SIMPLICIAL SETS

LUKAS VOKRINEK

Introduction

Triangulations of spaces. Simplicial sets originated as geometric objects. Geometres were usingtriangulations to define invariants of surfaces and higher dimensional manifolds (the invariants werethe genus and Betti numbers which later evolved into homology groups). They were used as acombinatorial substitute for geometric objects. One may also use them to algorithmically solvecertain questions of geometry and topology.

As an example, to algorithmically compute anything reasonable about a sphere S2 one has tomake some sort of finite presentation. Its defining equation is not of much use here. It is nottoo hard to see that S2 is homeomorphic to the surface of a 3-dimensional simplex. Say that thesimplex is centred at the origin. Then the homeomorphism ∂∆3 → S2 consists of normalizingeach vector, v 7→ v

|v| . Doing so progressively results in a homeomorphism ∆3 → D3. The moral

of this is that as long as only topological questions are concerned we may replace a ball by a3-dimensional simplex, i.e. a convex hull of 4 points in a general position. In fact any number ofpoints in R3 in general position will yield a convex hull homeomorphic to D3 but the 3-simplexis special in that its vertices are affine independent. Any 4-tuple of affine independent points inarbitrary vector space yield the very same (homeomorphic) convex hull. One may abstract thisand think of the 3-simplex determined by (an arbitrary, abstract) four element set.

Let us return now to the sphere S2 which is homeomorphic to the surface of the 3-simplex.This surface still has a very combinatorial nature. It is a union of the convex hulls of all the threeelement subsets. One may again abstract this and arrive at the notion of an abstract simplicialcomplex K. This is given by a finite set K0 of vertices together with a collection of its subsets(those which contribute an edge/face/higher dimensional thing) satisfying a simple axiom: thecollection is downward closed (a subset of something in the collection is also in the collection; alsoall singleton subsets should be in the collection). A (geometric) simplicial complex |K| is thenobtained by choosing arbitrarily a collection of affine independent points (one for each vertex) andtaking the union of the convex hulls of points corresponding to sets in the collection.

There is a metatheorem here: nice spaces are homeomorphic to simplicial complexes. We haveseen an instance of this for the 2-sphere and it is also true that arbitrary surface or indeed anysmooth manifold is homeomorphic to a simplicial complex. There is a more general statementthat applies almost universally and we will get to it later.

Triangulations of maps. So far everything looks very nice. The problem arises however as soonas we start considering maps between spaces. There are two kinds of problems here leading totwo different considerations. Both of these say that “there is too few simplicial maps” but for twodifferent reasons. Even the most reasonable continuous maps are not “simplicial”, i.e. determinedby combinatorial data on the underlying abstract simplicial complex. We first fix this somewhatand arrive at the notion of a simplicial set.

The constant map ∆1 → ∆0 is not simplicial, at least not in the obvious way1. The problem isthat while the image of both vertices is the only vertex of ∆0 there is no possible image for the edgeof ∆1 because ∆0 has none. Very much the same problem occurs when considering morphisms ofgraphs – by the way a 1-dimensional abstract simplicial complex is simply a graph. We will solve

Date: May 15, 2012.1I do not claim that it is impossible to define simplicial maps in a way that the constant map would be included

and indeed such a definition exists. After introducing parallel simplices it becomes more difficult and I do not knowof any good solution.

1

2 LUKAS VOKRINEK

this by introducing “degenerate” faces – roughly speaking these are faces of geometric dimensionless than the combinatorial one. In the subset terminology these would be subsets with somevertices repeated. It is also technically advantageous to allow multiple faces with the same vertexset. This is to make colimits geometrically correct. In graphs (abstract simplicial complexes) apushout of two edges along their common vertex set is again an edge while the desired thing is apair of parallel edges. Another example is a colimit consisting of identifying two endpoints of ∆1

to a single vertex. The colimit in graphs is a single vertex while the desired thing is a loop at thisvertex.

Remark. In the lecture I had an idea of thinking of the degenerate simplices as sticking out of theblackboard so that when one looks at the blackboard perpendicularly these are not seen but stillare present ready to be targets of simplicial maps.

By introducing multiple edges we lose some of the combinatorics. We have to give up the“collection of subsets” approach and rather associate to each n-dimensional simplex a collectionof its (n − 1)-dimensional faces. To make the combinatorics easy it is preferable to order thesimplicial complexes. This is to make the collection into an ordered collection which may thus bespecified by a tuple of maps. For graphs the ordering means simply an orientation and in thisway we arrive at the notion of an oriented multigraph; the easiest way to define a non-orientedmultigraph is to add symmetries to the oriented version (an edge is a pair of oriented edges goingin the opposite directions). Doing the very same thing for simplicial sets one arrives at the notionof a symmetric simplicial set. Historically however non-symmetric simplicial sets took over. I willhave a further categorical comment on this.

Definition. A simplicial set is a collection of sets Xn for n ≥ 0 together with “face” mapsXn → Xn−1 and “degeneracy” maps Xn → Xn+1 which satisfy a collection of axioms.

One can find the axioms in any reasonable text on simplicial sets (and it is intended to be hereat some point but I am too lazy at the moment). I will however explain what these mean. Thereare two ways of obtaining a given vertex of a triangle. In both these ways one first leaves out oneof the other vertices and gets an edge (a different one – there are precisely two which contain thegiven vertex). In the second step one leaves out the other vertex. These two ways of obtainingthe same vertex are then asked to be equal in the formal definition. The degeneracy maps behavein a similar way. The formal definition will follow in the next section.

We will now return to topological spaces and their maps. Let us believe now that we havemanaged to make the constant map simplicial and in fact any projection K × L → L will besimplicial. Still there is too few simplicial maps. This is obvious in the strict sense as thereare uncountably many maps K → L but only a finite number of simplicial ones (assuming herethat K and L are finite simplicial complexes or simplicial sets). There is a theorem called the“simplicial approximation”. It has two versions – either one may alter the map slightly (in somemetric sense) to obtain a simplicial map (in some weaker sense to be specified later) or one maydeform it slightly to obtain a simplicial map. More precisely the deformation is to be understoodas a homotopy and thus the claim is that every continuous map is homotopic to a simplicial map.As there is still an infinite number of homotopy classes of maps while there is only a finite numberof simplicial maps there has to be something non-trivial going on.

We illustrate this on the circle. With our definition we may picture it as a vertex with a singleoriented edge going from this vertex to itself. There is a whole sequence of (homotopy classes of)continuous maps from a circle to itself. They are specified by a degree (the number of times thesource circle wraps around the target circle). Viewing S1 as a subset of complex numbers the mapof degree d is simply z 7→ zd. We see by an easy inspection that there are only two simplicialmaps: the identity (of degree 1) and the constant map (of degree 0). To produce the others weneed to “subdivide” the source circle into d segments (and it is easy to do so).

There is a precise definition of a subdivision and the simplicial approximation theorem reallysays: if K is finite and f : K → L a continuous map there exists a subdivision K ′ of K such thatf is homotopic to a simplicial map K ′ → L.

SIMPLICIAL SETS 3

Therefore if one is willing to subdivide the domain simplicial maps provide a good model forhomotopy classes of continuous maps between simplicial complexes. From this perspective onehas to treat homotopic maps as the same. Let us try to indicate briefly what this does to spaces.Two spaces are called homotopy equivalent if there exist continuous maps going in the oppositedirections for which both composites are homotopic to the identity. If we think up to homotopysuch spaces are to be considered isomorphic (which for me reads “the same”). For example a ballis homotopy equivalent to a point. One may thus hope to solve certain homotopical problems(e.g. compute some invariants) via simplicial sets. Returning to the metatheorem above fromthis homotopical point of view we have a stronger statement: almost every space is homotopyequivalent to a simplicial complex. There is a further refinement of homotopy equivalence called aweak homotopy equivalence. Every space is weakly homotopy equivalent to a simplicial complex.

I would like to justify why it is useful to think “homotopically” even for geometers. Homotopicalmethods answer some completely geometric problems. For example the space of inner productson a vector space is homotopy equivalent to a point (it is convex). Considering a bundle of suchone may then show that it always has a section, in particular there is a Riemannian metric onevery smooth manifold (and is essentially unique). I will now describe some more serious problemswhich are of homotopical nature, some of them rather surprisingly.

(1) Classify all (vector or other) bundles over a space.(2) Classify all differentiable structures on a manifold (of dimension at least 5).(3) Classify all cobordism classes of smooth manifolds.(4) Characterize the sphere among manifolds (Poincare conjecture).(5) Classify all (smooth) manifolds (of dimension at least 5).

