ⊗1

#!

Fh′′⊗F(h′⊗h)

Fh′′⊗(Fh′⊗Fh)

1⊗χ �#!Fh′′⊗(Fh′⊗Fh) Gh′′⊗(Gh′⊗Gh)

α′′⊗(α′⊗α)�� Gh′′⊗(Gh′⊗Gh)

Gh′′⊗G(h′⊗h)

1⊗χ��

(Fh′′⊗Fh′)⊗Fh

Fh′′⊗(Fh′⊗Fh)

a��

(Gh′′⊗Gh′)⊗Gh

Gh′′⊗(Gh′⊗Gh)

a��

∼=

∼=

��


The 2-cells in the diagram are given by the mate of ωF on the left, ωG onthe right, a naturality isomorphism in the middle square, and unique coherencecells in the top and bottom middle regions. The 2-cell a� is defined similarly,and the unit and counit of this adjoint equivalence are given by the unit andcounit for a in X . The naturality isomorphisms are also given by the naturalityisomorphisms for a, a� in X , and it is a simple matter to check that this givesan adjoint equivalence in the appropriate functor-bicategory.

The unit (Ia, i) : a → a is defined by setting i equal to the composite below.

F Iaι�−→ IFa

ι−→ G Ia .

The left unit transformation l is defined to have component l(h,α) with firstcomponent lh and second component the composite below.

F(Ib⊗h) F Ib⊗Fhχ �

�� F Ib⊗Fh G Ib⊗Ghi⊗α �� G Ib⊗Gh G(Ib⊗h)

χ �� G(Ib⊗h)

Gh

Gl

��

F(Ib⊗h)

Fh

Fl

��Fh Gh

α��

F Ib⊗Fh

IFb⊗Fh

ι�⊗1��

IFb⊗Fh IGb⊗Gh1⊗α

�� IGb⊗Gh

G Ib⊗Gh

ι⊗1

#!

Fh

IFb⊗Fh

l��"��

IGb⊗Gh

Gh

l

!��

∼=

��

∼=

The upper left and upper right 2-cells are the mates of γ F and γ G , respec-tively, and the upper middle 2-cell is a unique coherence cell while the lowermiddle 2-cell is the mate of the naturality isomorphism for l. The naturalityisomorphism for l is given by the naturality isomorphism in X . A similar defi-nition gives l�, and the unit and counit of this adjoint equivalence are the sameas those for l in X . The same definitions give the adjoint equivalence r.

The modifications π,μ, λ, and ρ are given by those same modifications inX . A lengthy calculation shows that these are 3-cells in Eq(F,G). This dataobviously satisfies the axioms necessary for Eq(F,G) to be a tricategory asthey are the same axioms that hold in X . Thus we have given Eq(F,G) thestructure of a tricategory. It is immediate that we can choose the adjoint equiv-alence χ for the functor π to be the identity adjoint equivalence, similarly forι. The rest of the proof that we can equip π with the structure of a strict functoris trivial.

The pseudo-icon σ is constructed as follows. The adjoint equivalences σ aredefined by

σ(h,α) = α

σ �(h,α) = α�,

122 Basic structure

with the obvious units and counits defined by the units and counits of theadjoint equivalence α as well as those for l, r. For (β, ) : (h,α) → (h′,α′),we have the invertible 3-cell .

The 3-cell �(h′,α′),(h,α) is given by the pasting diagram below.

Fh′Fh F(h′h)χ �� F(h′h) F(h′h)F1 �� F(h′h)

Fh′Fh

χ ��

Fh′Fh

Gh′Gh

α′α��

Gh′Gh

G(h′h)

χ��

F(h′h) F(h′h)

1

�"Fh′Fh

Gh′Gh

α′α

��Gh′Gh G(h′h)χ

�� G(h′h) G(h′h)G1

��G(h′h) G(h′h)

1

!�Gh′Gh

Gh′Gh1 ��

Fh′Fh

Fh′Fh1 ��

∼=∼=

∼=

∼= ∼=

The 3-cell Ma is given by the pasting below.

I F Iι �� F I F I

F1 ��F I F I

1

��F I

I

ι�

��I

G I

ι

��

I

G I

ι

��G I G I

G1��G I G I

1

)'∼=

∼=∼=

These 3-cells give modifications between the appropriate transformationssince they are composed of modifications or naturality isomorphisms, and thepseudo-icon axioms follow from a very simple computation since �,M onlyconsist of constraints arising from the hom-bicategories.

The following lemma is straightforward to prove.

Lemma 7.18 Let α : F → G be a transformation between functors of tri-categories. Let βa : Fa → Ga be a family of 1-cells in the target indexed bythe objects of the source. Let ma : αa → βa be a family of adjoint equiva-lences indexed by the objects of the source. Then there is a transformation βwith components given by the cells βa and a modification m : α ⇒ β withcomponents given by the cells ma.

Corollary 7.19 Let α : F → G and β : G → H be transformations.Assume that F,G, H agree on objects, and assume that there are adjoint

7.4 Change of structure 123

equivalences ma between αa and IFa and na between βa and IGa. Then thereis an equivalence pseudo-icon γ : F → H.

Proof This follows immediately from the previous lemma and Lemma 7.13.

We can also use pseudo-icons to compare the two possible compositionlaws for strict functors. The difficulty in defining a category of tricategoriesand strict functors is that the composite of strict functors qua functors will nolonger be strict. We have already constructed a category Tricat1 whose objectsare tricategories and whose morphisms are virtually strict functors; we denotethis composition by ◦s . Given a composable pair of strict functors G, F , thestrict functor G ◦s F agrees with G ◦ F on cells, but is equipped with theunique constraint data making it strict. The following proposition is a simplecalculation which essentially reduces to repeated use of the unit constraints forthe target tricategory and the functors involved.

Proposition 7.20 If G, F are a composable pair of strict functors, then thereis an equivalence pseudo-icon ϕ : G ◦ F ⇒ G ◦s F.

Proposition 7.21 Let X be a category-enriched 2-graph, and let F :FX → T be a functor from a free tricategory into any tricategory. Then thereexists a strict functor G : FX → T and a pseudo-icon α : F → G whosecomponents at every 1-cell are equivalences.

Proof As in the bicategorical version of this proposition, let G be the strictfunctor FX → T whose restriction to X agrees with the restriction of F to X .Then the desired equivalence pseudo-icon is given by the following compositeof pseudo-icons.

Fr �−→ F ◦ (π ◦v ι) 1F ∗φ−→ F ◦ (π ◦ ι)a�−→ (F ◦ π) ◦ ι σ∗1ι−→ (G ◦ π) ◦ ιa−→ G ◦ (π ◦ ι) 1G∗φ�

−→ G ◦ (π ◦v ι)r−→ G.

7.4 Change of structure

This section will give three results, each of which explains how it is possibleto obtain new tricategory structures from known ones. The first result showshow to transport a tricategory structure along a map of its underlying data.

124 Basic structure

This is the first step towards showing that every tricategory is triequivalent to aparticular kind of semi-strict 3-category. The theorem given here will be usedrepeatedly to construct tricategory structures throughout this work. The secondand third result of this section show how to perturb a known tricategory struc-ture by altering its composition law. The result is a new tricategory structureon the same cells that is closely related to the original structure.

For the following theorem, we require the notion of a biadjoint biequiva-lence in a tricategory T . This is a technical requirement, and for the generaldefinition together with a discussion of the (lack of essential) differencebetween a biequivalence and a biadjoint biequivalence, we refer the readerto Gurski (2012).

Theorem 7.22 (Transport of Structure) Let T be a tricategory, and let S bea set. Let S(a, b) be an S×S-indexed set of bicategories. Given a function H0 :S → obT and an S × S-indexed set of biadjoint biequivalences (Hab, H �

ab),

Hab : S(a, b) → T (Ha, Hb),

there is a unique tricategory structure on S and a unique functor H thatagrees with H0 on objects and Hab on hom-bicategories such that the followingconditions hold.

(1) The functor ⊗ : S(b, c)× S(a, b) → S(a, c) is the composite

S(b, c)× S(a, b)H×H−→ T (Hb, Hc)× T (Ha, Hb)

⊗−→T (Ha, Hc)

H �−→ S(a, c).

(2) The transformation χ is

⊗T ◦ H × H = id ◦ ⊗T ◦ H × Hα�∗1−→ H H � ⊗T (H × H) = H ◦ ⊗S,

and the transformation χ � is

H ◦ ⊗S = H H � ⊗T (H × H)α∗1−→ id ⊗T (H × H) = ⊗T ◦ H × H.

The counit of this adjunction χ � χ � is the following composite.

H H � ⊗T (H × H) ⊗T (H × H)α∗1 �� ⊗T (H × H) H H � ⊗T (H × H)

α�∗1 ��

H H � ⊗T (H × H) H H � ⊗T (H × H)(α�α)∗1 ��

H H � ⊗T (H × H) H H � ⊗T (H × H)1 ��

−1∗1��

7.4 Change of structure 125

The unit is determined similarly, and a check shows that this gives anadjoint equivalence in the appropriate bicategory.

(3) The functor 1 → S(a, a) is the composite

1IHa−→ T (Ha, Ha)

H �−→ S(a, a).

(4) The transformation ι is

IHa = id ◦ IHaα�∗1−→ H H �IHa

and ι� is

H H �IHaα∗1−→ id ◦ IHa = IHa .

The counit of this adjunction ι � ι� is given by the composite below.

IHa H H �IHaα∗1 �� H H �IHa IHa

α�∗1 ��

IHa IHa(α�α)∗1 ��

IHa IHa1 ��

−1∗1��

The unit is determined similarly, and a check shows that this is an adjointequivalence in the appropriate bicategory.

(5) The modifications ω, γ , and δ for the functor H are all identities.

Proof We have provided the first four pieces of data directly. The rest of thedata for the tricategory S is determined by (7.22) as follows. The modificationπ is determined by the first functor axiom and the fact that each Hab is locallyfaithful, and the modification μ is determined by the second functor axiom.The second and third tricategory axioms then determine λ and ρ, and the firsttricategory axiom follows by applying H , using the tricategory axioms in T ,and then noting that each Hab is locally faithful.

Our next result shows how it is possible to change the composition law of atricategory to a new composition law.

Theorem 7.23 (Change of Composition) Let T be a tricategory with com-position ⊗. Let �abc : T (b, c) × T (a, b) → T (a, c) be a family of functorsindexed by triples of objects of T , and let sabc : ⊗ ⇒ � be a similarly indexedfamily of adjoint equivalences. Then there is a tricategory T� with

126 Basic structure

• obT� = obT ,

• T�(a, b) = T (a, b), and

• composition law �abc : T (b, c)× T (a, b) → T (a, c)

and a functor S : T → T� which is the identity on objects and on hom-bicategories.

Proof We need to provide the remaining data for T� and show that it satisfiesthe tricategory axioms. First, we specify that T� has the same unit as T . Thetransformation a� is given by

� ◦ (� × 1)s∗(s×1)−→ ⊗ ◦ (⊗ × 1)

a−→ ⊗ ◦ (1 × ⊗) s�∗(1×s�)−→ � ◦ (1 × �),

and a�� is given by

((s ∗ (1 × s)

) ◦ a�)

◦ (s� ∗ (s� × 1)

).

The unit and counit of this adjoint equivalence are the obvious composites ofunits and counits for a and s.

Similarly, l� and r� are defined by the diagrams below, where s� is theopposite adjoint equivalence of s.

T (a, b)T (b, b)× T (a, b) 31Ib × 1

T (a, b)

T (a, b)

1

��

T (b, b)× T (a, b)

T (a, b)

⊗

!��T (b, b)× T (a, b)

T (a, b)� /-

l��

s��

T (a, b)T (a, b)× T (a, a) 311 × Ia

T (a, b)

T (a, b)

1

��

T (a, b)× T (a, a)

T (a, b)

⊗

!��T (a, b)× T (a, a)

T (a, b)� /-

r��

s��

The modifications π�, μ�, λ�, ρ� are all obtained by pasting appropriateidentity modifications for the transformations s ×1×1, s ×1, s, s�, 1× s�, 1×1 × s� to the exterior of π,μ, λ, ρ after applying inverses of units for each ofthe adjoint equivalences 1 × s, s × 1, 1 × s × 1, s and unit isomorphisms (fromthe functor bicategories) where appropriate.

Using this definition and the fact that s is an adjoint equivalence, it is asimple matter to check the three tricategory axioms.

For the final claim, we need to give the constraint data for S. The adjointequivalence χ is the adjoint equivalence s, and the adjoint equivalence ι isthe identity adjoint equivalence. The component at h, g, f of the invertiblemodification ω is given by the diagram below.

7.5 Triequivalences 127

(h � g)� f (h ⊗ g)� fs�1 �� (h ⊗ g)� f (h ⊗ g)⊗ f

s �� (h ⊗ g)⊗ f h ⊗ (g ⊗ f )a ��(h � g)� f

(h � g)⊗ f

s

��(h � g)⊗ f

(h ⊗ g)⊗ f

s⊗1

��(h ⊗ g)⊗ f h ⊗ (g ⊗ f )a

�� h ⊗ (g ⊗ f ) h � (g ⊗ f )s�

�� h � (g ⊗ f ) h � (g � f )1�s�

�� h � (g � f )

h � (g ⊗ f )

1�s

#!h � (g ⊗ f )

h ⊗ (g ⊗ f )

s

#!

(h ⊗ g)⊗ f

(h ⊗ g)⊗ f

1

�"��h ⊗ (g ⊗ f )

h ⊗ (g ⊗ f )

1

�"��

∼=∼=

∼=

The left isomorphism is the composite of a unit isomorphism for the hom-bicategory with a naturality isomorphism for s, the middle isomorphism isthe composite of two unit isomorphisms for the hom-bicategory, and the rightisomorphism is the composite of inverses of counits and unit isomorphisms.The component of the invertible modification γ at f is given by composingthe isomorphism 1 � 1 ∼= 1 with a unit isomorphism in the hom-bicategory; δis defined similarly.

It is now easy to check the functor axioms using the fact that s is an adjointequivalence.

Finally, we introduce a result that allows one to alter the units in a tricate-gory, in much the same way that the previous result allowed a change in thecomposition law. We will not prove this, as the details are similar to those inthe previous proof.

Theorem 7.24 (Change of Units) Let T be a tricategory with units Ia :1 → T (a, a). Let Ia : 1 → T (a, a) be a collection of functors indexedby the objects of T , and let ra be a similarly indexed collection of adjointequivalences between Ia and Ia. Then there is a tricategory TI with

• obTI = obT ,• TI (a, b) = T (a, b), and• unit given by Ia : 1 → T (a, a)

and a functor R : T → TI that is the identity on objects and hom-bicategories.

7.5 Triequivalences

This section will introduce the notion of triequivalence. It is a direct categori-fication of the notion of equivalence of categories. We replace the condition ofthe functor being an isomorphism on hom-sets with being a biequivalence on

128 Basic structure

hom-bicategories, and replace essential surjectivity with the notion of triessen-tial surjectivity. This in turn relies on the notion of an internal biequivalence ina tricategory T .

Definition 7.25 1. A 1-cell f : a → b in a tricategory T is an internalbiequivalence if there exists a 1-cell g : b → a such that f ⊗ g is equivalentto idb in the bicategory T (b, b) and g ⊗ f is equivalent to ida in the bicategoryT (a, a).2. A specified biequivalence in a tricategory T consists of

• a pair of 1-cells f : a → b and g : b → a;• four 2-cells α : f ⊗ g ⇒ idb, α� : idb ⇒ f ⊗ g, β : g ⊗ f ⇒ ida , andβ� : ida ⇒ g ⊗ f ;

• and two specified equivalences (α, α�, ε f g, η f g) and (β, β�, εg f , ηg f ) inT (b, b) and T (a, a), respectively.

Remark 7.26 Note that a 1-cell f is a biequivalence if and only if there existsa biadjoint biequivalence containing f by the main theorem of Gurski (2012).

Definition 7.27 A functor H : T → T ′ is triessentially surjective if everyobject of T ′ is internally biequivalent to an object of the form Ha, a ∈ T .

Definition 7.28 A functor H : T → T ′ is a triequivalence if each Hab is abiequivalence and H is triessentially surjective.

Remark 7.29 The functors S : T → T�, R : T → TI constructed in theprevious section are triequivalences.

Theorem 7.30 Every tricategory T is triequivalent to a tricategory T ′ withthe same objects as T and T ′(a, b) a strict 2-category for all objects a, b.

Proof For each pair of objects a, b ∈ T , we can choose a biadjoint biequiva-lence T ′(a, b) → T (a, b)with T ′(a, b) a strict 2-category using the coherencetheorem for bicategories. By Proposition 7.22, we extend this to a tricategoryT ′ and a functor T ′ → T . It is clear that this is a triequivalence.

8

Gray-categories and tricategories

In this chapter, we will establish an important relationship between categoriesenriched over the monoidal category Gray and certain kinds of semi-stricttricategories. The first step is to define an intermediate notion, that of acubical tricategory. We will then show that strict, cubical tricategories areessentially Gray-categories. With this relationship in mind, we will thenprove a weak form of coherence that will be necessary later, namely thatevery tricategory is triequivalent to a cubical one. This intermediate theoremappears in Gordon et al. (1995), and the presentation here follows that oneclosely. Finally, we will show that the canonical strictification B → stB forbicategories extends to a functor of tricategories st : Bicat → Gray.

8.1 Cubical tricategories

This section is devoted to proving a weak form of the coherence theorem fortricategories. The theorem proved here will be used as a stepping stone to thestronger version of coherence. This weak form will introduce many of the con-cepts necessary to continue, and will be a simple consequence of a few resultsthat are important later.

Definition 8.1 A tricategory T is cubical if

(1) each bicategory T (a, b) is a strict 2-category,

(2) each functor Ia : 1 → T (a, a) is a cubical functor, and

(3) each functor ⊗ : T (b, c)× T (a, b) → T (a, c) is a cubical functor.

Remark 8.2 It should be noted that condition 2 above does not appear in thedefinition of cubical tricategory given by Gordon et al. (1995).

129

130 Gray-categories and tricategories

Remark 8.3 The definition of cubical tricategory could be reformulated asfollows. The monoidal category Gray is the category of 2-categories and2-functors equipped with the Gray tensor product. We also have the tricat-egory Gray which has the same objects and 1-cells, but takes into accountthe closed structure as well. We can also equip the tricategory Gray with amonoidal structure, extending the usual Gray tensor product. Then a categoryenriched in Gray, as a monoidal tricategory, is a cubical tricategory.

The main result of this section is the following theorem.

Theorem 8.4 For any tricategory T , there is a cubical tricategory stT and atriequivalence stT → T .

To prove this, we need to use the functor st : Bicat → 2Cat which wasexplored in Chapter 3. Recall that if X is a bicategory, stX has the same objects,a 1-cell f from x to y is a composable string of arrows

xf1→ x1

f2→ · · · fn→ y

(where for n = 0, we have a unique arrow in stX from x to x), and a 2-cell α :f ⇒ g is a 2-cell in X from e( f ) to e(g), where we define e( f ) inductively by

• e( f ) = idx if n = 0,

• e( f ) = f1 if n = 1, and

• e( f ) = e( f ′) ∗ f1 if n > 1, where f ′ is the 1-cell given by fn fn−1 · · · f2.

The “inclusion” X → stX sending each object to itself, each 1-cell f to thelength 1 composable string, and each 2-cell α to itself is a biequivalence, andthere is a distinguished retraction given by sending each object to itself, each1-cell f of stX to e( f ), and each 2-cell α to itself. It is easy to prove Theorem8.4 after we prove some preliminary results.

Proposition 8.5 Let X,Y be bicategories. Then there exists a cubical functorst : stX × stY → st(X × Y ) such that

1. st is the identity on objects and

2. there is an invertible icon ζ as pictured below.

stX × stY

st(X × Y )

st

��stX × stY X × Y

eX ×eY

��

st(X × Y )

X × Y

eX×Y

��

ζ��

8.1 Cubical tricategories 131

Proof We shall define st using Proposition 3.2, so we must define it with eachvariable held fixed and define a structure 2-cell satisfying certain axioms. Bynecessity, it is the identity on objects.

Note that identity 1-cells in stX are the length-zero composable stringswhich we shall write as 1x ; the identity 1-cell in the bicategory X will bewritten as idx . Thus we define st( f, 1) to be the composable string

( fn, idx )( fn−1, idx ) · · · ( f1, idx ),

and st(1, g) is defined similarly. Let I n be the 1-cell in X given bye(idx idx · · · idx ), where the identity appears n times. To define st on 2-cells(α, 1), where α : f ⇒ f ′ in stX , we must give a 2-cell in st(X × Y )

st((α, 1)

): st

(( f, 1)

)⇒ st

(( f ′, 1)

).

By definition, this is a 2-cell in X × Y

e(( fn, id)( fn−1, id) · · · ( f1, id)

)⇒ e

(( f ′

m, id)( f ′m−1, id) · · · ( f ′

1, id)).

Since composition in X × Y is componentwise, this 2-cell now has source

(e( f ), I n) and target (e( f ′), I m). Define st((α, 1)

)to be (α, γn,m) where

γn,m : I n ⇒ I m is the isomorphism given by structure constraints, uniqueby coherence. It is now easy to check that we have defined strict 2-functors stxand sty by holding each variable fixed separately.

The next step is to define the structure 2-cell � f,g for ( f, g) : (x, y) →(x ′, y′).

(x, y) (x, y′)st(1,g) �� (x, y′)

(x ′, y′)

st( f,1)

��

(x, y)

(x ′, y)

st( f,1)

��(x ′, y) (x ′, y′)

st(1,g)��

� f,g

EV $$$$$$$$$$$

$$$$$$$$$$$

By definition, this is a 2-cell in X × Y with

e(( fn, id) · · · ( f1, id)(id, gm) · · · (id, g1)

)as its source and

e((id, gm) · · · (id, g1)( fn, id) · · · ( f1, id)

)as its target. We define � f,g to be the unique isomorphism given by coherencebetween these 1-cells. There are now three axioms to be checked, but these allfollow from coherence and various naturality conditions.


For the second statement, let ( f, g) be a 1-cell in stX × stY . The componentζ( f,g) of this icon is an invertible 2-cell(

st(1, g) ∗ st( f, 1)) ⇒ (

e(g), e( f )).

By definition, this 2-cell has source given below.

e((id, gm)(id, gm−1) · · · (id, g1)

) ∗ e(( fn, id) · · · ( f1, id)

).

There is a unique coherence isomorphism between the above cell and(e(g), e( f )

)that provides the component of ζ at ( f, g); it is now trivial to

check that this defines an icon.

Remark 8.6 The previous result is the beginning of the proof that strictifi-cation is a monoidal functor from the 2-category of bicategories, functors, andicons to the 2-category of 2-categories, 2-functors, and icons. It is not the casethat this result holds on the level of categories because st is only natural up toan invertible icon.

We need two additional results before the main proof.

Proposition 8.7 Let F : B → C be a functor between bicategories. Thenthere is an invertible icon ω as pictured below.

stB Be ��stB

stC

stF

��stC Ce

��

B

C

F

��ωEV ///////

///////

Proof Since e is the identity on objects and stF agrees with F on objects,these composites agree on objects and hence we can define icons betweenthem. We need only provide the components at a 1-cell f of stB to completethe data for this icon. Let f be such a 1-cell, so that f is a composable string( fn, . . . , f1). If we write F f for the string (F fn, . . . , F f1), then ω f is a 2-cellin C of the form

F(e( f )

) ⇒ e(F f

).

There is a unique coherence isomorphism between these by the coher-ence theorem for functors, and the icon axioms follow immediately fromcoherence.

We do not provide an explicit proof of the next result. It can be derived fromthe results of Gurski (2012), or by using the correspondence between invertibleicons and adjoint equivalences given in Chapter 1.

8.1 Gray-categories 133

Proposition 8.8 The biequivalences e : stX → X and f : X → stX extendto give a biadjoint biequivalence between X and stX in the tricategory Bicat.

Remark 8.9 Note that this result can be tightened up to give an adjoint equiv-alence between the same bicategories but viewed as objects of the 2-categoryIcon.

We can now prove the main result of this section.

Proof of 8.4 Let T be a tricategory. The cubical tricategory stT will have thesame objects as T with (stT )(a, b) = st

(T (a, b)

). We apply the Transport of

Structure theorem to the identity function on the set of objects of T and thebiadjoint biequivalences e : stT (a, b) → T (a, b), f : T (a, b) → stT (a, b).Combining Propositions 8.7 and 8.5 gives an adjoint equivalence between

stT (b, c)× stT (a, b)e×e−→ T (b, c)× T (a, b)

⊗−→ T (a, c)

and

stT (b, c)× stT (a, b)st−→ st(T (b, c)× T (a, b))

st⊗−→ stT (a, c)e−→ T (a, c).

Taking the appropriate mate gives an adjoint equivalence between (st⊗) ◦ stand the composition functor used in the Transport of Structure theorem. Sim-ilarly, we can take the unit 1 → st(T (a, a)) to be the unique strict functorwhose image on the unique object is I . There is an adjoint equivalence betweenthis functor and the unit given by the Change of Structure theorem. By theChange of Composition and Change of Units theorems, this constructs the tri-category structure on stT with the desired composition and units, as well as atriequivalence stT → T .

Theorem 8.10 There is a triequivalence T → stT that is the identity on0-cells and is the inclusion f : T (a, b) → st

(T (a, b)

)on hom-bicategories.

We will not provide a proof of this theorem as it is completely straightfor-ward. All that remains is to identify the remaining constraint data and checkthe functor axioms; all of the data is obtained by pasting together units/counitsof the biadjoint biequivalence (e, f ) and the adjoint equivalences used in theprevious proof. The axioms are then simple to check.

8.2 Gray-categories

In this section, we will highlight the relationship between categories enrichedover Gray and tricategories. Since the final form of the coherence theo-rem for tricategories will state that every tricategory is triequivalent to a


Gray-category, we must first explain how Gray-categories are tricategories.We then go on to show how the strictification functor st can be extended toa functor of tricategories, and moreover that it embeds Bicat as a full sub-tricategory of Gray. This embedding is not a triequivalence, as pointed out byLack (2007), as there are 2-categories which are not strictly biequivalent to a2-category of the form stX for any bicategory X .

Definition 8.11 A Gray-category is a category enriched over the monoidalcategory 2Cat equipped with the Gray tensor product.

Theorem 8.12 1. Every strict 3-category is a Gray-category.2. The structure of a Gray-category determines, and is uniquely determined bythe structure of a strict, cubical tricategory.

Proof First, we note that every strict 2-functor A × B → C is also a cubicalfunctor. Thus the composition 2-functor for a strict 3-category X gives riseby the universal property to a composition 2-functor X (b, c) ⊗ X (a, b) →X (a, c). The rest of the Gray-category structure is simple to check.

For the second statement, it is a simple matter to directly compare data andaxioms. Note that the underlying data for a strict, cubical tricategory alwayssatisfies the tricategory axioms, so that the data for a Gray-category corre-sponds to the first four pieces of data for a strict, cubical tricategory, and theaxioms for a Gray-category correspond to the rest of the data for a strict,cubical tricategory.

Corollary 8.13 There is a strict, cubical tricategory Gray with objects strict2-categories and hom-2-categories Hom(A, B).

Proof Since Gray is a closed monoidal category with internal hom-functorHom, it is in particular enriched over Gray.

Remark 8.14 There are now two objects with the name Gray. First, thereis the monoidal category whose underlying category is 2Cat and whose ten-sor is the Gray-tensor product. And now, there is a tricategory whose objectsare 2-categories, and whose hom-2-categories consist of strict 2-functors,pseudo-natural transformations, and modifications between the 0-cell bound-ary 2-categories. These two structures contain essentially the same informa-tion, and it should be clear from context if we are refering to the monoidalcategory or the tricategory.

Remark 8.15 It should be remarked that Gray is not a small tricategory asit does not have a set of objects. The same will obviously be true of Bicat, butthis should not cause any concern. None of our constructions will ever result


in a tricategory-type structure that does not have small hom-bicategories, i.e.,hom-bicategories which have sets of 0-, 1-, and 2-cells.

We now turn to the task of proving that Bicat is triequivalent to a fullsub-tricategory of the tricategory Gray constructed in Corollary 8.13. Beforeproving this theorem, we need to establish the following local result whichis a consequence of the coherence theorem for functors and properties of thefunctor st.

Proposition 8.16 The function sending each functor of bicategories F :X → Y to the strict 2-functor stF : stX → stY extends to a biequivalence ofbicategories stXY : Bicat(X, Y ) → Hom(stX, stY ).

Proof First, we must define stXY on the 1-cells and 2-cells of Bicat(X,Y ).Given a transformation α : F ⇒ G, define stα to be the transformation withcomponent (stα)a = αa at a and with naturality constraint (stα) f given bythe commutativity of the following diagram, where f = ( fn, . . . , f1) and theunmarked isomorphisms come from coherence.

e(αb, F fn, . . . , F f1

)e(

G fn, . . . ,G f1, αa

)(stα) f ��e(αb, F fn, . . . , F f1

)

αb ◦ F(

e( f ))∼=

��

e(

G fn, . . . ,G f1, αa

)

G(

e( f ))

◦ αa

∼=��

αb ◦ F(

e( f ))

G(

e( f ))

◦ αaαe( f )��

The transformation axioms then follow from the fact that α is a transformationand coherence.

Now given : α � β, we construct st by giving it the component(st)a = a . Coherence implies that this is a modification.

It is clear that this is a functor on the relevant hom-categories since modifi-cations are composed componentwise. Now we define the structure constraintsand prove that they give a functor of bicategories. In each case, the relevantmodification has as its component at a the appropriate constraint isomorphism.The modification axioms are satisfied because of coherence.

Proving that stXY is a biequivalence requires proving that it is biessentiallysurjective and locally an equivalence of categories. To prove the first of theseclaims, recall that there are biequivalences f : X → stX and e : stY → Y .Given a 2-functor F : stX → stY , let F : X → Y be the composite eF f . Wewill show, using the transformationω from Proposition 8.7, that F is equivalentto st(F). The invertible icon ω is an isomorphism

F ◦ estX ∼= st(F) ◦ estY .


Since estX ∼= st(eX ) and e f ∼= 1, we get

F ∼= Fst(eX )st( fX )∼= FestX st( fX )∼= st(F)estY st( fX )∼= st(F)st(eX )st( fX )

= st(F).

To show that stXY is locally an equivalence of categories, let F,G : X → Ybe a pair of functors and α : stF ⇒ stG be a transformation. We define α bythe following formulas.

(α)a = αa

(α) f = α f .

Note that in the second formula, we are identifying the 1-cell f with the length-one string ( f ). In the notation of Proposition 5.1, α = e ∗ α ∗ f . There is anisomorphism between α and stα given by a modification all of whose compo-nents are constraint isomorphisms. Thus stXY is locally essentially surjective.To see that stXY is locally full and faithful, note that a modification is deter-mined by its components, so → st is injective. On the other hand, anymodification : stα � stβ gives rise to a modification with the samecomponents by restriction. It is immediate that is a modification and thatst = . Thus stXY is locally full and faithful, therefore a biequivalence.

Remark 8.17 Note that by the results of Gurski (2012), this biequivalencecould be extended to a biadjoint biequivalence. The weak inverse for stX,Y canbe chosen so that it sends F : stX → stY to e ◦ F ◦ f .

Definition 8.18 Let A, B be 2-categories. Then A and B are strictly biequiv-alent if there exist strict 2-functors F : A → B and G : B → A such that G Fis equivalent to 1A in Bicat(A, A) and FG is equivalent to 1B in Bicat(B, B).

Remark 8.19 Since A, B are strict 2-categories and the functorsFG,G F, 1A, and 1B are strict 2-functors, we could have demanded thatG F be equivalent to 1A in Gray(A, A), and similarly for FG, for a logi-cally equivalent definition. It is now easy to check that two strict 2-categoriesare strictly biequivalent if and only if they are internally biequivalent in thetricategory Gray.

Definition 8.20 Let Gray′ be the full sub-Gray-category of Gray deter-mined by all the strict 2-categories which are strictly biequivalent to a2-category of the form stB for some bicategory B.


Theorem 8.21 The tricategory Bicat is triequivalent to the tricategoryGray′.

Proof All that remains to be shown is that st extends to a functor of tricate-gories Bicat → Gray. The objects of Gray′ are precisely those in the imageof st, and we have already shown that the functors on hom-bicategories arebiequivalences.

For the adjoint equivalence χ , note that there is an invertible icon betweenthe two composite functors whose component at a pair of transformations(β, α) is a modification whose components are all 2-cells represented by iden-tities. We let this invertible icon induce the adjoint equivalence. For the adjointequivalence ι, note that st sends the identity on X to the identity on stX ;furthermore, st sends the identity transformation 1X ⇒ 1X to the identitytransformation 1stX ⇒ 1stX , so ι is the identity adjoint equivalence. Themodifications ω, γ, δ are all defined to be given by 2-cells represented byunique coherence isomorphisms, from which the functor axioms follow bycoherence.

Remark 8.22 It should be noted that the tricategory Bicat is not triequivalentto the tricategory Gray, as shown by Lack (2007). It is easy to see that theinclusion Gray′ ↪→ Gray is not a triequivalence, as the 2-category I with

• a single object x ,• a single idempotent f : x → x , and• only identity 2-cells

is not strictly biequivalent to any 2-category of the form stB. Lack uses a sim-ilar example to show that the inclusion Gray ↪→ Bicat is not a triequivalence,and then proves that any triequivalence Gray → Bicat would be forced to beequivalent to this inclusion. This produces an immediate contradiction, henceGray is not triequivalent to Bicat.

9

Coherence via Yoneda

This chapter will prove that every tricategory is triequivalent to aGray-category by a Yoneda-style argument. Such a proof proceeds in a numberof steps. First, we must study functor tricategories. Second, we must producea Yoneda embedding, and prove that it is actually an embedding. Finally, wemust identify a sub-object of the target of the Yoneda embedding as the desiredtriequivalent Gray-category. In the case of coherence for bicategories, thesewere the only steps required; here we require one more initial step, namelythat our tricategory T gets replaced by a cubical one.

The first goal is to establish the existence of a tricategory structure onthe collection of functors, transformations, modifications, and perturbationsbetween fixed source and target tricategories. We will not complete the fullproofs here, but we will establish the complete local structure – for tricat-egories S, T and functors F,G : S → T between them, we construct thehom-bicategory Tricat(S, T )(F,G). The full tricategory Tricat(S, T ) wouldrequire a number of additional calculations that we only choose to study in thecase that T is a Gray-category. We thus show that if T is a Gray-category,the bicategory Tricat(S, T )(F,G) is actually a 2-category, and then go on toproduce the remaining data for the tricategory Tricat(S, T ) and show that theresulting tricategory structure is also a Gray-category.

The second goal is to construct a Yoneda functor. A full tricategoricalYoneda lemma would express the existence of a functor

T → Tricat(T op,Bicat)

having certain properties; in particular, it should be a triequivalence whenthe target is appropriately restricted. We will not prove this theorem here,as it would require a large quantity of tedious calculations in constructingthe functor tricategory in the target. Instead, we will restrict ourselves to the

138

9.1 Local structure 139

case when T is a cubical tricategory, and then prove a similar result for thefunctor

T → Tricat(T op,Gray).

Since T is cubical, we can replace Bicat with Gray, and now the functortricategory in the target is itself a Gray-category.

The final goal is then to show that this Yoneda functor is an embeddingof T into the functor tricategory. The Gray-category triequivalent to T willthen be the essential image of the Yoneda functor. The general result, whenT is an arbitrary tricategory, is obtained by first replacing T with the cubicaltricategory stT and then applying these results.

9.1 Local structure

The first section will focus on local results that apply when S, T is any pairof tricategories. We will prove that if F,G : S → T is any pair of functorsbetween tricategories, then there is a bicategory Tricat(S, T )(F,G) whoseobjects are transformations α : F → G, whose 1-cells are the modificationsbetween these, and whose 2-cells are the perturbations between these.

Theorem 9.1 Let S, T be tricategories, and F,G : S → T be functors. Thenthere is a bicategory Tricat(S, T )(F,G) with 0-cells the transformations α :F → G, 1-cells the modifications m : α ⇒ β, and 2-cells the perturbationsσ : m � n.

Proof To define such a bicategory, we must give hom-categories, a composi-tion functor, associativity and unit isomorphisms, and then verify two axioms.The hom-category Tricat(S, T )(F,G)(α, β), hereafter abbreviated [α, β], isdefined to have objects the modifications m : α ⇒ β and morphisms the per-turbations σ : m � n. Composition of morphisms is given by defining thecomponent at a of the composite τ ◦ σ to be τa ◦ σa , where this compositionis the vertical composition of 2-cells in the appropriate hom-bicategory. Sim-ilarly, the identity arrow 1m : m � m has as its component at a the identity2-cell 1ma , once again taken in the appropriate hom-bicategory. It is immediatethat these are perturbations. It is easy to see that this does give the structure of acategory, as vertical composition of 2-cells in a bicategory is strictly associativeand strictly unital.

The next step in establishing the local bicategory structure is to provide acomposition functor

∗ : [β, γ ] × [α, β] → [α, γ ].

140 Coherence via Yoneda

On objects, we define n ∗ m to have as its component at a the compositena ∗ ma , where we now use the composition of 1-cells in the appropriatehom-bicategory. To give a modification, we must also provide an invertiblemodification (in the bicategorical sense). This consists of, for each f ∈T (a, b), a 2-cell

1G f ⊗ (na ∗ ma) ◦ α f ⇒ (nb ∗ mb)⊗ 1F f ◦ γ f .

This 2-cell is given by the pasting diagram below; the unmarked isomorphismsare unique constraint isomorphisms.

αb⊗F f G f ⊗αa

α f �� G f ⊗αa G f ⊗γa1⊗(nama) ��αb⊗F f

βb⊗F f

mb⊗1

'%((((((((((((

βb⊗F f G f ⊗βaβ f

��

G f ⊗αa

G f ⊗βa

1⊗ma

'%((((((((((((

G f ⊗βa

G f ⊗γa

1⊗na

��))))))))))))βb⊗F f

γb⊗F f

na⊗1

'%((((((((((((

αb⊗F f

γb⊗F f

(nama)⊗1

�� γb⊗F f

G f ⊗γa

γ f

.,

∼=

∼=m f

C� ))))))))))))

n f

�'((((((

((((((

It is immediate that this is invertible, and the modification axiom is trivial tocheck using that m f and n f give modifications.

We now define τ ∗ σ to have component τa ∗ σa at a, where this horizon-tal composite is formed in the appropriate hom-bicategory. Once again, it isstraightforward to show that this defines a perturbation. Functoriality of ∗ fol-lows since it is merely the interchange law for the hom-bicategories used inour constructions.

The next step is to define the associativity and unit structure constraints. Theassociativity constraint is given by the perturbation A : (p∗n)∗m � p∗(n∗m)having as its component at the object a the 2-cell

Aa : (pa ∗ na) ∗ ma ⇒ pa ∗ (na ∗ ma)

which is the associativity constraint in the appropriate hom-bicategory. Thesingle axiom for being a perturbation follows immediately as a consequenceof coherence. Similar definitions provide the left and right unit constraints, Land R, respectively. There are now two bicategory axioms to check, but thesefollow directly from the fact that they hold locally by coherence, i.e., in eachhom-bicategory separately.

Corollary 9.2 Let S be a tricategory and let T be a tricategory such thateach T (a, b) is a strict 2-category. Then for any pair of weak functors F,G :S → T , the bicategory Tricat(S, T )(F,G) is a strict 2-category.

9.2 Global results 141

Proof Since the associativity and unit constraints are given by the constraintsin the hom-bicategories of the target, the result is immediate.

9.2 Global results

For this section, S will be any tricategory and T will be any strict, cubicaltricategory, i.e., a Gray-category.

Theorem 9.3 (Cubical Composition) Under the above hypotheses, there isa cubical composition functor

⊗ : Tricat(S, T )(G, H)× Tricat(S, T )(F,G) → Tricat(S, T )(F, H)

such that β ⊗ α is the transformation defined by

• the component at the object a is given by

(β ⊗ α)a = βa ⊗ αa;• the adjoint equivalence β ⊗ α is given by

(1) (β ⊗ α) f is the composite

(βb ⊗ αb)⊗ F f=−→ βb ⊗ (αb ⊗ F f )

1⊗α f−→ βb ⊗ (G f ⊗ αa)=−→

(βb ⊗ G f )⊗ αaβ f ⊗1−→ (H f ⊗ βa)⊗ αa

=−→ H f ⊗ (βa ⊗ αa),

and(2) (β ⊗ α)�f is the composite

(1 ⊗ α�f ) ∗ (β�

f ⊗ 1),

(3) the counit of this adjunction is the obvious composite of counits, and theunit is the obvious composite of units;

• the invertible modification � is provided by the pasting diagram below,where we have written tensor as concatenation;

βcαc FgF f βcGgαb F f1αg1 �� βcGgαb F f Hgβbαb F f

βg11 �� Hgβbαb F f HgβbG f αa

11α f �� HgβbG f αa

HgH fβaαa

1β f 1��

HgH fβaαa

H(g f )βaαa

χ11��

βcαc FgF f

βcαc F(g f )

11χ

��βcαc F(g f ) βcG(g f )αa

1αg f

�� βcG(g f )αa H(g f )βaαaβg f 1

��

βcGgαb F f

βcGgG f αa

11α f ��

βcGgG f αa

HgβbG f αa

βg11

��HHHHHHHHHHHHHHβcGgG f αa

βcG(g f )αa

1χ1��

∼=

1⊗�αYW aaaaaaaa

aaaaaaaa

�β⊗1C� ))))))))))))


and• the invertible modification M is given by the pasting diagram below.

βaαa βaαa IFa= ��βaαa

βa IGaαa

=539999999999999βaαa

IHaβaαa

=

��

βaαa IFa βaαa F Ia11ιF �� βaαa F Ia

βa G Iaαa

1αIa��

βa G Iaαa

H Iaβaαa

βIa 1��

βa IGaαa βa G Iaαa1ιG 1

��

IHaβaαa H IaβaαaιH 11=ιH 1

��

1⊗Mα��

Mβ⊗1��

Proof To give a cubical functor as above, we first need to provide strict2-functors ⊗α and ⊗β which each hold one variable constant. First, note thatthe formulas above do indeed give a transformation β ⊗ α : F ⇒ H . Thus wehave defined the values of these functors on 0-cells, so we now extend them to1- and 2-cells. Here we give explicit formulas for ⊗β ; those for ⊗α are sim-ilar. For a modification m : α ⇒ α′, we define ⊗β(m) to be the followingtrimodification. The component at a is

⊗β(m)a = 1βa ⊗ ma,

where the identity 2-cell is taken in the relevant hom-bicategory. For each f :a → b in S, the modification ⊗β(m) is defined to have component at f givenby the following pasting diagram.

βbαb F f βbG f αa1⊗α f �� βbG f αa H fβaαa

β f ⊗1�� H fβaαa

H fβaα′a

1⊗1⊗ma

��

βbαb F f

βbα′b F f

1⊗mb⊗1

��βbα

′b F f βbG f α′

a1⊗α′f

�� βbG f α′a H fβaα

′aβ f ⊗1

��

βbG f αa

βbG f α′a

1⊗1⊗ma

��1⊗m fEV ///////

///////∼=

On 2-cells, we define ⊗β by the formula

⊗β(σ )a = 11βa⊗ σa .

The perturbation axiom is immediate.Now we check that ⊗β is a strict 2-functor. First, note that

⊗β(n)a ∗ ⊗β(m)a = ⊗β(n ∗ m)a

since T is a Gray-category. This shows that the components for objects of themodifications ⊗β(n) ∗ ⊗β(m) and ⊗β(n ∗ m) coincide, and it is straightfor-ward to check the same for the components for 1-cells. If m is the identitymodification, it is easy to check that ⊗β(m) is the identity as well. Finally, we

9.2 Global results 143

can check that ⊗β preserves all possible compositions of 2-cells, as well asidentity 2-cells, by similar arguments.

Finally, to define a cubical composition functor we must provide a structure2-cell and check that it satisfies three axioms. This perturbation will have as itscomponent at a the coherence cell

(na ⊗ 1α′a) ∗ (1βa ⊗ ma)

∼=⇒ (1β ′a⊗ ma) ∗ (na ⊗ 1αa )

given by the isomorphism � arising from the Gray-category structure on T .The perturbation axiom is a consequence of the naturality of the isomorphism� from the Gray-category structure on T . It is immediate that this satisfies thenecessary axioms to give the comparison cell for a cubical functor, as they aresatisfied locally by the Gray-category axioms in T .

We are now in a position to prove the main theorem of this section.

Theorem 9.4 (Gray-category structure) Let S be any tricategory and letT be a strict, cubical tricategory. Then there is a Gray-category Tricat(S, T )with

• objects weak functors F : S → T ,

• hom-2-categories Tricat(S, T )(F,G) as given above, and

• composition 2-functor

Tricat(S, T )(G, H)⊗ Tricat(S, T )(F,G) → Tricat(S, T )(F, H)

induced by the cubical functor in Theorem 9.3.

Proof All that remains is to provide a unit map 1 → Tricat(S, T )(F, F) andto prove that composition is strictly unital and associative. The unit is givenby the 2-funtor which sends the unique object to the identity transformation1F : F → F given by the following. The component at a is the 1-cell IFa

given by the unit in T . The adjoint equivalence

1F : (idFa)∗ ◦ F → (idFa)∗ ◦ F

is taken to be the identity (recall that T has strict units), and the invertiblemodifications are both the identity. The rest of the unit 2-functor is determinedsince it is a strict 2-functor. It is immediate that this gives ⊗ a strict unit by theproof of the previous theorem and the Gray-category axioms for �.

Finally, we check associativity. From the definition of β ⊗ α, we see that

(γ ⊗ β)⊗ α = γ ⊗ (β ⊗ α)


since the composition in T is strictly associative and unital. An easy computa-tion shows that the same holds for 1- and 2-cells. Since ⊗ is strictly associativeand unital, Tricat(S, T ) has been given the structure of a Gray-category.

Remark 9.5 Gordon et al. (1995) outline a strategy for providing a tricate-gory structure on the 3-globular set whose 0-cells are functors between fixedtricategories, whose 1-cells are transformations, whose 2-cells are modifica-tions, and whose 3-cells are perturbations. It would be a simple matter to usethe results above and the Transport of Structure theorem to realize that strat-egy, but we have refrained from doing so as it is not necessary for our proof ofthe coherence theorem for tricategories. Additionally, this tricategory structurewould not be the naive one with the composition functor

⊗ : Tricat(S, T )(G, H)× Tricat(S, T )(F,G) → Tricat(S, T )(F, H)

given by composition of transformations on 0-cells. This is analogous to thefact that the tricategories Bicat and B in the previous chapter do not coincide,but instead are only triequivalent.

Remark 9.6 Power (2007) gives the construction of a functor Gray-category[S,Gray] from a Gray-category S into Gray whose objects are Gray-functors, whose 1-cells are (fully weak) transformations, whose 2-cells aremodifications, and whose 3-cells are perturbations. This Gray-category is real-ized as the Eilenberg–Moore object of a Gray-monad on GrayobS . It is clearthat this is a sub-Gray-category of our Tricat(S,Gray). One should be able toconstruct a monad whose algebras are the weak rather than strict (i.e., Gray-)functors, but to do so would involve substantial calculations.

9.3 The cubical Yoneda lemma

This section will focus on the case when the target tricategory T is cubical, andthat assumption will now be made throughout this section. We proceed with anumber of calculational lemmata in order to make the proofs more digestible.

Most of the proofs in this section are unenlightening calculations. Manyfollow directly from the tricategory axioms, but some are quite involved.

Lemma 9.7 Let a be an object of T . Then there is a functor

T (−, a) : T op → Gray

whose value at b is the 2-category T (b, a).

9.3 The cubical Yoneda lemma 145

Proof Recall that the tricategory Gray has 0-cells strict 2-categories, 1-cellsstrict 2-functors, 2-cells transformations, and 3-cells modifications. First, wehave that T (−, a)(b) = T (b, a) which is a strict 2-category since T is cubical.If f : b → b′ is a 1-cell in T , then T (−, a)( f ) : T (b′, a) → T (b, a) (whichwe shall now call f ∗) is defined as follows.

• On the 0-cells of the hom-2-categories, f ∗(g) = g ⊗ f .• On the 1-cells α : g → h, f ∗(α) = α ⊗ 1 f .• On the 2-cells : α ⇒ β, f ∗() = ⊗ 11 f .

Since the hom-bicategories for T are strict 2-categories and the compositionfunctor is cubical, we have that

f ∗(β ∗ α) = (β ∗ α)⊗ 1 f

= (β ∗ α)⊗ (1 f ∗ 1 f )

= β ⊗ 1 f ∗ α ⊗ 1 f

= f ∗(β) ∗ f ∗(α),

so f ∗ strictly preserves composition. Composition being cubical also forcesf ∗ to strictly preserve units, thus proving that f ∗ is a strict 2-functor.

For α : f → f ′, we define the transformation T (−, a)(α) : f ∗ ⇒ f ′∗(now denoted α∗) as follows.

• For g : b′ → a, the component α∗g is 1g ⊗ α : g ⊗ f → g ⊗ f ′.

• For a 1-cell β : g → g′, we define the 2-cell α∗β to be the inverse of the

structure 2-cell for cubical composition.

g ⊗ f

g′ ⊗ fβ⊗1 ��

g ⊗ f

g ⊗ f ′1⊗α ��

g′ ⊗ f

g′ ⊗ f ′1⊗α

��

g ⊗ f ′

g′ ⊗ f ′

β⊗1

��

�−1β,α

��

• The transformation axioms follow immediately from the cubical functoraxioms.

For : α ⇒ α′, we define the modification T (−, a)() : α∗ � α′∗ (nowdenoted ∗) by the following.

• For a 0-cell g in the hom-2-category, the component g is 11g ⊗ .• The modification axiom is a result of the naturality axioms for the cubical

composition.

Now that we have defined T (−, a) on cells, we must show that it is a functorwhen equipped with appropriate constraint data. First, we check that it defines


a homomorphism of bicategories on the appropriate hom-bicategories. It isclear that composition of 3-cells is preserved strictly, as are identity 3-cells;therefore we have functors

T op(b, b′)(g, g′) → Gray(

T (b, a), T (b′, a))(g∗, g′∗).

Now let α : f → f ′ and α′ : f ′ → f ′′ be 1-cells in T op(b, b′). Then (α′ ∗α)∗has component at g

1g ⊗ (α′ ∗ α) = (1g ⊗ α′) ∗ (1g ⊗ α) = α′∗g ∗ α∗

g ⊗ α

by the same argument as above. By the characterization of cubical functors, itis easy to see that the 2-cells (α′ ∗ α)∗β and α′∗

β ∗ α∗β are equal as well. Thus we

see that on the hom-bicategories – which are actually strict 2-categories – wehave defined strict functors.

Next we construct the adjoint equivalence χ for T (−, a). This consists ofa pair of transformations and a pair of invertible modifications satisfying thetriangle identities. The transformation χ has component at h ∈ T op(x, y) theassociator ah f g : (h ⊗ f )⊗ g → h ⊗ ( f ⊗ g), so that the adjoint equivalenceχ is just the adjoint equivalence a (for T ) with two of the variables held fixed.

The adjoint equivalence ι is just the opposite of the adjoint equivalence rfor T . The invertible modification ω is a mate of π (for T ), and the invertiblemodifications γ and δ are mates of ρ and μ, respectively.

The first functor axiom follows from the first tricategory axiom, and thesecond functor axiom follows from the third tricategory axiom.

Lemma 9.8 Let f : a → a′ be a 1-cell of T . Then there is a transformationT (−, f ) : T (−, a) → T (−, a′) whose component at the object b is a functorwhich is g → f ⊗ g on objects.

Proof The component at an object b will be the strict 2-functor

f∗ : T (b, a) → T (b, a′)

defined by

• f∗(g) = f ⊗ g,• f∗(α) = 1 f ⊗ α, and• f∗() = 11 f ⊗ .

This is a 2-functor by the same arguments used to show that f ∗ is a 2-functor.Next we construct an adjoint equivalence

T(−, f) : ( f∗)∗ ◦ T (−, a) → ( f∗)∗ ◦ T (−, a′)


in the appropriate functor bicategory. First, we must define the transformationT (−, f ) to have a component at g : b → b′ (in T op); this component willbe a 1-cell in Gray(T (a, b), T (a′, b′)), that is, a transformation between strict2-functors. The source 2-functor is defined on objects by

j → f ⊗ ( j ⊗ g),

and the target 2-functor is defined on objects by

j → ( f ⊗ j)⊗ g.

The adjoint equivalence is then the opposite of the adjoint equivalence a (sincea is the associativity adjoint equivalence for T , this is actually the associativityadjoint equivalence for T op). The invertible modification� is the mate of π−1

with source a ∗ (a� ⊗1)∗a� and target a� ∗ (1⊗a). The invertible modificationM is the mate of ρ−1 with source a� ∗ (1 ⊗ r �) and target r �.

The first transformation axiom follows from the first tricategory axiom,the second is proved using the second tricategory axiom, and the third is animmediate consequence of the third tricategory axiom.

Lemma 9.9 Let α : f ⇒ f ′ be a 2-cell in T . Then there is a modifica-tion T (−, α) : T (−, f ) ⇒ T (−, f ′) whose component at the object b is atransformation whose component at g is

f ⊗ gα⊗1−→ f ′ ⊗ g.

Proof A transformation has as its data components at objects and naturalityisomorphisms for each 1-cell. The naturality isomorphism is the modificationwhich is given componentwise by the isomorphism �−1 given by the cubicalcomposition.

The invertible modification T (−, α) is defined to have its component at jbe the naturality isomorphism for a�.

The two modification axioms are consequences of the fact that � and Mgiven in the previous lemma are modifications.

Lemma 9.10 Let : α � α′ be a 3-cell in T . Then there is a perturba-tion T (−, ) : T (−, α) � T (−, α′) whose component at the object b is themodification whose component at g is

α ⊗ 1g⊗1=⇒ α′ ⊗ 1g.


Proof The single axiom is trivial using the naturality of the isomorphism γ−1

that is the naturality isomorphism for T (−, α).

Theorem 9.11 Let T be a cubical tricategory. Then there is a functor

y : T → Tricat(T op,Gray)

that is defined on cells as below.

a → T (−, a)f → T (−, f )α → T (−, α) → T (−, ).

Proof Now that we have defined y on cells, we must examine its functo-riality and provide constraints to give it the structure of a functor betweentricategories. For ease of notation, we will write the composition along0-cell boundaries in Tricat(T op,Gray) as �. First, we examine y on hom-bicategories, which in our case are strict 2-categories. It is immediate fromthe definition given in Lemma 9.10 that y strictly preserves identity 3-cellsand that

y( ◦ ) = y() ◦ y( ).

Finally, we need to compare y(α′ ∗ α) to y(α′) ∗ y(α), where we are writingthe composition of 1-cells in the hom-bicategories as concatenation. Since the

composition in T is cubical, the transformations y(α′ ∗α)b and(

y(α′)∗y(α))

bhave the same components at g; similarly, these transformations have the samenaturality isomorphisms by the cubical composition axioms. We now compare

the invertible modifications y(α′ ∗ α)g and(

y(α′) ∗ y(α))

g. It follows from

the fact that T is locally a 2-category and that its composition is cubical thatthese two modifications have the same components, hence are in fact equal. Itfollows similarly that if α is the identity, then so is y(α). Thus y is given thestructure of a strict 2-functor on each of the hom-2-categories.

Next, we must define an adjoint equivalence χ : � ◦ y × y ⇒ y ◦ �. For anobject of the source (g, f ), we need a 1-cell

y(g)� y( f ) → y(g ⊗ f ).

Such a 1-cell is a modification between transformations; the component at anobject b of T is the transformation a�. The required invertible modification isthe naturality isomorphism for a�. The adjoint χ � is defined similarly, and the


unit and counit for this adjunction are given by the inverses of the units andcounits for the adjoint equivalence a.

Next, we must determine the unit adjoint equivalence ι. The modification ιhas source 1y(a) and target y(Ia). Thus we define the component at b to be thetransformation l�. The required invertible modification is the naturality isomor-phism for l�. The rest of the definition is made in analogy with the definitionof χ .

The invertible modification ω is the mate of π−1 with source (a ⊗1)∗a�∗a�

and target a� ∗ (1 ⊗ a�). The invertible modification γ is the mate of λ withsource (l ⊗ 1) ∗ a� ∗ l� and target the identity; the invertible modification δ isdefined similarly.

The first functor axiom follows immediately from the first tricategoryaxiom. The second functor axiom then follows immediately from the secondtricategory axiom.

Theorem 9.12 (Cubical Yoneda lemma) Let T be a cubical tricategory, andy : T → Tricat(T op,Gray) be the functor constructed above. Then y is alocal biequivalence, i.e., each 2-functor

ya,a′ : T (a, a′) → Tricat(T op,Gray)(

T (−, a), T (−, a′))

is a biequivalence.

Proof We must show that this 2-functor is locally an equivalence and isbiessentially surjective.

1. The 2-functor ya,a′ is locally faithful.Let , : α ⇒ β be parallel 2-cells in T (a, a′), and assume that y() =y( ). Two perturbations are equal if and only if they have identical compo-nents for all objects. Thus we see that ⊗ 11g = ⊗ 11g for all g : b → a.In particular, taking b = a and g = Ia , we get that ⊗ 11I = ⊗ 11I . Thefollowing diagram commutes by the naturality of r .

f ⊗Ia f ′⊗Iaα⊗1 �� f ′⊗Ia

f ′

r f ′��

f ⊗Ia

f

r f

��f f ′α ��f f ′

β

��

rαG4 ��

��

f ⊗Ia f ′⊗Ia

α⊗1

��f ⊗Ia f ′⊗Ia

β⊗1�� f ′⊗Ia

f ′

r f ′��

f ⊗Ia

f

r f

��f f ′

β��

⊗1��

rβG4 ��

=


This gives the following equality of 2-cells in the 2-category T (a, b), using thesame diagram with instead of .

( ∗ 1r f ) ◦ rα = ( ∗ 1r f ) ◦ rα.

But since rα is invertible and r f is an equivalence 1-cell, this implies that = .

2. The 2-functor ya,a′ is locally full.Let α, β : f → f ′ be parallel 1-cells in T (a, a′), and let σ : y(α) ⇒ y(β)be a perturbation between them. Thus for each object b in T op, we have a3-cell σb : y(α)b � y(β)b in Gray. Such a 3-cell consists of a modificationbetween the transformations y(α)b and y(β)b. The modification σb has as itscomponent at the object g ∈ T (b, a) a 2-cell (σb)g : α ⊗ 1g ⇒ β ⊗ 1g . Thuswe obtain the 2-cell below, denoted σ , where we have taken appropriate matesof the naturality isomorphisms for r to obtain the unmarked cells.

f f ′

α

�f f ⊗I

r �� f ⊗I f ′⊗I

α⊗1I

)'f ⊗I f ′⊗I

β⊗1I

(& f′⊗I f ′r ��f f ′

β

*(

��

(σa)I��

��

We now claim that y(σ ) = σ . The perturbation y(σ ) has as its component at bthe modification with component at g given by σ⊗11g , and we must show thatthis is equal to (σb)g . Now the 3-cell σ ⊗ 11g is given by the pasting diagrambelow.

f ⊗g f ′⊗g

α⊗1

�f ⊗g ( f ⊗I )⊗g�� ( f ⊗I )⊗g ( f ′⊗I )⊗g

(α⊗1I )⊗1

)'( f ⊗I )⊗g ( f ′⊗I )⊗g

(β⊗1I )⊗1

(&( f ′⊗I )⊗g f ′⊗g��f ⊗g f ′⊗g

β⊗1

*(

��

(σa )I ⊗1��

��

r �⊗1 r⊗1


This is equal to the pasting diagram

f ⊗g

f ⊗g

1

�"��f ⊗g ( f ⊗I )⊗g

r �⊗1 ��f ⊗g

f ⊗g

1!��

f ⊗g

( f ⊗I )⊗g

r �⊗1

��

f ⊗g

( f ⊗I )⊗g

r �⊗1

#!

f ⊗g f ′⊗g

α⊗1

)'

( f ⊗I )⊗g ( f ′⊗I )⊗g

(α⊗1)⊗1

)'( f ⊗I )⊗g ( f ′⊗I )⊗g

(β⊗1)⊗1

(&

f ⊗g f ′⊗g

β⊗1

(&

f ′⊗g

( f ′⊗I )⊗g

r �⊗1

��

f ′⊗g

( f ′⊗I )⊗g

r �⊗1

#!

f ′⊗g

f ′⊗g

1

!��

f ′⊗g

f ′⊗g

1

�"��

( f ′⊗I )⊗g f ′⊗gr⊗1 ��

∼=

∼=

⇓(σa)I ⊗1

∼=

∼=

∼=

∼=

by expanding out the mates involved; note that we have used in an essentialway that T has hom-2-categories and not just hom-bicategories. The unmarkedisomorphisms are either naturality isomorphisms (for r �) tensored with anidentity or unit isomorphisms (for the adjoint equivalence r) tensored with anidentity.

Since each σb is a modification and σ is a perturbation, we have thefollowing equality of 3-cells in T ,

f ⊗g

( f ⊗I )⊗g

r �⊗1

��

f ⊗g f ′⊗g

α⊗1

)'

( f ⊗I )⊗g ( f ′⊗I )⊗g

(α⊗1)⊗1

)'( f ⊗I )⊗g ( f ′⊗I )⊗g

(β⊗1)⊗1

(&

f ′⊗g

( f ′⊗I )⊗g

r �⊗1��

f ⊗g

( f ⊗I )⊗g

r �⊗1

��

f ⊗g f ′⊗g

α⊗1

)'f ⊗g f ′⊗g

β⊗1

(&

( f ⊗I )⊗g ( f ′⊗I )⊗g

(β⊗1)⊗1

(&

f ′⊗g

( f ′⊗I )⊗g

r �⊗1��

=∼=

∼=⇓(σa)I ⊗1

⇓(σb)g

where once again the unmarked isomorphisms are naturality isomorphisms forr � tensored with identities. Combining the above pasting diagram with thisequality gives that (y(σ )b)g = (σb)g since the rest of the cells in the resultingdiagram are pairs of isomorphisms with their inverses.

3. The 2-functor ya,a′ is locally essentially surjective.To show this, let α : y( f ) → y( f ′) be a modification. We must show thatthere is a 2-cell α : f → f ′ and an invertible perturbation y(α) ∼= α.

The component of α at the object b in T op is a transformation αb withcomponent

(αb)g : f ⊗ g → f ′ ⊗ g


and naturality isomorphism shown below.

f ⊗g f ′⊗g(αb)g ��f ⊗g

f ⊗g′

1⊗β��

f ⊗g′ f ′⊗g′(αb)g′

��

f ′⊗g

f ′⊗g′

1⊗β��

(αb)β$G//////

//////

In particular, we also have the 2-cell in T shown below.

fr �−→ f ⊗ I

(αa)I−→ f ′ ⊗ Ir−→ f ′

We shall denote this 2-cell by α, and the claim is that y(α) ∼= α in the functortricategory. An invertible perturbation exhibiting such an isomorphism wouldhave data consisting of, for every object b in T op, an invertible modificationy(α)b � αb. This would consist of, for every g : b → a in T , an isomorphismbetween (αb)g and (y(α)b)g; since the composition in T is cubical, this is anisomorphism between (αb)g and

f ⊗ gr �⊗1−→ ( f ⊗ I )⊗ g

(αb)I ⊗1−→ ( f ′ ⊗ I )⊗ gr⊗1−→ f ′ ⊗ g

satisfying the axiom for being a modification. The data for α also gives, forevery j : b → b′ in T op, an invertible 3-cell α j in Gray. Such an invertiblemodification gives an isomorphism

(α j )g : ((αb)g ⊗ 1

) ∗ a� ⇒ a� ∗ (αb)g⊗ j .

Thus the required perturbation has its component at g given by the followingpasting diagram,

f ⊗ g ( f ⊗ I )⊗ gr �⊗1 �� ( f ⊗ I )⊗ g ( f ′ ⊗ I )⊗ g

(αb)I ⊗1 �� ( f ′ ⊗ I )⊗ g f ′ ⊗ gr⊗1 ��( f ⊗ I )⊗ g

f ⊗ (I ⊗ g)

a��

f ⊗ (I ⊗ g) f ′ ⊗ (I ⊗ g)(αb)I⊗g

��

( f ′ ⊗ I )⊗ g

f ′ ⊗ (I ⊗ g)

a��

f ⊗ (I ⊗ g)

f ⊗ g

1⊗l��

f ⊗ g f ′ ⊗ g(αb)g

��

f ′ ⊗ (I ⊗ g)

f ′ ⊗ g

1⊗l��

f ⊗ g

f ⊗ g

1

�� f ′ ⊗ g

f ′ ⊗ g

1

X��

IF KKKKKKKKKKKKKK

IF KKKKKKKKKKKKKK

��

where the triangular regions are μ and the appropriate mate of μ from left toright, the top square is the mate of (αg)I , and the bottom square is the nat-urality isomorphism for α. These 3-cells piece together to give an invertiblemodification.

The single perturbation axiom then holds since this is a modification.


4. The 2-functor ya,a′ is biessentially surjective.Let f : T (−, a) → T (−, a′) be any transformation. Then the component ata of this transformation gives a functor fa : T (a, a) → T (a, a′). Evaluationat Ia then gives fa(Ia) : a → a′, which we now write as f . The claim is thaty( f ) is equivalent to f .

We will construct a modification α : f ⇒ y( f ) that is an equivalence;for a modification to be an equivalence, it suffices that each component αx

is an equivalence 2-cell in the hom-bicategory of the target. Thus such anequivalence modification requires, for each object b in T , a transformationfb ⇒ y(( f )b that is an equivalence. Such a transformation has its componentat g : b → a an equivalence fb(g) → fa(Ia)⊗ g.

The transformation f gives, for every β : b → b′ in T , an adjointequivalence between the functors β∗ ◦ fb′ and fb ◦ β∗. Setting β = g andevaluating at Ia , we get an equivalence fb(Ia ⊗ g) → fg(b). Composing thiswith the equivalence fb(g) → fb(Ia ⊗ g) given by fb(l�), we produce thedesired component of the transformation. The naturality isomorphism and thetransformation axioms follow from those of f and l�.

The modification α also requires an invertible 3-cell αh in Gray for each1-cell h of T . This is easily constructed as the composite of� for the transfor-mation f , coherence isomorphisms from T , and naturality isomorphisms forthe transformation f . Coherence and the transformation axioms for f implythat αh is indeed a modification, and that the modification axioms hold for α.Thus y is locally biessentially surjective.

Remark 9.13 The proof given here is very similar to the proof byGordon et al. (1995), especially the first two parts. The third part differs in thatwe are required to check different axioms to ensure that the same constructionproduces the appropriate isomorphism. We have not avoided the calculationalwork of Gordon et al., rather we have used similar calculations to produce thefunctor tricategory and to show that our Yoneda embedding is a functor.

Our definition of tricategory should allow for a definition of the tri-category of prerepresentations Prep(T ) analogous to the one given byGordon et al. (1995), and there should be a forgetful functor

Tricat(T op,Gray) → Prep(T ).

Thus the proof given here should be seen as a lift of the proof fromGordon et al. to the functor tricategory.

We end with one final result which mildly strengthens the cubical Yonedalemma. Recall that a 2-functor F : X → Y is an internal biequivalence in


Gray if there exists a 2-functor G : Y → X such that FG and G F are pseudo-naturally equivalent to the identity 2-functors. In general, a 2-functor which isa biequivalence (in the sense that it is locally an equivalence and biessentiallysurjective, a condition which is logically equivalent to being a biequivalence inBicat) will not be a biequivalence in Gray. For example, if X is a 2-categorythen e : stX → X is a 2-functor and a biequivalence that is not (in general) abiequivalence in Gray.

Proposition 9.14 The functor y : T → Tricat(T op,Gray) is locally abiequivalence in Gray if T is cubical.

Proof We must construct a 2-functor

w : Tricat(T op,Gray)(

T (−, a), T (−, a′))

→ T (a, a′)

which is a weak inverse for ya,a′ . Since we already know that ya,a′ is abiequivalence, showing that w ◦ ya,a′ is equivalent to the identity suffices.For α : T (−, a) → T (−, a′), we define w(α) = αa(1a); since α is atransformation, its component at a is a 2-functor αa : T (a, a) → T (a, a′),which we evaluate at 1a to obtain the required 1-cell a → a′ in T . Simi-larly, w() = (a)1a , the component of the transformation a at the object1a ∈ T (a, a), and w(m) = (ma)1a , the component of the modification ma atthe object 1a . It is clear that this is a 2-functor, and the composite w ◦ ya,a′sends f : a → a′ to f ⊗ 1a , so is equivalent to the identity 2-functor using theright unit equivalence in T .

9.4 Coherence for tricategories

Here we finally give the coherence theorem for tricategories. The proof issimple using the results of the last section and Section 6.1.

Corollary 9.15 (Coherence for tricategories) For every tricategory T thereis a Gray-category T ′ and a triequivalence T → T ′ which is an isomorphismon objects.

Proof In Chapter 8, we constructed a triequivalence T → stT that is theidentity on objects. By Theorem 9.12, the functor

y : stT → Tricat(stT op,Gray)

9.4 Coherence for tricategories 155

is a local biequivalence. Thus we define T ′ to be the full sub-Gray-categorywith objects those y(a) in the functor tricategory Tricat(stT op,Gray) for alla ∈ T . By construction, the composite

T → stT → T ′

is the desired triequivalence.

Corollary 9.16 Every tricategory T with one object is triequivalent to amonoid in the monoidal category Gray.

Proof A monoid in Gray determines, and is determined by (up to the choiceof object), a one-object Gray-category.

10

Coherence via free constructions

In this chapter, we will prove a coherence theorem comparing free tricategoriesto free Gray-categories. This theorem states that the natural functor inducedby the universal property from the free tricategory to the free Gray-categoryon the same underlying data is a triequivalence. It is also a simple matter toprove a similar result comparing Gray-categories and strict 3-categories: thenatural functor induced by the universal property from the free Gray-categoryto the free strict 3-category on the same underlying data is a triequivalence.This latter result might seem surprising, as it is well-known that not every tri-category is triequivalent to a strict 3-category, but in fact these results onlyexpress that the maps of monads from the free tricategory monad to the freeGray-category monad to the free strict 3-category monad can be equipped withcontractions in the sense of Leinster (2004); this condition is one requirementfor a monad to be a reasonable monad for a theory of weak 3-categories. Asin the case of the coherence theory for bicategories, we can use this resultto prove that diagrams of constraint 3-cells of a certain type always com-mute. Our results differ from the analogous ones for bicategories in that onlysome diagrams commute for tricategories but all diagrams of constraint 2-cellscommute in a bicategory. As an example, we explicitly construct a diagramof constraint 3-cells that is not required to commute in general, and in factdoes not commute in example tricategories which arise from braided monoidalcategories.

With this coherence theorem in hand, we can mimic the proofs ofJoyal and Street (1993) to construct, for each tricategory T , a strictificationGrT and a triequivalence GrT → T . This strictification functor will havea distinguished pseudo-inverse, and both of these triequivalences will beused in later sections to explore the coherence theory for functors betweentricategories.

156


10.1 Coherence for tricategories

Let X be a category-enriched 2-graph. Then the inclusion X ↪→ FG(F2C X) ofX into the free Gray-category generated by X induces a strict functor

: F(FB X) → FG(F2C X)

by the universal property of the free tricategory. Thus our coherence theoremfor tricategories is as follows.

Theorem 10.1 (Coherence for tricategories) Let X be a category enriched2-graph. Then the strict functor

: F(FB X) → FG(F2C X)

is a triequivalence between the free tricategory on X and the free Gray-category on X.

Before proving this theorem, we need two results just as in the proof ofcoherence for bicategories. The first is that has a universal property, theproof of which follows immediately from the universal property of the freeGray-category functor.

Lemma 10.2 Let X be a category-enriched 2-graph, and let F : FX → Gbe a strict functor into a Gray-category G. Then there exists a unique strictfunctor Fs : FG(F2C X) → G such that F = Fs as maps of the underlyingBicat-graphs.

The second result we need is a simple construction which allows us to extendmaps of Bicat-graphs X → T , where X has hom-2-categories instead of themore general hom-bicategories, with T a tricategory to maps of Bicat-graphsFG X → T .

Lemma 10.3 Let f : X → T be a map of Bicat-graphs from abicategory-enriched graph X into a tricategory T , and assume that all of thehom-bicategories of X are 2-categories. Then it is possible to extend f to amap of bicategory-enriched graphs f : FG X → T such that the followingdiagram commutes in Gr(Bicat).

X FG X��X

T�� FG X

T��

Proof The object function f0 is the same as f0. Now let a, b be objects of X .We define

fa,b : FG X (a, b) → T ( f a, f b)

158 Coherence via free constructions

to be the weak functor given by the following data. On the object h = hn · · · h1,we define

f (h) = (· · · ( f hn ⊗ f hn−1)⊗ f hn−2)⊗ · · · ⊗ f h2)⊗ f h1.

We will temporarily adopt the notation that the basic 1-cell 1 jα1i−1 will bewritten 1αi 1 to indicated that α is in the i th position; the index j will besuppressed, as it does not contribute to the definition below in a meaningfulfashion. A similar notation will be adopted for 2-cells. On the basic 1-cell1αi 1, we define

f (1αi 1) = (· · · (1 ⊗ 1)⊗ · · · ⊗ f αi )⊗ · · · ⊗ 1)⊗ 1.

On the 1-cell α = (1αin 1, 1αin−1 1, . . . , 1αi1 1), we define

f (α) = ( · · · ( f (1αin 1) ∗ f (1αin−1 1)) ∗ · · · ∗ f (1αi2 1)

) ∗ f (1αi1 1);

we also define f of an identity cell to be a tensor of identity cells of the samelength as the source (and thus target). On a basic 2-cell 1i 1 : 1αi 1 ⇒ 1βi 1,we define

f (1i 1) = (· · · (1 ⊗ 1)⊗ · · · ⊗ f i )⊗ · · · ⊗ 1)⊗ 1.

We extend this to strings of basic 2-cells in analogy with how we defined fon strings of 1-cells. We define f (�αi ,β j ) to be the canonical isomorphismgiven by the functoriality constraint in T of the functor ⊗. This is extendedover composites of 2-cells in the obvious fashion, and clearly gives a map ofcategory-enriched 2-graphs.

Now we need to give structure constraints f (β) ∗ f (α) ∼= f (β ∗ α) andf (1h) ∼= 1 f h . The first of these is given by the associativity constraint inthe target bicategory and the second is given by the unique isomorphismbetween a tensor of identity 2-cells and the identity 2-cell on a tensor pro-vided by the functoriality of ⊗. Coherence for functors implies that the twoaxioms are satisfied, hence we have given a map of bicategory-enriched graphsFG X → T .

Proof of 10.1 First, note that is the identity on objects, so we need onlycheck that it is a local biequivalence.

1. The functor is 2-locally full, 2-locally essentially surjective, and locallybiessentially surjective.Let M be any 2-category-enriched graph. Note that we have the inclusioni : M → FM , thus the induced map i : FG M → FM of bicategory-enriched


graphs. We also have the strict functor K : FM → FG M given by theuniversal property of the free tricategory. It is then easy to check that

FG Mi−→ FM

K−→ FG M

is the identity in Gr(Bicat) using the fact that K is strict. This gives that forevery pair of objects a, b in M , the following composite is the identity in thecategory Bicat.

FG M(a, b)i−→ FM(a, b)

K−→ FG M(a, b).

Now if f is any object of FG M(a, b), then i( f ) is an object of FG M(a, b)that maps to f under K , so K is locally surjective on objects. If α : K f →K g is any 1-cell in FG M(a, b), then there are composites of the constraintsa, a�, l, l�, r, r � that give a (non-unique) 1-cell

c f : f → i K f,

since f and i K f differ only in association and by the presence of units fromthe definitions of i and K ; the same holds for g. Since K maps all of theseconstraints to identities, the image of

fc−→ i K f

iα−→ i K gc�−→ g

is α, so K is 2-locally essentially surjective. The same argument proves that Kis 2-locally full as in the proof of coherence for bicategories, once again usingthat i is locally surjective on the level of 2-cells.

We will now specialize to the case when M = F2C X for some category-enriched 2-graph X . Let l : FB X → F2C X be the locally strict localbiequivalence given by coherence for bicategories. We then have that factorsas the composite (in the category of bicategory-enriched graphs) K ◦F(l). ByTheorem 6.13, F(l) is a triequivalence. Therefore both K and F(l) are 2-locally full, 2-locally essentially surjective, and locally biessentially surjective,so is as well.

2. The functor is 2-locally faithful.First, we have a 2-locally faithful functor P : FX → G into a Gray-categoryG by the coherence theorem for tricategories. Thus we can produce a strictQ : FX → G and an equivalence pseudo-icon α : P → Q. The universalproperty of then gives a functor Qs with Qs = Q as maps of the underlyingBicat-graphs. We know that Q is 2-locally faithful since P is and there is anequivalence pseudo-icon α between, so must be 2-locally faithful as well.


10.2 Coherence and diagrams of constraints

An important type of coherence theorem is one stating that a certain large classof diagrams commutes. In this section, we develop one such theorem as it willbe necessary for constructing strictifications. In practice, this is perhaps themost useful form of coherence as it allows one to avoid checking diagramsby hand.

Before proving this theorem, we first recall how it is possible to prove thatevery diagram of constraint 2-cells that arises as a diagram of constraints in afree bicategory commutes using the fact that the strict functor FX → Fs X isa biequivalence between the free bicategory on a category-enriched graph andthe free 2-category on the same graph. Given such a diagram of constraint2-cells in a bicategory B, there is a locally discrete sub-category-enrichedgraph D of B for which the diagram in question is the image, under the strictfunctor FD → B, of a diagram in FD. Thus proving that the diagram com-mutes in B reduces to proving that it commutes in FD. Now the diagram inquestion is mapped to a composite of identities in Fs D, thus commutes there.But since the map FD → Fs D is a biequivalence, it is locally an equivalenceof categories and therefore the original diagram commutes in FD as well.

We follow an analogous strategy using the free tricategory and free Gray-category functors. The first step is proving that, in certain free Gray-categories,every diagram of 3-cells commutes. A simple definition is required beforeproving this.

Theorem 10.4 Let X be a category-enriched 2-graph. Then the natural func-tor from the free Gray-category on X to the free strict 3-category on X is atriequivalence. In particular, the free tricategory on X, the free Gray-categoryon X, and the free strict 3-category on X are all triequivalent.

Proof By a general result of Kelly (1980), the hom-2-categories in the freeGray-category on X are disjoint unions of 2-categories of the form

X (an, b)⊗ X (an−1, an)⊗ · · · ⊗ X (a, a1).

Since the induced functor from the universal property is bijective on objects,we only need to prove that the induced hom-functors are biequivalences. Thisfollows immediately from repeated applications of Corollary 3.22.

Definition 10.5 A category-enriched 2-graph X is 2-locally discrete if eachcategory X ( f, g) is a discrete category.

Corollary 10.6 Let X be a 2-locally discrete category-enriched 2-graph.Then in the free tricategory on X, FX, every diagram of 3-cells commutes.