1. Formal definitions

As described in the motivation a simplicial set X is a collection of sets Xn (of the so calledsimplices of X of dimension n) together with maps which associate to each such n-simplex itsvarious faces and degeneracies. We said that to make this technically simple we think of simplicesas being ordered. This is thought of as ordering vertices of each simplex so that we may indexthe faces of codimension one (i.e. of dimension one less) by the number of the vertex that is leftout. Analogously the degeneracies are indexed by the vertex that is repeated. The whole objectis best organized into a functor. By the above we should like to have objects the nonnegativeintegers which are typically written as [0], [1], . . . , [n], . . .. There are two ways of thinking of [n],either as an n-dimensional simplex from now on denoted |∆n|, i.e. a convex hull of n + 1 pointswhich are affine independent for simplicity denoted by 0, . . . , n or simply as the set of its vertices,[n] = {0, . . . , n}. The vertex sets are sufficient since all the face and degeneracy maps are inducedby maps of vertices. In both descriptions the ordering of the simplex consists of the ordering ofthe vertices which we depict by orienting the edges so that they point in the increasing order.

To describe morphisms we propose a very useful way of thinking of the n-simplices of X,i.e. the elements of Xn (here serving only as a motivation). Geometrically these are in bijectionwith simplicial maps σ : |∆n| → X as these are fully determined by the image of the maximalface (the ordering is essential – there is a unique way of making an order-preserving map whereasfor non-ordered simplicial complexes there would be n! such maps). The images of the faces of|∆n| are then given by the composition of the face inclusion of |∆n| with the “classifying map”σ and similarly the degeneracies are given by precomposing with a projection map which sendstwo vertices to a single one (in an ordered manner). In this way we get all maps in our categoryby composing the face inclusions and projections and it is a simple matter to show that one getsall order-preserving maps in this way (in fact a sharper result is given in Lemma 1.4). Thereforea morphism [n] → [m] is a map which preserves the ordering or alternatively the induced affinemap of the convex hulls. The resulting category is denoted ∆ and its part looks like

· · · [2]//

// [1] s0 //

d0oo

d1oo

d2oo

[0]d0

oo

d1oo

4 LUKAS VOKRINEK

We write di for the unique injective map whose image contains all vertices except the i-th and si

for the unique surjective map which apart from si(i) = si(i+ 1) is injective.Three interpretations of ∆ (objects in the first row and morphisms in the second):

combinatorial geometric categorical

[n] = {0, . . . , n} |∆n| = conver hull of [n] [n] = (0→ · · · → n)

order-preserving maps order- and vertex-preserving affine maps functors

Definition 1.1. A simplicial set X is a contravariant functor X : ∆op → Set. The category ofall simplicial sets is denoted by sSet = [∆op,Set]. In particular the morphisms, called simplicialmaps, are the natural transformations.

We write X([n]) = Xn. The reason why it is a contravariant functor should be clear byrealizing that the i-th face of x ∈ Xn is geometrically the pre-composition with the face inclusion,σ 7→ (di)∗σ = σdi which is of contravariant nature (e.g. (di)∗(dj)∗σ = σdjdi = (djdi)

∗σ). Theimage of di under X is typically denoted by di and the image of si by si. A simplex of X iscalled degenerate if it lies in the image of some si. When thinking geometrically we pretendthat degenerate simplices don’t exist. This is sometimes quite misleading as there might be non-degenerate simplices whose faces are degenerate and such are absolutely important to the theory.A simple picture is as follows

•v

•d1x

88

d2x &&

u x

•

s0v

KS

v

(the double arrow pretends to be at the same time the equal sign to denote that the edge isdegenerate and an arrow to describe the ordering). Recalling that di consists of skipping i,i.e. geometrically skipping the i-th vertex, we have u = d1d2x, v = d0d2x = d0d1x (plus two moreways of writing the same) and the degenerate edge s0v on the right is d0x.

The advantage of such a definition as opposed to the one with the face operators di and degen-eracy operators si satisfying relations is in that one does not need to remember these relations.They are simply given by (the opposites of those in) ∆ and as such are easily worked out – wewill do this later for relations between face operators.

We will now aim at the definition of the space corresponding to X called its geometric real-ization. We need to develop first the basic tool for working with simplicial sets. The principle issummarized in the statement: the category sSet is the free completion of ∆ under colimits. Firstlythere is a “Yoneda embedding” ∆→ sSet which sends

[n] 7→ ∆n = ∆(−, [n]),

the contravariant hom-functor determined by [n] which we call the standard n-simplex. This willbe our only example for some time. Every k-simplex of ∆n can be obtained from ιn = id[n] by theoperators, θ = θ∗ιn. Therefore ∆n is generated by the n-simplex ιn (see the algebraic descriptionahead). It is a content of the Yoneda lemma2,

sSet(∆n, X) ∼= Xn,

that it is in fact freely generated by ιn: any prescription of the image of ιn can be extended toa simplicial map in a unique way. Moreover the Yoneda isomorphism implies that the Yonedaemdedding is a full embedding, ∆ can be thought of as a full subcategory of sSet.

Regardless of its formal definition we should think of this as the n-dimensional simplex and insome sense it is – only it also contains the information about all its faces and their degeneracies.Now we claim that every simplicial set can be obtained as a colimit of the standard n-simplicesfor varying n. Geometrically this is almost evident. We may decompose the simplicial set intoits simplices (all of them) and then form a diagram from these so that its colimit will glue thesimplices back to their place to form the original simplicial set. The resulting diagram is therefore

2The prescriptions are: f 7→ f(ιn) and x 7→ (θ 7→ θ∗x).

SIMPLICIAL SETS 5

canonical and its objects are indexed by all the simplices of X. Also the morphisms are all themorphisms between these. Namely the simplex category ∆↓X has as objects all simplices of Xand for y ∈ Xm and x ∈ Xn we have

∆↓X(y, x) = {θ : [m]→ [n] | y = θ∗x}

Therefore morhpisms in ∆↓X are precisely those of the form θ∗xθ−→ x (those familiar with

Grothendieck fibrations should recognize one here). Since by the Yoneda lemma simplices of Xare in bijection with maps from the standard simplices we might reinterprent the category ∆↓X:its objects are simplicial maps ∆n → X (for various n) and their morphisms are maps between thestandard simplices for which the obvious triangle commutes. Such a category is called the commacategory and this is how ∆↓X is read: “∆ comma X”. There is a diagram PX : ∆↓X → sSetsending Xn 3 x 7→ ∆n. Alternatively PX can be viewed as a functor ∆↓X → ∆ since ∆ is a fullsubcategory of sSet via the Yoneda embedding. From this point of view, PX(x) = [n].

Theorem 1.2 (Canonical presentation). There is a canonical isomorphism colim∆↓X PX∼=−−→ X.

Before going into the actual proof I would like to comment on this theorem from the algebraicside. My aim in the beginning was to describe simplicial sets as geometric objects. They canbe also looked at from the algebraic side. Namely they are an example of a variety of universalalgebras (multisorted) as is in fact any presheaf category. Namely simplicial sets are given by setsXn and unary operations between these, di and si, which satisfy certain identities. As is quite oftenthe case one obtains a nicer description when instead of going for the minimal set or operationsand identities, one goes for the maximal such collections – this means consider all operations andall identities satisfied. This is exactly what the category ∆ does for us. Returning to the colimitdecomposition a presentation for an algebra is a set of generators and relations for which thealgebra is “universal”. Forgetting about the relations one considers the free algebra on the givenset of generators and the presented object is then obtained by the quotient by the congruencegenerated by the relations. There is a canonical presentation for every algebra – take all theelements as generators and all the relations between them. The result of taking the free algebraon all elements of X results in the disjoint union

∐[n]

∐x∈Xn

∆n. Indeed ∆n is the free algebra

on one generator in dimension n – this is the content of the Yoneda lemma sSet(∆n, X) ∼= Xn.Imposing relations then amounts to adding the arrows in the diagram.