10.3 A non-commuting diagram 161

10.3 A non-commuting diagram

In this section, we will give an example of a diagram of constraint 3-cells thatis not required to commute by the coherence theorem. This diagram will be acategorified version of the Eckmann–Hilton argument. Finally, we will see thatevery braided monoidal category gives rise to a tricategory in which for everypair of objects a, b such that the braiding γa,b is not equal to γ−1

b,a , there is adiagram of constraint 3-cells of the type to be detailed that does not commute.

Let T be a tricategory, and let x be an object of T . Assume that there are2-cells α, β : Ix ⇒ Ix . Consider the diagram of constraint 3-cells below, inwhich all the 1-cells are Ix .


The cells marked F arise from the functoriality of ⊗, the cells marked U arisefrom the isomorphism l ∼= r : I ⊗ I → I , and the cells marked N arisefrom the naturality of l and r . In general, this diagram does not arise in freetricategories, unless α and β are already composites of constraint 2-cells, andmoreover, composites that arises in a free tricategory. To see this, considerthe fourth 2-cell pasting along the right composite of 3-cells which has cellsr, α, β, l�. For this pasting to be the target of the correct naturality 3-cell in afree tricategory, the 2-cell r must go from a composite f ⊗ I to f , where I isthe unit 1-cell freely generated by the tricategory structure. On the other hand,in the fifth pasting, we have the 2-cell l which, via U , is isomorphic to r . Thesame argument shows that this must be a 2-cell l : I ⊗ g ⇒ g where I is onceagain the unit freely generated from the free structure. Inspecting sources andtargets then shows that the source and target units in both α and β must bethose freely generated, and the only 2-cells that one can freely generate withan I as its entire source or target are unit constraints. Thus if this diagramarises in a free tricategory, we must have that α and β are both compositesof constraint 2-cells. In particular, given a tricategory with non-trivial endo-2-cells of some unit Ia , the diagram produced above is not required to commuteby the coherence theorem for tricategories.

10.4 Strictifying tricategories

Here we will construct a strictification GrT for a tricategory T . The tricategoryGrT will be a Gray-category and will support a triequivalence GrT → T .

Definition 10.7 Let fn, fn−1, . . . , f1 be a sequence of composable 1-cells ina tricategory T , and let [n] denote the graph

0r1→ 1

r2→ · · · rn→ n

viewed as a category-enriched 2-graph with empty hom-categories. Then achoice of association for this sequence is a pair (γ, E) consisting of a 1-cellγ in the free tricategory on [n] which, under the functor from the free tri-category to the free strict 3-category induced by the universal property, mapsto rnrn−1 · · · r1, and the strict functor E : F[n] → T which sends ri tofi . The 1-cell E(γ ) in T is called an evaluated association of the string{ fn, fn−1, . . . , f1}.

The tricategory GrT has the same objects as T . The 2-category GrT (a, b)has for 0-cells strings of composable 1-cells of T , written { fi }. Note that the

10.4 Strictifying tricategories 163

identity for an object a is the unique empty string beginning and ending at a.A 1-cell α : { fi } → {g j } consists of composable strings of the following:

(1) three numbers k, l1, l2 with k ≤ l1, k ≤ l2 such that

• if m < k, then fm = gm , and

• if n > 0, then fl1+n = gl2+n if either side exists;

(2) a pair (σ, τ ), where σ = (σ, D) is a choice of association for the sub-string { fi }k≤i≤l1 and τ = (τ, E) is a choice of association for the substring{g j }k≤ j≤l2 ;

(3) a 1-cell α : D(σ ) → E(τ ) in T (a, b).

We additionally include the empty 1-cell, denoted ∅{ fi }, which is the identityon { fi }.

Before defining the 2-cells of GrT (a, b), we must define an evaluation func-tion e : GrT2 → T2 on the underlying 2-globular sets GrT2 and T2. On 0-cells,e is the identity function. On 1-cells,

e({ fi }) = (· · · ( fn ⊗ fn−1)⊗ fn−2)⊗ · · · ⊗ f2)⊗ f1;we write this particular association as [ fi ]. We also define the value of e on theempty 1-cell from a to a as Ia , so [ ] = Ia . For each pair of parallel 1-cellsγ, γ ′ in F[n] which map to the single 1-cell 0 → n in the free strict 3-categoryon [n], we choose one 2-cell γ ⇒ γ ′ in F[n]. We call this 2-cell aγ,γ ′ , and itinduces a 2-cell E(γ ) ⇒ E(γ ′) in T , also written aγ,γ ′ , for the associations(γ, E), (γ ′, E) for a fixed string of 1-cells { fn, . . . , f1}. In fact we can chooseγ ⇒ γ ′ to be part of an adjoint equivalence by coherence, which induces anadjoint equivalence aγ,γ ′ �eq a�

γ,γ ′ .We can now define e on the 2-cells of GrT2. A 2-cell of GrT2 is, as defined

above, a string of basic cells which each consist of a choice of substring, achoice of associations for the source and target, and an actual cell betweenthose associations. Let α be such a basic cell from { fi } to {g j } with notationas above. If we treat the associated substring D(σ ) as a single cell, then thereis a 1-cell

a : [ fi ] → [ fi+, D(σ ), fi−]where fi− is the string consisting of those cells with index less than k and fi+is the string consisting of those cells with index greater than l1. Note that wealso have a cell

a� : [ fi+, D(σ ), fi−] → [ fi ].


We define e(α) to be the cell

[ fi ] a−→ [ fi+, [ fi ]σ , fi−] (···(1⊗1)⊗···⊗α)⊗···⊗1−→ [ fi+, [g j ]τ , g j−] a�−→ [g j ],where we follow the convention that for unparenthesized strings of lengthgreater than two, we compose using a leftward bias so that

αnαn−1 · · ·α1

means the cell

(· · · (αn ∗ αn−1) ∗ αn−2) ∗ · · · ∗ α2) ∗ α1.

Any 1-cell α in GrT (a, b) is a string of such basic cells,

α = αnαn−1 · · ·α1.

We define e(α) to be the composite below, parenthesized according to ourconvention:

e(α) = e(αn)e(αn−1) · · · e(α1).

It is immediate that the difference between e(β) ∗ e(α) and e(β ∗ α) is merelyone of association, so these two cells differ by a unique isomorphism arisingfrom the associativity isomorphism in the hom-bicategory. We also define thevalue of e on the empty 1-cell to be (· · · (1 ⊗ 1)⊗ · · · 1)⊗ 1.

A 2-cell : α ⇒ β in GrT (a, b) is a 2-cell : e(α) ⇒ e(β) in T . Itis now necessary to equip GrT (a, b) with compositions and units, and thenshow that these choices give GrT (a, b) the structure of a 2-category. The 1-cell identities are the empty strings, and the 2-cell identities are obtained as theidentity 2-cells in T . The composition of 1-cells is given by concatenation ofstrings, and it is clearly associative and unital. Vertical composition of 2-cellsis inherited from T , and hence is strictly associative and unital. Horizontalcomposition is also inherited from T , in that we define ∗ to be the 2-cell

e(β ∗ α) ∼= e(β) ∗ e(α) ∗−→ e(β ′) ∗ e(α′) ∼= e(β ′ ∗ α′),

where the unlabeled isomorphisms are the unique cells given by the coherencetheorem. It follows by the uniqueness of the isomorphisms that composi-tion satisfies interchange and is strictly associative. Thus GrT (a, b) is a strict2-category.

To provide GrT with the structure of a Gray-category, we must construct acubical composition functor

� : GrT (b, c)× GrT (a, b) → GrT (a, c)


and show that it satisfies appropriate associativity and unit conditions. On0-cells, we define

{ fi } � {g j } = { fi , g j }

by concatenating lists. If (k, l1, l2, σ, τ, α) : { fi } → { f ′j } is a basic 1-cell and

{gh} is any other 0-cell such that { fi } � {gh} is defined, then there is a basic1-cell α �∅ given by

(k + H, l1 + H, l2 + H, σ, τ, α),

where H is the length of {gh}. This can be extended to an arbitrary 1-cellα = (αn, . . . , α1) by

α �∅ = (αn �∅, . . . , α1 �∅),

and we can similarly define ∅ � α when {gh} � { fi } is defined. Thus we defineβ � α to be the cell given by the string

(∅ � α) ∗ (β �∅),

where composition means concatenation of strings. We define ∅ �∅ = ∅.To define � on 2-cells, it suffices to define both 1 � : ∅ � α ⇒ ∅ � β and

� 1 : α � ∅ ⇒ β � ∅ for a 2-cell : α ⇒ β. We then extend this to adefinition of � : α � α′ ⇒ β � β ′ by the following formula.

� = (1 � ) ∗ ( � 1).

To begin, let α, β be 1-cells { fi } → {g j }, and let {hk} be another 0-cell suchthat {hk} � { fi } is defined. If α and β are basic 1-cells, then e(∅ � α) is the1-cell displayed below.

[hk, fi ] a−→ [hk, fi+, D(σ ), fi−] (···⊗α)⊗···⊗1−→[hk, fi+, E(τ ), fi−] a�−→ [hk, g j ].

This gives the following pasting diagram of isomorphism 2-cells in T (a, b),where the unlabeled 1-cells are given by our choice of associations and all the2-cell isomorphisms are the unique isomorphisms given by coherence.


[hk , fi ] [hk , fi−,D(σ ), fi+]�� [hk , fi−,D(σ ), fi+] [hk , fi−,E(τ ), fi+]�� [hk , fi−,E(τ ), fi+] [hk ,g j ]��[hk , fi ]

[hk ]⊗[ fi ]��

[hk , fi−,D(σ ), fi+]

[hk ]⊗[ fi−,D(σ ), fi+]��

[hk , fi−,E(τ ), fi+]

[hk ]⊗[ fi−,E(τ ), fi+]��

[hk ]⊗[g j ]

[hk ,g j ]#!

[hk ]⊗[ fi ] [hk ]⊗[ fi−,D(σ ), fi+]�� [hk ]⊗[ fi−,D(σ ), fi+] [hk ]⊗[ fi−,E(τ ), fi+]�� [hk ]⊗[ fi−,E(τ ), fi+] [hk ]⊗[g j ]��

a a�

a a�

(···⊗α)⊗···⊗1

(···(1⊗1)···⊗1)⊗(···⊗α···1)(···(1⊗1)···⊗1)⊗a (···(1⊗1)···⊗1)⊗a�

(···(1⊗1)···⊗1)⊗(

a�(···⊗α⊗···⊗1)a

)�"

∼=ZC �� ∼=

3K MMMMMMMMMMMMMMMMMMMMMMMMMM

∼= ��

∼=��

Thus we have a 2-cell isomorphism e(∅ � α) ⇒ a�(e(∅)⊗ e(α))a, so we candefine 1 � to be the composite

e(∅ � α) ⇒ a�(e(∅)⊗ e(α))a1∗(1⊗)∗1=⇒ a�(e(∅)⊗ e(β))a ⇒ e(∅ � β).

Now we extend this definition to strings of basic cells. Let

α = (αn, . . . , α1)

be a 1-cell in GrT . Using the above construction, we define a canonical iso-

morphism e(∅ � α) ∼= a�(

e(∅) � e(α))

a below (note that we suppress the

association of 2-cells here, by coherence for bicategories).

e(∅ � α) = e(∅ � αn, . . . ,∅ � α1) definition of ∅ �−= e(∅ � αn) · · · e(∅ � α1) definition of e∼= a�(e(∅)⊗ e(αn)

)a · · · a�(e(∅)⊗ e(α1)

)a by the above

∼= a�(e(∅)⊗ e(αn)) · · · (e(∅)⊗ e(α1)

)a counit of a �eq a�

∼= a�((

e(∅) · · · e(∅)) ⊗ (

e(αn) · · · e(α1)))

a unique coherence iso

∼= a�(

e(∅)⊗ e(α))

a unique coherence iso

We now make the same definition of 1� as above, using our canonical isomor-phism and its inverse. This immediately implies that 1� ◦1� = 1� ( ◦ )and 1 � 1 = 1.

Assume that and are 3-cells in GrT such that ∗ is defined. We nowshow that

(1 � ) ∗ (1 � ) = 1 � ( ∗ ).

Note that we have the diagram below in T , where we have focused on the laststep of the canonical isomorphism above with the omission of the associators.


(e(∅)···e(∅)

)⊗(

e(βm )···e(β1)e(αn)···e(α1)) (

e(∅)⊗e(β))◦(

e(∅)⊗e(α))

��(

e(∅)⊗e(β))◦(

e(∅)⊗e(α))

(e(∅)⊗e(β ′)

)◦(

e(∅)⊗e(α′))1⊗∗1⊗

��(e(∅)⊗e(β ′)

)◦(

e(∅)⊗e(α′))

e(∅)⊗e(β ′α′)

∼=��

(e(∅)···e(∅)

)⊗(

e(βm )···e(β1)e(αn)···e(α1))

e(∅)⊗e(βα)

∼=

��e(∅)⊗e(βα) e(∅)⊗e(β ′α′)

1⊗(∗ )�� e(∅)⊗e(β ′α′) e(∅)⊗e(β ′α′)∼=

��

∼=

All of the isomorphisms above are unique coherence isomorphisms. Thisdiagram commutes by the naturality of the various coherence isomorphismsinvolved. Writing out the composites that give (1�)∗ (1� ) and 1� ( ∗ ),we see that the two composites that make up this diagram appear, one in(1 � ) ∗ (1 � ) and one in 1 � ( ∗ ). Since the rest of the definitionsof these two cells are identical, we can conclude that they are in fact equal.This concludes the proof that ∅ � − is a 2-functor; a similar proof shows thesame of − �∅.

The final piece of data for the Gray-category structure of GrT is anisomorphism

�β,α : (β �∅) ∗ (∅ � α)∼==⇒ (∅ � α) ∗ (β �∅)

satisfying three axioms. This amounts to an isomorphism

e(β �∅)e(∅ � α)∼==⇒ e(∅ � α)e(β �∅)

in T . Assume first that α is a basic 2-cell α, and similarly for β. We then define� by the following pasting diagram of isomorphisms, where the compositearound the top and right is e(β �∅)e(∅ � α) and the composite around the leftand bottom is e(∅ � α)e(β �∅).

[gk , fi ] [gk+,[gk ]τ ,gk−, fi ]a �� [gk+,[gk ]τ ,gk−, fi ] [gk+,[g′

k′ ]τ ′ ,gk−, fi ]�� [gk+,[g′k′ ]τ ′ ,gk−, fi ] [g′

k′ , fi ]a�

�� [g′k′ , fi ]

[g′k′ , fi+,[ fi ]σ , fi−]

a

��[g′

k′ , fi+,[ fi ]σ , fi−]

[g′k′ , fi+,[ f ′

i ′ ]σ ′ , fi−]

(···⊗α)···⊗1

��[g′

k′ , fi+,[ f ′i ′ ]σ ′ , fi−]

[g′k′ , f ′

i ′ ]

a�

��

[gk , fi ]

[gk , fi+,[ fi ]σ , fi−]

a

��[gk , fi+,[ fi ]σ , fi−]

[gk , fi+,[ f ′i ′ ]σ ′ , fi−]

(···⊗α)···⊗1

��[gk , fi+,[ f ′

i ′ ]σ ′ , fi−]

[gk , f ′i ′ ]

a�

��[gk , f ′

i ′ ] [gk+,[gk ]τ ,gk−, f ′i ′ ]a

�� [gk+,[gk ]τ ,gk−, f ′i ′ ] [gk+,[g′

k′ ]τ ′ ,gk−, f ′i ′ ]�� [gk+,[g′

k′ ]τ ′ ,gk−, f ′i ′ ] [g′

k′ , f ′i ′ ]

a��

[gk+,[gk ]τ ,gk−, fi ]

[gk+,[gk ]τ ,gk−, fi+,[ fi ]σ , fi−]

a��

[gk+,[gk ]τ ,gk−, fi+,[ fi ]σ , fi−]

[gk+,[gk ]τ ,gk−, fi+,[ f ′i ′ ]σ ′ , fi−]

(···⊗α)···⊗1

��[gk+,[gk ]τ ,gk−, fi+,[ f ′

i ′ ]σ ′ , fi−]

[gk+,[gk ]τ ,gk−, f ′i ′ ]

a�

��

[gk+,[g′k′ ]τ ′ ,gk−, fi ]

[gk+,[g′k′ ]τ ′ ,gk−, fi+,[ fi ]σ , fi−]

a

��[gk+,[g′

k′ ]τ ′ ,gk−, fi+,[ fi ]σ , fi−]

[gk+,[g′k′ ]τ ′ ,gk−, fi+,[ f ′

i ′ ]σ ′ , fi−]

(···⊗α)···⊗1

��[gk+,[g′

k′ ]τ ′ ,gk−, fi+,[ f ′i ′ ]σ ′ , fi−]

[gk+,[g′k′ ]τ ′ ,gk−, f ′

i ′ ]a��

[gk , fi+,[ fi ]σ , fi−]

[gk+,[gk ]τ ,gk−, fi+,[ fi ]σ , fi−]a��

[gk+,[gk ]τ ,gk−, fi+,[ fi ]σ , fi−]

[gk+,[g′k′ ]τ ′ ,gk−, fi+,[ fi ]σ , fi−]

(···⊗β)···⊗1

53��

[gk+,[g′k′ ]τ ′ ,gk−, fi+,[ fi ]σ , fi−]

[g′k′ , fi+,[ fi ]σ , fi−]

a� ��MMM

[gk , fi+,[ f ′i ′ ]σ ′ , fi−]

[gk+,[gk ]τ ,gk−, fi+,[ f ′i ′ ]σ ′ , fi−]

a��MMM

[gk+,[gk ]τ ,gk−, fi+,[ f ′i ′ ]σ ′ , fi−]

[gk+,[g′k′ ]τ ′ ,gk−, fi+,[ f ′

i ′ ]σ ′ , fi−]

(···⊗β)···⊗1

539999999999

[gk+,[g′k′ ]τ ′ ,gk−, fi+,[ f ′

i ′ ]σ ′ , fi−]

[g′k′ , fi+,[ f ′

i ′ ]σ ′ , fi−]a� ��MMM

∼= ∼= ∼=

∼= ∼=∼=

∼= ∼= ∼=

(···⊗β)···⊗1

(···⊗β)···⊗1


All of these isomorphisms are unique coherence isomorphisms by coherence.From this, the naturality axiom for � follows immediately. The other twoaxioms follow by the uniqueness of the isomorphisms. Thus we have defined� when α and β are basic 2-cells, and the second and third axioms for theisomorphism � serve to define it in general.

Theorem 10.8 Let T be a tricategory. Then the definitions above serve togive GrT the structure of a Gray-category.

Proof Since we have already given the cubical composition functor, all thatremains is to show that it is strictly unital and associative. The unit condition istrivial as the unit 1-cell is the empty string. For associativity, first note that con-catenation of lists is strictly associative, so � is strictly associative on 1-cells.For 2-cells, we have the following computation.

δ � (β � α) = δ � (∅ � α ∗ β �∅)

= ∅ � (∅ � α ∗ β �∅) ∗ δ �∅

= ∅ �∅ � α ∗ ∅ � β �∅ ∗ δ �∅ �∅

= ∅ � α ∗ (∅ � β ∗ δ �∅) �∅

= (δ � β) � α.

A similar calculation, using the naturality of the associativity constraint in thehom-bicategory, shows that � is strictly associative on the 3-cells of GrT .

Theorem 10.9 The map e of the underlying 2-globular sets GrT2 → T2 canbe extended to a map of category-enriched 2-graphs by setting e() = . Thismap can then be given the structure of a functor GrT → T . This functor is atriequivalence.

Proof The first statement is trivial. For the second, we must construct theremaining data for the functor e. Restricting to the hom-bicategories, we havea map of category-enriched graphs

eab : GrT (a, b) → T (a, b);this is given the structure of a functor of bicategories by using the coherenceisomorphisms of the hom-bicategory for structure constraints and the definitionof e. Coherence for bicategories then immediately implies that the necessarydiagrams commute.

The transformation χ has component at {g j }, { fi } the 2-cell

[g j ] ⊗ [ fi ] a−→ [g j , fi ]chosen previously. For space purposes, we will write α for the cell abbreviated

(· · · ⊗ α)⊗ 1)⊗ · · · ⊗ 1


above. There is a unique coherence isomorphism

e(β)⊗ e(α) ∼= a� ⊗ a� ∗ β ⊗ α ∗ a ⊗ a

given by coherence for functors. Upon composition with the inverse of thisisomorphism, the naturality isomorphism for χ at the pair of basic 2-cells β, αis given by the pasting diagram below.

[g j ]⊗[ fi ] [g j , fi ]a �� [g j , fi ] [g j+,E(τ ),g j−, fi ]a �� [g j+,E(τ ),g j−, fi ]

[g j+,E ′(τ ′),g j−, fi ]

β

��[g j+,E ′(τ ′),g j−, fi ]

[g′j ′ , fi ]

a�

��[g′

j ′ , fi ]

[g′j ′ , fi+,D(σ ), fi−]

a

��[g′

j ′ , fi+,D(σ ), fi−]

[g′j ′ , fi+,D′(σ ′), fi−]

α

��[g′

j ′ , fi+,D′(σ ′), fi−][g′j ′ , f ′

i ′ ]a�

31

[g j ]⊗[ fi ]

[g j+,E(τ ),g j−]⊗[ fi+,D(σ ), fi−]

a⊗a

��[g j+,E(τ ),g j−]⊗[ fi+,D(σ ), fi−]

[g j+,E ′(τ ′),g j−]⊗[ fi+,D′(σ ′), fi−]

β⊗α

��[g j+,E ′(τ ′),g j−]⊗[ fi+,D′(σ ′), fi−]

[g′j ′ ]⊗[ f ′

i ′ ]

a�⊗a�

��[g′

j ′ ]⊗[ f ′i ′ ] [g′

j ′ , f ′i ′ ]a

��

[g j+,E(τ ),g j−]⊗[ fi+,D(σ ), fi−]

[g j+,E(τ ),g j−, fi ]

a

��HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH[g j+,E(τ ),g j−]⊗[ fi+,D(σ ), fi−]

[g j+,E ′(τ ′),g j−]⊗[ fi+,D(σ ), fi−]

β⊗1

��GGGGGGGGGGGGG

[g j+,E ′(τ ′),g j−]⊗[ fi+,D(σ ), fi−]

[g j+,E ′(τ ′),g j−]⊗[ fi+,D′(σ ′), fi−]

1⊗αGX��

[g j+,E ′(τ ′),g j−]⊗[ fi+,D′(σ ′), fi−]

[g′j ′ , fi+,D′(σ ′), fi−]

a

��GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG

[g j+,E ′(τ ′),g j−]⊗[ fi+,D(σ ), fi−]

[g j+,E ′(τ ′),g j−, fi ]

a

(&%%%%%%%%%%%%%%%%%[g j+,E ′(τ ′),g j−]⊗[ fi+,D(σ ), fi−]

[g′j ′ , fi+,D(σ ), fi−]

a

)'&&&&&&&&&&&&&&&&

∼=

∼=

∼= ∼=

∼=

∼=

The isomorphisms are all unique coherence isomorphisms. It is then clear howwe extend this definition when the cells involved are not basic 2-cells. Thetransformation χ � is defined in precisely the same fashion, using a� instead ofa; the unit and counit of this adjoint equivalence are given by those for a and a�.

Both Iea and e(Ia) are the identity 1-cell Ia . Thus we define the adjointequivalence ι to be the identity adjoint equivalence.

The modifications ω, γ , and δ are all given by unique coherence cells bycoherence. From this we also see that the required axioms hold.

Now we must show that e is a triequivalence. First, it is surjective on objects.Given objects a, b in GrT , we must show that each functor GrT (a, b) →T (a, b) is a biequivalence. It is surjective on 0-cells since each 0-cell f isthe image of the string { f }. Now let { fi } and {g j } be 0-cells with length I, J ,respectively. Any α : [ fi ] → [g j ] is the image of (0, I, J, [ ], [ ], α) (where [ ]


refers to our standard association) by definition. Finally, this functor is clearly2-locally full and faithful (i.e., locally full and faithful at the level of the 3-cells)by the definition of GrT .

Remark 10.10 In our construction of GrT , it was required that we make arbi-trary choices of 2-cell associators aγ,γ ′ . This construction depended on thesechoices, as did the construction of the constraint data for the triequivalence e. Ifwe denote the set of these associators by A, then our definitions are actually ofa Gray-category Gr(T, A) and a triequivalence eA : Gr(T, A) → T . For a dif-ferent set of associators A′, there is a strict triequivalence CA,A′ : Gr(T, A) →Gr(T, A′)which is the identity on 0-, 1-, and 2-cells and is compatible with theevaluation triequivalences eA and eA′ in the sense that there is an equivalencepseudo-icon α : eA → eA′CA,A′ whose components at 1-cells are all of theform r � ∗ l, and whose components at 2-cells are given by unique coherenceisomorphisms. From this point forward, we will assume that a single choice ofA has been made and that Gr(T ) means Gr(T, A) for this choice of A for alltricategories T .

Now we construct the (essentially obvious) pseudo-inverse to e, denoted fas in the case of bicategories.

Theorem 10.11 The map f : T → GrT of category-enriched 2-graphsgiven by

f (x) = xf (g) = {g}

f (α) = (0, 1, 1, [ ], [ ], α)f () = (1 ∗ ) ∗ 1

can be given the structure of a functor. This functor is a triequivalence.

Proof For the first claim, we need to give the rest of the data for f to be afunctor and check the required axioms. First, we need to give structure con-straints to make f a map of bicategory-enriched graphs. The compositionconstraint f (β) ∗ f (α) ∼= f (β ∗ α) is the unique coherence isomorphismin the hom-bicategory; the same is true of the constraint f (1g) ∼= 1 f (g). Thuswe have a map of bicategory-enriched graphs.

The adjoint equivalence χ is defined as follows. The component χhg isgiven by the cell (0, 2, 1, [ ], [ ], 1h⊗g), and χ �

hg is (0, 1, 2, [ ], [ ], 1h⊗g). Theunit and counit are given by the unique coherence isomorphisms in the hom-bicategory. The naturality isomophisms are given by the unique coherenceisomorphism from the coherence for functors theorem for bicategories using


the functoriality constraint of ⊗ and the constraints in the hom-bicategory. Itis now trivial to check the transformation axioms and that this is an adjointequivalence.

The adjoint equivalence ι has components defined by ιa = (0, 0, 1, [], [], 1Ia )

and ι�a = (0, 1, 0, [ ], [ ], 1Ia ). The unit and counit are given by the uniquecoherence isomorphisms in the hom-bicategory. The invertible 3-cells ι :ιa ∗ ia ⇒ f ia ∗ ιa and ι� are also given by unique coherence isomorphismsfrom the coherence for functors theorem. Once again, it is routine to check thetransformations axioms and that this is an adjoint equivalence.

The modifications ω, γ , and δ are all given by unique coherence isomor-phisms as above. These clearly give modifications, and the axioms for a functorare now immediate by the coherence theorem for functors.

For the second claim, first note that f is an isomorphism on objects. Thuswe need only prove that f is a local biequivalence to show that it is a triequiva-lence. It is straightforward to check that e f : T (a, b) → T (a, b) is equivalentto the identity in the functor bicategory, and since e is a biequivalence thatmeans f is as well.

10.5 Coherence for functors

In this section, we will establish a coherence result for functors between tricat-egories. The overall strategy remains the same as our proof of coherence forfunctors of bicategories. Our first goal is to prove analogues of the initial resultsin Section 3, Chapter 3. We begin by producing the free functor generated by amap of bicategory-enriched graphs. The following proposition constructs thisfunctor and provides its universal property.

Proposition 10.12 Let J : B → B ′ be a morphism in Gr(Bicats). Thenthere exists a tricategory FJ B ′, a map j : B ′ → FJ B ′ in Gr(Bicat), and alocally strict functor J : FB → FJ B ′ with the following properties.

(1) The square

B B ′J ��B

FB

i

��

B ′

FJ B ′j

��FB FJ B ′

J��

commutes in Gr(Bicats).


(2) Given a square

B B ′J ��B

S

K

��

B ′

T

L

��S T

F��

that is commutative in Gr(Bicat) with S, T tricategories and F a locallystrict functor between them, there exists a unique square

FB FJ B ′J ��FB

S

K��

FJ B ′

T

L��

S TF

��

such that

• L J = F K in Gr(Bicat),• K , L are strict functors,• K i = K and L j = L as morphisms of the underlying bicategory-

enriched graphs, and• L maps the adjoint equivalences χ and ι in FJ B ′ to the adjoint

equivalences of the same name in T .

Proof The tricategory FJ B ′ is constructed as follows. The 0-cells of FJ B ′are the 0-cells of B ′. The 1-cells of FJ B ′ are generated by new 1-cells Ia , the1-cells of B ′, and 1-cells J f for f ∈ FB1, subject to the relation that J f = f ′if f is a 1-cell of B such that J f = f ′ in B ′. The 2-cells of FJ B ′ are builtfrom the basic building blocks

(1) 2-cells α : f ⇒ g in B ′,(2) new 2-cells ia : Ia ⇒ Ia ,(3) the constraint cells l f , l�f , r f , r �

f , ahg f , and a�hg f ,

(4) the constraint cells χg f , χ�g f , ιa, and ι�a , and

(5) 2-cells Jα for α ∈ FB2

by tensoring along 0-cell boundaries and composing along 1-cell boundaries,subject to the relations (β)∗(α) = (β ∗α) (where here the left side is composi-tion in FJ B ′ while the right side is composition in B ′) and Jα = α′ if α ∈ B2

and Jα = α′ in B ′.The 3-cells are built similarly from the 3-cells of B ′, 3-cells J for ∈

FB3, constraint cells for the tricategory structure, and constraint cells for the


functor J , all subject to the required relations for both the tricategory structureon FJ B ′ and the functor J .

The functor J is defined on cells by the formula J (w) = Jw, where thecell Jw is one of the defining cells for FJ B ′. The constraint cells for J arethose given by the definition of FJ B ′, and the functor axioms hold by con-struction. The square in part 1 of the statement of the theorem then commutesautomatically.

For the second part of the statement, the strict functor K is determined bythe universal property of FB. The strict functor L is defined as follows. On0-cells, L agrees with L . The rest of the functor L is determined by strictness,local strictness, the relations K i = K , L j = L , and requiring L to map theconstraint cells in the definition of FJ B ′ to the constraint cells of the functorK . This gives the definition of L and immediately proves uniqueness.

Let J : X → Y be any map in 2Gr(Cat). We can apply the constructionof the free functor of bicategories between category-enriched graphs locally toproduce a locally strict map of bicategory-enriched graphs J l : FB X → Fl

J Y .This gives the commutative square in 2Gr(Cat) displayed below.

X YJ ��X

FB X��

Y

FlJ Y��

FB X FlJ Y

Jl��

Applying the universal property locally, we get a unique commutative squareof bicategory-enriched graphs

FB X FlJ Y

Jl��FB X

FG(F2C X)��

FlJ Y

FG(F2C Y )��

FG(F2C X) FG(F2C Y )FG (F2C J )

��

which when pasted with the previous square yields the square below.

X YJ ��X

FG(F2C X)��

Y

FG(F2C Y )��

FG(F2C X) FG(F2C Y )FG (F2C J )

��


We can consider the free functor generated by J l , denoted FJ l , and applyingthe universal property of FJ l we get the following square which commutes inGr(Bicat).

FX FJ l (FlJ Y )

J l ��FX

FG(F2C X)��

FJ l (FlJ Y )

FG(F2C Y )

��FG(F2C X) FG(F2C Y )

FG (F2C J )��

The left vertical map is the triequivalence from our coherence theorem fortricategories given by the universal property of the map X ↪→ FX . Thecoherence theorem for functors is now the following statement.

Theorem 10.13 (Coherence for functors) For all maps J : X → Y ofcategory-enriched 2-graphs, the strict functor : FJ l (Fl

J Y ) → FG(F2C Y ) isa triequivalence.

This proof requires the following lemma. The proof is the same as in thecase for bicategories, so it is omitted. It should be noted that the proof requiresthe ability to whisker pseudo-icons by functors, a precise formula for whichcan be found in Garner and Gurski (2009).

Lemma 10.14 Assume that the following squares commute in Gr(Bicat)where the Fi are functors between tricategories.

X YJ �� Y

B

Si

��

X

A

R

��A B

Fi

��

Let the following squares be those induced by the universal property.

FX FJ l (FlJ Y )

J l ��FX

A

R��

A BFi

��

FJ l (FlJ Y )

B

Si

��

Assume that the Si have the same object map, and that the Fi have the sameobject map. Then for every pair consisting of an equivalence pseudo-icon


α : F1 → F2 and an (obY × obY )-indexed collection of equivalencesβ : S1(y, y′) ⇒ S2(y, y′) such that

α ∗ 1R = β ∗ 1J

as (obX × obX)-indexed collections of pseudonatural equivalences, there isan equivalence pseudo-icon β : S1 → S2 such that

α ∗ 1R = β ∗ 1 J l

as pseudo-icons.

Proof of Theorem 10.13 Since l : FlJ Y → F2C Y is a local biequivalence,

there is a map of bicategory-enriched graphs going in the opposite directionwhich is a local pseudo-inverse and is defined by the following formulas.

x → xf → f

αn · · ·α1 → (· · · (αnαn−1)αn−2) · · · )α1

∅ f → 1 f

n · · ·1 → (· · · (n ∗ n−1) ∗ · · · ) ∗ 1.

The structure constraints are given either by associativity isomorphisms oridentities; it is simple to check the required axioms using coherence. If wewrite r for the composite of this map with the inclusion Fl

J Y → FJ l (FlJ Y ),

then we can produce a map of bicategory-enriched graphs r : FG(F2C Y ) →FJ l (Fl

J Y ) using Lemma 10.3. Using the strictness of : FJ l (FlJ Y ) →

FG(F2C Y ) and the definition of r , it is easy to check that r is the identityin the category of bicategory-enriched graphs. Using this fact and the samearguments used in the proof of the coherence theorem for tricategories, we seethat is locally biessentially surjective, 2-locally essentially surjective, and2-locally full.

By Proposition 7.21, there is a strict functor

S : FX → FJ l (FlJ Y )

and an equivalence pseudo-icon α : S → J l . The universal property then givesthe following commutative square in Gr(Bicats).

FX FJ l (FlJ Y )

J l ��FX

FX

1

��FX FJ l (Fl

J Y )S

��

FJ l (FlJ Y )

FJ l (FlJ Y )

E��


The identity square (which is induced by the obvious inclusion)

FX FJ l (FlJ Y )

J l ��FX

FX

1

��FX FJ l (F c

J Y )J l

��

FJ l (FlJ Y )

FJ l (FlJ Y )

1��

also satisfies the four conditions in the proposition. By the previous lemma andthe existence of α, we can conclude that E is 2-locally faithful.

The universal property of FJ l (FlJ Y ) also provides the square below.

FX FJ l (FlJ Y )

J l ��FX

FX

1

��FX FY

FJ��

FJ l (FlJ Y )

FY

1

��

The universal property of implies that ◦s 1 = (recall that ◦s is thecomposition of strict functors); since we know that is locally faithful, weneed only prove that 1 is as well. By the definition of FJ l (Fl

J Y ) and the factthat F is a left adjoint, there is a unique strict functor T such that the diagrambelow commutes.

FBY FJ l (FlJ Y )��FBY

FY��

FY

FJ l (FlJ Y )

T

��

It is now easy to check that S = T ◦s FJ using the definition of S given bythe construction in the previous chapter. But since S = T ◦s FJ , the followingsquare commutes in Gr(Bicat).

FX FJ l (FlJ Y )

J l ��FX

FX

1

��FX FJ l (Fl

J Y )S

��

FJ l (FlJ Y )

FJ l (FlJ Y )

T ◦s 1��

We now need to check that this square satisfies the four properties listed in thesecond part of Proposition 10.12 to conclude that T ◦s 1 = E ; then T ◦s 1

will be 2-locally faithful since E is, and thus 1 will be 2-locally faithful aswell. The first two properties are immediate. The third and fourth follow bydirect calculation using the fact that S and T are strict functors.

10.6 Strictifying functors 177

Corollary 10.15 Let J : X → Y be a map of category-enriched 2-graphs,and assume that X, Y are 2-locally discrete. Then in FJ l (Fl

J Y ), every diagramof 3-cells commutes.

10.6 Strictifying functors

In this section, we will use our coherence theorem to produce, from any functorF : S → T , a strict functor GrF : GrS → GrT . Note that, between Gray-categories, strict functors are the same as Gray-enriched functors.

The definition of GrF on objects is the same as that of the functor F onobjects. Since a 1-cell of GrS is either empty or a string { fi }, we can alsodefine GrF on 1-cells by the simple formulas below.

GrF(∅) = ∅

GrF({ fi }) = {F fi }.For the definition of GrF on the 1-cells of the hom-2-categories, we note thatit is only necessary to define GrF on basic 1-cells and then extend this tostrings by strict functoriality. Thus we need only define GrF on the basic 1-cell(k, l1, l2, σ, τ, α).

To establish notation, we have σ = (σ, D), τ = (τ, E). The 1-cell D(Fγ )is the evaluation of γ using the strict functor F D which agrees with the com-posite F D on cells. The 1-cell F D(γ ) is just F applied to the evaluation D(γ ).First, choose composites of constraint cells cσ : D(Fσ) → F D(σ ) for everyassociation σ just as we did for choosing associators aγ,γ ′ . These choices canbe equipped with the rest of the structure necessary to form an adjoint equiv-alence, so also give rise to cells c�

σ : F D(σ ) → D(Fσ). Thus we now defineGrF on the basic 1-cells of the hom-2-categories by

GrF(k, l1, l2, σ, τ, α) = (k, l1, l2, σ, τ, (c�τ Fα)cσ ).

We additionally define GrF(∅) = ∅.We will define GrF on 3-cells by using a canonical isomorphism that

we construct next. The 2-cell e((k, l1, l2, σ, τ, (c�

τ Fα)cσ ))

is given by thecomposite

a� ∗ c�τ ∗ Fα ∗ cσ ∗ a,

where we have written δ for the cell (· · · (1 ⊗ 1) ⊗ 1) · · · ⊗ δ) ⊗ · · · ⊗ 1, asin the proof of coherence. We thus have the isomorphism given by the pastingdiagram below, where each isomorphism is unique by our coherence theorem.


[F fi ] [F fi+,[F fi ]σ ,F fi−]a �� [F fi+,[F fi ]σ ,F fi−] [F fi+,F[ fi ]σ ,F fi−]cσ �� [F fi+,F[ fi ]σ ,F fi−]

[F fi+,F[g j ]τ ,F fi−]

Fα��


[F fi+,[Fg j ]τ ,F fi−]

c�τ

��[F fi+,[Fg j ]τ ,F fi−]

[Fg j ]a��

[F fi ]

F[ fi ]

c��

F[ fi ]

F[ fi+,[ fi ]σ , fi−]Fa

��F[ fi+,[ fi ]σ , fi−]

F[ fi+,[gi ]τ , fi−]

Fα��

F[ fi+,[gi ]τ , fi−] F[g j ]Fa�

�� F[g j ] [Fg j ]c�

��

F[ fi+,[ fi ]σ , fi−]

[F fi+,F[ fi ]σ ,F fi−]

c�

��

F[ fi+,[gi ]τ , fi−]


c�

��

∼=

∼=

∼=

It is then easy to see that the entire pasting is an isomorphism which is alsounique by coherence. Composing this with the composition constraint for Fgives a unique isomorphism

a� ∗ c�τ ∗ Fα ∗ cσ ∗ a

∼==⇒ c ∗ F(a� ∗ α ∗ a) ∗ c�.

It is easy to extend this isomorphism to when α and β are strings of basic2-cells. Now we define GrF() to be the composite

a�c�τ Fαcσa ∼= cF(a�αa)c� 1∗F()∗1=⇒ cF(a�βa)c� ∼= a�c�

τ Fβcσa.

Theorem 10.16 Let F : S → T be a functor between tricategories. ThenGrF as defined above is a strict functor between Gray-categories, i.e., a Gray-functor. Additionally, there are equivelance pseudo-icons as shown below.

GrS GrTGrF ��GrS

S

e

��

GrT

T

e

��S T

F��

ϕ

��

��

GrS GrTGrF

��

S

GrS

f

��

T

GrT

f

��

S TF ��

ψ

��

��

Proof For the first claim, we need to prove that GrF strictly preserves allcompositions and identities. This holds by definition for the 1-cells of GrS. Bydefinition GrF strictly preserves identity 2-cells and composition along 1-cellboundaries. Thus we need only check that

GrF(β � α) = GrF(β) � GrF(α).

The definitions of GrF and ∅ � α, β � ∅ make it clear that GrF(∅ � α) =∅ � GrF(α) and GrF(β � ∅) = GrF(β) � ∅. Thus we have the followingcalculation.

10.6 Strictifying functors 179

GrF(β � α) = GrF(∅ � α ∗ β �∅)

= GrF(∅ � α) ∗ GrF(β �∅)

= ∅ � GrF(α) ∗ GrF(β) �∅

= GrF(β) � GrF(α).

For 3-cells, it is obvious that GrF( ◦ ) = GrF()◦GrF( ) by interchangein the hom-2-categories and the functoriality of F . Similarly GrF(1) = 1 sinceF(1) = 1 by functoriality on 3-cells. Using the definition of ∗ in GrT ,it is routine to check that GrF( ∗ ) = GrF() ∗ GrF( ). To check thatGrF( � ) = GrF() � GrF( ), we only need to verify that this equationholds when either of or is the identity; the definition of � and thefact that GrF strictly preserves composition along 1-cells boundaries will thenensure that the equation holds in general. This is a simple calculation using thedefinition of 1 � and � 1, and Corollary 10.15.

Finally, we must show that GrF strictly preserves the Gray-structure iso-morphisms: GrF(�) = �. Since � is defined by a unique coherenceisomorphism, we need only show that GrF(�) is as well. This follows quicklyby the definition of the action of GrF on 3-cells and coherence. We have nowcompleted the proof that GrF is a Gray-functor between Gray-categories.

To define the equivalence pseudo-icon ϕ : e ◦ GrF → F ◦ e, we first definethe adjoint equivalence ϕ{ fi } to be the composite of the adjoint equivalencer�l and the adjoint equivalence c given by 1-cells c : [F fi ] → F[ fi ], c� :F[ fi ] → [F fi ] and the obvious unit and counit. The naturality isomorphismϕθ is the composite of naturality isomorphisms for c and r �l. The modifications� and M are given by unique coherence isomorphisms using Corollary 10.15,and the axioms follow immediately.

To define ψ : f ◦ F → GrF ◦ f , we first note that these functors agree on1-cells. Thus we can define ψ f = ∅F f and ψ �

f = ∅F f with identity unit andcounit. The naturality isomorphism ψθ is the identity. Once again, � and Mare given by unique coherence isomorphisms and the transformation axiomsfollow immediately.

Remark 10.17 Note that for ψ , the only non-trivial data are � and M . Thatis because f ◦ F = GrF ◦ f as maps of bicategory-enriched graphs.

With the proof of Theorem 10.16, we have shown how to replace tricate-gories and functors between them with Gray-categories and Gray-functors,up to triequivalence. This furthers the coherence theory begun by Gordonet al. (1995) and gives a rigorous justification to the use of Gray-functorsinstead of functors of tricategories as appropriate maps (for the purposes ofthree-dimensional category theory) between Gray-categories.

Part III

Gray-monads

11

Codescent in Gray-categories

The study of codescent in Gray-categories is the study of certain kinds ofcolimits. These colimits are a generalization of coequalizers, and we shallsee that they naturally appear in the study of algebras for Gray-monads. TheGray-category which represents codescent diagrams is here denoted G forits close connection with the simplicial category. In fact, the underlying cat-egory of G is the free category on the subcategory of op with objects[0], [1], [2], [3], and morphisms consisting of all face maps together withall degeneracy maps whose source is [0] or [1]. The definition given belowactually uses the objects [1], [2], [3], [4], identifying this category with a sub-category of instead (and here one should use the algebraist’s whichincludes the empty ordinal [0]). This presentation is not coincidental, as G

should be viewed as a kind of three-dimensional version of which hasthe universal property of being the free strict monoidal category generatedby a monoid. While we do not pursue this perspective any further, comput-ing the higher dimensional analogues of the strict monoidal category is aninteresting open problem.

The Gray-category G produces the lax version of the notion of codes-cent diagram. It is a Gray-computad, meaning that cells are added freely onedimension at a time, but that at each dimension we impose the conditions nec-essary to ensure that the resulting structure is a Gray-category. At dimensionone, this means we adjoin 1-cells but then force them to satisfy the categoryaxioms. At dimension two, we then adjoin 2-cells whose source and target areboth allowed to be arbitrary composites of 1-cells, but we do not require thefull strength of the 2-category axioms; horizontal composition of 2-cells is notdefined, and the composites (J ⊗1)∗ (1⊗ K ), (1⊗ K )∗ (J ⊗1) are not forcedto be equal. At dimension three we once again adjoin cells (this time whosesource and target are pastings of 2-cells), including coherence isomorphismsfrom the Gray-category structure. These constraints satisfy the Gray-category

183

184 Codescent in Gray-categories

axioms and naturality, but no other axioms are imposed on the 3-cells. In factit is straightforward to show that the Gray-category G is 2-locally posetal, sothat there exists at most one 3-cell with a given 2-cell source and 2-cell target.

We then go on to define the Gray-category Gps which represents the

pseudo-strength version of a codescent diagram. This object can be definedas a quotient of a Gray-computad as there are equations between pastingsof 3-cells. These equations are generated by the equations necessary to forcethe generating 3-cells to be invertible and the generating 2-cells to each comeequipped with the rest of the structure of an adjoint equivalence. Thus G

pscould be interpretted as an instance of the groupoidal version of a Gray-computad. In addition, we show how G

ps can be constructed from G in amanner akin to that of the construction of relative cell complexes in modelcategory theory (Hovey 1999).

Finally, we define lax codescent objects over lax codescent diagrams, andcodescent objects over codescent diagrams. These are the specific colimits thatwill be used in the study of algebras over Gray-monads. We give the explicitconstruction of these objects and their universal properties here, and we leavethe construction of these as weighted colimits until the next chapter.

11.1 Lax codescent diagrams

First we must define codescent diagrams. To do this, we must first define aGray-category G which gives the abstract shape of a codescent diagram.Note that this is slightly more general that is strictly necessary for our purposes,but a natural level of generality.

Definition 11.1 We define a Gray-category G as follows. The 0-cells of G are the symbols [1], [2], [3], [4]. The 1-cells of G are morphisms inthe free category on the graph whose vertices are the 0-cells given above andwhose edges are

• dk : [i] → [i − 1] for 0 ≤ k < i and• sk : [i] → [i + 1] for 0 ≤ k < i < 3.

The 2-cells of G are generated by six basic 2-cells, along with identity 2-cells, under two operations, all subject to certain relations. The basic 2-cellsare (when both source and target 1-cells are defined)

• Ai j : di d j ⇒ d j−1di when i < j ,• L j : d j s j ⇒ id,• R j : id ⇒ d j+1s j ,

11.1 Lax codescent diagrams 185

• N ds : d0s1 ⇒ s0d0,

• N sd : s0d1 ⇒ d2s0, and

• N s : s0s0 ⇒ s1s0.

We now define the two compositions used to generate all 2-cells in G .

• Given a composable string of three 1-cells h, g, f , and a 2-cell J : g ⇒ g′,there are 2-cells J ⊗ 1 : g f ⇒ g′ f and 1 ⊗ J : hg ⇒ hg′.

• Given a pair of composable 2-cells fK⇒ f ′ J⇒ f ′′, there is a 2-cell J ∗ K :

f ⇒ f ′′.

These two operations are required to satisfy the relations shown below.

• The operation ⊗ is associative, meaning that

(J ⊗ 1 f )⊗ 1g = J ⊗ (1 f ⊗ 1g),

1 f ⊗ (1g ⊗ J ) = (1 f ⊗ 1g)⊗ J,1 f ⊗ (J ⊗ 1g) = (1 f ⊗ J )⊗ 1g.

• The operation ⊗ is unital, meaning that

J ⊗ 1id = J,1id ⊗ J = J.

• The operation ∗ is associative and unital in the usual sense.

• The operation ⊗ is functorial in each variable separately, meaning that

(J ⊗ 1 f ) ∗ (K ⊗ 1 f ) = (J ∗ K )⊗ 1 f ,

(1 f ⊗ J ) ∗ (1 f ⊗ K ) = 1 f ⊗ (J ∗ K ),1 f ⊗ 1g = 1 f g.

The 3-cells of G are generated by three kinds of basic 3-cells, along withidentity 3-cells and isomorphisms required for the Gray-category structure,under three different operations, all subject to the Gray-category axioms. Thebasic 3-cells are shown below.

• There are four 3-cells, πi jk for i < j < k, whose 0-cell source is [4] andwhose 0-cell target is [1].


di d j dk

d j−1di dk

Ai, j ⊗1[Ybbbbbbb

d j−1di dk d j−1dk−1di1⊗Ai,k�� d j−1dk−1di

dk−2d j−1di

A j−1,k−1⊗1

MMWWWWWWW

di d j dk

di dk−1d j

1⊗A j,kMMWWWWWWW

di dk−1d j dk−2di d jAi,k−1⊗1

�� dk−2di d j

dk−2d j−1di

1⊗Ai, j

[Ybbbbbbb

⇓ πi jk

• There are two 3-cells, μi for i = 0, 1, whose 0-cell source is [2] and whose0-cell target is [1].

di

di di+1si

1⊗Ri

[Ybbbbbbbb

di di+1si di di siAi,i+1⊗1�� di di si

di

1⊗Li

MMWWWWWWWW

di di1��

⇓ μi

• There are four 3-cells, νli , ν

ri for i = 0, 1, whose 0-cell source is [2] and

whose 0-cell target is [1].

d0d1s1

d0d0s1

A01⊗1

[Ybbbbbbbb

d0d0s1 d0s0d01⊗N ds

�� d0s0d0

d0

L0⊗1

MMWWWWWWWW

d0d1s1 d01⊗L1

��

⇓ νl0

d0s0d1

d0d2s0

1⊗N sd

[Ybbbbbbbb

d0d2s0 d1d0s0A02⊗1 �� d1d0s0

d1

1⊗L0

MMWWWWWWWW

d0s0d1 d1L0⊗1��

⇓ νl1

d0

d0d2s1

1⊗R1

[Ybbbbbbbb

d0d2s1 d1d0s1A02⊗1 �� d1d0s1

d1s0d0

1⊗N ds

MMWWWWWWWW

d0 d1s0d0R0⊗1��

⇓ νr0

11.1 Lax codescent diagrams 187

d1

d1s0d1

R0⊗1

[Ybbbbbbbb

d1s0d1 d1d2s01⊗N sd

�� d1d2s0

d1d1s0

A12⊗1

MMWWWWWWWW

d1 d1d1s01⊗R0

��

⇓ νr1

• There is one 3-cell, νs whose 0-cell source is [1] and whose 0-cell targetis [2].

d0s0s0

d0s1s0

1⊗N s

Q<bbbbbbb

bbbbbbb

d0s1s0 s0d0s0N ds

01 ⊗1�� s0d0s0

s0

1⊗L0

NZWWWWWWWW

WWWWWWWW

d0s0s0 s0L0⊗1

��

⇓ νs

The 3-cell isomorphisms required for the Gray-category structure are thefollowing. Let f, f ′ : x → y and g, g′ : y → z be 1-cells in G , and letα : f ⇒ f ′, β : g ⇒ g′ be 2-cells. Then there is a 3-cell isomorphism

γβ,α : (β ⊗ 1 f ′) ∗ (1g ⊗ α) � (1g′ ⊗ α) ∗ (β ⊗ 1 f )

with 1-cell source g ⊗ f and 1-cell target g′ ⊗ f ′. This 3-cell is required to benatural in α and β, and to be the identity 3-cell if either α or β is the identity.Finally, these isomorphisms are required to satisfy two more “cubical” axiomsin order for G to be a Gray-category.

There are three different composition operations on 3-cells.

• Given 3-cells : α � β and ′ : β � δ, there is the composite ′ ◦ :α � δ.

• Given 2-cells α, α′, β, δ with

– α and α′ parallel,– the source of α equaling the target of δ, and– the target α equaling the source of β,

there are composite 3-cells ∗1δ : α∗δ � α′ ∗δ and 1β ∗ : β∗α � β∗α′.• Given 2-cells α, α′ : g ⇒ h; 1-cells f, f ′ such that the source of g equals

the target of f and the source of f ′ equals the target of g; and a 3-cell : α � α′, there are composite 3-cells ⊗ 1 f : α ⊗ 1 f � α′ ⊗ 1 f and1 f ′ ⊗ : 1 f ′ ⊗ α � 1 f ′ ⊗ α′.

Each of these compositions is associative and unital in the obvious fashion.

Definition 11.2 A lax codescent diagram in a Gray-category K is a Gray-functor G → K.


Proposition 11.3 In the Gray-category G, there is at most one 3-cellbetween any pair of parallel 2-cells. Hence every diagram of 3-cells commutes.

Proof First, consider the full sub-Gray-category generated by the same dataexcept for the cells πi jk , so that the 3-cells are generated by only cells of theform μ, ν, and the Gray-category isomorphisms. The 2-cell 1d0 ⊗ R0 onlyappears as a source or target 2-cell in one of these 3-cells, and the same holdsfor all the other cells of the form 1 ⊗ R, R ⊗ 1, 1 ⊗ L , L ⊗ 1. This implies thatany composite of a pair of 3-cells is necessarily of the form

(1t () ∗ )( ∗ 1s( )).

Thus, since every diagram of the Gray-category structure cells commutes,each 3-cell is entirely determined by its 2-cell source and target, and inparticular any parallel 3-cells are equal.

The exact same argument works for the full Gray-category G , as in factevery 2-cell that appears in the composite making up the source or target ofa cell πi jk does not appear in either the source or the target of any othergenerating 3-cell.

11.2 Codescent diagrams

There is also the notion of a codescent diagram which is the “pseudo” strengthversion as opposed to the lax version above. It can be defined in a similar way,by first specifying a classifying Gray-category G

ps . The Gray-category Gps

is defined analogously to G , but the basic 2-cells are now each required to bepart of a specified adjoint equivalence, and the basic 3-cells are now requiredto be invertible.

Definition 11.4 The Gray-category Gps is defined as follows. The 0-cells

are the symbols [1], [2], [3], [4]. The 1-cells of Gps are the same as those of

G , generated by di and s j . The 2-cells are generated by twelve basic 2-cells.

Ai j : di d j ⇒ d j−1di A�i j : d j−1di ⇒ di d j , i < j

L j : d j s j ⇒ id L�j : id ⇒ d j s j

R j : id ⇒ d j+1s j R�j : d j+1s j ⇒ id

N ds : d0s1 ⇒ s0d0 N ds� : s0d0 ⇒ d0s1

N sd : s0d1 ⇒ d2s0 N sd� : d2s0 ⇒ s0d1

N s : s0s0 ⇒ s1s0 N s� : s1s0 ⇒ s0s0.

These generate all of the 2-cells of Gps using the same generating operations

and relations as in G . The 3-cells are generated by

11.2 Codescent diagrams 189

• the same basic 3-cells πi jk, μi , νli , ν

ri , νs as G , but now with the require-

ment that they are invertible;

• invertible unit and counit 3-cells making Ai j the left adjoint part of anadjoint equivalence Ai j �eq A�

i j , and similarly with the other five pairs ofgenerating 2-cells; and

• the Gray-category structure isomorphisms.

These 3-cells generate all of the 3-cells of Gps using the same generating

operations and relations as in G .

Definition 11.5 A codescent diagram in a Gray-category K is a Gray-functor G

ps → K.

We can also build Gps by starting with G in the following way. First, for

a 2-category A let A denote the Gray-category with two objects 0, 1 andhom-2-categories given by

A(0, 0) = ∗,A(0, 1) = A,A(1, 0) = ∅,A(1, 1) = ∗.

Let 1 denote the arrow category, viewed as a discrete 2-category, and let Adjdenote the 2-category with a pair of objects a, b together with an adjoint equiv-alence f �eq g, f : a → b, between them. There is an inclusion i : 1 → Adjsending the single arrow to f , as well as a Gray-functor a01 : 1 → G

sending the single 2-cell (which arises from the single arrow in 1) to A01. Thuswe can form the pushout below in GrayCat.

1 Ga01 �� G

G01

��

1

Adj

i��

Adj G01

��

We can form such a pushout for each generating 2-cell, and taking the coprod-uct of these diagrams will give a new pushout where we have written g : 1 → G for a generic classifying map of a generating 2-cell.

∐1 G

∐g �� G

G1

��

∐1

∐Adj

∐i��∐Adj G

1��


If we write 2 for the 2-category with a single 2-cell between a pair of parallel1-cells, and Iso for the 2-category with a single invertible 2-cell between a pairof parallel 1-cells, there is an inclusion 2 ↪→ Iso. Just as in the 2-cell case,we can use the maps classifying the generating 3-cells 2 → G

1 and take thepushout along these inclusions. Taking the coproduct of the generating mapsfor each generating 2-cell and each generating 3-cell and forming the pushout,the result is G

ps .

11.3 Codescent objects

Here we will define the notion of a lax codescent object of a lax codescentdiagram, and then define the notion of a codescent object of a codescentdiagram.

To begin, let X : G → K be a lax codescent diagram in K; we will denoteX ([i]) by Xi , and will omit mention of X when referring to the image of abasic 2-cell or 3-cell.

Definition 11.6 A lax codescent object for X consists of the following data,subject to three axioms, which are universal in the sense that we describebelow.

Data: The data for a lax codescent object consists of

• an object X0 of K,• a 1-cell x : X1 → X0,• a 2-cell ε : xd1 ⇒ xd0,• a 3-cell M , and

xd1d2

xd0d2

ε⊗1[Ybbbbbbb

xd0d2 xd1d01⊗A02 �� xd1d0

xd0d0

ε⊗1

MMWWWWWWW

xd1d2

xd1d1

1⊗A12 MMWWWWWWW

xd1d1 xd0d1ε⊗1

�� xd0d1

xd0d0

1⊗A01

[Ybbbbbbb

⇓ M

• a 3-cell U .

x xd1s01⊗R0 �� xd1s0 xd0s0

ε⊗1 �� xd0s0

x

1⊗L0

��

x

x

1

��

⇓ U

11.3 Codescent objects 191

Axioms: The lax codescent object must satisfy the following three axioms.(We have omitted the ⊗ symbol to conserve space.)

xd1d2d3 xd0d2d3ε11 �� xd0d2d3 xd1d0d3

1A021 �� xd1d0d3 xd0d0d3ε11 �� xd0d0d3

xd0d2d0

11A03

��xd0d2d0

xd1d0d0

1A021

��xd1d0d0

xd0d0d0

ε11

��

xd1d2d3

xd1d2d2

11A23

��xd1d2d2

xd1d1d2

1A121

��xd1d1d2

xd0d1d2

ε11

��xd0d1d2 xd0d1d1

11A12

�� xd0d1d1 xd0d0d11A011

�� xd0d0d1 xd0d0d011A01

��

xd1d2d3

xd1d1d3

1A121

'%""""""""""""""

xd1d1d3 xd0d1d3ε11 �� xd0d1d3

xd0d0d3

1A011

��##############xd1d1d3

xd1d2d1

11A13

��xd1d2d1

xd1d1d1

1A121

��xd1d1d1

xd0d1d1

ε11

��

xd1d1d2

xd1d1d111A12

��ccccccc

xd1d2d1 xd0d2d1ε11

��

xd0d1d3

xd0d2d1

11A13

��xd0d2d1

xd1d0d1

1A021

��xd1d0d1

xd0d0d1

ε11

��

xd1d0d1

xd1d0d0

11A01

��TTTTTTT

⇓M1

∼=

⇓M1

⇓1π123

∼=

⇓1π013

∼=

=

=

xd1d2d3 xd0d2d3ε11 �� xd0d2d3 xd1d0d3

1A021 �� xd1d0d3 xd0d0d3ε11 �� xd0d0d3

xd0d2d0

11A03

��xd0d2d0

xd1d0d0

1A021

��xd1d0d0

xd0d0d0

ε11

��

xd1d2d3

xd1d2d2

11A23

��xd1d2d2

xd1d1d2

1A121

��xd1d1d2

xd0d1d2

ε11


11A12

�� xd0d1d1 xd0d0d11A011

�� xd0d0d1 xd0d0d011A01

��

xd0d2d3

xd0d2d2

11A23

��xd0d2d2

xd1d0d2

1A021

��xd1d0d2

xd0d0d2

ε11

��

xd1d2d2

xd0d2d2

ε11

��TTTTTTT

xd0d1d2

xd0d0d2

1A011

��##############

xd1d0d3

xd1d2d0

11A03

��xd1d2d0

xd1d1d0

1A121

��xd1d1d0

xd0d1d0

ε11

��

xd1d0d2 xd1d1d011A02 ��

xd0d0d2 xd0d1d011A02

��

xd1d2d0

xd0d2d0ε11 ��ccccccc

xd0d1d0

xd0d0d0

1A011

'%""""""""""""""

∼= ∼=

⇓1π023

∼=

⇓1π012

⇓M1 ⇓M1


Universality: The lax codescent object (X0, x, ε,M,U ) has the followinguniversal property, given in the three parts below.

1. Given

• an object Y of K,

• a 1-cell y : X1 → Y ,

• a 2-cell ε : y ⊗ d1 ⇒ y ⊗ d0,

• a 3-cell M , and

y ⊗ d1 ⊗ d2

y ⊗ d0 ⊗ d2

ε⊗1*(��

y ⊗ d0 ⊗ d2 y ⊗ d1 ⊗ d01⊗A02 �� y ⊗ d1 ⊗ d0

y ⊗ d0 ⊗ d0

ε⊗1

��

y ⊗ d1 ⊗ d2

y ⊗ d1 ⊗ d1

1⊗A12 ��

y ⊗ d1 ⊗ d1 y ⊗ d0 ⊗ d1ε⊗1

�� y ⊗ d0 ⊗ d1

y ⊗ d0 ⊗ d0

1⊗A01

*(��

⇓ M


• a 3-cell U ,

y y ⊗ d1 ⊗ s01⊗R0�� y ⊗ d1 ⊗ s0 y ⊗ d0 ⊗ s0

ε⊗1 �� y ⊗ d0 ⊗ s0

y

1⊗L0

��

y

y

1

!��

⇓ U

satisfying the three lax codescent axioms (where x is replaced with y, and theother cells are replaced with their overlined versions), there exists a unique1-cell f : X0 → Y such that

• f ⊗ x = y,

• 1 f ⊗ ε = ε,

• 1 f ⊗ M = M , and

• 1 f ⊗ U = U .

2. Given


• a pair of 1-cells g1, g2 : X0 → Y ,

• a 2-cell α : g1 ⊗ x ⇒ g2 ⊗ x , and

• an invertible 3-cell

X2

X1d1

��

X1 X0x �� X0

Y

g1

��

X2

X1d0 ��

X1 X0x�� X0

Y

g2

��X1

X0

x

��

⇓ ε

⇓ α

X2

X1d1

��

X1 X0x �� X0

Y

g1

��

X2

X1d0 ��

X1 X0x�� X0

Y

g2

��

X1

X0

x

��

⇓ ε

⇓ α�


satisfying the following two axioms,

g1xd1d2

g1xd0d2

1ε1

RQRRRRRRR

g1xd0d2

g1xd1d011A02 �"��g1xd1d0

g1xd0d0

1ε1!�DDDD

g1xd0d0

g2xd0d0

α11

+)PPP

PPPP

g1xd1d2

g2xd1d2

α11+)P

PPPP

PP

g2xd1d2

g2xd1d111A12

!�DDDD

g2xd1d1

g2xd0d1

1ε1

�"��g2xd0d1

g2xd0d0

11A01

RQRRRRRRRg2xd1d2

g2xd0d2

1ε1

RQRRRRRRR

g2xd0d2

g2xd1d0

11A02

��$$$$$

g2xd1d0

g2xd0d01ε1 !�JJJJJ

g1xd0d2

g2xd0d2

α11+)P

PPPP

PP

g1xd1d0

g2xd1d0

α11

\[ddd

ddd

⇓1

∼= ⇓1

⇓M1

g1xd1d2

g1xd0d2

1ε1

RQRRRRRRR

g1xd0d2

g1xd1d011A02 �"��g1xd1d0

g1xd0d0

1ε1!�DDDD

g1xd0d0

g2xd0d0

α11

+)PPP

PPPP

g1xd1d2

g2xd1d2

α11+)P

PPPP

PP

g2xd1d2

g2xd1d111A12

!�DDDD

g2xd1d1

g2xd0d1

1ε1

�"��g2xd0d1

g2xd0d0

11A01

RQRRRRRRRg1xd1d1

g2xd1d1

α11

\[ddd

dddg1xd1d1

g1xd0d11ε1 ��$$$$$

g1xd1d2

g1xd1d1

11A12

!�JJJJJ g1xd0d1

g1xd0d0

11A01

RQRRRRRRRg1xd0d1

g2xd0d1

α11

+)PPP

PPPP

⇓1M

∼= ⇓1

∼=

g1x

g1xd1s0

11R0

PPZZZZZZZZ

g1xd1s0

g1xd0s01ε1 �"��

g1xd0s0

g2xd0s0

α11!��

g2xd0s0

g2x

11L0

\[ddd

dddd

d

g1x g2xα

��g1x

g1x

1

�"��

g1xd0s0

g1x

11L0

��g1x

g2xα !��

=⇓1U ∼=

g1x

g1xd1s0

11R0

PPZZZZZZZZ

g1xd1s0

g1xd0s01ε1 �"��

g1xd0s0

g2xd0s0

α11!��

g2xd0s0

g2x

11L0

\[ddd

dddd

d

g1x g2xα

��g1x

g2x

α

��

g2x

g2x1 ��g2x

g2xd1s0

11R0

#!g1xd1s0 g2xd1s0α11

�� g2xd1s0 g2xd0s01ε1

��

=∼= ⇓1U

⇓1

there exists a unique 2-cell α : g1 ⇒ g2 such that α ⊗ 1x = α and is thecanonical isomorphism from the Gray-category structure, �α,ε.

3. Given


• a pair of 1-cells g1, g2 : X0 → Y ,

• a pair of 2-cells α1, α2 : g1 ⇒ g2, and

• a 3-cell

g1x g2x

α1⊗1

]Zg1x g2x

α2⊗1

QG⇓


such that the following axiom holds,

g1xd1 g2xd1α111 �� g2xd1

g2xd0

1ε

��

g1xd1

g1xd0

1ε

��g1xd0 g2xd0

α211

$#g1xd0 g2xd0

α11153

∼=

⇓1

g1xd1 g2xd1

α11153

g1xd1 g2xd1

α211

$#g2xd1

g2xd0

1ε

��

g1xd1

g1xd0

1ε

��g1xd0 g2xd0

α211��

∼=

⇓1

there is a unique 3-cell : α1 � α2 such that ⊗ 1 = .

Definition 11.7 Let X : Gps → K be a codescent diagram in K. A codescent

object for X consists of the following data, subject to three axioms, which areuniversal in the sense that we describe below.

Data: The data for a codescent object consists of

• an object X0 of K,• a 1-cell x : X1 → X0,• an adjoint equivalence ε with ε : xd1 ⇒ xd0, and• invertible 3-cells M,U with source and target given by the same cells as in

the lax case.

Axioms: The codescent object satisfies three axioms which are identical tothose in the lax case.

Universality: The codescent object has the following universal property,given in three parts below.

1. Given the same data as in the first part of the universal property of a laxcodescent together with an adjoint equivalence ε �eq ε

�, such that M,U areboth invertible, there exists a unique 1-cell f : X0 → Y such that

• f x = y,• 1 f ⊗ ε = ε as adjoint equivalences,• 1 f ⊗ M = M , and• 1 f ⊗ U = U .

2. The codescent object is universal as described in the second part of theuniversal property of the lax codescent object.

3. The codescent object is universal as described in the third part of theuniversal property of the lax codescent object.

12

Codescent as a weighted colimit

This chapter will focus on how to interpret lax codescent objects and codescentobjects as weighted colimits. This result will be useful later when we want toapply more of the general machinery of weighted colimits in enriched cate-gories to the particular coherence problems we will study. The diagram overwhich these colimits will be taken is the same as in the previous chapter, G

for lax codescent objects and Gps for codescent objects, so we need only estab-

lish the weight. This can often be done abstractly by the use of the Yonedalemma, and that strategy will be employed here once we compute some otherweights first. The weights we will compute explicitly will give other colimits,and we will show that lax codescent objects and codescent objects can be builtup from these simpler weighted colimits. This will prove that every Gray-category K with all colimits of a few simpler types will also have all codescentobjects; in particular, showing that a Gray-functor preserves codescent objectscan then be reduced to showing it preserves these simpler colimits. As code-scent objects are a kind of three-dimensional coequalizer, all of these simplercolimits will also have a coequalizer-like nature.

Since we are enriching over Gray, the weighted colimits required will allhave weights of the form W : Cop → Gray, where C is a suitable indexingGray-category. Here “op” means the opposite category in the usual enrichedsense; concretely, that means we reverse the direction of 1-cells but leave2- and 3-cells alone. For the weighted colimits that we compute explicitly,these weights are not complicated, although any Gray-functor into Gray willnecessarily involve a large amount of data.

We begin this chapter with a general review of weighted colimits in Gray-categories, followed by some examples that will be useful in showing thatcodescent and lax codescent objects are weighted colimits.

196

12.0 Weighted colimits in Gray-categories 197

12.1 Weighted colimits in Gray-categories

A weighted colimit in a Gray-category K requires two inputs: the diagramF : C → K over which we will take the colimit, and the weight J : Cop →Gray. Given these, the weighted colimit J ∗ F is defined as a representingobject, and in particular requires knowledge of the “enriched functor category”[Cop,Gray]. Here we will make precise all of the necessary concepts beforemoving on to specific examples.

We will begin by describing a Gray-category [A, B] whose objects areGray-functors F : A → B and whose 1-cells are Gray-natural transforma-tions between them. This is part of the closed structure on GrayCat arisingfrom the symmetric monoidal closed structure on Gray. (See Crans (1999)for a discussion on making GrayCat closed for the purposes of developingsemi-strict 4-categories.)

Definition 12.1 Let A, B be Gray-categories, and F,G : A → B Gray-functors between them. The 2-category [A, B](F,G) is defined as follows.

• The 0-cells are the Gray-natural transformations α from F to G. Thus a0-cell consists of 1-cells αa : Fa → Ga in B for all objects a of A suchthat for any cell κ of A with 0-cell source a and 0-cell target a′,

Gκ ⊗ αa = αa′ ⊗ Fκ;note that in the expression above, we are interpreting composition in B usingthe formulation of the Gray tensor product which is a 2-functor in eachvariable when the other is held constant.

• The 1-cells s : α → β consist of a family of 2-cells sa : αa ⇒ βa inB, indexed by the objects of A, such that for a 1-cell f : a → a′ in A,1G f ⊗ sa = sa′ ⊗ 1F f .

• The 2-cells ρ : s ⇒ t consist of a family of 3-cells ρa : sa � ta inB, indexed by the objects of A, such that for a 1-cell f : a → a′ in A,1G f ⊗ ρa = ρa′ ⊗ 1F f .

This concludes the description of the cells of [A, B](F,G). We leave it tothe reader to verify that, with componentwise compositions and units, this isindeed a 2-category.

Proposition 12.2 The 2-category [A, B](F,G) defined above is the end∫a∈A

B(Fa,Ga)

in the category of 2-categories and 2-functors.

198 Codescent as a weighted colimit

Proof Since we are working in a complete category, this end can be computedas the equalizer

∏a∈A B(Fa,Ga)

∏a,a′∈A Gray

(A(a, a′), B(Fa,Ga′)

)ρ ��

λ��

where ρ is defined on a family of 0-cells {αa} to be the family

{A(a, a′) G−→ B(Ga,Ga′) −⊗αa−→ B(Fa,Ga′)},and λ is defined on a family of 0-cells {αa} to be the family

{A(a, a′) F−→ B(Fa, Fa′)αa′⊗−−→ B(Fa,Ga′)};

the definition of ρ and λ on 1- and 2-cells is similar. It is known that theobjects of

∫a∈A B(Fa,Ga) are the enriched transformations, so we only need

to compute the 1- and 2-cells of this end.A family {sa} of 1-cells of

∏a∈A B(Fa,Ga) is a 1-cell of the equalizer

precisely when the pseudo-natural transformation

A(a, a′) B(Ga,Ga′)G �� B(Ga,Ga′) B(Fa,Ga′)

−⊗αa

]ZB(Ga,Ga′) B(Fa,Ga′)

−⊗βa

QG⇓−⊗sa

equals the pseudo-natural transformation

A(a, a′) B(Fa, Fa′)F �� B(Fa, Fa′) B(Fa,Ga′).

αa′⊗−

]ZB(Fa, Fa′) B(Fa,Ga′).

βa′⊗−

QG⇓sa′⊗−

Since F and G are both Gray-functors, they both preserve ⊗ as well as theGray-category structure isomorphisms for interchange. The transformations−⊗sa and sa′ ⊗− are given by whiskering, so the naturality isomorphisms arethese structure isomorphisms. Therefore these pseudo-natural transformationsare equal if and only if they have equal components on objects. This showsthat a family {sa} is an element of the equalizer exactly when it satisfies theequation in the definition above. The argument showing that the 2-cells of theequalizer are those satisfying the formula given previously is analogous.

12.1 Weighted colimits in Gray-categories 199

Although we will not need this for our discussion of weighted colimits, wewill complete the description of the Gray-category structure on [A, B]. Wehave already given the objects and the hom-2-categories, all that remains is togive the units, the composition, and check the axioms. The unit for the Gray-functor F is a 2-functor ∗ → [A, B](F, F)which is determined completely byits value on the single object, which we define to be the identity Gray-naturaltransformation.

Proposition 12.3 There is a 2-functor

[A, B](G, H)⊗ [A, B](F,G) → [A, B](F, H)

which is defined on objects by sending β ⊗ α to the Gray-natural transfor-mation whose component at a is βa ⊗ αa, and this defines the structure of aGray-category on [A, B] which has objects the Gray-functors F : A → B andhom-2-categories as defined earlier.

Proof This functor is defined on higher cells in the following manner. On agenerating 1-cell of the form s ⊗1α , it is given by the family of 2-cells sa ⊗1αa

which is easily confirmed to be a 1-cell in [A, B](F, H); a symmetric formulagives the image of this functor on a generating 1-cell of the form 1β ⊗ t . On agenerating 2-cell of the form ρ⊗1α , it is given by the family of 2-cells ρa ⊗1αa

which is also easily confirmed to be a 2-cell in [A, B](F, H), and similarly fora generating 2-cell of the form 1β ⊗σ . This functor sends the generating 2-cellisomorphisms

(s ⊗ 1α′) ∗ (1β ⊗ t) ∼= (1β ′ ⊗ t) ∗ (s ⊗ 1α)

to the 2-cell isomorphism whose component at a is the Gray-categorystructure isomorphism

(sa ⊗ 1α′a) ∗ (1βa ⊗ ta) ∼= (1β ′

a⊗ ta) ∗ (sa ⊗ 1αa )

in B. As before, it is easy to check that these components give a valid2-cell in [A, B](F, H). These assignments are well-defined on equivalenceclasses of generating 2-cells in [A, B](G, H) ⊗ [A, B](F,G), and thus givea well-defined composition 2-functor. Since all of the axioms are checkedcomponentwise, they follow from the Gray-category structure on B.

When A, B are both small Gray-categories, then [A, B] exists as a smallGray-category as well. If B is only locally small, then [A, B] exists as a locallysmall Gray-category; this is the case when B = Gray.

Definition 12.4 Let C be a small Gray-category, and let K be a Gray-category. Given Gray-functors J : Cop → Gray and F : C → K, the


J-weighted colimit of F is an object J ∗ F of K together with a Gray-naturalisomorphism

K(J ∗ F, X) ∼= [Cop,Gray](J,K(F−, X)).

12.2 Examples: coinserters and coequifiers

Here we will compute three different kinds of weighted colimits. We will giveboth the weighted version, and the explicit universal properties in terms ofcells and factorizations. Two of these colimits will be of the coinserter vari-ety, one that inserts a 2-cell between parallel 1-cells and one that inserts a3-cell between parallel 2-cells. Finally, the third colimit will be a coequifierthat makes a parallel pair of 3-cells equal.

Example: Co-2-inserters. The first kind of weighted colimit we will discussis a co-2-inserter. This is the weighted colimit that universally inserts a 2-cellbetween a pair of parallel 1-cells. Thus the diagram over which one is takingthe colimit is indexed by the Gray-category C with two objects 0, 1 and hom-2-categories given by

• C(0, 1) is the discrete 2-category with two objects which we will call fand g,

• C(1, 0) is empty, and• both C(0, 0) and C(1, 1) are discrete with a single object.

The weight J : Cop → Gray is defined by

• J (0) is the 2-categorya•−→b•,

• J (1) is discrete with a single object,• J ( f ) : J (1) → J (0) maps the single object to a, and• J (g) : J (1) → J (0) maps the single object to b.

We will now determine the 2-category [Cop,Gray](J,K(F−, X)) in orderto describe the co-2-inserter of a diagram as a collection of cells with certainproperties. The Gray-functor F : C → K will be identified with its imagewhich we will write as f, g : A → B.

An object of [Cop,Gray](J,K(F−, X)) is a Gray-natural transformationα from J to K(F−, X). There are two components coming from the objectsof C ,

12.2 Examples: coinserters and coequifiers 201

α0 : J (0) → K(A, X),α1 : J (1) → K(B, X).

These two components amount to a pair of 1-cells m, n : A → X and a2-cell π : m ⇒ n in the case of the J (0)-component, and a single 1-cellh : B → X in the case of the J (1)-component. Naturality of α then givesthat m = h f and n = hg. Thus we have a bijection between objectsof [Cop,Gray](J,K(F−, X)) and pairs (h, π) where h : B → X andπ : h f ⇒ hg.

A 1-cell s : α → α′ of [Cop,Gray](J,K(F−, X)) consists of pseudo-natural transformations si : αi ⇒ βi for each object of C , satisfying oneaxiom. Write the object α as the pair (h, πα), and the object β as the pair(k, πβ). The transformation s1 amounts to a 2-cell s : h ⇒ k since J (1) isdiscrete with one object. The transformation s0 consists of a pair of 2-cellsh f ⇒ k f, hg ⇒ kg and an invertible 3-cell S.

h f k f�� k f

kg

πβ��

h f

hg

πα ��hg kg��

∼= S

The single axiom then shows that the unmarked 2-cells in the previous diagramare both of the form s ⊗ 1.

A 2-cell ρ : s ⇒ t of [Cop,Gray](J,K(F−, X)) consists of modificationsρi : si � ti for each object of C , satisfying one axiom. The modification ρ1

amounts to a single 3-cell ρ : s � t , just as s1 and t1 each amount to a single2-cell. The modification ρ0 consists of a pair of 3-cells

h f k f,

s⊗1

�h f k f,

t⊗1

*(⇓ ρ f hg kg

s⊗1

�hg kg

t⊗1

*(⇓ ρg

which commute with the invertible 3-cells S, T in the obvious fashion. Thesingle axiom then shows that ρ f = ρ ⊗ 1 f and ρg = ρ ⊗ 1g .

These calculations produce the following structure for the cells of K(J ∗F, X).

• The set of 1-cells α : J ∗ F → X is isomorphic to the set of pairs (h, π)where h : B → X and π : h f ⇒ hg.

• The set of 2-cells s : α ⇒ β is isomorphic to the set of pairs (s, S) wheres : h ⇒ k and S is an invertible 3-cell πβ ∗ (s ⊗ 1 f ) ∼= (s ⊗ 1g) ∗ πα .