Proof. The actual proof is simple. There is a cone PX ⇒ X whose component at x ∈ Xn,

λx : PXx = ∆n → X,

is simply the map corresponding to x under the Yoneda isomorphism, i.e. θ 7→ θ∗x. A refinedstatement is: the map colim∆↓X PX → X induced by the universal property of the colimit is anisomorphism. Firstly every x lies in the image as x = λx(id). Moreover if it is the image of any

other element x = λy(θ) this means x = θ∗y in which case there is an arrow xθ−→ y in ∆↓X and

therefore id and θ are identified in the colimit. This means that the map is also injective. �

Another way of thinking of a simplicial set is as this presentation – a presentation of a geometricobject to be glued from simplices. This will become even more concrete when we define (shortly)the geometric realization of a simplicial set. One may then view the simplicial set as a prescriptionof a way to glue the topological simplices in order to obtain this space (a kind of a recipe). There isa way of going from the presentation back to the simplicial set – one can already see the simplicesas objects and operators as labels for morphisms. There are however presentations which arenot associated to simplicial sets – one might say that these are not complete. The completenessmeans that a face/degeneracy of a generator is again a generator. In terms of the functor PXthe completeness is expressed in this functor being a discrete Grothendieck fibration. These non-complete presentations in fact turn out to be very useful. One may give finite presentations ofsimplicial sets. We note here that all simplicial sets are infinite objects (as soon as they are non-empty) as every vertex yields through its degeneracies a simplex in every dimension. One may

6 LUKAS VOKRINEK

however present a simplicial set by a finite diagram3. For example a simplicial set

• //

y•

• //

OO ??

•

OO

x

(which is in fact ∆1 × ∆1) may be specified by the following presentation: it is given by twogenerators x, y in dimension two satisfying one relation d1x = d1y. In terms of a diagram ofstandard simplices this simplicial set is the colimit of

∆2 d1

←−− ∆1 d1

−−→ ∆2

which once again can be specified by a finite amount of data even though the resulting simplicialset is an infinite object.

We give now some applications now of the presentation result. Typically to define/prove some-thing for all simplicial sets one does it for the standard simplices and then extend to all simplicialsets using the canonical presentation. Our first example of this phenomenon is the geometricrealization. This is the topological space that we would like to associate to a simplicial set. Withfinite simplicial complexes this was rather straightforward. For simplicial sets this is slightly moreabstract – after all it is harder to draw a multigraph, one cannot use straight lines for this pur-pouse. The geometric idea is however quite clear. The simplicial set is composed from simplicesglued together via the canonical presentation. A geometric realization of the standard n-simplexis the topological n-simplex. The whole geometric realization is then glued together from theseaccording to the very same presentation.

|X| = colim∆↓X

|PX | = colim∆n→X

|∆n|

This of course relies on the fact that not only we are able to form topological simplices but alsorealize the operators between them, i.e. there is a functor

|∆•| : ∆→ Top.

and |PX | is the composition ∆↓X PX−−−→ ∆|∆•|−−−−→ Top. We have used the embedding |∆•|

previously when motivating the definition of ∆ and this is one of the ways to imagine it – as thecategory of ordered topological simplices together with vertex- and order-preserving affine mapsbetween them, i.e. as a subcategory of Top.

Example. So far we have defined only the standard n-simplices ∆n. We defined the geometricrealization by a cocontinuous (i.e. colimit preserving) extension starting from the association∆n 7→ |∆n|. We will verify that the two possible definitions of |∆n| agree, namely that theabove colimit for X = ∆n is really |∆n|. The category ∆↓∆n has a terminal object, the identityid : ∆n → ∆n, as every ∆k → ∆n factors uniquely through it (alternatively every simplex θadmits a unique map θ to id). Therefore the colimit is easy to compute as the image of theterminal object,

colim∆↓∆n

P∆n ∼= |P∆n(id)| = |∆n|.

Quite formally our formula implies that the geometric realization is a left adjoint. Let us denotefor a topological space T by ST the “singular simplicial set” of T given by SnT = Top(|∆n|, T ).Then it follows

Top(|∆n|, T ) = SnT ∼= sSet(∆n, ST )

which is natural in [n]. Taking a limit over the simplex category ∆↓X one obtains

Top(|X|, T ) ∼= sSet(X,ST ).

3This could be compared to finitely generated abelian groups – in general they are infinite but may be specified

and dealt with algorithmically using presentations. In this way Zn is the abelian group with n generators and no

relations. This specification takes a finite amount of data. Similarly to define a homomorphism is the same as togive a matrix (again a finite object). One may compute from this its kernel/image etc.

SIMPLICIAL SETS 7

One of our aims will be to show that this adjunction is in fact some sort of equivalence – moreconcretely in the “up to homotopy” sense that we were discussing in the beginning. Let us describethe counit explicitly. It is the canonical map |ST | → T given by mapping the topological n-simplexcorresponding to x ∈ SnT to T via the map x : |∆n| → T . The space |ST | is thus the collectionof all “singular simplices” of X packed together; every simplex has a “tautological” way of gettingmapped to T . Part of the claimed adjoint equivalence is therefore that this map is a (weak)homotopy equivalence. This is the approximation result we mentioned in the introduction: everytopological space is (weakly) homotopy equivalent to a (geometric realization of a) simplicial set.We will prove this claim later.

There is a more standard description of ∆ namely as a subcategory of Cat consisting of thenon-empty finite ordinals and all order preserving maps between them, i.e. ∆ is a full subcategoryof Cat. In very much the same way as in the case of the geometric realization we obtain anadjunction

Cat(τX, C) ∼= sSet(X,NC)where again NnC = Cat([n], C) – the set of all chains of composable n-tuples of morphisms. Thissimplicial set NC is called the nerve of the category C. The left adjoint is again the uniquecocontinuous functor extending ∆n 7→ [n]. We will not comment much on this except that insome sense again sSet and Cat are equivalent, the homotopy equivalence in Cat is however muchless intuitive than the one in Top. In fact it is classically defined via sSet and thus the result isnot all that surprising. There is however a special case of the nerve construction which is useful(in fact NC is very important on its own). Let C be a category with one object, i.e. a monoid(typically a group). Then NC (or rather its geometric realization) is called the classifying spaceof the group G and denoted BG. Its topological invariants are important, namely its homologyand cohomology groups are called the homology and cohomology groups of the group G. Theydepend solely on the group G. Of course there is an ultimate topological invariant of G, namelythe (homotopy type of) the topological space BG itself.

We have described how a simplicial set can be built from free simplicial sets, i.e. the standardsimplices. This was a one-step process – we built the whole simplicial set from one diagram. Thereis an alternative approach which is useful in other occasions. One may build any simplicial setinductively by adding in the n-th step only simplices of dimension n. The resulting stage is calledthe n-skeleton of X and is formally defined as follows.

Definition 1.3. The n-skeleton sknX of a simplicial set X is a simplicial subset generated by allthe simplices of dimension at most n.

Here we are using the algebraic nature of simplicial sets (to make sense of “generating”). It isobvious that k-simplices of sknX are exactly those x ∈ Xk of the form θ∗y where y ∈ Xl withl ≤ n and θ : [k] → [l]. We will see shortly that sknX consists precisely of those simplices whichare (iterated) degeneracies of simplices of dimension at most n, i.e. that we may restrict θ in theabove to surjective maps. Since geometrically we ignore the degenerate simplices from this pointof view we are throwing away all simplices of dimension bigger than n. A nice definition startswith restricting the canonical presentation to simplices of dimension at most n. Then sknX is thecolimit of this restricted diagram as we will see shortly. In any case we first need to understandmaps in ∆ to prove any of the claims.

Lemma 1.4. In ∆ every map θ factors uniquely as a composition θ = θm◦θe where θm is injectiveand θe is surjective. Moreover every injective map factors (non-uniquely) as a composition of facemaps and every surjective map as a composition of degeneracy maps.

Proof. The first claim is simply the image factorization. The second claim is easy. �

Consequently a simplicial set X is generated by a set M of its simplices if every simplex of Xis an (iterated) degeneracy of an (iterated) face of some element of M . Applied to the n-skeletonsknX of X,

sknX = {θ∗x | dimx ≤ n} = {θ∗eθ∗mx | dimx ≤ n} = {θ∗ey | dim y ≤ n}consists precisely of the (iterated) degeneracies of simplices of dimension at most n.

8 LUKAS VOKRINEK

Example 1.5. First recall that ∆n is generated by a single element ιn of dimension n andis therefore equal to its n-skeleton (we say that it is n-skeletal). The “boundary of ∆n” is asimplicial set ∂∆n = skn−1 ∆n. It consists precisely of all maps [m] → [n] that factor throughsome [k] with k < n. These are precisely the non-surjective maps. We will give a presentationof ∂∆n which will be important later. The point is that every such map factors through somedi : [n− 1]→ [n] and if it factors through di and dj then it factors through a certain [n− 2]→ [n],namely that which avoids i and j. These form a commutative square where we assume i < j.

[n− 2]di //

dj−1

��

[n− 1]

dj

��

∆n−2 di //

dj−1

��

∆n−1

dj

��

[n− 1]di

// [n] ∆n−1

di// ∂∆n

Therefore one obtains a presentation: ∂∆n is generated by di subject to the relations as above,dj−1d

i = didj . The diagram that gives the presentation has n+1 copies of ∆n−1, one for each face

di of ιn, and copies of ∆n−2 indexed by pairs i < j. The morphisms in the diagram are exactlythe same as those in the top and left part of the above square which in fact depicts (part of) thediagram together with a cone to ∂∆n which we claim to be universal.

The fact that every simplex of ∂∆n factors through some di implies that the map from thecolimit to ∂∆n is surjective. For injectivity if two simplices of the colimit get mapped to the samesimplex of ∂∆n, say θ coming from the i-th copy of ∆n−1 and η coming from the j-the copy, thentheir common image in ∂∆n is in fact a chain [m]→ [n] which avoids both i and j and therefore θavoids j−1 and η avoids i. The resulting chains in [n−2] obtained by omitting these two elementsclearly have to agree. But this means exactly that there is a simplex of ∆n−2 which maps to θand η proving that these agree in the colimit4.

We will deduce now that the geometric realization of ∂∆n is the boundary of |∆n|, i.e. theunion of the faces of |∆n|. The idea is roughly the same – there is a map

|∂∆n| → ∂|∆n|

where the left hand side can be computed as the colimit of a diagram of topological simplicesobtained from the presentation above by applying | − |. This map is surjective because eachface of |∆n| is hit by one of |∆n−1| in the diagram and injective because if two points of thecolimit get mapped to a common point in ∂|∆n| then in fact these two points must lie in thefaces of the corresponding spaces |∆n−1| and come from a copy of |∆n−2| (more precisely viadi : |∆n−2| → |∆n−1| which parametrizes the appropriate intersection). Since the spaces arecompact Hausdorff the above map is a homeomorphism.

In the proceeding we will need a result about general n-skeletal simplicial sets. The idea issimple: in such simplicial sets everything is determined by what happens in dimension up to n.First we prove a lemma about general simplicial sets.

Lemma 1.6. Every simplex can be expressed uniquely as a degeneracy of a non-degenerate simplex.

Proof. The existence of such an expression is obvious. Let θ∗1x1 = θ∗2x2 where both θ1, θ2 aresurjective and both x1, x2 non-degenerate. Assume first that θ1 6= θ2 or more concretely5 thatthere exist i, j for which θ2(i) 6= θ2(j) but θ1(i) = θ1(j). In ∆ every surjective map has a section,in fact many sections. We obtain a section σ of θ2 with θ1σ non-injective by choosing σ(θ2(i)) = iand σ(θ2(j)) = j. Then

x2 = σ∗θ∗2x2 = σ∗θ∗1x1 = (θ1σ)∗x1

is degenerate, a contradiction. Therefore θ1 = θ2 and in the above the right hand side is x1. �

4An alternative proof (with the same ingredients) goes like this: the diagram in question is a subdiagram of thecanonical one associated with ∂∆n. One proves that they induce an isomorphism on colimits by showing that the

inclusion functor is right cofinal.5In an element free way θ2 does not factor through θ1.

SIMPLICIAL SETS 9

Lemma 1.7. A simplicial map f : X → Y between n-skeletal simplicial sets is an isomorphismif and only it is an isomorphism in all dimensions up to n.

Proof. Let θ∗y be a simplex of Y written as a θ-degeneracy of a non-degenerate simplex y. Wewill describe now its possible preimages. Suppose that θ∗y = f(η∗x) for some surjective η andnon-degenerate x. Since f preserves non-degenerate simplices and is a bijection on them the onlypossibility is θ = η and x = f−1(y). �

Remark. There is a more conceptual proof of Lemma 1.6: for θi as in the proof form the pushoutsquare

[n]θ1 // //

θ2����

[m1]

η2

����

[m2]η1

// // [k]

in ∆ and prove that it is an “absolute pushout”, i.e. one preserved by an arbitrary functor. Herewe are mostly concerned with the universal example of such a functor, the Yoneda embedding,which sends it to

∆n //

��

∆m1

��

∆m2 // ∆k

The universal property of this pushout square gives a more precise statement: if a simplex is botha θ1-degeneracy and a θ2-degeneracy then it is in fact a (ηiθi)-degeneracy.

At this point we can prove our claim that sknX can be computed by restricting the canonicalpresentation to simplices of dimension at most n and computing its colimit. It is easily observedthat n-skeletal simplicial sets are preserved by colimits. Therefore we can restrict our attention tosimplices of dimension at most n and there the proof is exactly the same as that for Theorem 1.2.

We would like to understand now the way sknX is obtained from skn−1X. Obviously this isby adding all the non-degenerate n-simplices. We are also interested in the relations. We claimthat there is a pushout square ∐

x∈Xnnon−deg

∂∆n //

��

skn−1X

��∐x∈Xn

non−deg

∆n // sknX

This is easy to check in dimensions lower than n since both vertical maps are isomorphisms.In dimension n the difference between the top and the bottom is exactly in the non-degeneraten-simplices and is the same on the two sides. Since all simplicial sets involved (including thepushout) are n-skeletal the result follows.

Now that we understand the passage from the (n − 1)-skeleton to the n-skeleton we observethat X is the colimit

X ∼= colim

(sk0X → sk1X → · · · → sknX → · · ·

)(in fact it is the union of this chain).

Proposition 1.8. A geometric realization of any simplicial set is a CW-complex.

Proof. The geometric realization of ∆n is |∆n| the n-dimensional ball while |∂∆n| is its boundary,the (n− 1)-dimensional sphere. �

10 LUKAS VOKRINEK

There is a relative version. Every monomorphism A→ X of simplicial sets can be obtained bysuccesively adding n-simplices with n increasing. In the above pushout restrict to x ∈ Xn non-degenerate not contained in A. The preceding proposition then generalizes to |A| → |X| being aninclusion of a subcomplex. Quite easily one can deduce that the geometric realization preservesequalizers. In fact it preserves all finite limits; we are left with the case of binary products.

Proposition 1.9. The geometric realization preserves finitary products, |X × Y |∼=−−→ |X| × |Y |,

at least when one forms the product on the right in the category of compactly generated Hausdorffspaces.

Proof. The point of the category of compactly generated Hausdorff spaces (or any of its variations)is that the product distributes over colimits (because it is a left adjoint in each variable). Thewhole proposition then reduces to the case of simplices via

|X| × |Y | ∼= | colim∆n→X

∆n| × | colim∆m→Y

∆m| ∼= colim∆n→X,∆m→Y

|∆n| × |∆m|

and

|X × Y | ∼= | colim∆n→X

∆n × colim∆m→Y

∆m| ∼= colim∆n→X,∆m→Y

|∆n ×∆m|.

Therefore one needs to show that |∆n ×∆m| → |∆n| × |∆m| is an isomorphism (by the way thiswould not hold without degeneracies). The idea is to produce a simple presentation for ∆n ×∆m

using categories and the fact that the nerve functor N preserves limits. Namely

∆n ×∆m = Cat(−, [n])× Cat(−, [m]) ∼= Cat(−, [n]× [m]) = N([n]× [m]),

the nerve of the category [n]× [m] which looks somewhat like

• //2 • //• //•

• //

OO

1 • //

OO

• //

OO

•

OO

• //

OO

00

//

• //

OO

1• //

OO

2•

OO

3

Clearly every simplex (i.e. a chain of composable morphisms) is obtained via the face operatorsfrom one of the maximal non-degenerate ones – those which start at (0, 0) and end at (n,m) andalways go either one step to the right or one step up. These are the generators. The relationsdescribe when a chain lies in two such maximal non-degenerate chains. This happens when it liesin a subchain formed by omitting one of the vertices which is at the corner of the path determinedby the chain of the morphisms. There is then a relation expressing this as a suchain of a maximalnon-degenerate chain with the omitted vertex replaced by the other possibilitiy. As an example achain indicated in the above diagram lies in three maximal chains and these are connected by azig-zag in the presentation

RRUUR

∆5 d2

←−− ∆4 d2

−−→RURUR

∆5 d3

←−− ∆4 d3

−−→RUURR

∆5

In any case all these chains determine easily certain simplices embedded into |∆n| × |∆m|. Infact a chain (i0, j0) → · · · → (ik, jk) gives rise to the k-simplex spanned by vertices (il, jl) (herethought of as points in |∆n| × |∆m| where we use the elements of [n] to denote also the vertices of|∆n|). It is a combinatorial exercise to show that the incidence relations they satisfy are exactlythose coming from this presentation.