• The set of 3-cells ρ : s � t is isomorphic to the set of 3-cells ρ : s � tsuch that the following equation of 3-cells holds.

h f

k f

s⊗1

��

k f

kg

πβ

��

h f

hg

πα

��

hg

kg

s⊗1

��hg

kg

t⊗1

^\⇓ S

⇓ρ⊗1

= h f

k f

t⊗1

��h f

k fs⊗1OO k f

kg

πβ

��

h f

hg

πα

��

hg

kg

t⊗1

��

⇓ T

⇓ρ⊗1

These calculations allow us to reformulate the notion of a co-2-inserter asfollows. Given f, g : A → B, the co-2-inserter is an object C equipped with a1-cell h : B → C and a 2-cell π : h f ⇒ hg. It is universal in the followingsense.

• Given another object C , a 1-cell u : B → C , and a 2-cell π : u f ⇒ ug,there exists a unique 1-cell u : C → C such that u = uh and π = 1u ⊗ π .

• Let (u, π) and (v, ν) be as above, and let u = uh, v = vh be the uniquefactorizations. Given a pair consisting of a 2-cell s : u ⇒ v and an invertible3-cell S : ν ∗ (s ⊗ 1 f ) ∼= (s ⊗ 1g) ∗ π , there exists a unique 2-cell s :u ⇒ v such that s = s ⊗ 1h and S is the canonical isomorphism from theGray-category structure on K.

• Let (u, π) and (v, ν) be as above, and let u = uh, v = vh be the uniquefactorizations. In addition, let (s, S) and (t, T ) be as in the previous point,with factorizations s = s ⊗1h and t = t ⊗1h . Given a 3-cell : s � t suchthat the equality of pasting diagrams below holds,

u f

v f

s⊗1

��

v f

vg

ν

��

u f

ug

π

��

ug

vg

s⊗1

��ug

vg

t⊗1

^\⇓ S

⇓⊗1

= u f

v f

t⊗1

��u f

v fs⊗1OO v f

vg

ν

��

u f

ug

π

��

ug

vg

t⊗1

��

⇓ T

⇓⊗1

there is a unique : s � t such that = ⊗ 1h .


Example: Co-3-inserters. Our next example will be that of a co-3-inserter.This is the weighted colimit which universally inserts a 3-cell between a paral-lel pair of 2-cells. The diagram C over which we will take this colimit has twoobjects, 0 and 1, and the hom-2-categories are given by

• C(0, 1) consists of two objects, f, g, and two 1-cells α, β : f → g;• C(1, 0) is empty; and• both C(0, 0) and C(1, 1) are discrete with a single object.


• J (0) is the 2-category

a b,

r

]Za b,

s

QG⇓ δ

• J (1) is discrete with a single object,• J ( f ) : J (1) → J (0) maps the single object to a,• J (g) : J (1) → J (0) maps the single object to b,• J (α) : J ( f ) ⇒ J (g) is the pseudo-natural transformation whose only

component is r , and• J (β) : J ( f ) ⇒ J (g) is the pseudo-natural transformation whose only

component is s.

We will now give the structure of the 2-category [Cop,Gray](J,K(F−, X)),but leave the details of the computation to the reader. We remind the reader thatthe Gray-functor F : C → K has image

A B.

f

��A B.

g

��α⇓ ⇓β

• An object of [Cop,Gray](J,K(F−, X)) consists of a pair (h, ) whereh : B → X and : 1h ⊗ α � 1h ⊗ β.

• A 1-cell (s, S) : (h, ) → (k, ) consists of a 2-cell s : h ⇒ k togetherwith invertible 3-cells

h f hg1⊗α �� hg

kg

s⊗1��

h f

k f

s⊗1��

k f kg1⊗α

��

⇓ Sα

h f hg1⊗β �� hg

kg

s⊗1��

h f

k f

s⊗1��

k f kg1⊗β

��

⇓ Sβ


such that the following diagram commutes.

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ β) ∗ (s ⊗ 1)

∗1

��

(s ⊗ 1) ∗ (1 ⊗ α)

(s ⊗ 1) ∗ (1 ⊗ β)

1∗ ��

(s ⊗ 1) ∗ (1 ⊗ β) (1 ⊗ β) ∗ (s ⊗ 1)Sβ

��

• A 2-cell ρ : (s, S) ⇒ (s′, S′) consists of a 3-cell ρ : s � s′ such that thefollowing diagram (and one with α’s replaced with β’s) commutes.

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ α) ∗ (s′ ⊗ 1)

1∗(ρ⊗1)

��

(s ⊗ 1) ∗ (1 ⊗ α)

(s′ ⊗ 1) ∗ (1 ⊗ α)

(ρ⊗1)∗1

��(s′ ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s′ ⊗ 1)

S′α

��

These calculations allow us to reformulate the notion of a co-3-inserteras follows. Given 1-cells f, g : A → B and 2-cells α, β : f ⇒ g, theco-3-inserter is an object C equipped with a 1-cell h : B → C and a 3-cell : 1h ⊗ α � 1h ⊗ β. It is universal in the following sense.

• Given another object C , a 1-cell u : B → C , and a 3-cell : 1u ⊗ α �1u ⊗ β, there exists a unique 1-cell u : C → C such that u = uh and = 1u ⊗ .

• Let (u, ) and (v, ) be as above, and let u = uh, v = vh be the uniquefactorizations. Given a pair (s, S) consisting of a 2-cell s : u ⇒ v andinvertible 3-cells Sα, Sβ such that

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ β) ∗ (s ⊗ 1)

∗1��

(s ⊗ 1) ∗ (1 ⊗ α)

(s ⊗ 1) ∗ (1 ⊗ β)

1∗ ��

(s ⊗ 1) ∗ (1 ⊗ β) (1 ⊗ β) ∗ (s ⊗ 1)Sβ

��

commutes, there is a unique s : u ⇒ v such that s = s ⊗1h and both Sα, Sβare the canonical isomorphism 3-cells from the Gray-category structureon K.

• Let (u, ) and (v, ) be as above, and let u = uh, v = vh be the uniquefactorizations. In addition, let (s, S) and (t, T ) be as in the previous point,


with factorizations s = s ⊗ 1h and t = t ⊗ 1h . Given a 3-cell ρ : s � t suchthat the following diagram (and one with α’s replaced with β’s) commutes,

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ α) ∗ (t ⊗ 1)

1∗(ρ⊗1)

��

(s ⊗ 1) ∗ (1 ⊗ α)

(t ⊗ 1) ∗ (1 ⊗ α)

(ρ⊗1)∗1

��(t ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (t ⊗ 1)

T α

��

there is a unique 3-cell ρ : s � t such that ρ = ρ ⊗ 1h .

Example: Coequifiers. The final example is that of a coequifier, which is theweighted colimit that universally makes a pair of parallel 3-cells equal. Thediagram C over which we will take this colimit has two objects, 0 and 1, andthe hom-2-categories are given by

• C(0, 1) consists of two objects f and g, two 1-cells α, β : f → g, and two2-cells , : α ⇒ β;

• C(1, 0) is empty; and• both C(0, 0) and C(1, 1) are discrete with one object.


• J (0) is the 2-category

a b,

r

]Za b,

s

QG⇓ δ

• J (1) is the discrete 2-category with a single object,• J ( f ) maps the single object to a,• J (g) maps the single object to b,• J (α) is the transformation whose only component is r ,• J (β) is the transformation whose only component is s, and• J ( ) = J () is the modification whose single component is δ.

We will now give the structure of the 2-category [Cop,Gray](J,K(F−, X)),but leave the details of the computation to the reader. We remind the reader thatthe Gray-functor F : C → K has image

A B.

f

��A B.

g

��α��

β��

��


• An object of [Cop,Gray](J,K(F−, X)) consists of a 1-cell h : B → Xsuch that 1h ⊗ = 1h ⊗ .

• A 1-cell (s, S) : (h, ) → (k, ) consists of a 2-cell s : h ⇒ k togetherwith invertible 3-cells

h f hg1⊗α �� hg

kg

s⊗1��

h f

k f

s⊗1��

k f kg1⊗α

��

⇓ Sα

h f hg1⊗β �� hg

kg

s⊗1��

h f

k f

s⊗1��

k f kg1⊗β

��

⇓ Sβ

such that the following diagram commutes.

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ β) ∗ (s ⊗ 1)

(1h⊗)∗1

��

(s ⊗ 1) ∗ (1 ⊗ α)

(s ⊗ 1) ∗ (1 ⊗ β)

1∗(1h⊗ )��

(s ⊗ 1) ∗ (1 ⊗ β) (1 ⊗ β) ∗ (s ⊗ 1)Sβ

��

• A 2-cell ρ : (s, S) ⇒ (s′, S′) consists of a 3-cell ρ : s � s′ such that thefollowing diagram (and one with α’s replaced with β’s) commutes.

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ α) ∗ (s′ ⊗ 1)

1∗(ρ⊗1)

��

(s ⊗ 1) ∗ (1 ⊗ α)

(s′ ⊗ 1) ∗ (1 ⊗ α)

(ρ⊗1)∗1

��(s′ ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s′ ⊗ 1)

S′α

��

These calculations allow us to reformulate the notion of a coequifier asfollows. Given 1-cells f, g : A → B, 2-cells α, β : f ⇒ g, and 3-cells , : α � β, the coequifier is an object C equipped with a 1-cell h : B → Csuch that 1h ⊗ = 1h ⊗ . It is universal in the following sense.

• Given another object C and a 1-cell u : B → C such that 1u ⊗ = 1u ⊗,there exists a unique 1-cell u : C → C such that u = uh.

• Let u and v be as above, and let u = uh, v = vh be the unique factorizations.Given a pair (s, S) consisting of a 2-cell s : u ⇒ v and invertible 3-cellsSα, Sβ such that

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ β) ∗ (s ⊗ 1)

(1v⊗)∗1

��

(s ⊗ 1) ∗ (1 ⊗ α)

(s ⊗ 1) ∗ (1 ⊗ β)

1∗(1u⊗ )��

(s ⊗ 1) ∗ (1 ⊗ β) (1 ⊗ β) ∗ (s ⊗ 1)Sβ

��

12.3 Codescent 207

commutes, there is a unique s : u ⇒ v such that s = s ⊗1h and both Sα, Sβare the canonical isomorphism 3-cells from the Gray-category structureon K.

• Let u and v be as above, and let u = uh, v = vh be the unique factor-izations. In addition, let (s, S) and (t, T ) be as in the previous point, withfactorizations s = s ⊗ 1h and t = t ⊗ 1h . Given a 3-cell ρ : s � t such thatthe following diagram (and one with α’s replaced with β’s) commutes,

(s ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (s ⊗ 1)Sα �� (1 ⊗ α) ∗ (s ⊗ 1)

(1 ⊗ α) ∗ (t ⊗ 1)

1∗(ρ⊗1)

��

(s ⊗ 1) ∗ (1 ⊗ α)

(t ⊗ 1) ∗ (1 ⊗ α)

(ρ⊗1)∗1

��(t ⊗ 1) ∗ (1 ⊗ α) (1 ⊗ α) ∗ (t ⊗ 1)

T α

��

there is a unique 3-cell ρ : s � t such that ρ = ρ ⊗ 1h .

12.3 Codescent

We are finally able to show that codescent objects are weighted colimits. Thiswill be proven by showing that the codescent object can be built out of coequi-fiers and the two kinds of coinserters we constructed previously. The usualenriched version of the Yoneda lemma automatically constructs the weight;we will remind the reader how the argument proceeds, but will not explicitlyconstruct the entire weight.

Proposition 12.5 If a Gray-category K has all co-2-inserters, co-3-inserters,and coequifiers, then it has lax codescent objects for all lax codescentdiagrams.

Proof Let X : G → K be a lax codescent diagram. Let (Y1, h, π) be theco-2-inserter of d1 and d0. Then, let (Y2, h′, ) be the co-3-inserter of

hd1d2π⊗1−→ hd0d2

1⊗A02−→ hd1d0π⊗1−→ hd0d0,

hd1d21⊗A12−→ hd1d1

π⊗1−→ hd0d11⊗A01−→ hd0d0.

Next, let (Y3, h′′, ) be the co-3-inserter of

h′h 1⊗R0−→ h′hd1s01⊗π⊗1−→ h′hd0s0

1⊗L0−→ h′h

and the identity 2-cell on h′h. To ensure that these cells satisfy the axioms fora lax codescent object, we will take three separate coequifiers. To do this, wemake some preliminary definitions. Define x : X1 → Y3 to be h′′h′h, ε to be


1h′′h′ ⊗ π , and M to be 1h′′ ⊗ . Now let (Y4, j) be the coequifier of the twopastings in the first lax codescent object axiom where we replace x with x , εwith ε, M with M , and U with . Then let (Y5, j ′) be the coequifier of the twopastings in the second lax codescent object axiom where we replace x with1 j ⊗ x , ε with 1 j ⊗ ε, M with 1 j ⊗ M , and U with 1 j ⊗. Finally, let (Y6, j ′′)be the coequifier of the two pastings in the third lax codescent object axiomwhere we replace x with 1 j ′ j ⊗ x , ε with 1 j ′ j ⊗ ε, M with 1 j ′ j ⊗ M , and Uwith 1 j ′ j ⊗ . We will show that Y6, equipped with appropriate cells, is a laxcodescent object for F .

First, the 1-cell x : X1 → Y6 is j ′′ j ′ jh′′h′h. The 2-cell ε : xd1 ⇒ xd0 is1 j ′′ j ′ jh′′h′ ⊗ π . The 3-cell

M : (ε ⊗ 1) ∗ (1 ⊗ A02) ∗ (ε ⊗ 1) � (1 ⊗ A01) ∗ (ε ⊗ 1) ∗ (1 ⊗ A12)

is 1 j ′′ j ′ jh′′ ⊗ . The 3-cell U : (1⊗ L0)∗(ε⊗1)∗(1⊗ R0) � 1x is 1 j ′′ j ′ j ⊗.By construction, this data satisfies the axioms for a lax codescent object, andwe leave the straightforward verification that this data has the correct universalproperty to the reader.

Remark 12.6 The same trick as in the two-dimensional case shows that ifK has all co-2-inserters, co-3-inserters, and coequifiers, then it has codescentobjects for all codescent diagrams.

13

Gray-monads and their algebras

In this chapter, we will construct a variety of Gray-categories. In each case,we begin with a Gray-category K and a Gray-monad T on it. We begin witha discussion of the Gray-category of strict T -algebras and strict higher cellsbetween them. This is constructed as the enriched category of algebras, KT ,which is the Eilenberg–Moore object of the monad T in the 2-category ofGray-categories, Gray-functors, and enriched transformations. The existenceof such an object exists for abstract reasons, but we identify the cells of KT

explicitly.The second Gray-category we construct is Lax-T -Alg. This Gray-category

has objects which are lax T -algebras, 1-cells which are lax T -functors,2-cells which are T -transformations, and 3-cells which are T -modifications.It should be noted that the 2-cells here are not the lax version (called laxT -transformations, and also defined here for completeness), but instead thepseudo-strength version. This choice is made to ensure that the codescentarguments needed later work.

The third and final Gray-category we will construct is Ps-T -Alg. This is thepseudo-strength version of Lax-T -Alg whose objects are pseudo-T -algebras,whose 1-cells are T -functors, and whose 2- and 3-cells are the same as thosein the lax case. There are inclusions of KT into the other Gray-categories,and there is an embedding of Ps-T -Alg into Lax-T -Alg. It would not be fairto call this an inclusion, as the 0- and 1-cells in the pseudo case have strictlymore data than in the lax case as we require that some 2-cells are not merelyequivalences but come as part of a specficied adjoint equivalence. Neverthe-less, one could identify Ps-T -Alg as a sub-Gray-category of Lax-T -Alg, up totriequivalence.

209

210 Gray-monads and their algebras

13.1 Enriched monads and algebras

We begin by recalling the basic notions of enriched monad theory, and special-izing to the case when enriching over Gray. For the purposes of this section,V is a closed symmetric monoidal category with all finite limits.

Definition 13.1 A V-monad (T, μ, η) on a V-category X is a monad in the2-category V-Cat on the object X . Thus (T, μ, η) consists of a V-functor T :X → X and V-natural transformations μ : T 2 ⇒ T, η : 1 ⇒ T satisfying theusual monad axioms.

Given a monad T on X , one goal is to understand the object of algebras,X T ; this exists by a general theorem of Street’s (1972). The V-category X T isdefined, up to isomorphism, by the property that there is a natural isomorphismof categories

V-Cat(A, X T ) ∼= V-Cat(A, X)T∗ ,

where the monad T∗ is an ordinary monad in Cat which acts on the categoryV-Cat(A, X) by composition with T . In particular, this shows that the under-lying category of X T is isomorphic to the category of algebras for T∗ acting onthe underlying category of X .

We now specialize to the case where V = Gray and compute the Gray-category X T explicitly. Since the underlying category of X T is isomorphicto the category of algebras for T∗, we conclude that the objects of X T arethe algebras for T in the usual sense: an object x ∈ X together with a 1-cell m : T x → x satisfying the standard algebra axioms. Given two algebras(x,m), (y, n), an algebra 1-cell f : (x,m) → (y, n) consists of a 1-cell f :x → y in X satisfying n ⊗ T f = f ⊗ m, the standard axiom for a mapbetween algebras. Given two 1-cells f, g : (x,m) → (y, n), an algebra 2-cellα : f ⇒ g consists of a 2-cell α : f ⇒ g in X such that 1n ⊗ Tα = α ⊗ 1m .Finally, given two 2-cells α, β : f ⇒ g, an algebra 3-cell : α � β consistsof a 3-cell : α � β in X such that 1n ⊗ T = ⊗ 1m .

Proposition 13.2 The cells above are the data for a Gray-category whichsatisfies the defining property of X T stated above.

Proof The composition of a pair of composable 1-cells g and f is definedto be g ⊗ f in X . Since T is a Gray-monad, g ⊗ f is another algebra 1-cell. The identity 1x is easily seen to be an algebra 1-cell 1x : (x,m) →(x,m), and it is the unit 1-cell in X T ; similarly, identity 2- and 3-cells are

13.1 Enriched monads and algebras 211

algebra cells, and these are the unit 2- and 3-cells. Given algebra 2-cells α, α′which are composable along 1-cell boundaries, it is easy to check that α′ ∗ αis another algebra 2-cell. Given algebra 3-cells ,′ which are composablealong 2-cell boundaries, it is also easy to check that ′ is another algebra3-cell. Thus it remains to define composition for algebra 2- and 3-cells alonglower dimensional boundaries.

Given algebra 2-cells α : f ⇒ g, β : f ′ ⇒ g′ with s( f ′) = t ( f ), thealgebra 2-cell (β⊗1 f )∗ (1g′ ⊗α) is defined to be the cell of the same name inX (note that this defines both whiskerings simultaneously, as the computationsbelow can easily be broken into two separate computations showing that β⊗ 1and 1 ⊗α are algebra 2-cells). If f : (x,m) → (y, n) and g : (y, n) → (z, p),then (β ⊗ 1 f ) ∗ (1g′ ⊗ α) satisfies the algebra 2-cell axiom by the calculationbelow.

1p ⊗ T((β ⊗ 1 f ′) ∗ (1g ⊗ α)

)= 1p ⊗

(T (β)⊗ 1T f ′

)∗

(1T g ⊗ T (α)

)=

(1p ⊗ Tβ ⊗ 1T f ′

)∗

(1p ⊗ 1T g ⊗ Tα

)=

(β ⊗ 1n ⊗ 1T f ′

)∗

(1g ⊗ 1n ⊗ Tα)

=(β ⊗ 1 f ′ ⊗ 1m

)∗

(1g ⊗ α ⊗ 1m)

=((β ⊗ 1 f ′) ∗ (1g ⊗ α)

)⊗ 1m .

Given algebra 2-cells α, α′ : g ⇒ g′ with an algebra 3-cell : α ⇒ α′, andan algebra 1-cell f such that g ⊗ f exists, the cell ⊗ 1 f is an algebra 3-cellby the following calculation.

1n ⊗ T ( ⊗ 1 f ) = 1n ⊗ T ⊗ 1T f

= ⊗ 1m ⊗ 1T f

= ⊗ 1 f ⊗ 1m .

Given an algebra 1-cell h such that h ⊗ g exists, a similar calculation showsthat 1h ⊗ is also an algegra 3-cell. Given an algebra 2-cell β : g′ ⇒ g′′, thecell 1β ∗ is an algebra 3-cell by the following calculation.

1n ⊗ T (1β ∗ ) = 1n ⊗ (1Tβ ∗ T)= (1n ⊗ 1Tβ) ∗ (1n ⊗ T)= (1β ⊗ 1m) ∗ ( ⊗ 1m)

= (1β ∗ )⊗ 1m .


Finally, a similar calculation shows that ∗ 1β ′ is also an algebra 3-cell.Let�β,α : (β⊗1 f ′)∗(1g ⊗α) � (1g′ ⊗α)∗(β⊗1 f ) be the Gray-category

structure isomorphism. Then

1p ⊗ T�β,α = 1p ⊗�Tβ,Tα

= �1p⊗Tβ,Tα

= �β⊗1n ,Tα

= �β,1n⊗Tα

= �β,α⊗1m

= �β,α ⊗ 1m

shows that � is an algebra 3-cell. Since composition, units, and the structure3-cells are the same as those in X , we have proven that X T is a Gray-categoryand that the forgetful functor π : X T → X is a Gray-functor.

Finally, we must show that X T satisfies the universal property of theEilenberg–Moore object. There is a canonical Gray-natural transformationalg : Tπ ⇒ π whose component at the object (x,m) is the 1-cell m : T x → x .Thus we get a functor of ordinary categories

GrayCat(A, X T ) → GrayCat(A, X)T∗

by sending F : A → X T to πF equipped with the structure map alg ∗ 1F .Now assume that G is an object of GrayCat(A, X)T∗ , so that G : A → Xis equipped with a Gray-natural μ : T G ⇒ G. By precomposing G withfunctors ∗ → A, we see that every object of the form Ga for a ∈ A comesequipped with a T -algebra structure, (Ga, α : T Ga → Ga). Precomposing Gwith functors whose domains are any of

• •��

• •�B• •@?⇓

• •�B• •@?⇓ ⇓�

shows that the images of higher dimensional cells are algebra cells. Thus wecan define a Gray-functor G : A → X T by sending objects a to (Ga, α :T Ga → Ga) and just applying G to higher dimensional cells. These twofunctors are inverses, so X T is the Eilenberg–Moore object of T .

13.2 Lax algebras and their higher cells 213

Definition 13.3 The objects of X T as defined above are called strict T -algebras.

13.2 Lax algebras and their higher cells

Here we will give definitions of lax algebras and the higher cells between, aswell as the pseudo-strength versions. Throughout, K will be a Gray-categoryand T a Gray-monad on it.

Definition 13.4 A lax T -algebra X consists of

• an object X of K,

• a 1-cell x : T X → X ,

• 2-cells m : x ⊗ T x ⇒ x ⊗ μ, i : 1x ⇒ x ⊗ η, and

• 3-cells π, λ, ρ (where equalities are marked to indicate if they are naturalityor a monad axiom)

x ⊗ T x ⊗ T 2x

x ⊗ μ⊗ T 2xm⊗1

��

x ⊗ μ⊗ T 2x x ⊗ T x ⊗ μTnat. x ⊗ T x ⊗ μT

x ⊗ μ⊗ μT

m⊗1

��

x ⊗ T x ⊗ T 2x

x ⊗ T x ⊗ Tμ1⊗T m ��

x ⊗ T x ⊗ Tμ x ⊗ μ⊗ Tμm⊗1

�� x ⊗ μ⊗ Tμ

x ⊗ μ⊗ μT

monad

��

��

⇓ π

1 ⊗ x

x ⊗ η ⊗ x

i⊗1

��

x ⊗ η ⊗ x x ⊗ T x ⊗ ηTnat. x ⊗ T x ⊗ ηT

x ⊗ μ⊗ ηT

m⊗1

��

1 ⊗ x x ⊗ 1x ⊗ 1 x ⊗ μ⊗ ηTmonad

⇓ λ

x ⊗ 1

x ⊗ T x ⊗ Tη

1⊗T i

��

x ⊗ T x ⊗ Tη x ⊗ μ⊗ Tηm⊗1 �� x ⊗ μ⊗ Tη

x ⊗ 1

monad

��

��

x ⊗ 1 x ⊗ 1

⇓ ρ


subject to the following axioms.

xT xT 2xT 3x xμT 2xT 3xm11 �� xμT 2xT 3x xT xμT T 3xxT xμT T 3x xμμT T 3x

m11 �� xμμT T 3x xμTμT 3xxμTμT 3x

xμT 2xTμTxμT 2xTμT

xT xμT TμTxT xμT TμT

xμμT TμT

m11

��

xμT 2xT 3x

xμT 2xT 2μ

11T 2m��

xμT 2xT 2μ

xT xμT T 2μxT xμT T 2μ

xμμT T 2μ

m11��

xT xT 2xT 2μ xμT 2xT 2μm11

��

xT xμT T 2μ xT xTμμT 2

xT xμT T 3x

xT xT 2xμT 2xT xT 2xμT 2

xT xTμμT 2

1T m1��

xT xTμμT 2

xμTμμT 2

m11��

xT xT 2xμT 2 xμT 2μT 2m11

��

xμμT T 3x

xμT 2μT 2xμT 2μT 2

xT xμTμT 2xT xμTμT 2

xμμTμT 2

m11

��

xT xμTμT 2 xT xμT TμT

xT xT 2xT 3x

xT xT 2xT 2μ

11T 2m��

xT xT 2xT 2μ

xT xTμT 2μ

1T m1��

xT xTμT 2μ

xμTμT 2μ

m11��

xμTμT 2μ xμμT T 2μxμμT T 2μ xμTμμT 2xμTμμT 2 xμμTμT 2xμμTμT 2 xμμT TμT

∼=

⇓π1

=

=

=

⇓π1

=

=

xT xT 2xT 3x xμT 2xT 3xm11 �� xμT 2xT 3x xT xμT T 3xxT xμT T 3x xμμT T 3x

m11 �� xμμT T 3x xμTμT 3xxμTμT 3x

xμT 2xTμTxμT 2xTμT

xT xμT TμTxT xμT TμT

xμμT TμT

m11

��

xT xT 2xT 3x

xT xT 2xT 2μ

11T 2m

��xT xT 2xT 2μ

xT xTμT 2μ

1T m1

��xT xTμT 2μ xT xTμTμTxT xTμTμT xμTμTμT

m11�� xμTμTμT xμμT TμT

xT xT 2xT 3x

xT xTμT 3x

1T m1 539999999999999

xT xTμT 3x

xμTμT 3x

m11

��MMMMMMMMMMMMMMMMMMMMMMMMMMMMxT xTμT 3x

xT xT 2xTμTxT xT 2xTμT

xT xTμTμT

1T m1

��

xT xT 2xTμT

xμT 2xTμT

m11

��MMMMMMMMMMMMMMMMMMMMMMMMMMM

⇓1Tπ

⇓π1

=

⇓π1

xT x1

xT xT 2xT 2η

11T 2i

;9ZZZZZ

xT xT 2xT 2η xμT 2xT 2ηm11 �� xμT 2xT 2η xT xTημxT xTημ xμTημ

m11 �� xμTημ

xμ

dddd

d

dddd

d

xT x1

xT x

ddddd

ddddd

xT x

xμ

m

(&

xT x

xT xTμT 2η��

��

xT xT 2xT 2η

xT xTμT 2η

1T m1�BEEEE

xT xTμT 2η

xμTμT 2η

m11�BEEEE

xμT 2xT 2η

xT xμT T 2η

========

xT xμT T 2η

xT xTημ<<<<<<<<<<

xT xμT T 2η

xμμT T 2η

m11 �====

xμTμT 2η xμμT T 2ηxμμT T 2η

xμTημ<<<<<<<<<<<<<

<<<<<<<<<<<<<⇓1Tρ

⇓π1= =

=

xT x1

xT xT 2xT 2η

11T 2iRQQQQQ

xT xT 2xT 2η xμT 2xT 2ηm11 �� xμT 2xT 2η xT xTημxT xTημ xμTημ

m11 �� xμTημ

xμ

,,,,

,,,,

xT x1

xT x

,,,,

,,,,

xT x

xμ

m

(&xT x1 xμ1m1

�� xμ1 xμxμ x1μx1μ xμxμ1

xμT 2xT 2η

11T 2i.,2222

x1μ

xT xTημ

1T i1_]eeee∼= = ⇓ρ1

=


1xT x

xηxT x

i11

#!xηxT x

xT xηT T x��

xT xηT T x

xT xT 2xηT 2 xT xT 2xηT 2

xμT 2xηT 2

m11��xμT 2xηT 2

xT xμT ηT 2

��

xT xμT ηT 2

xμμT ηT 2

m11

��xμμT ηT 2

xμ

1xT x

xμm

��

1xT x 1xμ1m

�� 1xμ xμηTμxμηTμ xμTμηT 2xμTμηT 2 xμμT ηT 2

xηxT x xηxμ11m

�� xηxμ xT xηTμxT xηTμ xT xTμηT 2

xT xT 2xηT 2

xT xTμηT 2

1T m1 ��

xT xTμηT 2

xμTμηT 2

m11

��

1xμ

xηxμ

i11

`^ffffff

xT xηTμ

xμηTμ

m11S5f

ffff

f

∼= ⇓λ1 =

=

⇓π1

=

1xT x

xηxT x

i11

#!xηxT x

xT xηT T x��

xT xηT T x

xT xT 2xηT 2 xT xT 2xηT 2

xμT 2xηT 2

m11��xμT 2xηT 2

xT xμT ηT 2

��

xT xμT ηT 2

xμμT ηT 2

m11

��xμμT ηT 2

xμ

1xT x

xμm

��

1xT x xT xxT x xμηT T x

xT xηT T x

xμηT T x

m11

��

xμηT T x

xμT 2xηT 2��

�� =

=⇓λ1

x1T x

xT xTηT x

1T i1#!xT xTηT x

xT xT 2xTηT��

xT xT 2xTηT xμT 2xTηTm11 �� xμT 2xTηT

xT xμT TηT

��

��

xT xμT TηT

xμμT TηT

m11��

x1T x xT xxT x xT xTμTηTxT xTμTηT xμTμTηTm11

�� xμTμTηT xμμT TηT

xT xT 2xTηT

xT xTμTηT

1T m1

A@ggggggggggg

⇓1Tλ ⇓π1

x1T x

xT xTηT x

1T i1#!xT xTηT x

xT xT 2xTηT��

xT xT 2xTηT xμT 2xTηTm11 �� xμT 2xTηT

xT xμT TηT

��

��

xT xμT TηT

xμμT TηT

m11��

x1T x xT xxT x xT xTμTηTxT xTμTηT xμTμTηTm11

�� xμTμTηT xμμT TηT

xT xTηT x xμTηT xm11 �� xμTηT x

xT x

xμTηT x

xμT 2xTηTTTTTTTTTTTTTT

TTTTTTTTTTTTT

⇓ρ1

==

Definition 13.5 A lax T -functor f : X → Y consists of

• a 1-cell f : X → Y in K,• a 2-cell F : f ⊗ x ⇒ y ⊗ T f , and• 3-cells h,m


f ⊗ 1

f ⊗ x ⊗ η1⊗iX

��

f ⊗ x ⊗ η

y ⊗ T f ⊗ η

F⊗1��

f ⊗ 1

1 ⊗ f"""""

"""""

1 ⊗ f y ⊗ η ⊗ fiY ⊗1

�� y ⊗ η ⊗ f

y ⊗ T f ⊗ η

nat.

##########

⇓ h

f ⊗ x ⊗ T x

y ⊗ T f ⊗ T xF⊗1

��

y ⊗ T f ⊗ T x y ⊗ T y ⊗ T 2 f1⊗T F�� y ⊗ T y ⊗ T 2 f

y ⊗ μ⊗ T 2 f

m⊗1

��

f ⊗ x ⊗ T x

f ⊗ x ⊗ μ

1⊗m ��

f ⊗ x ⊗ μ y ⊗ T f ⊗ μF⊗1

�� y ⊗ T f ⊗ μ

y ⊗ μ⊗ T 2 f

nat.

��

��

⇓ m


f xT xT 2x yT f T xT 2xF11 �� yT f T xT 2x yT yT 2 f T 2x

1T F1�� yT yT 2 f T 2x yT yT 2 yT 3 f11T 2 F�� yT yT 2 yT 3 f yμT 2 yT 3 f

ε11 �� yμT 2 yT 3 f

yT yμT T 3 fyT yμT T 3 f

yT yT 2 f μTyT yT 2 f μT

yμT 2 f μT

ε11��

f xT xT 2x

f xT xTμ

11T ε

��f xT xTμ

f xμTμ

1ε1

��f xμTμ f xμμTf xμμT yT f μμT

F11�� yT f μμT yμT 2 f μT

f xT xT 2x

f xμT 2x

1ε1 " ��

f xμT 2x yT f μT 2xF11

�� yT f μT 2x yμT 2 f T 2x

yT yT 2 f T 2x

yμT 2 f T 2x

ε11

" ��

yμT 2 f T 2x

yμT 2 yT 3 f

11T 2 F

��f xμT 2x

f xT xμT

YYYY

YY

YYYY

YY

f xT xμT

f xμμT

1ε1%�Y

YYYY

Yf xT xμT yT f T xμTF11

�� yT f T xμT yT yT 2 f μT1T F1

��

yT f μT 2x

yT f T xμT

SSSSSSSS

SSSSSSSS

⇓m1 ∼=

⇓1π = =

⇓m1

f xT xT 2x yT f T xT 2xF11 �� yT f T xT 2x yT yT 2 f T 2x

1T F1�� yT yT 2 f T 2x yT yT 2 yT 3 f11T 2 F�� yT yT 2 yT 3 f yT yμT T 3 f

ε11 �� yT yμT T 3 f

yT yμT T 3 fyT yμT T 3 f

yT yT 2 f μTyT yT 2 f μT

yμT 2 f μT

ε11��

f xT xT 2x

f xT xTμ

11T ε

��f xT xTμ

f xμTμ

1ε1

��f xμTμ

f xμμTf xμμT yT f μμTF11

�� yT f μμT yμT 2 f μT

yT f T xT 2x

yT f T xTμ

11T ε��

f xT xTμ yT f T xTμF11

VThhhhhh yT f T xTμ yT yT 2 f Tμ1T F1

VThhhhh yT yT 2 f Tμ yT yTμT 3 fhhhhh hhhhh

yT yT 2 yT 3 f

yT yTμT 3 f

1T ε1 ��yT yTμT 3 f

yμTμT 3 f

ε11 ��

yT yT 2 f Tμ

yμT 2 f Tμ

ε11��

yμT 2 f TμyμTμT 3 f= /-iiiiii

f xμTμyT f μTμ

F11/-iiiiiii yT f μTμ

yμT 2 f Tμiiiiii iiiiiiyμTμT 3 f

yμμT T 3 fyμμT T 3 f

yμT 2 f μT

99999999

99999999

yT yμT T 3 f

yμμT T 3 f

ε11X��

∼= ⇓1T m⇓π1

⇓m1=

= =


f 1x f xηx1i1 �� f xηx yT f ηx

F11�� yT f ηx

yη f xyη f x

yηyT f

11F��yηyT f

yT yηT T fyT yηT T f

yμηT T f

ε11��yμηT T f

yT f

f 1x

1 f x1 f x

1yT f

1F ��1yT f

yT f

��

��

1 f x yη f xi11

��

1yT f yηyT fi11

��

⇓h1

∼=

⇓λ1

f 1x f xηx1i1 �� f xηx yT f ηx

F11 �� yT f ηx yη f xyη f x

yηyT f

11F��

yηyT f

yT yηT T fyT yηT T f

yμηT T f

ε11��

yμηT T f

yT f

f 1x

f xf x

yT f

F��

f xηx

f xT xηTf xT xηT

f xμηT

1ε1 ��f xμηT

yT f μηT

F11��

f xμηT

f x��

��

yT f ηx

yT f T xηTyT f T xηT

yT yT 2 f ηT

1T F1��yT yT 2 f ηT

yμT 2 f ηT

ε11��

f xT xηT yT f T xηTF11

��

yT f μηT yμT 2 f ηTyμT 2 f ηT

yμηT T f

GGGGGGGG

⇐1λ

=

⇓m1 ∼=

=

f x1 yT f 1F1 �� yT f 1 yT f T xTη

1T i �� yT f T xTη

yT yT 2 f Tη

1T F1

��yT yT 2 f Tη

yμT 2 f Tη

ε1��yμT 2 f Tη

yT f μTη

f x1

f xf x yT fF

�� yT f yT f μTη

yT f 1

y1T fy1T f

yT yTηT f

1T i1��yT yTηT f yT yT 2 f TηyT yTηT f

yμTηT f

ε1USd

ddd

yμTηT f

yT f

y1T f

yT f

=⇓ρ1

⇓1T h

=

=

f x1 yT f 1F1 �� yT f 1 yT f T xTη

1T i �� yT f T xTη

yT yT 2 yTη

1T F1��

yT yT 2 yTη

yμT 2 f Tη

ε1��

yμT 2 f Tη

yT f μTη

f x1

f xf x yT fF

�� yT f yT f μTη

f x1

f xT xTη

1T i

]Zjjjjjjj

f xT xTη

yT f T xTη

F1��

f xT xTη

f xμTη

1ε1��

f xμTη

f xkkkkkkk

kkkkkkkf xμTη yT f μTη

F1�� yT f μTη yμT 2 f Tη

⇐1ρ

∼=

⇓m1

=

We include the following definition for completeness, although it will notfeature in any of our results or constructions. It seems plausible that studyingenriched monads in the context of the lax version Gray-tensor product wouldmake use of lax T -transformations, but in order to construct a Gray-categoryof lax T -algebras we will have to restrict to the case of T -transformations asthey are defined later.

Definition 13.6 A lax T -transformation α : f ⇒ g consists of

• a 2-cell α : f ⇒ g in K and

• a 3-cell A

f ⊗ x y ⊗ T fF �� y ⊗ T f

y ⊗ T g

1⊗Tα

��

f ⊗ x

g ⊗ x

α⊗1

��g ⊗ x y ⊗ T g

G��

⇓ A



f 1

f xη

1i��

f xη yT f ηF1 �� yT f η

yT gη

1Tα1

��

f 1

g1α1 MMlllll

g1

1gJJJJJJJJJJJJ

1g

yηg

i1

(&KKKKK

yηg

yT gηaaaaa

aaaaag1

gxη

1i

��)))))))

f xη

gxη

α1�

��

gxη yT gηG1

��∼=

⇓A1

⇓hG

= f 1

f xη

1i��

f xη yT f ηF1 �� yT f η

yT gη

1Tα1

��

f 1

1 f

lllll

lllll

1 f

1g1α )'JJJJJJ

1g

yηg

i1

(&KKKKK

yηg

yT gηaaaaa

aaaaa1 f

yη fi1

��))))))

yη f

yT f η��

��yη f

yηg

1α

'%((((((

⇓hF

=

∼=

f xT x

yT f T x

F1

RQRRRRRRR

yT f T x

yT yT 2 f1T F �"��

yT yT 2 f

yμT 2 f

m1!�DDD

yμT 2 f

yμT 2g

1T 2α

+)PPPPPPP

f xT x

gxT x

α1+)P

PPPPPP

gxT x

gxμ1m !�DDDDD

gxμ

yT gμ

G1

�"��

yT gμ

yμT 2gRRRRRRR

RRRRRRRgxT x

yT gT x

G1

RQRRRRRRR

yT gT x

yT yT 2g

1T G

��$$$$$

yT yT 2g

yμT 2gm1 !�JJJJJ

yT f T x

yT gT x

1Tα1+)P

PPPPPP

yT yT 2 f

yT yT 2g

1T 2α

\[ddd

ddd

⇓A1

⇓1T A ∼=

⇓mG

f xT x

yT f T x

F1

RQRRRRRRR

yT f T x

yT yT 2 f1T F �"��

yT yT 2 f

yμT 2 f

m1!�DDD

yμT 2 f

yμT 2g

1T 2α

+)PPPPPPP

f xT x

gxT x

α1+)P

PPPPPP

gxT x

gxμ1m !�DDDDD

gxμ

yT gμ

G1

�"��

yT gμ

yμT 2gRRRRRRR

RRRRRRRf xμ

gxμ

α1

\[ddd

dddf xμ

yT f μF1 ��$$$$$

f xT x

f xμ

1m!�JJJJJJ yT f μ

yμT 2 fRRRRRRR

RRRRRRRyT f μ

yT gμ

1Tα1

+)PPPPPPP

⇓mF

∼= ⇓A1

=

Definition 13.7 A T -modification : α � β consists of a 3-cell : α � β

in K subject to the following axiom.

(1y⊗Tα)∗F (1y⊗Tβ)∗F(1⊗T)∗1 �� (1y⊗Tβ)∗F

G∗(β⊗1x )

B

��

(1y⊗Tα)∗F

G∗(α⊗1x )

A

��G∗(α⊗1x ) G∗(β⊗1x )

1∗(⊗1)��

We can now present the pseudo-strength version of these cells.

Definition 13.8 A pseudo-T -algebra X consists of an underlying lax T -algebra X equipped with adjoint equivalences m �eq m�, i �eq i � such thatπ, λ, ρ are invertible 3-cells.

Definition 13.9 A pseudo-T -functor f : X → Y consists of an underlyinglax T -functor equipped with an adjoint equivalence F �eq F � such that H,Mare invertible 3-cells.

13.3 Total structures 219

Definition 13.10 A T -transformation α : f ⇒ g consists of an underlyinglax T -transformation such that A is an invertible 3-cell.

13.3 Total structures

We are now ready to prove that lax algebras and the appropriate higher cellsbetween them form a Gray-category. After doing so, we will show the samefor the pseudo-strength version of these concepts. We break this proof into anumber of smaller propositions.

Proposition 13.11 Let X,Y be lax T -algebras for the Gray-monad T . Thereis a 2-category, Lax-T -Alg(X,Y ), whose objects are lax T -functors f : X →Y , whose 1-cells are T -transformations between them, and whose 2-cells areT -modifications between those.

Proof The identity cell on f : X → Y consists of the 2-cell 1 f and the 3-cell1F ; it is easy to check that this is a T -transformation. Given T -transformations(α, A) : f ⇒ g, (β, B) : g ⇒ h, the composite T -transformation (β, B) ∗(α, A) consists of the 2-cell β ∗ α and the 3-cell shown below.

f x yT fF �� yT f

yT g

1Tα

��yT g

yT h

1Tβ

��

f x

gx

α1

��gx

hx

β1

��hx yT h

H��

gx yT gG

��

⇓ A

⇓ B

It is straightforward to check the T -transformation axioms.Given T -modifications : α � β, : β � δ, it is simple to check

that the 3-cell defines a T -modification α � δ. Similarly, if we haveT -modifications : α � β, : α′ � β ′ where α′ ∗ α (and hence β ′ ∗ β)exists, then the 3-cell ∗ defines a T -modification α′ ∗ α � β ′ ∗ β. Theidentity T -modification on α is the identity 3-cell. It is now a simple matter tocheck that composition is associative and unital for both the T -transformationsand the T -modifications, and that the middle four interchange law holds, so thisis a 2-category.

Proposition 13.12 Let X,Y, Z be lax T -algebras. Then there is a 2-functor

� : Lax-T -Alg(Y, Z)⊗ Lax-T -Alg(X,Y ) → Lax-T -Alg(X, Z)


such that the underlying cell of �(g⊗ f ), which we now denote by g� f , is theunderlying cell of f composed with the underlying cell of g via the composition⊗ in K.

Proof To complete the description of g � f , we must give a 2-cell and a pairof 3-cells. The structure 2-cell is

g ⊗ f ⊗ x1⊗F−→ g ⊗ y ⊗ T f

G⊗1−→ z ⊗ T g ⊗ T f.

The 3-cell hg� f is the composite below.

g f 1

g f xη

1i

��

g f xη gyT f η1F1 �� gyT f η

zT gT f η

G1

��

g f 1

g1 f

llllll

llllll

g1 f

1g fJJJJJJJJJJJJ

1g f

zηg f

i1

(&KKKKKK

zηg f

zT gT f ηaaaaaa

aaaaaag1 f

gyη f

1i1��

gyη f

gyT f η��

��gyη f

zT gη fG1 �'''

zT gη f

zηg f

''''''

⇓1h f

⇓hg1

=

The 3-cell mg� f is the composite below.

g f xT x

gyT f T x

1F1��******

gyT f T x

zT gT f T xG1 ��

zT gT f T x zT gT yT 2 f1T F �� zT gT yT 2 f

zT zT 2gT 2 f

1T G1��

zT zT 2gT 2 f

zμT 2gT 2 f

m1

��++++++

g f xT x

g f xμ1m ��

g f xμ

gyT f μ1F1 ��

gyT f μ

zT gT f μ

G1

�� zT gT f μ

zμT 2gT 2 f$$$$$$$$$$$$

gyT f T x gyT yT 2 f1T F

�� gyT yT 2 f

zT gT yT 2 f

G1

��gyT yT 2 f

gyμT 2 f

1m1+),

,,,,

gyμT 2 f

gyT f μ

gyμT 2 f zT gμT 2 fG1 �� zT gμT 2 f zμT 2gT 2 f

∼=

⇓1m f

⇓mg1

=

To show that g � f is a lax T -functor, we must check three axioms. Thisrequires three tedious, but completely straightforward diagram chases usingthe lax T -functor axioms and the Gray-category axioms.

Next we define �(α ⊗ 1 f ), or α � 1 f as we will write it from now on, fora T -transformation α : g ⇒ g′. The cell α � 1 f has source g � f and targetg′ � f , and it consists of the 2-cell α ⊗ 1 f and the 3-cell shown below.


g f x gyT f1F �� gyT f zT gT fG1 �� zT gT f

zT g′T f

1Tα1

��

g f x

g′ f x

α1

��g′ f x g′yT f

1F�� g′yT f zT g′T f

G ′1��

gyT f

g′yT f

α1

��∼= ⇓ A1

The two axioms now follow from the Gray-category axioms and theT -transformation axioms for α. The definition of �(1g ⊗ β), which we writeas 1g � β, is analogous.

Finally, we define �( ⊗ 1 f ), now written � 1 f , where is aT -modification with source α and target α′. The underlying 3-cell of � 1 f

is ⊗ 1 f . We must check that this is a T -modification α � 1 f � α′ � 1 f ,but that follows immediately from naturality of the structure isomorphisms ina Gray-category. The definition of �(1g ⊗ ), now written 1g � , is similar.

We have defined � on the generating 1-cells of Lax-T -Alg(Y, Z) ⊗Lax-T -Alg(X,Y ), and it is clear that the assignments above preserve compo-sition and identities on the 1-cell level. We have also defined � on whiskered2-cells, and it is clear that � is a 2-functor if we hold either variable fixed. Wemust now define � on the structure 2-cells

�β,α : (β ⊗ 1 f ′) ∗ (1g ⊗ α) � (1g′ ⊗ α) ∗ (β ⊗ 1 f ).

By functoriality, �(�β,α) must be a 2-cell

(β � 1 f ′) ∗ (1g � α) � (1g′ � α) ∗ (β � 1 f ),

thus we need to give a T -modification as displayed above. The claim is that the3-cell structure isomorphism �β,α in K is a T -modification with this sourceand target, and this is easily verified using the naturality of the cells �β,α .Therefore we can define �(�β,α) to be the T -modification �β,α , completingthe construction of the 2-functor �.

Theorem 13.13 The following data defines a Gray-category Lax-T -Alg:

• the objects of Lax-T -Alg are lax T -algebras,

• the hom-2-categories are the 2-categories Lax-T -Alg(X,Y ) defined inProposition 13.11,

• the composition 2-functor is the 2-functor � defined in Proposition 13.12,and

• the unit ∗ → Lax-T -Alg(X, X) is the 2-functor defined by sending theunique object of the source to the lax T -functor 1 with f = 1, F = 1,H = 1,M = 1.


Proof First, note that the unit defined above is actually a lax T -functor sinceall of the axioms reduce to the identities π = π, ρ = ρ, λ = λ. Second, wehave defined all of the structure on hom-2-categories, so we only have to checkthat � gives a unital and associative composition.

To check the unit conditions, we start with g � 1. The underlying cell isg ⊗ 1 = g, and the cells hg�1,mg�1 are equal to

hg ⊗ 11 = hg, mg ⊗ 11 = mg,

respectively, so g � 1 = g; the same argument shows that 1 � f = f , so� is unital on 1-cells. We must also show that 11 � β = β, α � 11 = α, andsimilarly for 3-cells, but these equalities are also immediate from the definitionof these whiskered cells. Thus � is unital.

Finally, we must show that � is associative. Since this holds if and only if �is associative on generating cells and structure cells, we will only check thesecases. First, we must show that (h � g)� f equal h � (g � f ). The underlyingcells are equal, as they are both h ⊗ g ⊗ f . The structure 2-cell for (h � g)� f is

hg f x11F−→ hgyT f

1G1−→ hzT gT fH11−→ wT hT gT f,

and it is easy to check that this is the structure 2-cell for h � (g � f ) as wellby the Gray-category axioms. Finally, we must check that structure 3-cells foreach of these lax T -functors agree. The 3-cell h(h�g)� f is

hg f 1

hg f xη

1i

��)))))))))))))))

hg f xη hgyT f η1F1 �� hgyT f η

hzT gT f η

1G1

'%((((((

hzT gT f η

wT hT gT f η

H1

'%((((((

hg f 1

hg1 f

======

======

hg1 f

h1g f

======

======

h1g f

1hg f

======

======

1hg f

wηhg f

i1

��////////////////

wηhg f

wT hT gT f η��

hg1 f

hgyη f

1i1

��<<<<<<<<<<<<<<<<

hgyη f

hgyT f η%%%%%%%%%%%%%%

hgyη f

hzT gη f

1G1 ��

hzT gη f

wT hT gη f

H1" 11wT hT gη f

wηhg f

555555

h1g f

hzηg f

1i1

QGkkkkkk

hzηg f

hzT gη f��

��hzηg f wT hηg f

H1��mmmm wT hηg f wηhg fmmm mmm

⇓1h f

⇓1hg1

⇓hh1

=

=


and the 3-cell hh�(g� f ) is the cell shown below.

hg f 1

hg f xη

1i

[Yaaaaaaaaaaaaaaaaaaaa

hg f xη hgyT f η1F1 �� hgyT f η

hzT gT f η

1G1

MMllllllll

hzT gT f η

wT hT gT f η

H1

MMllllllll

hg f 1

hg1 f

======

======

hg1 f

h1g f

======

======

h1g f

1hg f

======

======

1hg f

wηhg f

i1

��////////////////

wηhg f

wT hT gT f η��

hg1 f

hgyη f

1i1

��..................

hgyη f

hzT gη f

1G1'%"""

hzT gη f

hzηg f

""""""

hzηg f

wT hηg f

H1'%"""

wT hηg f

wηhg f

��

hzηg f

hzT gT f η��

hgyη f

hgyT f η��

��

h1g f

hzηg f

1i1

��

⇓1h f

⇓1hg1

⇓hh1

=

=

These two 3-cells are clearly equal. To complete the proof that � is associativeon 1-cells, we must also prove that m(h�g)� f = mh�(g� f ), but this calculationproceeds in exactly the same fashion as the one above.

We must also check that � is associative on 2- and 3-cells. To prove this, weneed only prove it on generating cells which reduces to checking the followingthree cases, where we have written ϕ for a generic 2- or 3-cell.

(ϕ � 1)� 1 = ϕ � (1 � 1) (1)(1 � ϕ)� 1 = 1 � (ϕ � 1) (2)(1 � 1)� ϕ = 1 � (1 � ϕ) (3)

For (1) in the case when ϕ is a 2-cell, the left side has underlying 2-cell ϕ ⊗1g ⊗ 1 f while the right side has underlying 2-cell ϕ ⊗ 1g f , and these cells areequal since 1g ⊗ 1 f = 1g f . The structure 3-cells are also equal by the Gray-category axioms. The same argument shows that equation (1) holds when ϕ isa 3-cell, and also that equation (3) holds for 2- and 3-cells. For equation (2)in the case of a 2-cell ϕ, the underlying 2-cells on both sides are once againthe same and equal to 1g ⊗ ϕ ⊗ 1 f ; the same holds when ϕ is a 3-cell. Thestructure 3-cells, in the case that ϕ is once again a 2-cell, are also equal bythe associativity of composition on 3-cells. Thus in each case above, the cellson both sides of the equation have the same data, and since their sources andtargets are equal by the associativity of � on 1-cells, these cells are thereforeequal and � is associative on generating higher cells as well.


We also must check the associativity axioms on the structure cells �β,α .Once again there are a few cases to check, and these are

• �δ,β ⊗ 1α ,• �1⊗β,α , and• �δ⊗1,α .

Note that we do not have a case for 1δ ⊗�β,α because of the choice of startingparenthesization. In each case, we must check that the two different maps(

Lax-T-Alg(Z ,W )⊗ Lax-T-Alg(Y, Z))⊗ Lax-T-Alg(X,Y )

→ Lax-T-Alg(X,W )

send these cells to the same cell. This is straightforward to check in each case,finishing the associativity axiom.

Proposition 13.14 The Gray-category KT of strict algebras embeds as a2-locally full sub-Gray-category of Lax-T -Alg.

Proof First, every strict T -algebra can be regarded as a lax T -algebra with thecells m, i, π, λ, ρ all being identity cells; note that for π, λ, ρ to be identities,we require that they have the same source and target, which in this case will bean identity 2-cell since m, i are identities. Second, we can regard every algebra1-cell (the 1-cells in KT ) as a lax T -functor with F, h,m all being identitycells. Third, an algebra 2-cell α : f ⇒ g can be regarded as a T -transformationwith A = 1 since we have that the 2-cells F,G from the algebra 1-cells areboth identities. Finally, an algebra 3-cell : α � β produces a T -modificationsince F,G, A, B are all identity cells. This gives the assignment on cells of aputative Gray-functor KT → Lax-T -Alg.

To show that this is actually a Gray-functor, we must check that it preservescomposition, units, and the Gray-category structural isomorphisms. First, thisassignment is a 2-functor on the hom-2-categories. This claim is easy to checkby verifying that the three different possible composites (β ∗α, , and ∗)of strict algebra cells are preserved under being mapped into Lax-T -Alg, whichreduces to the fact that composites of identity cells are again identity cells.Identity 2- and 3-cells are also mapped to identities, as the identity cells inLax-T -Alg have components which are identities. Second, this assignmentpreserves composition along 0-cell boundaries. To prove this, we must showthat if f, g are composable algebra 1-cells then their composite is given byg � f in Lax-T -Alg, as well as similar conditions for whiskered 2- and 3-cells.Once again, this follows from noting that composites of identities are identitiestogether with the Gray-category axioms which are required to show that the


isomorphism 3-cells in some pasting diagrams arising from the Gray-categorystructure are in fact identities. Finally, we must show that this assignment pre-serves the Gray-category structure isomorphisms, but in each case these aregiven by �β,α , and since the function on underlying cells KT → Lax-T -Algsends every algebra 3-cell to the T -modification with the same underlying cell,this is immediate.

To conclude the proof, we must show that the Gray-functor KT →Lax-T -Alg we have just constructed is a 2-locally full embedding. It is clearthat if two different cells (of the same dimension) in KT get mapped to thesame cell in Lax-T -Alg, then they were in fact the same cell in KT since thefunctor does not change underlying data. Now let : α � β be any T -modification between 2-cells in the image of KT → Lax-T -Alg; note that the0- and 1-cell boundaries of these 2-cells are also in the image of this functor.In equational form, the single T -modification axiom is

B((1 ⊗ T) ∗ 1F

)=

(1G ∗ ( ⊗ 1)

)A.

Since the lower dimensional cells all arise from cells in KT , we have thatA, B, F,G are all identities, and this equation reduces to

1 ⊗ T = ⊗ 1

which shows that is an algebra 3-cell, so the embedding KT → Lax-T -Algis 2-locally full.

Proposition 13.15 There is a Gray-category Ps-T -Alg whose objects arethe pseudo-T -algebras. The assignment sending each pseudo-T -algebra to itsunderlying lax T -algebra is a Gray-functor

U : Ps-T -Alg → Lax-T -Alg

which is a 2-local isomorphism.

Proof The objects of Ps-T -Alg are pseudo-T -algebras, and the 2-categoriesPs-T -Alg(X,Y ) have 0-cells which are pseudo-T -functors, 1-cells which areT -transformations, and 2-cells which are T -modifications. The compositionlaws and units are the same for Ps-T -Alg(X,Y ) as for Lax-T -Alg(X,Y ), not-ing that the structure 3-cell for (β, B)∗ (α, A) is invertible when A, B are bothinvertible.

The next step in establishing a Gray-category structure is to define acomposition functor,

�ps : Ps-T -Alg(Y, Z)⊗ Ps-T -Alg(X,Y ) → Ps-T -Alg(X, Z).


Given composable pseudo-T -functors ( f,F, h f ,m f ), (g,G, hg,mg), wherehere we have denoted the two adjoint equivalences by the bold F,G, we definethe composite to have the following data. The underlying 1-cell is g ⊗ f .The adjoint equivalence has left adjoint (G ⊗ 1) ∗ (1 ⊗ F) and right adjoint(1 ⊗ F �) ∗ (G� ⊗ 1); the unit is

11⊗ηF=⇒ (1 ⊗ F �) ∗ (1 ⊗ F)

1∗(ηG⊗1)∗1=⇒ (1 ⊗ F �) ∗ (G� ⊗ 1) ∗ (G ⊗ 1) ∗ (1 ⊗ F),

and the counit is defined similarly. The 3-cells hg�ps f ,mg�ps f are defined bythe same pasting diagrams as in the lax case, and it should be noted that theseare invertible because each cell in the pasting is invertible. This data gives apseudo-T -functor because this data extends that of g � f which is a lax T -functor. The definition of �ps on higher cells is then exactly the same as �.The only part of the Gray-category structure that is not immediately apparentfrom these definitions is associativity of �ps , but all that needs to be checkedhere is that the adjoint equivalences for (h �ps g)�ps f and h �ps (g �ps f )coincide, but this follows easily from the Gray-category axioms for K.

Finally, it is obvious that the assignment of the underlying lax data toany of the pseudo-strength objects gives a Gray-functor U : Ps-T -Alg →Lax-T -Alg, and since both Gray-categories have the same definitions of 2- and3-cells this functor is 2-local isomorphism.

Corollary 13.16 The Gray-category KT of strict algebras is a 2-locallyfull sub-Gray-category of Ps-T -Alg, and the inclusion of KT into Lax-T -Algfactors as

KT ↪→ Ps-T -AlgU−→ Lax-T -Alg.

Proof This follows immediately from the previous two propositions by not-ing that the identity 2-cell F can be equipped with the structure of the identityadjoint equivalence.

14

The reflection of lax algebras intostrict algebras

So far, we have studied three kinds of algebras for a Gray-monad T . Themost basic are the strict algebras, which are just algebras in the usual senseof enriched category theory. The second kind are lax algebras, and strict alge-bras are just lax algebras satisfying the condition that certain data consists ofidentity cells; we have seen that this gives an inclusion of Gray-categories

KT ↪→ Lax-T -Alg.

The third kind of algebra is a pseudo-algebra, which is a lax algebra equippedwith extra structure demonstrating that some of the data is invertible in theappropriate sense. We know that every strict algebra can be equipped withthe structure of a lax algebra by adding in extra data given by identity cells.The same process allows us to equip each strict algebra with the structure ofa pseudo-algebra in which all of the adjoint equivalences are identity adjointequivalences, producing the inclusion

KT ↪→ Ps-T -Alg.

Our goal in this section is to construct left adjoints to both of these functorsusing the notion of codescent objects.

The strategy is as follows. Given a lax algebra (X, x,m, i, π, λ, ρ) overT , we construct a canonical lax codescent diagram X . This diagram willbe in the Gray-category KT of strict T -algebras, and will consist of cellswhich are either free (i.e., in the image of T ) or constructed from the struc-ture cells μ : T 2 ⇒ T, η : 1 ⇒ T for the Gray-monad T . In factthe Gray-category G was constructed precisely to accomodate such a dia-gram. We then require that KT has lax codescent objects for lax codescentdiagrams; there are a variety of assumptions on K, T separately which canensure these, but assuming they exist is enough to prove the first theorem.

227

228 The reflection of lax algebras into strict algebras

Assuming we can take lax codescent objects, the assignment of the lax algebra(X, x,m, i, π, λ, ρ) to the lax codescent object of X is the object-part of aGray-functor Lax-T -Alg → KT , and we show that this is the left Gray-adjoint to the inclusion. Additionally, it is easy to give a partial converse to thisresult: if the inclusion KT ↪→ Lax-T -Alg has a left Gray-adjoint, then KT

must have lax codescent objects for the lax codescent diagrams of the formX . Additionally, the proofs can all be adjusted to account for pseudo-algebrasinstead of lax algebras using codescent objects instead of lax codescent objects.This is the Gray-enriched version of the proof given by Lack (2002a).

14.1 The canonical codescent diagram of a lax algebra

Beginning with a lax algebra (X, x,m, i, π, λ, ρ), we will construct a canoni-cal lax codescent diagram

X : G → KT .

After doing so, we will explain how to modify this construction to construct acodescent diagram from a pseudo-T -algebra as well.

We define the action of X on the objects of G by X([i]) = T i X . The1-cells of G are generated by basic 1-cells

dk : [i] → [i − 1] 0 ≤ k < i,sk : [i] → [i + 1] 0 ≤ k < i.

We define X(dk) : T i X → T i−1 X by

X(dk) is T kμT i−k−2 : T i X → T i−1 X, 0 ≤ k < i − 1,X(di−1) is T i−1x : T i X → T i−1 X.

We define X(sk) : T i X → T i+1 X to be T k+1ηT i−k−1 : T i X → T i+1 X .There are six different kinds of basic 2-cells, and we define the action of X

on each of them now.

• The cell X(Ai j ) : X(di d j ) ⇒ X(d j−1di ) is

– T m when i = 1, j = 2, and the 0-cell source is T 3 X ;

– T 2m when i = 2, j = 3, and the 0-cell source is T 4 X ; and

– the identity otherwise, arising from either the monad axioms for T or thenaturality of μ.

• The cell X(L j ) : X(d j s j ) ⇒ 1 is the identity, arising from the unit axiomsfor the monad T .

14.1 The canonical codescent diagram of a lax algebra 229

• The cell X(R j ) : 1 ⇒ X(d j+1s j ) is

– T i when j = 0 and the 0-cell source is T X ;– the identity when j = 0 and the 0-cell source is T 2 X , arising from the

unit axioms for the monad T ; and– T 2i when j = 1 and the 0-cell source is T 2 X .

• The cell X(N ds) is the identity, arising from the naturality of μ.• The cell X(N sd) is the identity, arising from the naturality of η.• The cell X(N s) is the identity, arising from the naturality of η.

There are four different kinds of basic 3-cells, and we define the action of Xon each of them now.

• There are four 3-cells πi jk , where i < j < k. The 2-cell sources and targetsof both X(π012) and X(π013) are all identity 2-cells, so we define both ofthese 3-cells to be identities as well. The source and target of X(π023) areequal by naturality, so we define this to be the identity 3-cell as well. Wedefine X(π123) to be the 3-cell Tπ from the lax algebra structure on X .

• There are two 3-cells μi , where i = 0, 1. The 2-cell source and target ofX(μ0) are both identity 2-cells, so we define X(μ0) to be the identity 3-cell.We define X(μ1) to be Tρ.

• There are four 3-cells νli , ν

ri , where i = 0, 1. The 2-cell sources and targets

of the cells X(νli ) are all identity 2-cells, so we define both of these 3-cells

to be identities as well. The source and target of νr0 are equal by naturality,

so we define this to be the identity 3-cell. We define νr1 to be Tλ.

• Finally, there is the 3-cell νs , and we define X(νs) to be the identity by thenaturality of η.

We have defined X on the generating cells of G , and since the rest of thisGray-category is generated freely from this data, our assignments determinea Gray-functor G → K. Our goal was to construct a lax codescent diagramin KT , and since all of the cells in the definition are either structure cells fromthe monad or are of the form T a for some cell a, we have proven the followingproposition.

Proposition 14.1 For any lax T -algebra X, X defines a lax codescentdiagram in KT .

Now assume that X = (X, x,m, i, π, λ, ρ) is a pseudo-T -algebra. Apply-ing the functor U above, we produce a lax algebra U X . This gives rise to alax codescent diagram, U X . By definition, U X sends every generating 2-cellof G to either an identity 2-cell in KT , or to a 2-cell T m, T 2m, T i , T 2i .Identity cells, m, and i are all part of adjoint equivalences, so every generating


2-cell of G is mapped to an equivalence in K. Since T is a Gray-functor, itmaps adjoint equivalences to adjoint equivalences. This shows that T m, T 2m,T i , T 2i are actually part of canonical adjoint equivalences in KT obtained byapplying T or T 2 to the underlying adjoint equivalences in K. Additionally, Tsends invertible 3-cells to invertible 3-cells, so all of the 3-cell data for U X isinvertible. Combining these facts with the above proposition, we have proventhe following.

Proposition 14.2 For any pseudo-T -algebra X, U X can be extended to acodescent diagram, X ps , in KT using the adjoint equivalences obtained byapplying T, T 2 to m and i .

14.2 The left adjoint, lax case

There is a forgetful functor KT ↪→ Lax-T -Alg which we will write asi ; our goal in this section is to show that it has a left Gray-adjoint. LetX = (X, x,m, i, π, λ, ρ) be a lax T -algebra, and let Y be a strict T -algebra. Aleft Gray-adjoint to i consists of a Gray-functor L : Lax-T -Alg → KT and aGray-natural isomorphism

KT (L X,Y ) ∼= Lax-T -Alg(X, iY ).

We will give the construction of L first, and then the required naturalisomorphism.

For the rest of this section, we will require the existence of certain weightedcolimits. In particular, we will need lax codescent objects of lax codescentdiagrams in KT . We identify four possible assumptions.

C1 The Gray-category KT has lax codescent objects of lax codescentdiagrams.

C2 The Gray-category K has lax codescent objects of lax codescent dia-grams, and T preserves them.

C3 The Gray-category K has co-2-inserters, co-3-inserters, and coequifiers,and T preserves them.

FC The Gray-category K is cocomplete, and T preserves α-filtered colimitsfor some regular cardinal α.

Since lax codescent objects can be built from co-2-insterters, co-3-inserters,and coequifiers, assumption C3 implies assumption C2. KT is the enrichedcategory of T -algebras in the usual sense, and it always has any weightedcolimit which exists in K and is preserved by T , so C2 implies C1. Assump-tion FC also implies C1 since it actually implies that KT is cocomplete as a

14.2 The left adjoint, lax case 231

Gray-category. From this point on, we will generally state theorems using C1,but specific examples are often best examined using FC or C3. When assum-ing any of these, we also implicitly assume that every diagram has a chosencolimit.

Theorem 14.3 Let X = (X, x,m, i, π, λ, ρ) be a lax T -algebra, and let Xbe its associated lax codescent diagram. Under assumption C1, the assigmentwhich sends X to the lax codescent object of X is the object part of a Gray-functor L : Lax-T -Alg → KT .

We will prove this theorem in smaller pieces, first defining L on the highercells and then showing it is a functor.

Lemma 14.4 Let f = ( f, F, H,M) be a lax T -algebra map X → X ′. Thecomposite

T XT f−→ T X ′ π ′−→ L X ′

factors through a 1-cell L f : L X → L X ′ in KT .

Proof By the first part of the universal property of the lax codescent object,we need only construct certain data satisfying alternate versions of the threelax codescent object axioms to construct a unique such factorization satisfyingthose axioms. This data consists of a 1-cell y, a 2-cell ε, and a pair of 3-cellsM,U , and we have already given the 1-cell datum as y = π ′ ⊗ T f . The 2-celldatum is a cell

ε : y ⊗ d1 ⇒ y ⊗ d0.

This is defined to be the pasting diagram of 2-cells given below.

T 2 X

T X

T x

[Ybbbbbbbb

T X T X ′T f �� T X ′

L X ′

π ′

MMWWWWWWWW

T 2 X

T X

μ

MMWWWWWWWW

T X T X ′T f

�� T X ′

L X ′

π ′

[Ybbbbbbbb

T 2 X T 2 X ′T 2 f

�� T 2 X ′

T X ′

T x ′

a FFFFFFFFT 2 X ′

T X ′μ

BACCCCCCCC

⇓ T F

=⇓ ε′


The 3-cell M is defined to be the pasting diagram below.

π ′T f T xT 2x

π ′T x ′T 2 f T 2x

1T F1

RQQQQQQQQ

π ′T x ′T 2 f T 2x

π ′μT 2 f T 2x

ε′��

π ′μT 2 f T 2x

π ′T f μT 2x��

π ′T f μT 2x π ′T f T 2xμTπ ′T f T 2xμT

π ′T x ′T 2 f μT

1T F1!��

π ′T x ′T 2 f μT

π ′μT 2 f μT

ε′

��

π ′μT 2 f μT

π ′T f μμT

,,,,,,,,

,,,,,,,,

π ′T f T xT 2x

π ′T f T xTμ

11T mA@gggggggg

π ′T f T xTμ

π ′T x ′T 2 f Tμ1T F1��π ′T x ′T 2 f Tμ π ′μT 2 f Tμ

ε′�� π ′μT 2 f Tμ

π ′T f μTμ�� π ′T f μTμ

π ′T f μμT!!!!!!!!!

!!!!!!!!!

π ′μT 2 f T 2x

π ′μT 2xT 3 f11T 2 F

!��

π ′μT 2xT 3 f π ′T x ′μT T 3 fπ ′T x ′μT T 3 f

π ′T x ′T 2 f μT��

π ′T x ′T 2 f T 2x

π ′T x ′T 2x ′T 3 f11T 2 F ��

π ′T x ′T 2x ′T 3 f

π ′μT 2xT 3 f

ε′

��π ′T x ′T 2x ′T 3 f

π ′T x ′TμT 3 f

1T m′1 ��

π ′T x ′TμT 3 f π ′μTμT 3 fε′11

�� π ′μTμT 3 f

π ′μT 2 f Tμ

π ′μTμT 3 f

π ′μμT T 3 f��

��

π ′T x ′μT T 3 f

π ′μμT T 3 f

ε′11��

π ′μμT T 3 f

π ′μT 2 f μT��

��

π ′T x ′TμT 3 f

π ′T x ′T 2 f Tμ

∼=

⇓1T m

⇓M ′1

=

=

=

The 3-cell U is defined to be the pasting diagram below.

π ′T f

π ′T f T xTη

11T i��$$$$$$$$$$

π ′T f T xTη π ′T x ′T 2 f Tη1T F1 �� π ′T x ′T 2 f Tη π ′μT 2 f Tη

ε′11 �� π ′μT 2 f Tη

π ′T f μTη

��

��

π ′T f μTη

π ′T f

π ′T f

π ′T f

π ′T f π ′T x ′TηT f1T i ′1 �� π ′T x ′TηT f

π ′T x ′T 2 f Tη$$$$$$$$$$

$$$$$$$$$$π ′T x ′TηT f π ′μTηT f

ε′11�� π ′μTηT f

π ′μT 2 f Tη$$$$$$$$$$

$$$$$$$$$$π ′μTηT f

π ′T f999999999999999999999

999999999999999999999

⇓1T h

⇓U ′1

=

To produce the factorization required, it now suffices to show that the datagiven above satisfies three axioms corresponding to the three lax codescentobject axioms. In each case, the pasting diagrams are quite large, but themajority of the cells are identities. The first axiom (“associativity”) follows byapplying the first lax T -functor axiom, then the first lax codescent object axiomfor X ′, then naturality of the Gray-category isomorphism structure cells. Thetwo unit axioms are proved in precisely the same fashion, using the appro-priate unit axiom for the lax T -functor f and the lax codescent object axiomfor X ′.

Lemma 14.5 Let f1, f2 : X → X ′ be lax T -algebra maps, and let α =(α, A) be a T -transformation between them. Then there is a 2-cell Lα : L f1 ⇒L f2 in KT such that 1π ′ ⊗ Tα = Lα ⊗ 1π .


Proof Once again, we will define this cell using the universal property of thelax codescent object. To construct Lα, we will give a 2-cell δ : L f1 ⊗ π ⇒L f2 ⊗ π and an invertible 3-cell

T 2 X

T XT x

��

T X L Xπ �� L X

L X ′

L f1

��

T 2 X

T Xμ ��

T X L Xπ�� L X

L X ′

L f2

��T X

L X

π

��

⇓ ε

⇓ δT 2 X

T XT x

��

T X L Xπ �� L X

L X ′

L f1

��

T 2 X

T Xμ ��

T X L Xπ�� L X

L X ′

L f2

��

T X

L X

π

��

⇓ ε

⇓ δ �

satisfying two axioms. The universal property of the lax codescent object thengives that there is a unique 2-cell Lα such that both Lα⊗ 1π = δ and is thecanonical isomorphism given by the Gray-category structure. By the definitionof L on 1-cells, we have that

L fi ⊗ π = π ′ ⊗ T fi ,

so we define δ = 1π ′ ⊗ Tα and define to be the pasting below.

π ′T f1T x π ′T x ′T 2 f11T F1 �� π ′T x ′T 2 f1 π ′μT 2 f1

ε′1 �� π ′μT 2 f1 π ′T f1μπ ′T f1μ

π ′T f2μ

1Tα1

��

π ′T f1T x

π ′T f2T x

1Tα1

��π ′T f2T x π ′T x ′T 2 f21T F2

�� π ′T x ′T 2 f2 π ′μT 2 f2ε′1

�� π ′μT 2 f2 π ′T f2μ

π ′T x ′T 2 f1

π ′T x ′T 2 f2

11T 2α

��

π ′μT 2 f1

π ′μT 2 f2

11T 2α

��⇓ 1T A ∼= =

The two axioms then both follow from the T -transformation axioms togetherwith naturality and the Gray-category axioms.

Remark 14.6 The proof of the previous lemma is what forced us to useT -transformations as the 2-cells in Lax-T -Alg instead of the more general laxT -transformations. In order to use the universal property of the lax codescentobject, we must eventually conclude that the 3-cell constructed above isinvertible, as it will be equal to a Gray-category constraint 3-cell. The onlyway to ensure this is to require that the 3-cell A is also invertible, which meansthat (α, A) is a T -transformation instead of merely a lax one. We could haveused lax T -transformations if we were enriching over the original, lax versionof the Gray-tensor product

Lemma 14.7 Let f1, f2 : X → X ′ be lax T -algebra maps, let α1, α2 bea pair of T -transformations between them, and let : α1 � α2 be a T -modification. Then there is a 3-cell L : Lα1 ⇒ Lα2 in KT such that 11π ′ ⊗T = L ⊗ 11π .


Proof Once again, we will define this cell using the universal property of thelax codescent object. To use the universal property, we must construct a 3-cell : Lα1 ⊗ 1π � Lα2 ⊗ 1π satisfying one axiom, and then the third partof the universal property produces L = . We have constructed Lαi suchthat Lαi ⊗ 1π = 1π ′ ⊗ Tαi , so define to be 11π ′ ⊗ T. The single axiomthen follows from the T -modification axiom of and the fact that the Gray-category isomorphism involving αi and ε can be expressed using T Ai by thesecond part of the universal property of the lax codescent object.

Proof of Theorem 14.3 We have already defined L on cells, and it remains toshow that L preserves composition, units, and the Gray-category structure. Ineach case, we will check that the cells in question satisfy the same universalproperties and so must be equal. We begin with the axioms on 1-cells.

Let 1X : X → X be the identity lax T -functor for a lax algebra X . Thedefinition of L1X gives a factorization π = L1X ⊗π induced by the universalproperty of the lax codescent object. It is easy to check that the 1-cell 1L X

satisfies the four conditions in the first universality axiom, so L1X = 1L X .Now let g, f be a composable pair of lax T -functors. We must show that

Lg ⊗ L f = L(g ⊗ f ). To do this, we can check that Lg ⊗ L f satisfies the fourconditions in the first universality axiom that determine the 1-cell L(g ⊗ f ).The first condition is that

L(g ⊗ f )⊗ πX = πZ ⊗ T (g ⊗ f ).

Computing Lg ⊗ L f ⊗ πX , we have

Lg ⊗ L f ⊗ πX = Lg ⊗ πY ⊗ T f= πZ ⊗ T g ⊗ T f= πZ ⊗ T (g ⊗ f )

by using the definition of L f , the definition of Lg, and the fact that T is aGray-functor. Therefore L(g ⊗ f ) ⊗ πX = Lg ⊗ L f ⊗ πX , so Lg ⊗ L fsatisfies the first condition in the universal property. The second condition isthat 1L(g⊗ f ) ⊗ εX is equal to the pasting diagram below.

T 2 X

T X

T x

��

T X T YT f �� T Y T Z

T g �� T Z

L Z

πZ

��

T 2Y

T Y

T y

��T 2 X

T X

μ

��

T X T YT f

�� T Y T ZT g

�� T Z

L Z

πZ

��

T 2 X T 2YT 2 f

�� T 2Y T 2 ZT 2g

�� T 2 Z

T Z

μ

��T 2 Z

T Z

T z

��T 2Y

T Y

μ

��

⇓ T F ⇓ T G

⇓ εZ

= =


By the definition of Lg, this pasting diagram is equal to the one below.

T 2 X

T X

T x

��

T X T YT f ��

T 2 X T 2YT 2 f

�� T 2Y

T Y

T y

��⇓ T F

=T 2 X

T X

μ

��

T X T YT f

��

T 2Y

T Y

μ

��

T Y

LY

πY

��

T Y

LY

πY

��

LY L ZLg ��⇓ εY

And now by the definition of L f , this is equal to the pasting diagrambelow.

T 2 X

T X

T x

��

T X

L X

πX

��

T 2 X

T X

μ

��

T X

L X

πX

��

⇓ εX L X LYL f �� LY L Z

L f ��

Thus Lg⊗L f satisfies the second of the four conditions in the first universalityaxiom for L(g ⊗ f ).

There are two more conditions to check, those related to the 3-cells M,U ,and the proof proceeds in the same fashion as above; we leave it to the readerto conclude the proof that L(g ⊗ f ) = Lg ⊗ L f .

For 2- and 3-cells, the proofs are analogous: in each case, we check that thecomposite of two cells in the image of L satisfy the conditions for the universalproperty of L of the composite, and similarly for identities.

The final requirement to check is that L preserves the Gray-categorystructure isomorphisms, or that L�β,α = �Lβ,Lα . The 3-cell L�β,α is deter-mined by a single 3-cell satisfying one axiom, in which case L�β,α ⊗1 = . In this case, the single axiom is the equality of pasting dia-grams below, where the unmarked isomorphisms are from the Gray-categorystructure.


By the Gray-category axioms, = �Lβ,Lα ⊗ 1 satisfies this axiom, provingthat L�β,α = �Lβ,Lα and completing the proof that L is a Gray-functor.

Theorem 14.8 The Gray-functor L : Lax-T -Alg → KT is the left Gray-adjoint to the inclusion i : KT → Lax-T-Alg.