SIMPLICIAL SETS 11

I will give a detailed proof for ∆1 ×∆1. It has two maximal 2-simplices corresponding to thetwo indicated paths in

• //1 •

• //

OOOO//

00

•

OO

1

Denoting the one going through the bottom right corner by x and the one through the top leftcorner by y we have a relation d1x = d1y. In the geometric realization we get two 2-simplicesmeeting along their common (first) face. The picture looks very much the same as that of [1]× [1],

• //

y•

• //

OO ??

•

OO

x

except now labeled with the images of the geometric realization of the 2-simplices x and y. Ageometrically minded reader might try to do the same for ∆2 × ∆1. One gets three 3-simplicesforming a subdivision of the prism |∆2| × |∆1|.

We describe a general point of |∆n| × |∆m| through its pair of barycentric coordinates. If |∆n|is the convex hull of points v0, . . . , vn then a point with barycentric coordinates (t0, . . . , tn) is theconvex combination t0v0 + · · · + tnvn, i.e. we assume here that all ti are non-negative and addup to 1. Another possibility is to say that |∆n| is the convex hull of the generators e0, . . . , en ofthe vector space Rn+1 and then barycentric coordinates are the actual coordinates of the point inRn+1. Let now P = ((t0, . . . , tn), (s0, . . . , sm)) be a point of the product. We will try to expressit as a convex combination of some vertices of the product which form a simplex of our claimedtriangulation. First we may suppose by symmetry that s0 ≤ t0 and further that s0 > 0 forotherwise the point P actually lies in |∆n| × d0|∆m|. It is quite easy to see that any simplex ofthe claimed triangulation is spanned by (e0, e0) and vertices (ei, ej) where at least one of i, j isnon-zero. Therefore in any such convex combination we must have (e0, e0) with the coefficientmin(s0, t0). Assume that s0 ≤ t0. Then

P = s0(e0, e0) + (1− s0)((t′0, . . . , t′n), (0, s′1, . . . , s

′m))

and the second point in the combination lies in |∆n|×d0|∆m| and by induction it can be expresseduniquely as a combination of points forming a chain. Therefore this expression is unique for P andthis means that |∆n| × |∆m| is a disjoint union of the interiors of the simplices from our claimeddecomposition. But that means that it is an actual decomposition. �

2. The homotopy theory in simplicial sets

We start by showing that sSet is a closed category, i.e. that there exists not only a set ofsimplicial maps between two simplicial sets X,Y but again a simplicial set of such which we willdenote by hom(X,Y ). The requirement is

sSet(K, hom(X,Y )) ∼= sSet(K ×X,Y ). (1)

Since a simplicial set can be reconstructed from the knowledge of maps into it out of the standardsimplices (by Yoneda lemma) we might “compute” hom(X,Y ) by substituting K = ∆n,

hom(X,Y )n ∼= sSet(∆n, hom(X,Y )) ∼= sSet(∆n ×X,Y )

Definition 2.1. The function complex hom(X,Y ) is the simplicial set defined by

hom(X,Y ) = sSet(∆• ×X,Y )

The definition itself supplies an isomorphism (1) for K = ∆n and in fact one which is naturalin ∆ ⊆ sSet (via Yoneda embedding). As any simplicial set is a colimit of standard simplices in acanonical way one may prolong this isomorphism to arbitrary K since both sides turn colimits inthe K-variable to limits.

12 LUKAS VOKRINEK

We may now define homotopies of simplicial maps in the same manner as for topological spaces,i.e. as simplicial maps

h : ∆1 ×X → Y,

or in other words as 1-simplices of hom(X,Y ). In this situation we of course say that h is ahomotopy from d1h to d0h. There is a small technical problem – homotopies cannot be composed.We are therefore led to the following definition

Definition 2.2. Two simplicial maps are called homotopic if they can be connected by a zig-zagof homotopies (of arbitrary direction).

Another way of formulating this is: consider on pairs (f, g) a relation “there exists a homotopyfrom f to g”. It is neither symmetric nor transitive and the real homotopy relation on maps is thesymmetric transitive hull. We will now describe a third point of view which is important. Thereis a fundamental construction called the “set of components” of a simplicial set. We denote byπ0X the quotient of X by the equivalence relation generated by the set d1x ∼ d0x for x ∈ X1.The set of homotopy classes of maps from X to Y is then [X,Y ] = π0 hom(X,Y ).

We can define a homotopy equivalence in an obvious way as a map f : X → Y which admits amap g : Y → X with both gf and fg homotopic to the respective identity maps. There is howeverone problem with this definition: it does not give the topologically correct notion. Consider asimplicial map

• BB��• // • e}}v

sending the bottom arrow to the non-trivial loop e and the top arrow to the degenerate s0v. Theonly maps going in the opposite direction are constant (in the sense that they factor through thepoint ∆0, here concretely e is mapped to a degenerate simplex). There is no hope that this wouldbe a homotopy inverse since we have:

Proposition 2.3. Geometric realizations of homotopic maps are homotopic.

Proof. We have already seen that the geometric realization preserves products. Therefore

|h| : |∆1| × |X| ∼= |∆1 ×X| −→ |Y |is a homotopy from |d1h| to |d0h|. �

Returning to the example above both geometric realizations are circles (exercise: find some nicepresentations and work out the geometric realization) and the map is homotopic to a homeomor-phism hence is a homotopy equivalence. On the other hand the constant map is not a homotopyequivalence (since S1 is not contractible).

We will now correct this. Let us first characterize homotopy equivalences in a slightly strangeway whose significance will lie in motivating the definition of a weak equivalence.

Lemma 2.4. A map (simplicial or continuous) f : X → Y is a homotopy equivalence if and onlyif f∗ : [Y,Z]→ [X,Z] is a bijection.

Proof. If f is a homotopy equivalence with a homotopy inverse g then the inverse of f∗ is simplyg∗. In the other direction assuming that f∗ is a bijection take Z = X and let g : Y → X be suchthat f∗[g] = [id]. Then one checks easily (hint: set Z = Y ) that g is a homotopy inverse to f . �

There is a dual statement which is more suited for topological spaces. We will see this later.Let us say that a simplicial map f : X → Y is a realization homotopy equivalence6 if the geometricrealization |f | : |X| → |Y | is a homotopy equivalence.

Remark. For those who know a bit of algebraic topology I would like to prove that both the unitand the counit of the adjunction | − | a S are weak homotopy equivalences and thus simplicialsets and topological spaces are essentially the same concept as long as homotopy properties areconcerned. Starting from a topological space T we want to compare it to the space |ST |. Thecounit is a map εT : |ST | → T and we want to show that it is a weak homotopy equivalence.

6Joyal-Tierney use the name geometric homotopy equivalence.

SIMPLICIAL SETS 13

We will postpone a bit the proof that εT induces an isomorphism on π1. We compute the (local)homology groups. The point with |ST | is that we may use the simplicial homology groups for thiscomputation. The simplicial chain complex of |ST | is however exactly the same as the singularchain complex of T ,

id : C∆∗ (|ST |) ∼−−→ C∗(|ST )) −→ C∗(T ).

Next we will deal with the fundamental groupoid of |ST | and T . It is rather simple to constructa simplicial version Π1(ST ) ⊆ Π1(|ST |) (quickly: Π1 is similar to τ : sSet→ Cat above but ratherstarting from Π1(∆n) = Gpd([n]), the groupoidification of [n]). The inclusion Π1(ST ) ↪→ Π1(|ST |)is an equivalences of groupoids while Π1(ST ) → Π1(T ) is even an isomorphism (both claims are2-dimensional therefore easy). In the opposite direction we want to show that ηX : X → S|X| isa realization homotopy equivalence. Therefore apply | − | to the map and obtain the first map inthe sequence

|X| |ηX |−−−−→ |S|X||ε|X|−−−−→ |X|

whose composition is identity. The second map is a weak homotopy equivalence hence so is thefirst. I will give later an alternative proof which relies on the homotopy properties of simplicialsets.

Applying the above lemma we see that f is a realization homotopy equivalence precisely when[|Y |, T ] → [|X|, T ] is a bijection or equivalently when [Y, ST ] → [X,ST ] is a bijection. Thereforerealization homotopy equivalences are detected by the singular simplicial sets ST of topologicalspaces. As I would like to avoid as much topology as possible I define a new concept that we willbe working with where we replace the collection of ST by a bigger class called Kan complexeswhich are of tremendous importance regardless of my goal of avoiding topology. Kan complexesare especially useful when talking about homotopies. It is possible to compose (higher) homotopiesof maps with targets Kan complexes.

Definition 2.5. A simplicial set X is called a Kan complex if it satisfies the following extensionproperty

Λk∆n //

��

X

∆n

77

where Λk∆n is the simplicial subset of ∆n generated by all its (n− 1)-dimensional faces with theexception of dkιn.

The diagram from the definition reads: for every Λk∆n → X there exists a (non-unique)extension ∆n → X. It will be the case for all the extension problems that we will encounter thatthe extension will not be unique. We will not emphasize this later.

Lemma 2.6. A singular simplicial set ST of a topological space T is a Kan complex.

Proof. By adjunction the extension problem translates to

|Λk∆n|f

//

��

T

|∆n|

77

where |Λk∆n| is the union of all the (n − 1)-dimensional faces of |∆n| with the exception of theone opposite the k-th vertex. Topologically this is just a hemisphere of the ball |∆n|. There existsa retraction r : |∆n| → |Λk∆n| which squashes the whole ball onto this hemisphere. The requiredextension is then for example fr. �

There is another series of examples of Kan complexes. A simplicial group is a simplicial setG with a group structure on each Gn in such a way that all operators θ∗ : Gn → Gm are group

14 LUKAS VOKRINEK

homomorphisms. Every simplicial group is a Kan complex. I do not know of any non-technicalproof and we will not need this fact, so let us move to the main definition.

We would like to define a weak homotopy equivalence to be a map f : K → L such that forevery Kan complex X the induced map of homotopy classes of maps [L,X]→ [K,X] is a bijection.For technical reasons one we prefer to make a fibrewise version of this definition.

First we generalize the notion of a Kan complex. We say that a simplicial map f : X → Y is aKan fibration if in every solid arrow commutative diagram

Λk∆n //

��

X

f

��

∆n //

77

Y

the dashed simplicial map exists making both triangles commute. As a special case the uniquemap X → ∆0 is a Kan fibration if and only if X is a Kan complex. In this situation we say thatthe inclusions Λk∆n → ∆n have the left lifting property against Kan fibrations.

Definition 2.7. A simplicial map f : K → L is said to be a weak homotopy equivalence if forevery Kan fibration X → Y and every square

K //

f

��

&fX

����

L //

88

Y

there exists a dashed simplicial map L → X such that the lower triangle commutes strictly andthe upper triangle commutes up to a homotopy K ×∆1 → X whose composition with X → Y isconstant (we say that the homotopy is fibrewise).

Remark. There is a second possibility – require X → Y to be a Kan fibration between Kancomplexes. The proof that these are the same requires that every Kan fibration is a pullback ofone whose codomain is a Kan complex (namely the fibrant replacement of the codomain of theoriginal Kan fibration). See Joyal-Tierney – they use the theory of minimal fibrations to establishthe claim. With this definition at hand one should be able to prove the 2-out-of-3 property moredirectly. Moreover f : K → L a weak homotopy equivalence should then be easily seen to beequivalent to f∗ : [L,X]→ [K,X] bijective for all Kan complexes X.

We will see shortly that an injective map is a homotopy equivalence if and only if it has a strictleft lifting property against Kan fibrations, i.e. the required homotopy can be always taken to beconstant. With this definition it is somewhat complicated to prove even the basic properties. Wewill need to understand better the injective homotopy equivalences.

3. Anodyne extensions

We say that a map j is a pushout of i if there exists a pushout square

• //

i

��

•j

��• // •

A transfinite composition of a chain

• // • // • // · · ·

SIMPLICIAL SETS 15

is the map from the initial object to the colimit. Finally we say that j is a retract of i if thereexists a diagram

• //

j

��

id++• //

i

��

•j

��• //

id

33• // •

Let us say that a class of maps is saturated if it closed under coproducts, pushouts, transfinitecompositions and retracts.

Theorem 3.1. The saturation of the set of all boundary inclusions ∂∆n → ∆n consists preciselyof all injective maps.

Proof. Every injective map can be decomposed into a transfinite composition of (relative) skeletoninclusions. Each inclusion is then a pushout of a coproduct of boundary inclusions. �

Proposition 3.2. The class of maps which have a left lifting property against Kan fibrations issaturated. Its elements are called anodyne extensions.

Proof. This is classical... (add a proof of one of the saturation properties) �

Proposition 3.3. Every anodyne extension is a retract of a transfinite composition of pushoutsof coproducts of horn inclusions. In particular the class of anodyne extensions is the saturation ofthe horn inclusions.

Proof. This is the so called small object argument. Its basic idea is simple. Start with f : K → Land try to make it into a Kan fibration. This means whenever there is a diagram

Λk∆n //

��

K

f

��

∆n // L

we want to add a simplex as a solution of this problem. Treating all these at the same timeamounts to taking ∐

Λk∆n //

��

K

f

��∐∆n // L

where the coproduct runs over all squares as above and the components of the horizontal mapsare the horizontal maps in the respective squares. Now to add a solution to this problem is toform a pushout X1. In this way we factor f into f : K → X1 → L where the first map is ananodyne extension. This produces new lifting problems so appyle the same to the map X1 → Land factor it as X1 → X2 → L etc. Now define X = colimXn to obtain a factorization

f : K → X → L.

Still the map K → X is an anodyne extension. We will see now that X → L is a Kan fibration.Any lifting problem takes place in some Xi → L by finiteness of Λk∆n,

Λk∆n //

��

Xi //

��

Xi+1

��

∆n //

44

L L

Since this problem has a solution added in Xi+1 the original problem has a solution.

16 LUKAS VOKRINEK

There is a final step. With this factorization consider

K //

f

��

X

����

L

88

L

where the dashed arrow expresses f as a retract of the anodyne extension K → X,

K

f

��

K

��

K

f

��

L //

id

66X // L

�

We are interested in certain types of maps that lie in the saturation of the horn inclusions. Itwill be convenient to write injective maps A → B in the form of a “relative space” (B,A). Weintroduce a product of relative spaces

(B,A)× (L,K)def= (B × L, (A× L) ∪A×K (B ×K))

it is the so called pushout corner map in the square

A×K //

��

B ×K

��

A× L // B × Li.e. the map from the pushout to the bottom right corner.

Lemma 3.4. The class of anodyne extensions is the saturation of the set of all inclusions

(∆n, ∂∆n)× (∆1, e)

where e is any of the two vertices of ∆1.

Proof. We will only prove that the maps from the statement are anodyne extensions. The pointis to factor the map into a sequence of maps where to get from one simplicial set to the otherconsists of adding precisely one generating (n+1)-simplex and that is attached via a horn inclusion.Namely we set (∆n×∆1)(−1) to be the subset from the statement where we choose for definitnesse = 1. We have a sequence

(∆n ×∆1)(−1) ↪→ (∆n ×∆1)(0) ↪→ · · · ↪→ (∆n ×∆1)(n) = ∆n ×∆1

The step (−)(k) is obtained by adding the generating (n+ 1)-simplex given by the chain

(0, 0)→ · · · → (k, 0)→ (k, 1)→ · · · → (n, 1).

With the exception of the faces dk and dk+1 all lie in ∂∆n×∆1 and the face dk lies in the previousstage. Therefore there is a pushout square

Λk+1∆n+1 //

��

(∆n ×∆1)(k−1)

��

∆n+1 // (∆n ×∆1)(k)

For the other containment it is true that (∆n,Λk∆n) is a retract of (∆n, ∂∆n)× (∆1, e) (wheree is 0 for k < n and 1 for k > 0). For details see [GJ], Proposition I.4.2. �

An important observation is that the class of all (B,A) for which (B,A)×(∆1, {e}) is anodyne issaturated. This is because via the product-hom adjunction the following conditions are equivalent

• (B,A)× (L,K) has a left lifting property against X → Y and

SIMPLICIAL SETS 17

• (L,K) has a left lifting property against hom(B,X)→ hom(B, Y )×hom(A,Y ) hom(A,X).

By the previous lemma this class contains (∆n, ∂∆n) and thus it must contain all pairs. We willgeneralize this in the second variable.

Theorem 3.5. If (B,A) is any pair and (L,K) is anodyne then so is (B,A)× (L,K).