Proof To show that L is the left adjoint of i , we will construct a Gray-naturalisomorphism

KT (L X,Y ) ∼= Lax-T -Alg(X, iY )

for a lax T -algebra X and a strict T -algebra Y . We will begin by constructinga 2-functor R : Lax-T -Alg(X, iY ) → KT (L X,Y ). Let f : X → iY be alax T -functor. To produce a strict T -functor R f : L X → Y , we will use theuniversal property of the lax codescent object. Thus we must produce a 1-cellg, a 2-cell ε, and two 3-cells M,U in KT satisfying three axioms. The 1-cellg is y ⊗ T f which is in KT since T f is strict by definition and y is the algebrastructure map for a strict algebra, hence a strict T -algebra map. The 2-cell ε is

yT f T x1T F−→ yT yT 2 f = yμT 2 f = yT f μ,

where the first equality is an algebra axiom for Y and the second is naturalityof μ. The 3-cell M is the pasting below.

yT f T xT 2x

yT yT 2 f T 2x

1T F1[Yaaaaa

yT yT 2 f T 2x

yμT 2 f T 2x$$$$$$$$$$

yμT 2 f T 2x

yT f μT 2xKKKK KKKK

yT f μT 2x yT f T xμTyT f T xμT

yT yT 2 f μT

1T F1)'JJJJ

yT yT 2 f μT

yμT 2 f μT

��

yμT 2 f μT

yT f μμT

llllll

llllll

yT f T xT 2x

yT f T xTμ11T m ��

yT f T xTμ

yT yT 2 f Tμ1T F1��yT yT 2 f Tμ

yμT 2 f Tμ�� yμT 2 f Tμ

yT f μμT��

yT yT 2 f T 2x

yT yT 2 yT 2 f11T 2 F ��

yT yT 2 yT 2 f

yμT 2 f Tμ

��

��yT yT 2 yT 2 f

yμT 2 yT 3 f$$$$$$$$$$$$

yμT 2 yT 3 f

yT yT 2 f μT$$$$$$$$$$$$

yμT 2 f T 2x

yμT 2 yT 3 f

11T 2 F

��

⇓1T m

=

=

=


The 3-cell U is the pasting below.

yT f yT f T xTη11T i �� yT f T xTη yT yT 2 f Tη

1T F1 ��yT f

yT yTηT f

DDDDDDDDDDD

DDDDDDDDDDD

yT yTηT f

yT yT 2 f Tη��

��

yT yT 2 f Tη yμT 2 f TηyμT 2 f Tη

yT f μTηyT f μTη

yT f

⇓1T h

We must now check three axioms, and then the universal property of the laxcodescent object ensures the existence of a unique strict T -functor g : L X →Y such that y ⊗ T f = g ⊗ πX , and we define R f = g. We leave verificationof the three axioms to the reader, noting that in each case it is simply a matterof applying naturality, the Gray-category and Gray-functor axioms, togetherwith the lax T -functor axioms for f .

Let β : f1 ⇒ f2 be a T -transformation with 0-cell source X and 1-cell targetiY . To produce a 2-cell Rβ : R f1 ⇒ R f2 in KT , we use the second part of theuniversal property of the lax codescent object. Thus we must produce a 2-cellα and an invertible 3-cell satisfying two axioms, and the universal propertywill ensure the existence of a unique 2-cell α (satisfying certain properties)which we will define to be Rβ.

The 2-cell α we must exhibit has source R f1πX and target R f2πX , so it is a2-cell yT f1 ⇒ yT f2 by the definition of R f . We define α to be 1 ⊗ Tβ whichis a 2-cell in KT by definition. Using this definition of α, the invertible 3-cell is the pasting diagram below.

yT f1T x yT yT 2 f11T F1 �� yT yT 2 f1 yμT 2 f1yμT 2 f1 yT f1μyT f1μ

yT f2μ

1Tβ1

��

yT f1T x

yT f2T x

1Tβ1

��yT f2T x yT yT 2 f21T F2

�� yT yT 2 f2 yμT 2 f2yμT 2 f2 yT f2μ

yT yT 2 f1

yT yT 2 f2

11T 2β

��

yμT 2 f1

yμT 2 f2

11T 2β

��⇓1T B = =

Finally, there are two axioms to check, but these follow directly from the T -transformation axioms.

Let : β1 � β2 be a T -modification between T -transformations, where has 0-cell source X and 0-cell target iY . To produce a 3-cell R : Rβ1 � Rβ2

in KT , we will use the third part of the universal property of the lax codescentobject. To do so, we must give a 3-cell , which in this case has 2-cell source1y ⊗ Tβ1 and 2-cell target 1y ⊗ Tβ2, thus we define = 1⊗ T. There is oneaxiom to check in order to apply the universal property, but it follows imme-diately from naturality and the T -modification axiom for . This completesthe assignment on cells of the putative 2-functor R : Lax-T-Alg(X, iY ) →KT (L X,Y ). We must also check that these assignments strictly preserve unitsand composition, but this is entirely straightforward. As an example, the 2-cell


R(β ′ ∗ β) is determined by the 2-cell 1 ⊗ T (β ′ ∗ β), together with a sin-gle invertible 3-cell. On the other hand, Rβ ′ ∗ Rβ is determined by the 2-cell(1 ⊗ Tβ ′) ∗ (1 ⊗ Tβ), together with a single invertible 2-cell, and this datamatches that for R(β ′ ∗ β) since T is a Gray-functor. We leave checking therest of functoriality to the reader.

We will now begin construction of the inverse of R, the 2-functor

S : KT (L X,Y ) → Lax-T-Alg(X, iY ).

Let g : L X → Y be a 1-cell in KT which is determined by the following data:

• g : T X → Y ,

• ε : gT x ⇒ gμ,

• the 3-cell M shown below

gT xT 2x

gμT 2x

ε1��

gμT 2x gT xμTgT xμT

gμμT

ε1

��

gT xT 2x

gT xTμ

1T m��

gT xTμ gμTμε1

�� gμTμ

gμμT��

��

⇓ M

• and the 3-cell U shown below.

g gT xTη1T i �� gT xTη gμTη

ε1 �� gμTη

g

g

g��

��

⇓ U

We define the lax T -functor Sg as follows. The 1-cell X → Y in K is g ⊗ η.The 2-cell datum, which we write as SG, is the composite

gηx = gT xηTε1−→ gμηT = gμTη = yT gTη,

where the last equality follows from the fact that g is a 1-cell in KT , hence amap of strict algebras. The 3-cell h is given by the pasting diagram below, inwhich the unmarked 2-cells are all equalities.


gη1

gηxη

11i

#!gηxη

gT xηT η��

��

gT xηT η gμηT ηε11 �� gμηT η

gμTηη

��

��

gμTηη

yT gTηηgη1

1gη��

��

1gη yηgηyηgη

yT gTηη��

��

gη1 gT 1ηgT 1η

gT xTηη

1T i1

#!gT xTηη

gT xηT η

gT xTηη

gμTηη

ε11

VTiiiiiiiiiiiiiiiiiiiiiiiiii

gT 1η

gμTηη

⇓U1

The 3-cell m is given by the pasting diagram below, in which the two largeunmarked regions are equalities by naturality, the monad axioms, the algebraaxioms for Y , or the algebra 1-cell axioms for g.

gηxT x gT xηT T xgT xηT T x gμηT T xε11 �� gμηT T x gμTηT xgμTηT x yT gTηT xyT gTηT x yT gT 2TηTyT gT 2TηT

yT gTμTηT

1T ε1

��yT gTμTηT

yT gTμT 2ηyT gTμT 2η

yT yT 2gT 2ηyT yT 2gT 2η

yμT 2gT 2η

gηxT x

gηxμ

11m

��gηxμ

gT xηTμgT xηTμ

gμηTμ

ε11

��gμηTμ gμTημgμTημ yT gTημyT gTημ yμT 2gT 2η

gT xηT T x

gT xT 2xηT 2gT xT 2xηT 2 gμT 2xηT 2ε11 �� gμT 2xηT 2 gT xμT ηT 2gT xμT ηT 2

gμμT ηT 2

ε11

��

gT xT 2xηT 2

gT xTμηT 2

1T m1

��gT xTμηT 2 gμTμηT 2

ε11�� gμTμηT 2 gμμT ηT 2gμμT ηT 2

yT gμT ηT 2yT gμT ηT 2

yT gTημ

⇓M1

The three axioms are then consequences of the axioms in the first part of theuniversal property of the lax codescent object.

Now let α : g1 ⇒ g2 be a 2-cell in KT with 0-cell source L X and 0-celltarget Y which is determined by the following data:

• 2-cell α : g1πX ⇒ g2πX and• invertible 3-cell with 2-cell source (α ⊗ 1μ) ∗ (1g1 ⊗ ε) and 2-cell target(1g2 ⊗ ε) ∗ (α ⊗ 1T x )

Define the T -transformation Sα as follows. The underlying 2-cell has as itssource the 1-cell S(g1πX ) = g1πXη and target S(g2πX ) = g2πXη, so wedefine Sα to be α⊗ 1η. The 3-cell datum, which we now write as S A, is givenby the pasting diagram below.


g1πXηx g1πX T xηTg1πX T xηT g1πXμηT1ε1 �� g1πXμηT g1πXμTηg1πXμTη yT g1TπX TηyT g1TπX Tη

yT g2πXη

1T α1

��

g1πXηx

g2πXηx

α11

��g2πXηx g2πX T xηTg2πX T xηT g2πXμηT

1ε1�� g2πXμηT g2πXμTηg2πXμTη yT g2πXη

g1πX T xηT

g2πX T xηT

α11

��

g1πXμηT

g2πXμηT

α11

��

g1πXμTη

g2πXμTη

α11

��= ⇓1 = =

There are now two axioms to check, and these both follow from the axioms inthe second part of the universal property for the lax codescent object.

Finally, let : α1 � α2 be a 3-cell in KT with 0-cell source L X and 0-celltarget Y , and assume that this is the 3-cell uniquely determined by the 3-cellpictured below.

T X Y

g1πX

BAT X Y

g2πX

a α11

��α21

��

��

In order to define a T -modification S , we must give the 3-cell in the diagrambelow.

X iY

g1πXη

BAX iY

g2πXη

a α111

��α211

��

��

Define this 3-cell to be ⊗ 1η. The single axiom for a T -modification isthen just an instance of naturality of the Gray-category structure isomorphism,using the fact that the 3-cell in the definition of S A is actually one suchstructure isomorphism.

Thusfar we have defined maps of underlying 2-globular sets (it should beclear that these functions respect source and target from the definitions)

R : Lax-T-Alg(X, iY ) → KT (L X,Y ),

S : KT (L X,Y ) → Lax-T-Alg(X, iY ).

Now we must show that these are 2-functors which are inverse to each other.Checking functoriality is trivial, and we leave this to the reader. Checkingthat these two functors are mutual inverses is a simple matter of repeated


applications of naturality and the monad and algebra axioms, which we alsoleave to the reader. For example, to show that S R = 1 on 0-cells, we com-pute that RS( f ) for f : X → iY , and we see that its underlying 1-cell isyT f η = yη f = f since Y is a strict algebra; the rest follows in a similarfashion.

We have constructed an isomorphism of 2-categories

Lax-T-Alg(X, iY ) ∼= KT (L X,Y ),

and to complete the proof all that remains is to prove Gray-naturality ineach variable. We begin by examining S. Naturality in the variable Y is thecommutativity of the following square.

KT (Y,Y ′)⊗ KT (L X,Y ) Lax-T-Alg(iY, iY ′)⊗ Lax-T-Alg(X, iY )i⊗S�� Lax-T-Alg(iY, iY ′)⊗ Lax-T-Alg(X, iY )

Lax-T-Alg(X, iY ′)��

KT (Y,Y ′)⊗ KT (L X,Y )

KT (L X,Y ′)��

KT (L X,Y ′) Lax-T-Alg(X, iY ′)S

��

On 0-cells, this means checking that k ⊗ S f = S(k ⊗ f ) for k : Y → Y ′ inKT and f : L X → Y . Using the universal property of L X , assume that f isdetermined by the data ( f , ε,M,U ). Then k ⊗ f is determined by the data(k ⊗ f , 1k ⊗ ε, 1 ⊗ M, 1 ⊗ U ). Using this, it is easy to check that k ⊗ S f =S(k ⊗ f ) using the definition of S.

Since the Gray-tensor product of a pair of 2-categories has two differentkinds of generating 1-cells, we need to check two different equations for nat-urality on 1-cells. The first equation to check is that S(α ⊗ 1 f ) = α ⊗ 1S f ,where α : k ⇒ k′ in KT and f : L X → Y as before. In this case, the 2-cellα ⊗ 1 f in KT is determined uniquely by the data α ⊗ 1 : k f ⇒ k′ f togetherwith the Gray-category structure isomorphism �α1,ε. Then both S(α ⊗ 1 f )

and α ⊗ 1S f have underlying 2-cell α ⊗ 1 f η, and both have 3-cell data givenby the Gray-category structure isomorphism, so these T -transformations areequal. Similarly, we must also check that S(1k ⊗ β) = 1k ⊗ Sβ. This followsfrom similar reasoning, using the associativity of the Gray-tensor product.

There are three kinds of generating 2-cells in the Gray-tensor product of apair of 2-categories, two similar to the different kinds of 1-cells and then thestructure isomorphisms�β,α . The proofs that S(⊗1 f ) = ⊗1S f and S(1k ⊗ ) = 1k ⊗ S are simpler versions than the arguments in the 1-cell cases, sowe will not repeat them. For the structure isomorphism �β,α , its image alongthe top and right of the square is the T -modification �iβ,Sα . Tracing the imagealong the left and bottom we get S�β,α = �β,α ⊗ 1η by definition. It is easyto show, using the universal property, that �β,α is �iβ,α . Since Sα = α ⊗ 1η,


these 2-cells are equal by the Gray-category axioms, completing the proofof naturality in S in the variable Y . The standard argument that the inverseof a natural transformation is also natural shows that R is Gray-natural in Yas well.

In order to prove Gray-naturality in X for both R and S, once again it suf-fices to prove it for one of them. Proving this for R, for example, requireschecking that the following square commutes.

Lax-T-Alg(X,iY )⊗Lax-T-Alg(X ′,X) KT (L X,Y )⊗KT (L X ′,L X)R⊗L �� KT (L X,Y )⊗KT (L X ′,L X)

KT (L X ′,Y )��

Lax-T-Alg(X,iY )⊗Lax-T-Alg(X ′,X)

Lax-T-Alg(X ′,iY )��

Lax-T-Alg(X ′,iY ) KT (L X ′,Y )R

��

One the level of 0-cells, this requires checking that R( f ⊗k) = R f ⊗Lk for f :X → iY, k : X ′ → X lax T -functors. The arguments are analogous to thoseused for naturality in Y in which one checks the conditions of the universalproperty to conclude that these cells are equal, and we omit the details.

Corollary 14.9 Assume that i : KT ↪→ Lax-T-Alg has a left Gray-adjoint.Then KT has lax codescent objects for all lax codescent diagrams of theform X.

Proof The previous proof demonstrated that lax T -functors f : X → iY arein bijection with the data in the first part of the universal property of the laxcodescent object of X . Now assuming that i has a left Gray-adjoint L , considerthe unit of this adjunction. This is a lax T -functor X → i L X , which thencorresponds under this bijection to a 1-cell π : T X → L X in KT , togetherwith a 2-cell ε and a pair of 3-cells M,U . The computations in the previousproof then show that this equips L X with the structure of the lax codescentobject for X .

14.3 The left adjoint, pseudo case

We now turn from lax algebras to pseudo-algebras. The proofs required in thissection are similar to the ones in the previous section, and thus we omit thedetails. The goal is the same, to construct a left Gray-adjoint to the forgetfulfunctor

i : KT ↪→ Ps-T-Alg.

14.3 The left adjoint, pseudo case 243

This will involve constructing codescent objects of codescent diagrams. Thuswe identify two different assumptions about the Gray-categories K and KT .

C1(ps) The Gray-category KT has codescent objects for codescent diagrams.C2(ps) The Gray-category K has codescent objects for codescent diagrams,

and T preserves them.

As before, assumption C2(ps) implies C1(ps). From the previous list of axiomsconcerning the existence of lax codescent objects, we also have assumptionsC3 and FC, and these both imply C1(ps). As before, we state the main the-orem using C1(ps), but examples are often better approached using anotherhypothesis.

Theorem 14.10 Let X = (X, x,m, i, π, λ, ρ) be a pseudo-T -algebra, andlet X ps be its associated codescent diagram. Under assumption C1(ps), theassignment which sends X to the codescent object of X ps is the object part ofa Gray-functor L : Ps-T-Alg → KT . The functor L is the left Gray-adjoint tothe inclusion i : KT ↪→ Ps-T-Alg.

We also have the same partial converse as in the lax case.

Proposition 14.11 Assume that i : KT ↪→ Ps-T-Alg has a left Gray-adjoint.Then KT has codescent objects for all codescent diagrams of the form X ps .

15

A general coherence result

This final chapter considers one last aspect of a general coherence theorem foralgebras over Gray-monads. We have established conditions under which aleft adjoint to the inclusion of strict algebras into either lax or pseudo-algebrasexists, so now we turn to determining when pseudo-algebras can be strictifiedto strict algebras. The coherence theorem in the previous chapter was aimedat getting a classifying object for weak maps, while the coherence theorem inthis chapter is aimed at strictifying weak algebras into strict ones. While theproblem of strictifying certain kinds of structures on categories (monoidal orsymmetric monoidal, for example) is what is most commonly referred to ascoherence, both it and the question of classifying maps are crucially importantfor the overall theory.

The results in the previous chapter all centered aroung the idea of (possiblylax) codescent objects. When such objects exists, at least for a limited class ofcodescent diagrams, in KT , we could produce a left adjoint to the inclusion ofstrict algebras into weak or lax algebras. On the other hand, given a pseudo-T -algebra X , it appears that X itself might be a kind of codescent object forthe canonical diagram X ps . It is not a codescent object in KT since X is notan object of KT , but one might ask if it is a codescent object in K of the samediagram. Studying this question requires examining one final kind of colimit,the tricategorical version of the codescent object.

This chapter consists of two sections. In the first, we introduce the Gray-category of codescent diagrams in K, and the concept of weak codescentobjects. Here we prove two important results. The first is that every codescentobject is also a weak codescent object, and the second is that every pseudo-algebra X is the weak codescent object of the diagram X ps . In the secondsection, we prove the main result, characterizing precisely those objects X forwhich the unit 1-cell ηX : X → i L X is an internal biequivalence in Ps-T -Alg.

244

15.1 Weak codescent objects 245

15.1 Weak codescent objects

Definition 15.1 Let K,K′ be Gray-categories, and F,G : K → K′Gray-functors between them. A 1-strict transformation α : F ⇒ G is a tri-transformation such that all of the components of the modifications �,M areidentity 3-cells.

Definition 15.2 Let K be a Gray-category. The Gray-category Codsc K isdefined to be the sub-Gray-category of the functor tricategory Tricat( G

ps,K)

with

• 0-cells the Gray-functors X : Gps → K,

• 1-cell f : X → Y the 1-strict transformations,• 2-cells α : f ⇒ g the trimodifications, and• 3-cell : α � β the perturbations.

Note that Codsc K is neither the full functor tricategory, since its objects and1-cells are somewhat strict, nor the Gray-functor category [ G

ps,K], since itshigher cells are somewhat weak. We do have the following proposition though.

Proposition 15.3 Let K be a Gray-category, and X,Y codescent diagramsin K. Then the inclusion

Codsc K(X,Y ) ↪→ Tricat( Gps,K)(X,Y )

is a 2-equivalence.

In order to prove this, we will need to use the particular structure of Gps .

Lemma 15.4 Let A be a Gray-category which is freely generated by a set of0-cells, 1-cells between those, 2-cells between composites of those, and 3-cellsbetween composites of those, perhaps with relations between composites of3-cells. If K is any Gray-category and F,G : A → K are Gray-functors, thento give a 1-strict transformation α : F ⇒ G it suffices to give the data

• for each object a ∈ A, a 1-cell αa : Fa → Ga in K;• for each generating 1-cell f : a → b in A, a 2-cell αb ⊗ F f ⇒ G f ⊗ αa

in K;• for each generating 2-cell θ : f ⇒ g in A with source f = fn⊗ fn−1⊗· · ·⊗ f1

and target g = gm ⊗ · · · ⊗ g1, a 3-cell

αb ⊗ Ff Gf ⊗ αaαf �� Gf ⊗ αa

Gg ⊗ αa

Gθ⊗1

��

αb ⊗ Ff

αb ⊗ Fg

1⊗Fθ

��αb ⊗ Fg Gg ⊗ αaαg

��

⇓ αθ

246 A general coherence result

where we define the component of α

– at an identity 1-cell to be the identity, and– at a composite inductively by the requirement that αg⊗ f is the composite

below:

αc FgF fαg1−→ Ggαb F f

1α f−→ GgG f αa

subject to the requirement that the 3-cells αθ are natural with respect to thegenerating 3-cells of A. If a pair of generating 3-cells (, ) is required to be apair of inverse isomorphisms, then it suffices to check naturality with respect to only (resp., only). Furthermore, if a quadruple (γ, δ, , ) consisting of apair of generating 2-cells γ, δ and a pair of generating 3-cells , exhibitingan adjoint equivalence γ �eq δ, then it suffices to define αγ only (resp., αδonly).

Proof Since A is freely generated as a Gray-category at each dimension(with relations only at the level of 3-cells) and all of the components of �,Mare the identity, the conditions above clearly are equivalent to giving a 1-stricttransformation α : F ⇒ G. To see the last claim, note that the naturalitycondition forces the definition of αδ once αγ is determined, and vice versa.

Proof of Proposition 15.3 Since these 2-categories have the same hom-categories by definition, we need only show that every tritransformation α :X → Y is isomorphic to a 1-strict one. To begin, we will construct a 1-stricttransformation α, and afterwards give the isomorphism α ∼= α.

Define the components at objects [i] for α to be the same as those for α:

α[i] = α[i].

For a generating 1-cell f : [i] → [ j], define the adjoint equivalenceα f �eq α

�f to be the same as the one for α once again. By Lemma 15.4, this

data, together with the definition that α I is the identity adjoint equivalence,determines all of the adjoint equivalences α f for any arbitrary 1-cell f .

For a generating 2-cell θ : f ⇒ g, we define αθ as the following composite.First, instances of � (more explanation below) for the transformation α areused to change αg into αg . Then αθ is used. Finally, instances of �−1 changeα f into α f . Thus αθ is the composite below, where we have written multipleinstanes of � or �−1 as a single cell.

αg ∗(

1 ⊗ X (θ))

�∗1−→ αg ∗(

1 ⊗ X (θ))

αθ−→(

Y (θ)⊗ 1)

∗ α f

1∗�−1−→(

Y (θ)⊗ 1)

∗ α f .


To make this precise, it remains to give a rigorous definition of the phrase“instances of �” used above. When g (or f , resp., for the �−1 case) is theidentity, then we define this to be the cell M−1 for the transformation α. Wheng is itself a generating 1-cell, we define this to be the identity. When g = g1g2

with each gi a non-identity generating 1-cell, we define this to be�g1,g2 for thetransformation α. Finally, when g = g1 · · · gn , then using the transformationaxioms for α there is a unique 3-cell given by composites of whiskerings of�’s from αg to αg . Since every 1-cell in G

ps has a well-defined length, this3-cell is uniquely determined by the recipe above, and hence defines αθ for θa generating 2-cell; Lemma 15.4 then gives αθ for an arbitrary 2-cell.

The only thing to check is that αθ is natural in θ . This follows from atedious yet straightforward diagram chase using the transformation axioms forα together with some basic naturality arguments. This finishes the constructionof the 1-strict transformation α.

To define an isomorphism p : α ∼= α, we let the component of p at eachobject be the identity. For each 1-cell f of G

ps , we must give an isomorphismp f : α f ⇒ α f satisfying two axioms. When f is an identity 1-cell, we definep f to be M for the transformation α. When f is a generating 1-cell, p f isthe identity. Finally, when f is a composite of generators, we define it usinginstances of �−1 in precisely the same way that we used instances of � inthe definition of αθ above. The first modification axiom is an associativityaxiom, and it follows directly from the first transformation axiom for α. Thesecond modification axiom is a unit axiom, and in this case trivially states thatM = M . Since all of the components on objects are isomorphisms, p is anisomorphism, completing the proof.

Notation. Let x ∈ K. We write 〈x〉 for the object of Codsc K which is givenby the constant codescent diagram which takes the value x on every object of G

ps and identities on all higher cells.

Lemma 15.5 Let K be a Gray-category. The assignment x → 〈x〉 is theobject-function of a Gray-functor K → Codsc K.

Proof To define 〈 f 〉 for a 1-cell f : x → y is the transformation with allcomponents on objects 〈 f 〉[i] = f while all higher cells in the definition beingthe identity. The same works for 2- and 3-cells, and it is easy to see that thisgives an inclusion of Gray-categories.

Definition 15.6 Let X ∈ Codsc K. The weak codescent object of X consistsof an object C together with a 1-cell

p : X → 〈C〉


in Codsc K such that the functor

K(C, A)〈−〉→ Codsc K

(〈C〉, 〈A〉

)p∗

−→ Codsc K(X, 〈A〉)is a Gray-natural biequivalence in Gray for all objects A.

Remark From this definition and the tricategorical Yoneda lemma, the weakcodescent object of a codescent diagram is unique up to internal biequivalencein K.

Proposition 15.7 Let X : Gps → K be a codescent diagram, and let its

codescent object be given by (X0, x, ε,M,U ). There there is a 1-cell κ : X →〈X0〉 in Codsc K whose component at the object [i] is the composite

Xid0−→ Xi−1

d0−→ · · · X1x−→ X0.

Proof We will use Lemma 15.4 to define κ , so we must define its componentsonly on generating 1- and 2-cells and check naturality against the generating 3-cells. All of the components on generating 1-cells are the identity except thoselisted below.

• The component κd1 for d1 : [2] → [1] is ε : xd1 ⇒ xd0.• The component κd1 for d1 : [3] → [2] is

xd0d11A01−→ xd0d0.

• The component κd2 for d2 : [3] → [2] is

xd0d21A02−→ xd1d0

ε1−→ xd0d0.


xd0d0d111A01−→ xd0d0d0.


xd0d0d211A02−→ xd0d1d0

1A011−→ xd0d0d0.


xd0d0d311A03−→ xd0d2d0

1A021−→ xd1d0d0ε11−→ xd0d0d0.

• The component κs0 for s0 : [1] → [2] is

xd0s01L0−→ x .


xd0d0s011L0−→ xd0.



xd0d0s111N ds−→ xd0s0d0

1L01−→ xd0.

We now define the components of κ on the generating 2-cells. Once again,we only list those components which are not the identity.

• The 3-cell κA12 , where A12 is the 2-cell below,

[3] [2]d2 �� [2]

[1]d1��

[3]

[2]d1

��[2] [1]

d1

��

⇓ A12

is defined to be M−1.


[4] [3]d2 �� [3]

[2]d1��

[4]

[3]d1

��[3] [2]

d1

��

⇓ A12

is defined to be 1x ⊗ π−1012.


[4] [3]d3 �� [3]

[2]d1��

[4]

[3]d1

��[3] [2]

d2

��

⇓ A13

is defined to be the 3-cell pasting below.

xd0d1d3

xd0d2d1

11A13

[Ybbbbbbbb

xd0d2d1 xd1d0d11A021 �� xd1d0d1

xd1d0d0

11A01

MMWWWWWWWW

xd0d1d3

xd0d0d3

1A011MMWWWWWWWW


�� xd0d2d0

xd1d0d0

1A021

[Ybbbbbbbb

1⇓π−1013

xd1d0d1 xd0d0d1ε11 �� xd0d0d1

xd0d0d0

11A01

MMWWWWWWWW

xd1d0d0 xd0d0d0ε11

��

∼=



[4] [3]d3 �� [3]

[2]d2��

[4]

[3]d2

��[3] [2]

d2

��

⇓ A23


xd0d2d3 xd0d2d211A23 �� xd0d2d2 xd1d0d2

1A021 �� xd1d0d2 xd0d0d2ε11 �� xd0d0d2

xd0d1d0

11A02

��xd0d1d0

xd0d0d0

1A011

��

xd0d2d3

xd1d0d3

1A021

��xd1d0d3

xd0d0d3

ε11


11A03

�� xd0d2d0 xd1d0d01A021

�� xd1d0d0 xd0d0d0ε11

��


�� xd1d2d0 xd1d1d01A121

�� xd1d1d0 xd0d1d0ε11

��

xd1d0d2

xd1d1d0

11A02

��xd1d2d0

xd0d2d0

ε11

��

⇓1π−1023

∼=

∼= ⇓M−11

• The 3-cell κL1 , where L1 is the 2-cell below,

[2]

[3]s1

��

[3]

[2]

d1

��

[2] [2]1

$#

⇓ L1

is defined to be 1x ⊗ (νl0)

−1.

• The 3-cell κR0 , where R0 is the 2-cell below,

[1]

[2]s0

��

[2]

[1]

d1

��

[2] [2]1

53

⇓ R0

is defined to be U .


• The 3-cell κR0 , where R0 is the 2-cell below,

[2]

[3]s0

��

[3]

[2]

d1

��

[3] [3]1

53

⇓ R0

is defined to be 1x ⊗ μ0.• The 3-cell κR1 , where R1 is the 2-cell below,

[2]

[3]s1

��

[3]

[2]

d2

��

[3] [3]1

53

⇓ R1


xd0 xd0d2s111R1 �� xd0d2s1 xd1d0s1

1A021 �� xd1d0s1 xd0d0s1ε11 �� xd0d0s1

xd0s0d0

11N ds

��xd0s0d0

xd0

1L01

��

xd1d0s1

xd1s0d0

11N ds

��xd1s0d0 xd0s0d0

ε11��

xd0

xd1s0d0

1R01��xd0

xd0

1

��

∼=⇓1νr

0

⇓U1

• The 3-cell κN sd is defined to be the 3-cell pasting below.

xd0s0d1 xd0d2s011N sd

�� xd0d2s0 xd1d0s01A021 �� xd1d0s0 xd0d0s0

ε11 �� xd0d0s0

xd0

11L0

��

xd1d0s0

xd1

11L0

��xd1 xd0ε

��

xd0s0d1

xd1

1L01��

∼=⇓1νl

1

• The 3-cell κN s is defined to be the 3-cell pasting below.

xd0d0s0s0 xd0d0s1s0111N s

�� xd0d0s1s0 xd0s0d0s011N ds 1 �� xd0s0d0s0 xd0s0

1L011 �� xd0s0

x

1L0

��

xd0s0d0s0

xd0s0

111L0

��xd0s0 x

1L0

��

xd0d0s0s0

xd0s0

11L01��

∼=⇓1νs


For the final step of the proof, we must check naturality axioms for thegenerating 3-cells, of which there are eleven, not counting inverses.

• Naturality with respect to each of π012, π013 is immediate as both axiomsconsist of showing that some diagram of 3-cells is the identity, and writ-ing these diagrams out shows that they consist only of pairs of inverseisomorphisms which cancel.

• Naturality with respect to π023 is similar to the above cases, where pairs ofinverse isomorphisms cancel out to leave exactly the same 3-cells on eachside of the naturality equation.

• Naturality with respect to π123 follows from the first codescent object axiomafter some simple rearrangements of cells using inverses.

• Naturality with respect to μ0 is immediate as both sides of the naturalityequation are identical.

• Naturality with respect to μ1 follows from the third codescent object axiom.

• Naturality with respect to each of νli is immediate as, after some cancellation

of mutual inverses, both sides of the naturality equations are identical; thesame also holds for the naturality axioms for νr

0 and νs .

• Naturality with respect to νr1 follows from the second codescent object

axiom.

Given a codescent diagram X in K, we can form its codescent object as aweighted colimit, or its weak codescent object as a higher dimensional, conicalcolimit. The next theorem relates these two concepts.

Theorem 15.8 Let K be a Gray-category, X a codescent diagram in K, andlet X0 be a codescent object for X. Then X0 is also a weak codescent objectfor X.

Proof In order to show that X0 is a weak codescent object for X , we mustconstruct a map of codescent diagrams X → 〈X0〉 and show that it satisfies auniversal property. This map of codescent diagrams is the map κ constructedin the previous proposition, and we must now show that it induces a Gray-biequivalence in the diagram below.

K(X0, A)〈−〉→ Codsc K

(〈X0〉, 〈A〉

)κ∗−→ Codsc K(X, 〈A〉).

To do so, we begin by constructing a 2-functor Q : Codsc K(X, 〈A〉) →K(X0, A).


Let g : X → 〈A〉 be a map of codescent diagrams. This consists of

• for 1 ≤ i ≤ 4, a 1-cell gi : Xi → A in K,• adjoint equivalences

gd j : gi−1d j �eq gi : g�d j

gs j : gi s j �eq gi+1 : g�s j

whenever d j : [i] → [i − 1] or s j : [i] → [i + 1] is defined, and• invertible 3-cells gα for each 2-cell α : h ⇒ k of the form

gt ⊗ X (h) gsgh �� gs

gs

gt ⊗ X (h)

gt ⊗ X (k)

1⊗X (α)

��gt ⊗ X (k) gsgk

��

⇑ gα

where h, k : s → t ,• all subject to the condition that the gα are natural in α.

In order to construct Qg : X0 → A, we use the universal property of thecodescent object. Such a 1-cell is uniquely determined by a 1-cell a : X1 → A,an adjoint equivalence ε : ad1 ⇒ ad0, and a pair of invertible 3-cells M,U ,satisfying certain axioms. We define these cells as follows.

• The 1-cell a is g1 : X1 → A.• The adjoint equivalence ε is the composite of the adjoint equivalence gd1

and the opposite of the adjoint equivalence gd0 :

g1d1gd1−→ g2

g�d0−→ g1d0.

• The invertible 3-cell M is given by the pasting below in which each cell isactually a mate, or an inverse of a mate, of the one listed.

g1d1d2

g2d2

gd1 1��

g2d2

g1d0d2

g�d0

1��

g1d0d2 g1d1d01A02 �� g1d1d0

g2d0

gd1 1

��

g2d0

g1d0d0

g�d0

1

��

g1d1d2

g1d1d1

1A12��

g1d1d1 g2d1gd1 1

�� g2d1 g1d0d1g�

d01

�� g1d0d1

g1d0d0

1A01

��

g2d2

g3

gd2

!��

g3

g2d0

g�d0

�"��g3

g2d1

g�d1

��

⇓gA12 ⇓gA01

⇓gA02


• The invertible 3-cell U is given by the pasting below, where the cell markedgL0 is actually the inverse of its mate.

g1 g1d1s01R0 �� g1d1s0 g2s0

gd1 1�� g2s0 g1d0s0

g�d0

1�� g1d0s0

g1

1L0

��

g1

g1GGGGGGGGGGGGGGGGGGGGGG

GGGGGGGGGGGGGGGGGGGGGG

g1 g1

g2s0

g1

gs0

��

⇓gR0 ⇓gL0

In order to show that this collection of data determines a 1-cell Qg, we arerequired to check that the cells above satisfy analogues of the codescent objectaxioms. In each case, these axioms follow from the Gray-category axiomstogether with the naturality axioms for the cells gAi j , gLi , gRi , and we omit theproofs here.

Now, let g, h : X → 〈A〉 be maps of codescent diagrams, and let α : g ⇒h be a 2-cell between them in Codsc K. We will define Qα : Qg ⇒ Qhusing the second part of the universal property of the codescent object. Thus toconstruct Qα, we must give

• a 2-cell α : Qg ⊗ x ⇒ Qh ⊗ x together with

• an invertible 3-cell : (α ⊗ 1d0) ∗ (1Qg ⊗ ε) � (1Qh ⊗ ε) ∗ (α ⊗ 1d1)

satisfying two axioms. Define the 2-cell to be the component α[1], and define to be the pasting below.

g1d1 g2gd1 �� g2 g1d0

g�d0 �� g1d0

h1d0

α11

��

g1d1

h1d1

α11

��h1d1 h2hd1

�� h2 h1d0h�

d0

��

g2

h2

α2

��⇓ αd1

⇓ αd�0

There are two axioms to check in order to satisfy the second part of theuniversal property of the codescent object and thus induce a unique 2-cell Qα :Qg ⇒ Qh. These two axioms can be easily proved using the fact that thecomponents of α are the data for a modification. A consequence of this factis the following equation, where the cells marked gA12, h A12 are inverses ofmates of those cells.


g1d1d2

g2d2

gd1 1

b_NNNNNNN

g2d2 g3gd2 �� g3

g2d1

g�d1

S5fff

ffff

g2d1

h2d1

α2186QQQQ

g1d1d2

h1d1d2

α111 +),,,

,

h1d1d2 h1d1d11A12

�� h1d1d1 h2d1hd1 1

��

g1d1d2 g1d1d11A12 �� g1d1d1 g2d1

gd1 1��g1d1d1

h1d1d1

α111��

∼= ⇓αd1

⇓gA12

g1d1d2

g2d2

gd1 1

b_NNNNNNN

g2d2 g3gd2 �� g3

g2d1

g�d1

S5fff

ffff

g2d1

h2d1

α2186QQQQ

g1d1d2

h1d1d2

α111 +),,,

,

h1d1d2 h1d1d11A12

�� h1d1d1 h2d1hd1 1

��h1d1d2

h2d2

hd1 1

&$XXXXXX

h2d2 h3hd2

�� h3

h2d1

h�d1 A@BBBBBB

g2d2

h2d2

α21

MMllllllg3

h3

α3

c`aaaaaa

=

⇓αd2

⇓αd1⇓αd�

1

⇓h A12

Applying this type of axiom three times (once each for A01, A02, A12) yieldsthe first axiom for the universal property of the codescent object, and applyingthis type of axiom twice (once each for L0, R0) yields the second axiom forthe universal property of the codescent object. Therefore by the second part ofthe universal property, we get a unique 2-cell Qα : Qg ⇒ Qh.

Now let g, h : X → 〈A〉 be maps of codescent diagrams, α, β : g ⇒ h be2-cells between them, and let : α � β be a 3-cell between those in Codsc K.We will define Q : Qα � Qβ using the third part of the universal propertyof the codescent object. To do so requires that we produce a single 3-cell

Qg ⊗ x Qh ⊗ x

Qα1��

Qg ⊗ x Qh ⊗ x

Qβ1

��⇓

satisfying a single axiom. We define this cell to be the component [1], andthe single axiom follows immediately from two applications of the perturbationaxiom for .

So far, we have constructed a map of 2-globular sets (it is immediate that thedefinitions above respect source and target)

Q : Codsc K(X, 〈A〉) → K(X0, A).

Now we must show that Q is a 2-functor. Once again, we use the universalproperty of the codescent object to show that Q preserves units and com-position. In each case, these calculations are immediate consequences of theGray-category structure together with the composition laws for trimodifica-tions and perturbations. As an example, to show that Qα ∗ Qβ = Q(α ∗ β),we must check that both of these sides are equal when − ⊗ 1x is applied tothem and that a certain pasting diagram of 3-cells is one of the Gray-categorystructure isomorphisms. The first of these is the calculation below.

(Qα ∗ Qβ)⊗ 1x = (Qα ⊗ 1x ) ∗ (Qβ ⊗ 1x )

= α1 ∗ β1

= Q(α ∗ β)⊗ 1x .


The second of these calculations is a direct consequence of the Gray-categoryaxioms. The rest of the proof follows in a similar manner, and we leave it tothe reader.

One should also note at this point that the 2-functor Q is Gray-natural inthe variable A. Showing this amounts to checking that the following diagramcommutes.

K(A, B)⊗ Codsc K(X, 〈A〉) K(A, B)⊗ K(X0, A)1⊗Q A �� K(A, B)⊗ K(X0, A)

K(X0, B)��

K(A, B)⊗ Codsc K(X, 〈A〉)

Codsc K(〈A〉, 〈B〉)⊗ Codsc K(X, 〈A〉)〈−〉⊗1

��Codsc K(〈A〉, 〈B〉)⊗ Codsc K(X, 〈A〉)

Codsc K(X, 〈B〉)��

Codsc K(X, 〈B〉)K(X0, B)

Q B

��

It is straightforward to check that this commutes on 0-cells. Applying the topcomposite to f ⊗ g produces f ⊗ Qg, while applying the bottom compositeproduces Q(〈 f 〉 ⊗ g); examining the data that determines these cells via theuniversal property shows that both are obtained by applying f ⊗ − to the datadetermining Qg. The case for generating 1- and 2-cells of the form w ⊗ 1 or1 ⊗ w follows from an analogous argument. For the generating 2-cells of theform �α,β , this reduces to showing that

�α,Qβ = Q(�〈α〉,β).