Proof. The class of all (L,K) for which (B,A)× (L,K) is anodyne is again saturated. It containsthe generating set of anodyne extensions from the above lemma by the associativity of the productof pairs

(B,A)× ((∆n, ∂∆n)× (∆1, {e})) ∼= ((B,A)× (∆n, ∂∆n))× (∆1, {e})and the observation preceding the theorem. Thus contains all anodyne extensions. �

4. Characterizing weak homotopy equivalences

We can now reinterpret these results in terms of the homotopy theory of Kan complexes. Firstwe prove that the homotopy relation for maps into a Kan complex is actually an equivalencerelation. If there are given homotopies from f0 to f1 and from f1 to f2 they can be organizedinto a single map Λ1∆2 ×A→ X. Since (A, ∅)× (∆2,Λ1∆2) is anodyne there exists an extension∆2 × A → X. Its restriction to the first face of ∆2 is a homotopy from f0 to f2. The proof ofsymmetry is similar starting with a homotopy from f0 to f1 at the second face and the constanthomotopy at f0 at the first face. After extending the zeroth face is a homotopy from f1 to f0. Weremark here that exactly the same applies in the fibrewise situation – the fibrewise homotopy isan equivalence relation.

We will now show that injective maps have the so called homotopy extension property againstfibrations. This means that if the homotopy in a solution

Af

//��

i

��

&fX

����

B //

g

88

Y

is fibrewise then g is fibrewise homotopic to a strict diagonal. The data from the solution may beorganized into

(A×∆1) ∪ (B × 1) //

��

∼��

X

����

B ×∆1 //

55

Y

where the map on the left is an anodyne extension. Therefore an extension exists and restrictingit to B ∼= B × 0 yields a strict lift. This implies easily the following theorem.

Theorem 4.1. Anodyne extensions are exactly injective weak homotopy equivalences.

The construction above is called the mapping cylinder. If more generally f : K → L is anysimplicial map then its mapping cylinder cyl f is the pushout

K × 1f

//

��

∼��

L

��

K ×∆1 // cyl f = (K ×∆1) ∪K L

Since the map on the left is an anodyne extension so is the one on the right. Moreover the mappingcylinder admits an inclusion of K (through K × 0 ⊆ K ×∆1) and a projection onto L obtainedby squashing K × ∆1 onto its end K × 1 (which gets identified with L). This projection is infact a deformation retraction – there exists a homotopy from the identity to the composition ofthe projection with the inclusion given on L by the constant homotopy and on K × ∆1 by the

homotopy (K ×∆1)×∆1 idK×h−−−−−→ K ×∆1 where h : ∆1 ×∆1 → ∆1 maps the vertex (0, 0) to 0and the remaining ones to 1.

18 LUKAS VOKRINEK

The anodyne extensions are closed under pushouts. Non-injective weak homotopy equivalencesare also closed under certain pushouts. The property is called the left properness.

Theorem 4.2 (left properness). The category of simplicial sets is left proper. Given a pushoutsquare

A // //

∼��

B

��

K // L

with A → K a weak homotopy equivalence its pushout B → L is a weak homotopy equivalenceprovided that A→ B is injective.

Proof. Let there be given a homotopy lifting problem for B → L as in the definition

A // //

∼��

B �O **44

��

X

����

K //

::

L // Y

There exists a lift up to homotopy A × ∆1 → X that is fibrewise over Y . By the homotopyextension property of A → B one may extend this to a homotopy B × ∆1 → X as indicated inthe diagram. The pair of dashed arrows constitute a simplicial map L→ X that has the requiredproperty (the homotopy was already constructed). �

We may now characterize general weak homotopy equivalences through anodyne extensions.

Proposition 4.3. A map f : K → L is a weak homotopy equivalence if and only if the inclusion

f : K → (K ×∆1) ∪K L of K into the mapping cylinder of f is an anodyne extension.

Proof. This may be proved directly without big problems and the reader is encouraged to do so7.We will present a different and more abstract proof based on the left properness of simplicial setsof Theorem 4.2. To prove first the implication “⇒” consider

Kf

//��

��

L

��

K ×∆1 // (K ×∆1) ∪K L

∼

&&K;;

∼;;

f

44

f// L

If f on the top is a weak homotopy equivalence then so is its pushout and finally also f which

is moreover injective hence an anodyne extension. If on the other hand f is a weak homotopy

equivalence then so is f as a composition of f with a weak homotopy equivalence. �

In its dual form Theorem 3.5 says that for an arbitrary pair (B,A) and a Kan fibration X → Ythe map

hom(B,X)→ hom(B, Y )×hom(A,Y ) hom(A,X)

is again a Kan fibration. This is especially useful when (B,A) = (∆1, ∂∆1) and the projection isY → ∗ with Y a Kan complex. In this case

ev : hom(∆1, Y )→ hom(∂∆1, Y ) = Y × Y

7Write out the lifting problem for f explicitly, realize that it is slightly more general than the homotopy lifting

problem for f , use the homotopy lifting for Kan fibrations to replace it by an ordinary lifting problem and thencorrect the solution to provide one for the original problem.

SIMPLICIAL SETS 19

is the “evaluation map”, on the 0-simplices given by f 7→ (f(0), f(1)). Using the fact that it isa fibration we obtain other fibrations by taking pullbacks. Assume that f : X → Y is any mapbetween Kan complexes and consider

X∼ //

f

��

hom(∆1, Y )×Y X //

����

hom(∆1, Y )

ev����

Y ×Xid×f

//

����

Y × Y

Y

where the projection Y × X → Y is a Kan fibration as a pullback of X → ∗. The map X →hom(∆1, Y ) ×Y X is given by x 7→ (cst f(x), x) and is a homotopy equivalence whose homotopyinverse is given by projection onto X. In this way one may factor arbitrary map between Kancomplexes into a homotopy equivalence followed by a fibration in a very geometric way. This isuseful because of the following observation.

Theorem 4.4. A map f : X → Y between Kan complexes is a weak homotopy equivalence if andonly if it is a homotopy equivalence.

Proof. Factoring f as above one obtains a diagram

X //

id

((

��

)ihom(∆1, Y )×Y X //

����

'gX

fww

Y

55

Y

where the homotopy in the triangle on the right is given essentially by the projection on the firstcomponent, hom(∆1, Y )×Y X → hom(∆1, Y ) (its adjunct is a map (hom(∆1, Y )×Y X)×∆1 → Y ).The homotopy inverse of f is easily seen to be the composition of the diagonal with the projectiononto X.

For the converse factor f as a trivial cofibration followed by a fibration

X //∼i // Z

p// // Y

and observe that as Z is also fibrant we may conclude from the first part that i is a homotopyequivalence and consequently so is p. We take any homotopy inverse s′ and by the homotopyextension property we may assume that ps′ = id. We still have s′p ∼ id but the problem is thatit is not a fibrewise homotopy. Nevertheless consider this homotopy and project it to a homotopyh : Z × ∆1 → Y between p ∼ p. Now use again the homotopy lifting property to lift this toa homotopy id ∼ t starting at id and ending at some map t. Composing with s′p we obtain ahomotopy s′p ∼ ts′p lying over h and so does s′p ∼ id that we started with. Together they forma map

Z × Λ0∆2 → Z

whose projection to Y can be extended to s1h : Z × ∆2 → Y that restricts to Z × ∂0∆2 to aconstant homotopy at p. Lifting to Z produces therefore a fibrewise homotopy ts′p ∼ id. Changingthe section s′ to s = ts′ therefore yields ps = id and sp ∼ id by a fibrewise homotopy. Considernow a homotopy lifting problem

Zg

//

p

��

&fA

����

Y //

gs

88

B

It is solved by gs as indicated. It remains to show that a composition of two weak homotopyequivalences is again a weak homotopy equivalence which is an easy exercise. �

20 LUKAS VOKRINEK

We record here that in the second part of the previous proof we in fact showed that a Kanfibration p : X → Y is a homotopy equivalence if and only if it is a weak homotopy equivalence.

Definition 4.5. A map p is called a trivial fibration if it is both a Kan fibration and a (weak)homotopy equivalence.

Another characterization explicitly obtained in the previous proof is that trivial fibrations arethose Kan fibrations that admit a section s : Y → X together with a fibrewise homotopy sp ∼ id (itis a fibrewise contraction onto this section so that one might say that Y is fibrewise contractible).As a corollary we obtain:

Theorem 4.6. Trivial fibrations are exactly those maps having the right property against allinjective maps (or equivalently all boundary inclusions ∂∆n → ∆n).

Proof. Let p : X → Y be a trivial fibration with a fibrewise deformation to a section s. Then anyhomotopy lifting problem

Af

//

i

��

&fX

p

����

Bg

//

88

Y

has a solution sg. Hence if i : A→ B is moreover injective we may find a strict solution.If on the other hand p : X → Y has the right lifting property against all injective maps then in

particular it is a Kan fibration (horn inclusions are injective). Replacing p by its mapping cylinder

X

��

p

ww

X

p

����

Y //

id

77cyl p //

77

g'

Y

we obtain a homotopy inverse dually to the argument in the previous theorem. �

Theorem 4.7. Any weak homotopy equivalence is a realization homotopy equivalence.