It is easy to check, however, that both of these cells are determined, via thethird part of the universal property of the codescent object, by the cell �α,β1 ,so must be equal.

To complete the proof, we must now show that the composite 2-functors

K(X0, A)〈−〉−→ Codsc K(〈X0〉, 〈A〉) κ∗−→ Codsc K(X, 〈A〉) Q−→ K(X0, A)

Codsc K(X, 〈A〉) Q−→ K(X0, A)〈−〉−→ Codsc K(〈X0〉, 〈A〉)

κ∗−→ Codsc K(X, 〈A〉)are both equivalent to identity functors. We will begin with the first of thesewhich is an endo-2-functor of K(X0, A). On 0-cells, this sends f : X0 → Ato Q(〈 f 〉 ⊗ κ). This 1-cell is determined uniquely by the following data:

• the 1-cell (〈 f 〉 ⊗ κ)1 = f x ,

• the 2-cell 1 f ⊗ ε : f xd1 ⇒ f xd0,

• the 3-cell 1 f ⊗ M , and

• the 3-cell 1 f ⊗ U .


The 1-cell f is clearly the 1-cell determined by this data, so Q(〈 f 〉 ⊗ κ) = f.We now compute the action of this endo-2-functor on 1-cells. Let α :

f ⇒ g be a 1-cell in K(X0, A). We must now compute Q(〈α〉 ⊗ 1κ). Thisis determined by one 2-cell and one invertible 3-cell. The 2-cell is

(〈α〉 ⊗ 1κ)1 = 〈α〉1 ⊗ 1κ1 = α ⊗ 1x ,

and it is easy to check from the definitions that the invertible 3-cell is the Gray-category structure isomorphism. Once again, the 2-cell α is determined by thisdata, so Q(〈α〉 ⊗ 1κ) = α.

Finally, we compute the action of the endo-2-functor on 2-cells. Let :α � β be a 2-cell in K(X0, A). We compute Q(〈〉 ⊗ 1κ) using the universalproperty once again, and it is determined by a single 3-cell. This 3-cell is thecomponent (〈〉 ⊗ 1κ)[1] which is easily seen to be ⊗ 1x . Once again, thisdatum determines , so Q(〈〉 ⊗ 1κ) = . Thus we have shown that thecomposite 2-functor

K(X0, A)〈−〉−→ Codsc K(〈X0〉, 〈A〉) κ∗−→ Codsc K(X, 〈A〉) Q−→ K(X0, A)

equals the identity.We now turn to computing the composite 2-functor

Codsc K(X, 〈A〉) Q−→ K(X0, A)〈−〉−→ Codsc K(〈X0〉, 〈A〉)

κ∗−→ Codsc K(X, 〈A〉).On 0-cells, this sends f : X → 〈A〉 to 〈Q f 〉 ⊗ κ . The components on objectsof this map of codescent diagrams are

• Q f ⊗ x = f1 for the object [1],• f1d0 for the object [2],• f1d0d0 for the object [3], and• f1d0d0d0 for the object [4].The components for 1- and 2-cells are essentially the same as those for κ , butwith f1 replacing x .

The next step is to compute 〈Qα〉 ⊗ 1κ . To do so requires that we givea component for each object, a component for each 1-cell, and then checkvarious axioms. The components on objects are

• α1 for the object [1],• α1 ⊗ 1d0 for the object [2],• α1 ⊗ 1d0d0 for the object [3], and• α1 ⊗ 1d0d0d0 for the object [4].


In each case, the component of 〈Qα〉 ⊗ 1κ at a given 1-cell is the appropriateGray-category structure isomorphism.

Finally, we can compute that 〈Q〉⊗1κ has components which are all of theform 1 ⊗ 1, where 1 here means the identity on i − 1 copies of d0 composedtogether, as in the computations for the action on 0- and 1-cells. Now that wehave computed that action of this 2-functor on cells, it is time to construct apseudo-natural equivalence between it and the identity functor. To do so, wemust give components on objects and 1-cells of Codsc K(X, 〈A〉), and thencheck three axioms.

We will begin by constructing an equivalence σ f : 〈Q f 〉 ⊗ κ → f in the2-category Codsc K(X, 〈A〉). The components of σ f at each object are givenbelow.

σf

1 = 1 f1 : f1 ⇒ f1 σf

2 = fd0 : f1d0 ⇒ f2

σf

3 = fd0d0 : f1d0d0 ⇒ f3 σf

4 = fd0d0d0 : f1d0d0d0 ⇒ f4.

The components of σ f for each generating 1-cell are given below, and weextend to composites and identities by taking the trimodification axioms as thedefinition once the components on generators are given.

• The component at any 1-cell d0 is the identity.

• The component at d1 : [2] → [1] is the isomorphism fd0 f �d0

fd1∼= fd1 from

the counit of the adjoint equivalence fd0 �eq f �d0

.

• The component at d1 : [3] → [2] is f A01 .

• The component at d2 : [3] → [2] is f A02 composed with the counit offd0 �eq f �

d0.

• The component at d1 : [4] → [3] is a Gray-category structure isomorphismcomposed with f A01 with source f2d0d1.

• The component at d2 : [4] → [3] is the composite of f A01 ⊗ 1d0 , a Gray-category structure isomorphism, and f A02 with source f2d0d2.

• The component at d3 : [4] → [3] is the composite of a counit of fd0 �eq f �d0

,f A02 ⊗ 1d0 , a Gray-category structure isomorphism, and f A03 with sourcef2d0d3.

• The component at s0 : [1] → [2] is fL0 .

• The component at s0 : [2] → [3] is a Gray-category structure isomorphismcomposed with fL0 with source f2d0s0.

• The component at s1 : [2] → [3] is the composite of fL0 ⊗ 1d0 , a Gray-category structure isomorphism, and fN ds with source f2d0s1.


These components are required to be natural in the generating 2-cells of Gps ,

and to satisfy the two trimodification axioms. Naturality follows from the nat-urality of the cells f Ai j , fL , fR, fN , and the trimodification axioms hold bydefinition.

The components of σ at a 1-cell α : f ⇒ g are isomorphisms

αi ∗ σ fi

∼= σgi ∗ (〈Qα〉 ⊗ 1κ).

We define these to be the identity when i = [1], and the component of α at di−10

in general. These components are required to assemble to give a perturbation,so there is one axiom to check. There is one equality of pasting diagrams tocheck per generating 1-cell, and in each case this is trivial using the definitionsof Q and κ , so we leave this to the reader.

This completes the definition of all the data required to define a pseudo-natural equivalence σ : Qκ∗〈−〉 ⇒ 1; now we must check the twotransformation axioms. The unit axiom is immediate, as the naturality squaresfor an identity trimodification are identities and the Gray-category structureisomorphism �α,β is an identity when either α = 1 or β = 1. The composi-tion axioms follow similarly, as the components for 1-cells in a composite arethe pasted components of the individual trimodifications. Thus we have con-structed a pseudo-natural transformation σ which is an equivalence since allof its components on objects are equivalences, completing the proof that thecodescent object is also a weak codescent object.

Proposition 15.9 Let X = (X, x,m, i, π, λ, ρ) be a pseudo-T -algebra in K.Then there is a 1-cell ζ : X ps → 〈X〉 in Codsc K whose component at theobject [i] is the composite

T i Xμ−→ T i−1 X

μ−→ · · · T Xx−→ X.

Proof Once again, we use Lemma 15.4 to define ζ , so we must define itscomponents only on generating 1- and 2-cells and check naturality against thegenerating 3-cells. All of the components on generating 1-cells are the identityexcept those listed below.

• The component ζd1 for d1 : [2] → [1] is m : x ⊗ T x ⇒ x ⊗ μ.• The component ζd2 for d2 : [3] → [2] is the composite

x ⊗ μ⊗ T 2xnat= x ⊗ T x ⊗ μT

m1⇒ x ⊗ μ⊗ μT .

• The component ζd3 for d3 : [4] → [3] is the composite

xμμT ⊗ T 3xnat= xT xμTμT 2

m11⇒ xμμTμT 2 .


We now define the components of ζ on the generating 2-cells. Once again,we only list those components which are not the identity.

• The 3-cell ζA12 , where A12 is the 2-cell below,

[3] [2]d2 �� [2]

[1]d1��

[3]

[2]d1

��[2] [1]

d1

��

⇓ A12

is defined to be π−1.

• The 3-cell ζA23 , where A23 is the 2-cell below,

[4] [3]d3 �� [3]

[2]d2��

[4]

[3]d2

��[3] [2]

d2

��

⇓ A23


xμT 2T 3x

xμT 2T 2μ

11T 2m

��

xμT 2T 2μ xT xμT T 2μxT xμT T 2μ xμμT T 2μm11 �� xμμT T 2μ

xμTμμT 2

��

��

xμTμμT 2

xμμTμT 2

xμT 2T 3x

xT xμT T 3xxT xμT T 3x

xμμT T 3x

m11��

xμμT T 3x xμT 2xμT 2xμT 2xμT 2 xT xμTμT 2xT xμTμT 2

xμμTμT 2

m11

��

xT xμT T 3x xT xT 2xμT 2xT xT 2xμT 2

xμT 2xμT 2

m11

��

xT xμT T 2μ

xT xTμμT 2

��

��

xT xTμμT 2 xμTμμT 2m11

��

xT xT 2xμT 2

xT xTμμT 2

1T m1

��

==

=⇓π−11

• The 3-cell ζR0 , where R0 is the 2-cell below,

[1] [1][1]

[2]s0

��

[2]

[1]

d1

��

⇓ R0

is defined to be the 3-cell ρ.


• The 3-cell ζR1 , where R1 is the 2-cell below,

[2] [2][2]

[3]s1

��

[3]

[2]

d2

��

⇓ R1

is defined to be the pasting diagram below.

xμ1 xμT 2xT 2η11T 2i �� xμT 2xT 2η xT xμT T 2ηxT xμT T 2η

xμμT T 2η

m11

��xμμT T 2η

xμ1

xμ1

x1μ��

��

x1μ xT xTημ1T i1

�� xT xTημ

xT xμT T 2η��

��xT xTημ

xμTημ

m11

��xμTημ

x1μ

x1μ

x1μ

"""""""""""""""""""

"""""""""""""""""""

xμTημ

xμ1��

��

x1μ xμ1

=

=

=

⇓ρ1

For the final step of the proof, we must check naturality axioms for thegenerating 3-cells.

• Naturality with respect to each of π012, π013 is immediate as all of the 3-cellsin the diagrams are identities.

• Naturality with respect to π023 is clear, as both of the 3-cells pastings areequal whiskerings of π−1 ⊗ 1.

• Naturality with respect to π123 follows from the associativity axiom for thepseudo-T -algebra structure on X .

• Naturality with respect to μ0 is immediate as all of the 3-cells in thediagrams are identities.

• Naturality with respect to μ1 follows from the right unit axiom for thepseudo-T -algebra structure on X .

• Naturality with respect to each of νli is immediate as all of the 3-cells in the

diagrams are identities.

• Naturality with respect to each of νr0 is clear, as both of the 3-cells pastings

are equal whiskerings of ρ ⊗ 1.

• Naturality with respect to νr1 follows from the fourth pseudo-T -algebra

axiom for X .

• Naturality with respect to νs is immediate as all the cells involved areidentities.


Theorem 15.10 Let X = (X, x,m, i, π, λ, ρ) be a pseudo-T -algebra in K.Then ζ : X ps → 〈X〉 exhibits X as the weak codescent object, in K, of X ps .

Proof To show this, we must construct a 2-functor

S : Codsc K(X ps, 〈A〉) → K(X, A)

and show that it is a pseudo-inverse to the composite 2-functor

K(X, A)〈−〉−→ Codsc K(〈X〉, 〈A〉) ζ ∗

−→ Codsc K(X ps, 〈A〉).Define S to be the following composite.

Codsc K(X ps, 〈A〉) ev1−→ K(T X, A)η∗

−→ K(X, A).

It is easy to compute that the 2-functor

K(X, A)〈−〉−→ Codsc K(〈X〉, 〈A〉) ζ ∗

−→ Codsc K(X ps, 〈A〉) S−→ K(X, A)

is equal to η∗⊗ζ ∗1 = (x ⊗η)∗. Using the adjoint equivalence i : xη �eq 1X , we

see that this composite 2-functor is pseudo-naturally equivalent to the identityon K(X, A).

Now we turn to showing that

Codsc K(X ps ,〈A〉) S−→K(X,A)K(X,A)〈−〉−→Codsc K(〈X〉,〈A〉) ζ

∗−→Codsc K(X ps ,〈A〉)

is pseudo-naturally equivalent to the identity on Codsc K(X ps, 〈A〉). We willproceed as before, first by computing the action of the composite 2-functor oncells and then constructing an explicit transformation whose components onobjects will be equivalences. On 0-cells, this composite sends f : X ps → 〈A〉to 〈 f1η〉 ⊗ ζ , on 1-cells it sends α to 〈α1 ⊗ 1η〉 ⊗ 1ζ , and on 2-cells it sends to 〈1 ⊗ 1η〉 ⊗ 1ζ .

We start by constructing equivalences ω f : 〈 f1η〉 ⊗ ζ ⇒ f which will thenbe the components on objects of our transformation ω. Define the componentω

f1 to be the composite

f1ηx = f1T xηTfd1−→ f2ηT

f �d0−→ f1μηT = f1.

The other components are then defined using ω f1 as shown below.

• ωf2 is f1ηxμ

ωf1 1−→ f1μ

fd0−→ f2.

• ωf3 is f1ηxμμT

ωf1 11−→ f1μμT

fd0d0−→ f3.

• ωf4 is f1ηxμμTμT 2

ωf1 111−→ f1μμTμT 2

fd0d0d0−→ f4.


We must now define the components of ω f on the generating 1-cells, follow-ing the same strategy we employed in the definition of σ f . The componentsare given below.

• ωfd0

= 1 for all d0.

• ωfd1

for d1 : [3] → [2] is shown below.

f1ηxμTμ f1ηxμμTf1ηxμμT

f1μμT

ωf1 11

��f1μμT

f3

fd0d0��

f1ηxμTμ

f1μTμ

ωf1 11

��f1μTμ

f2Tμ

fd0��

f2Tμ f3fd1

��

f1μTμ f1μμT

=

⇓ f A01

• ωfd1


f1ηxμμT TμT f1ηxμμTμT 2f1ηxμμTμT 2

f1μμTμT 2

ωf1 111

��f1μμTμT 2

f4

fd0d0d0��

f1ηxμμT TμT

f1μμT TμT

ωf1 111

��f1μμT TμT

f3TμT

fd0d0��

f3TμT f4fd1

��

f1μμT TμT f1μμTμT 2

=

⇓ f1A01

• ωfd2


f1ηxμμT T 2μ f1ηxμTμμT 2f1ηxμTμμT 2 f1ηxμμTμT 2f1ηxμμTμT 2

f1μμTμT 2

ωf1 111

��f1μμTμT 2

f2μTμT 2

fd0 11��

f1ηxμμT T 2μ

f1μμT T 2μ

ωf1 111

��f1μμT T 2μ

f2μT T 2μ

fd0 11��

f2μT T 2μ

f3T 2μ

fd0 1��

f3T 2μ f4fd2

��

f1ηxμTμμT 2

f1μTμμT 2

ωf1 111

��f1μTμμT 2

f2TμμT 2

fd0 11��f2TμμT 2

f3μT 2

fd1 1 ��f3μT 2

f4

fd0 ��

f2μTμT 2

f3μT 2 fd0 1Ha@@@@@@@@@@@

f1μμT T 2μ f1μTμμT 2f1μTμμT 2 f1μμTμT 2

f2μT T 2μ

f2TμμT 2@@@@@@@@@@ @@@@@@@@@@

= =

=

⇓ f A02

⇓ f A01 1

• The components ω fdi−1

for di−1 : [i] → [i − 1] are the most complicated,and we will not give the full pastings here. In each case, naturality for η iscombined with the cells f Ai j (and their mates and/or inverses).

– For i = 2, this is a pasting using f A12 ⊗ 1 with source f1T xT 2xηT 2 ,f A01 ⊗ 1 with source f1μTμηT 2 , and f A02 ⊗ 1 with source f1μT 2xηT 2 .


– For i = 3, this is a pasting using f A01 ⊗ 1 with source f1μTμηT 2μT ,f A12 ⊗ 1 with source f1T xT 2xηT 2μT , and a counit together with itsinverse for fd0 �eq f ·

d0.

– For i = 4, this is a pasting using f A01 ⊗1 with source f1μTμηT 2μTμT 2 ,f A12 ⊗ 1 with source f1T xT 2xηT 2μTμT 2 , f A02 ⊗ 1 with source givenby f1μT 2xηT 2μTμT 2 , f A02 ⊗ 1 with source f1μT 2xμT 2 , and f A03 withsource f2μT T 3x .

• ωfs0 for s0 : [1] → [2] is fL0 with source f1μTη, pasted with some identity

cells coming from the naturality of η.

• ωfs0 for s0 : [2] → [3] is fL0 with source f2μT TηT , pasted with some

identity cells coming from the naturality of η.

• ωfs1 for s1 : [2] → [3] is fL0 ⊗ 1 with source f1μTημ composed with fN ds ,

pasted with identities coming from the naturality of η and μ.

Just as before, these 3-cells are required to satisfy naturality conditions aswell as the trimodification axioms. The naturality conditions follow from thecorresponding naturality conditions for f , and the trimodification axioms areimmediate, just as in the definition of σ f .

The components of ω at a 1-cell α : f → g are natural isomorphisms

αi ∗ ω fi

∼= ωgi ∗ (〈α1 ⊗ 1η〉 ⊗ 1ζ ).

We define this component when i = 1 to be the pasting below.

f1ηx g1ηxα111 ��f1ηx

f1T xηTf1T xηT

f2ηT

fd1 1��

f2ηT

f1μηT

f �d0

1��

f1μηT

f1f1 g1α1

��

f1T xηT g1T xηTα111

��

f2ηT g2ηTα21

��

f1μηT g1μηTα111

��

g1ηx

g1T xηTg1T xηT

g2ηT

gd1 1��

g2ηT

g1μηT

g�d0

1��

g1μηT

g1

=

⇑ αd1 1

⇑ αd�01

=


f1ηx f1T xηTf1T xηT f2ηTfd1 1

�� f2ηT f1μηT

f �d0

1�� f1μηT f1f1

g1

α1

��

f1ηx

g1ηx

α111

��g1ηx g1T xηTg1T xηT g2ηT

gd1 1�� g2ηT g1μηT

g�d0

1�� g1μηT g1

f1T xηT

g1T xηT

α111

��

f2ηT

g2ηT

α21

��

f1μηT

g1μηT

α111

��

= ⇓ αd11 ⇓ αd�01 =

When i = 2, we define this component to be the pasting below.

f1ηxμ g1ηxμα1111 �� g1ηxμ

g1μ

ωg1 1

��g1μ

g2

gd0��

f1ηxμ

f1μ

ωf1 1

��f1μ

f2

fd0��

f2 g2α2

��

f1μ g1μα11

��

⇓ ωα1 1

⇓ αd0

For i = 3, 4, the same shape pasting as the previous one is used, but withαd0 changed to αd0d0 for i = 3 and αd0d0d0 for i = 4. The naturality of theseisomorphisms is immediate, as they come from the definition of the transfor-mation α. This concludes the construction of all of the data necessary to givea pseudonatural transformation ω : ζ ∗〈−〉S ⇒ 1.

We must now check the two transformation axioms. The unit axiom is triv-ial, as ωid

i is the identity 3-cell, and the rest of the cells in the unit axiomare identities by virtue of the fact that we are dealing with 2-categories and2-functors. For the associativity axiom, we must check that

(ωβi ∗ 1)(1 ∗ ωαi ) = ω

β∗αi

for each choice of i . In each case, it is easy to check that both sides are equalby interchange.

Thus we have constructed a transformation ω : ζ ∗〈−〉S ⇒ 1. In each case,the components of this transformation are equivalences, so ω itself is an equiv-alence. Therefore we have shown that ζ exhibits X as a weak codescent objectof X ps .

Remark. The proof above is essentially a proof that X ps together with Xis a split codescent diagram, together with a demonstration that every splitcodescent diagram has a weak codescent object. In this example, many of thecells are identities, so we chose to work directly rather than develop the gen-eral theory which requires even larger pasting diagrams and more complicatedarguments to check all of the relevant axioms.


15.2 Coherence for pseudo-algebras

Lemma 15.11 The forgetful functor U : Ps-T-Alg → K reflects internalbiequivalences.

Proof Let X,Y be pseudo-T -algebras, and let f : X → Y be a pseudo-T -functor such that U f : U X → UY is an internal biequivalence in K. Wemust show that f is an internal biequivalence in Ps-T-Alg. We will now writeU f as just f , as it will be clear when we are referring to the entire pseudo-T -functor. Recall that the pseudo-T -functor f consists of a 1-cell f , an adjointequivalence F �eq F � with F : f x ⇒ yT f , and two invertible 3-cells h,m .Since f is an internal biequivalence in K, we fix a biadjoint biequivalenceg �bieq f in K using the results of Gurski (2012). This consists of

• an adjoint equivalence α �eq α� with α : 1 ⇒ f g,

• an adjoint equivalence β �eq β� with β : g f ⇒ 1, and

• invertible 3-cells

g g f g1α �� g f g

g

β1

��

g

gGGGGGGGGGGGGGGGGGGGGGG


⇓ �

f f g fα1 �� f g f

f

1β

��

f

fGGGGGGGGGGGGGGGGGGGGGG


⇓ �

satisfying two axioms.We will now equip the 1-cell g with the structure of a pseudo-T -functor. The

2-cell G : gy ⇒ xT g is defined to be

gy11Tα−→ gyT f T g

1F �1−→ g f xT gβ11−→ xT g;

we extend this to an adjoint equivalence in the obvious fashion. The invert-ible 3-cell hg is given by the pasting diagram below, where h is the mate of htogether with naturality for η.

g

gyη

1i

��

gyη gyT f T gη11Tα1 �� gyT f T gη g f xT gη

1F �11 �� g f xT gη

xT gη

β111

" ��

g

gDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDD

1g xηgi1

�� xηg

xT gη��

��

gyη

gyη f g111α53DDDDD

gyη f g

gyT f T gη/// ///

g g f g1α �� g f g

gyη f g

1i1

(&KKKKKg f g

g

β1

NNnnnnnnng f g g f xηg

1i1�� g f xηg

xηg

β111

USPPPPPPPg f xηg

g f xT gη��

��∼=

⇓�

=⇓h

∼=

=

15.2 Coherence for pseudo-algebras 267

The invertible 3-cell mg is given by the pasting diagram below.

gyT y

gyT f T gT y

11Tα1

#!gyT f T gT y

g f xT gT y

1F �11

&$!!!!!!!!!!

g f xT gT y

xT gT yβ111

�$$$$$$$$$

xT gT y xT gT yT 2 f T 2g111T 2α�� xT gT yT 2 f T 2g

xT gT f T xT 2g

11T F �1" ��

xT gT f T xT 2g

xT xT 2g

1Tβ11

A@gggggggggg

xT xT 2g

xμT 2g

m1��

gyT y

gyμ

1m ��

gyμ

gyT f T gμ11Tα1 !��

gyT f T gμ g f xT gμ1F �11

�� g f xT gμ

xT gμ

β111

�"��

xT gμ

xμT 2g��

��

gyT y gyT yT 2 f T 2g111T 2α�� gyT yT 2 f T 2g

gyT f T xT 2g

11T F �1=<ooooo

gyT f T xT 2g

gyT f T gT f T xT 2g

11Tα111=<ooooo

gyT f T gT f T xT 2g gyT f T xT 2g111Tβ11 �� gyT f T xT 2g

g f xT xT 2g

1F �11NNn

nnnn

g f xT xT 2g

g f xμT 2g

11m1NNn

nnnn

g f xμT 2g xμT 2gβ111 ��gyT yT 2 f T 2g

gyμT 2 f T 2g

1m11

��

gyμ gyμT 2 f T 2g111T 2α�� gyμT 2 f T 2g

gyT f T gμ

eeeee

e

g f xT gμ

g f xμT 2g.............

.............

gyT f T xT 2g

gyT f T xT 2g

⇓11T�11

⇓m

!

∼=

=

=

The cell marked “!” is a unique isomorphism from the Gray-category struc-ture, and the cell marked m is a mate of m together with naturality for μ.There are three axioms to check, but they each follow from the correspondingaxiom for f together with the triangle identities and the biadjoint biequivalenceaxioms.

Next, we must show that α, α�, β, β� are all T -transformations. We will giveproofs for α and β as their adjoint pseudo-inverses follow in an analogousfashion. For α, we must give an invertible 3-cell A as shown below.

y yy

yT f T g

1Tα

��

y

f gy

α1

��f gy yT f T g

FG��

⇓ A

We define this to be the pasting below.

y

f gy

α1��

f gy

f gyT f T g

111Tα��

f gyT f T g

f g f xT g

11F �1 ��f g f xT g f xT g

1β11�� f xT g yT f T g

F1��

y

yT f T g

1Tα

" ��

yT f T g

f gyT f T g

α111

97$$$$$$$yT f T g

yT f T gIIIIIIIIIIIIIII

IIIIIIIIIIIIIIIyT f T g

f xT g

F �1��f xT g

f g f xT gα11197$$$$$$$

f xT g

f xT g

nnnn

nnnn

∼=

∼=∼=⇐

�−111


Finally, for β, we must give an invertible 3-cell B fitting in the diagram below.

g f x xT gT fG F �� xT gT f

x

1Tβ

��

g f x

x

β1

��x x

⇓ B

We define B to be the follow pasting.

g f x gyT f1F �� gyT f gyT f T gT f11Tα1 �� gyT f T gT f

g f xT gT f

1F �11��

g f xT gT f

xT gT f

β��

xT gT f

x

1Tβ��

gyT f

gyT f

gyT f T gT f

gyT f

111Tβ97��

g f xT gT f

g f x

111Tβ

FF))))))))))))))))))gyT f

g f x

1F �

��g f x x

β1��

g f x

g f x

lllllllllllllllllllllll

lllllllllllllllllllllll

∼=⇐

11T�

∼=

∼=

In both of these cases, the axioms follow from the triangle identities forthe adjoint equivalence F �eq F � together with the biadjoint biequivalenceaxioms.

Finally, we must show that �,� are invertible T -modifications. This onlyrequires checking one additional axiom as the only data is the 3-cell itself. Inthe case of �, this axiom is immediate by naturality, but the axiom for � doesrequire both of the biadjoint biequivalence axioms. Thus we have shown thatthe pseudo-T -functor f is an internal biequivalence in Ps-T-Alg by exhibitinga pseudo-T -functor g and showing that both f g and g f are equivalent to theidentity in Ps-T-Alg.

Lemma 15.12 Let X0 be the codescent object of the codescent diagram X,let Y be any object of K, and let g1, g2 : X0 → Y be a pair of 1-cells in K.Assume that the pair (α, ), with α : g1x ⇒ g2x, induce a 2-cell α : g1 ⇒ g2.If α is part of an adjoint equivalence α �eq α

�, then α is part of an adjointequivalence α �eq α

�.

Proof First, we must construct the 2-cell α�. To do so, we must give a 2-celltogether with an invertible 3-cell . The 2-cell will be α�. Taking the mate ofthe invertible 3-cell gives an invertible 3-cell

(1g1 ⊗ ε) ∗ (α� ⊗ 1d1) � (α� ⊗ 1d0) ∗ (1g2 ⊗ ε).


Define to be the inverse of this 3-cell. must satisfy two axioms in order touse the second part of the universal property to induce a unique 2-cell α�, butchecking these is an easy exercise in the use of mates.

To show that we have an adjoint equivalence α �eq α�, we must construct

an invertible unit and counit. The counit c : α ∗ α� � 1 will be determinedby a 3-cell with source (α ∗ α�)⊗ 1x and target 1g2x ; we choose this to be thecounit c : α ∗ α� � 1 of the original adjoint equivalence, using that

(α ∗ α�)⊗ 1x = (α ⊗ 1x ) ∗ (α� ⊗ 1x ) = α ∗ α�.

There is now a single axiom to check, but it is immediate by the triangle iden-tities and the fact that ⊗ 1x is the Gray-category structure isomorphism.Constructing the unit, and showing that the unit and counit are invertible,follows the same pattern, completing the proof.

Theorem 15.13 Let T be a Gray-monad on the Gray-category K, andassume that the inclusion i : KT ↪→ Ps-T-Alg has a left Gray-adjoint L.Then the component ηX : X → i L X is an internal biequivalence in Ps-T-Algif and only if the forgetful functor U : KT → K preserves the codescent objectL X of X ps as a weak codescent object.

Proof First, assume that the components ηX : X → i L X are internalbiequivalences in Ps-T-Alg. By Theorem 15.10, we know that ζ : X ps → 〈X〉exhibits X as a weak codescent object, so 〈UηX 〉 ⊗ ζ exhibits L X as a weakcodescent object as well.

Now assume that U preserves the codescent object L X as a weak codescentobject. We have the map ζ : X ps → 〈X〉, so by the universal property of theweak codescent object, there is a 1-cell k : L X → X such that ζ is equivalentto 〈k〉 ⊗ κ in Codsc K(X ps, 〈X〉); fix an adjoint equivalence between these,

δ �eq δ�, δ : 〈k〉 ⊗ κ → ζ.

Evaluating this on components at the object [1], we get an equivalence k⊗π �x . Now the component ηX : X → i L X is the composite

Xη−→ T X

π−→ L X.

Using the equivalence δ1, we get a composite equivalence

k ⊗ π ⊗ ηδ1−→ x ⊗ η

i �−→ 1,

so k is a left pseudo-inverse for ηX . To complete the proof, we will show thatit is a right pseudo-inverse as well.


In order to construct a 2-cell 1 ⇒ π ⊗ η ⊗ k, we will use the universalproperty of the codescent object L X . Therefore we must construct a 2-cella : π ⇒ πηkπ together with an invertible 3-cell satisfying two axioms. This2-cell is the composite below.

π = πμηTε�1−→ πT xηT = πηx

11δ�1−→ πηkπ.

We must also construct an invertible 3-cell as shown below,

T 2 X

T XT x

��$$$$$$

T X

L X

π

��

T 2 X

T Xμ ��

T X

L Xπ

��$$$$$$T X

L X

πηkπ

db⇓ε

⇓a

T 2 X

T XT x

��$$$$$$

T X

L Xπ ��

T 2 X

T Xμ ��

T X

L X

π

��$$$$$$

L X L Xπηk

��

T X

L X

π

�⇓ε

⇓aA�

and then check it satisfies two axioms. In order to do so, note that we have theinvertible 3-cells δd0, δd1 below.

kπμ kπμ1 �� kπμ

xμ

δ2��

kπμ

xμ

δ11��

xμ xμ1

��

⇓ δd0

kπT x kπμ1ε �� kπμ

xμ

δ2��

kπT x

xT x

δ11��

xT x xμm

��

⇓ δd1

Combining these gives an invertible 3-cell (δ1⊗1μ)∗(1k ⊗ε) ∼= m∗(δ1⊗1T x ),and taking mates gives the invertible 3-cell below.

xT x kπT xδ�11

�� kπT x

kπμ

1ε��

xT x

xμ

m��

xμ kπμδ�11

��

⇓ D

We also have the 3-cell M from the codescent object L X .

πT xT 2x

πμT 2 X

ε⊗1[Ybbbbbbb

πμT 2 X πT xμTπT xμT

πμμT

ε⊗1

MMWWWWWWWW

πT xT 2x

πT xTμ

1⊗T mMMWWWWWWWW

πT xTμ πμTμε⊗1

�� πμTμ

πμμTbbbbbbbb

bbbbbbbb

⇓ M


Taking mates here produces the invertible 3-cell shown below.

πμT 2 X πT xT 2xε�1 �� πT xT 2x πT xTμ

1T m ��πμT 2 X

πT xμT

��

πT xμT πμμTε1

�� πμμT πμTμπμTμ

πT xTμ

ε�1

��⇓ N

Using the 3-cells D, N just constructed, we define the required 3-cell A tobe the pasting below, where each of the regions marked with an equal signcommute by naturality or the monad axioms.

πT x πμηT T xπμηT T x πT xηT T xε�11 �� πT xηT T x πηxT xπηxT x πηkπT x

11δ�11�� πηkπT x

πηkπμ

111ε

��

πμηT T x

πμT 2xηT 2πμT 2xηT 2 πT xT 2xηT 2ε�11

��

πT xηT T x

πT xT 2xηT 2πT xT 2xηT 2

πT xTμηT 21T m1 '%(((

πT xTμηT 2 πηxμπηxμ πηkπμ11δ�11

��

πηxT x

πηxμ

11m

��

πT x

πμ

ε

��πμ πμμT ηT 2

πμT 2xηT 2

πT xμT ηT 2πT xμT ηT 2

πμμT ηT 2

ε11'%((((

πμμT ηT 2 πμTμηT 2πμTμηT 2

πT xTμηT 2

ε�11

��πμ

πμηTμ

��

��

πμηTμ πT xηTμε�11

�� πT xηTμ

πηxμ)))))))))))))))

)))))))))))))))

=

= = ⇑11D−1

⇑N−11

=

Now we must check that A satisfies two axioms, one concerning compati-bility with M and one concerning compatibility with U (we remind the readerthat M,U are the structure 3-cells for the codescent object L X ). Since thepasting diagram defining A is already quite large, we will not give these com-putations in full, but only explain how they can be reproduced. The axiom forM follows from the first codescent object axiom for L X and the naturalityaxiom for D in the form shown below. (Note here that the 3-cell denoted πX isthe 3-cell π from the algebra structure on X , not to be confused with the 1-cellπ : T X → L X .)

xT xT 2x

xμT 2x

m1

RQRRRRRRR

xμT 2x

xT xμT�� xT xμT

xμμT

m1!�DDDD

xμμT

kπμμT

δ�111

+)PPPPPPP

xT xT 2x

kπT xT 2x

δ�111+)P

PPPP

PP

kπT xT 2x

kπT xTμ1T m!�DDDD

kπT xTμ

kπμTμ

1ε1

�"��kπμTμ

kπμμTRRRRRRR

RRRRRRRkπT xT 2x

kπμT 2x

1ε1

RQRRRRRRR

kπμT 2x

kπT xμT$$$$$$$$

kπT xμT

kπμμT1ε1 !�JJJJJ

xμT 2x

kπμT 2x

δ�111+)P

PPPPP

xT xμT

kπT xμT

δ�111

\[ddd

ddd

⇓D−11

= ⇓D−11

⇓M

xT xT 2x

xμT 2x

m1

RQRRRRRRR

xμT 2x

xT xμT�� xT xμT

xμμT

m1!�DDDD

xμμT

kπμμT

δ�111

+)PPPPPPP

xT xT 2x

kπT xT 2x

δ�111+)P

PPPP

PP

kπT xT 2x

kπT xTμ1T m!�DDDD

kπT xTμ

kπμTμ

1ε1

�"��kπμTμ

kπμμTRRRRRRR

RRRRRRRxT xTμ

kπT xTμ

δ�111

\[ddd

dddxT xTμ

xμTμm1 ��$$$$$

xT xT 2x

xT xTμ

1T m!�JJJJJ xμTμ

xμμTRRRRRRR

RRRRRRRxμTμ

kπμTμ

δ�111

+)PPP

PPPP

⇓πX

∼= ⇓D−11

=


The axiom for U follows from the third codescent object axiom and thenaturality axiom for D in the form shown below.

x

xT xTη

1T i

��

xT xTη xμTηm1 �� xμTη

kπμTη

δ�111

��

x kπδ�1

�� kπ kπμTη

xT xTη

kπT xTη

δ�111��

kπ

kπT xTη11T i ��///kπT xTη

kπμTη1ε153DDD∼= =

⇓1U

⇓D−11

x

xT xTη

1T i

��

xT xTη xμTηm1 �� xμTη

kπμTη

δ�111

��

x kπδ�1

�� kπ kπμTηx

xμTηHHHHHHHHHHHHHHHHHH

HHHHHHHHHHHHHHHHHH

⇓ρ=

By the second part of the universal property of the codescent object, we get aunique 2-cell a : 1 ⇒ πηk such that a = a ⊗ 1π and A is the Gray-categorystructure isomorphism for a and ε.

Finally, we must show that a is an equivalence, but this follows imme-diately from Lemma 15.12 and the fact that a is visibly part of an adjointequivalence being the composite of two equivalences. Thus we have shownthat ηX : X → i L X has both left and right pseudo-inverses in K. By Lemma15.11, the forgetful functor U reflects internal biequivalences, so ηX is actuallya biequivalence in Ps-T-Alg.

Corollary 15.14 Assume that K has codescent objects of codescent dia-grams, and that T preserves them. Then the inclusion i : KT ↪→ Ps-T-Alghas a left adjoint L and each component ηX : X → i L X of the unit of thisadjunction is a biequivalence in Ps-T-Alg.

Proof Since U : KT → K preserves the weighted colimits that T preserves,this follows immediately from Theorem 15.13 together with Theorem 15.8.

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Index

algebra for a Gray-monadlax, 213lax functor, 215lax transformation, 217modification, 218pseudo, 218pseudo-functor, 218strict, 213, 224transformation, 218

codescent diagram, 188canonical, 228–230lax, 184–188

codescent object, 195as a weighted colimit, 207, 208lax, 190–195, 207weak, 247

coherencebicategories

for functors, 31strictification, 22the functor st, 27–29, 33via free constructions, 24

T-algebrasexistence of a left adjoint, 236, 242strictifying pseudo-algebras, 269–272

tricategoriesdiagrams of constraints, 160functors, 174strictification, 154, 162the functor Gr, 162, 177via free constructions, 157

cubical functor, 38nudging, 46of three variables, 41of two variables, 39

universal, 44, 54

factorization system, see also orthogonalfactorization system

free 2-category, 24free bicategory, 23free functor

bicategories, 29tricategories, 171

free Gray-category, 103free tricategory, 100functor

lax, 69lax homomorphism, 70locally strict, 70strict, 70trihomomorphism, 66

composition of, 107

graphbicategory-enriched, 97category-enriched, 23category-enriched 2-graph, 97

Gray tensor productas a left adjoint, 46closed symmetric monoidal structure, 48universal property, 44via factorization system, 55via generators and relations, 36

Gray-category, 134as a tricategory, 134of lax T-algebras, 221of pseudo-T-algebras, 225

Gray-monad, 210category of algebras, 210

277

278 Index

icon, 16internal biequivalence, 128

Lax-T-Alg, 221

mate, 17–20multicategory, 43

orthogonal factorization system, 49

perturbation, 77Ps-T-Alg, 225pseudo-icon, 114

equivalence, 115pseudo-T-algebra, 218

strictly biequivalent 2-categories, 136

T-algebralax, 213pseudo, 218strict, 213, 224

T-functorlax, 215pseudo, 218

T-modification, 218T-transformation, 218

lax, 217tricategory, 59

category of tricategories, 99, 109cubical, 129functor tricategory, 139–144Gray-categories as tricategories, 134opposite, 65strict, 63the tricategory st T, 130

triequivalence, 128trimodification, 76

lax, 77tritransformation, 71

composition of, 110lax, 75

weighted colimit, 199co-2-inserter, 200co-3-inserter, 203codescent object, 208coequifier, 205lax codescent object, 207

Yoneda lemmafor bicategories, 22for cubical tricategories, 148

201 Coherence in Three-Dimensional Category Theory

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