Proof. Translating the homotopy left lifting property to topological spaces we have a solution ofany problem

K //

��

&f

SU

����

≡

|K| //

��

&f

U

��

L //

88

ST |L| //

88

T

where U → T is any continuous map for which SU → ST is a Kan fibration. There is a specialclass of such maps, the Serre fibrations. They are defined exactly like Kan fibrations using thegeometric realization of the horn inclusions

Λk∆n //

��

SU

����

≡

|Λk∆n| //

��

U

��

∆n //

77

ST |∆n| //

77

T

This equivalence yields precisely that the singular simplicial set of a Serre fibration is a Kanfibration. We would like to apply the above homotopy lifting property to |K| → |L| itself. Sinceit is not a Serre fibration we have to replace it by one. This is done exactly in the same way as

SIMPLICIAL SETS 21

for simplicial sets this time without any fibrancy requirements

|K| //

id

((

��

'g

map(I, |L|)×|L| |K| //

����

'g

|K|

ww|L| //

77

|L|

We conclude that |K| → |L| is a homotopy equivalence as before. �

5. The relation to topological spaces

We will now show that every simplicial set A is weakly homotopy equivalent to the Kan complexS|A|. Our main tool will be the subdivision functor. First we will describe the subdivision of astandard simplex ∆n. We define it to be the nerve

sd ∆n def= NP0[n]

of the poset P0[n] of all non-empty subsets of [n]. There is a canonical simplicial map sd ∆n → ∆n

which sends a chain of subsets (objects of P0[n]) to the chain of their maximal elements (objectsof [n]). There is also a map ∆n → sd ∆n going in the opposite direction given by sending i ∈ [n]to the subset {0, . . . , i}. Although not natural it will be very important because it gives a sectionof the canonical map sd ∆n → ∆n. A picture of sd ∆2 looks like

{0} {1}

{2}

{0,1}

{1,2}{0,2}

with the vertex in the middle being {0, 1, 2}. The 2-simplex spanned by {0}, {0, 1}, {0, 1, 2} getsmapped to the generating 2-simplex of ∆2 while the others get mapped to degenerate 2-simplices.

Lemma 5.1. The inclusion ∆n → sd ∆n is an anodyne extension.

Proof. The inclusion is factored into a sequence where in each step one adds a single simplexcontaining the interior vertex {0, . . . , n}. These are ordered according to their dimensions, therest is immaterial.

In other words one may view sd ∆n as a cone on sd ∂∆n and ∆n sits inside as a cone on aparticular (n − 1)-simplex of sd ∂∆n. One may build sd ∂∆n from that simplex using boundaryinclusions and if one cone all of them one can build sd ∆n from ∆n using horn inclusions (alwaysthe maximal ones). �

We define a subdivision of a simplicial set A using the canonical presentation

sdAdef= colim

∆n→Asd ∆n.

Again this depends on the fact that [n] 7→ sd ∆n is actually a functor ∆ → sSet. This is ratherobvious since any [n]→ [m] induces an obvious P0[n]→ P0[m]. The order preserving map P0[n]→[n] is in fact a natural transformation and as such yields a natural transformation ε : sdA ⇒ A.As explained in the case of the geometric realization all functors constructed as cocontinuousextensions are left adjoints. The right adjoint of sd is denoted by Ex and is given by

ExA = sSet(sd ƥ, A).

22 LUKAS VOKRINEK

The adjunct to the natural transformation sdA → A is a natural transformation A → ExA.Iterating the Ex functor we therefore obtain a chain

A→ ExA→ Ex2A→ · · ·and define Ex∞A = colimn ExnA. It is also true that the natural transformation A→ Ex∞A isa weak homotopy equivalence with fibrant codomain but we will not prove this.

Lemma 5.2. The functor sd preserves injective maps and anodyne extensions.

Proof. The first claim follows from the injectivity of sd ∂∆n → sd ∆n, the second will follow fromthe fact that each sd Λk∆n → sd ∆n is anodyne. For simplicity put k = 0 and start addingsimplices according to their dimensions d. Within each dimension choose a simplex in sd ∂0∆n

that does not lie on the boundary. It is given by a chain S1 ⊆ · · · ⊆ Sd of subsets of {1, . . . , n}.Within this choice of a simplex add simplices in the order

{0} ⊆ {0} ∪ S1 ⊆ · · · ⊆ {0} ∪ SdS1 ⊆ {0} ∪ S1 ⊆ · · · ⊆ {0} ∪ Sd

· · ·S1 ⊆ S2 ⊆ · · · ⊆ Sd ⊆ {0} ∪ Sd

In other words in the same way one builds Λk∆n from its vertex using anodyne extensions onemay build the cone over it from the base giving a part of sd ∆n. Then one proceeds to build therest of sd ∆n which is the cone on ∂k∆n.

�

Proposition 5.3. The map sdK → K is a weak homotopy equivalence.

Proof. At this point we only give a proof for finite simplicial complexes. We will prove the generalcase later when we have stronger tools at our disposal. Observe that if K ⊆ L is a simplicialsubcomplex then L can be built from K in such a way that the simplices are attached alonginjective maps from ∂∆n.

We start from sd ∆n → ∆n being a weak homotopy equivalence (it has a section that is ananodyne extension). If L is obtained from K by attaching a single cell of dimension n then wehave pushout squares

sd ∂∆n //��

��

sdK∼ //

��

��

K

��

sd ∆n // sdL ∼// sd(L,K)

As the composition across the top row also factors through ∂∆n we may rewrite sd(L,K) as apushout in

sd ∂∆n ∼ //��

��

∂∆n // //��

��

K

��

∆n ∼ //

id ..

sd ∆n ∼ // • // //

��

sd(L,K)

��

∆n // L

By Lemma 5.4 the map • → ∆n is a weak homotopy equivalence (the map ∆n → • is injective sinceit has a left inverse) hence so is sd(L,K) → L. Composed with the weak homotopy equivalencesdL→ sd(L,K) above yileds the inductive step. �

Lemma 5.4 (a weak 2-out-of-3 property). Given Af−−→ B

g−−→ C the following holds. If both fand g are weak homotopy equivalences then so is gf . If f is an anodyne extension and gf is aweak homotopy equivalence then so is g. In particular the projection (K×∆1)∪K L→ L is alwaysa weak homotopy equivalence.

SIMPLICIAL SETS 23

Proof. The first is obvious by observing that vertical homotopies compose. For the other let therebe given a homotopy lifting problem on the left

Bl //

g

��

&fX

p

����

Alf

//

gf

��

&fX

p

����

Cm

//

88

Y Cm

//

d

88

Y

By composition with f we obtain the problem on the right which admits a solution as gf is a weakhomotopy equivalence. As the pair (B,A) × (∆1, ∂∆1) is anodyne we may extend the fibrewisehomotopy between lf ∼ dgf to a fibrewise homotopy l ∼ dg and d therefore serves as a solutionalso in the original homotopy lifting problem.

For the mapping cylinder projection we use K → (K × ∆1) ∪K L → L whose compositionis id and where the first map is a pushout of K → K × ∆1 that is the anodyne extension(K, ∅)× (∆1, 1). �

An important application is to the sets of homotopy classes of maps. If f : K → L is a weakhomotopy equivalence and X is a Kan complex then by definition every map K → X extendsup to homotopy to a map L → X. We will show now that this extension is in fact unique up tohomotopy. Assume that g0, g1 : L→ X are two maps such that g0f, g1f are homotopic. This datais organized into a map (K ×∆1) ∪K×∂∆1 (L× ∂∆1) −→ X whose domain fits into a diagram

K //��

∼��

L∼

++

��

∼��

(K ×∆1) ∪K L // (K ×∆1) ∪K×∂∆1 (L× ∂∆1) // L×∆1

By Lemma 5.4 the map on the right is a weak homotopy equivalence. Therefore an extension ofthe data could be found to a homotopy L × ∆1 → X between g0 and g1 (or in fact some mapshomotopic to them). This means precisely that the map

f∗ : [L,X]→ [K,X]

sending [g] 7→ [gf ] is a bijection. One may express this as weak homotopy equivalences beingviewed from X as homotopy equivalences.

Theorem 5.5 (Simplicial approximation). Let A be a finite simplicial complex thought of as asimplicial set and f : A→ S|K| a simplicial map (i.e. a continuous map |A| → |K|). Then thereexists k ≥ 0 and a homotopy commutative diagram

sdk A //

��

K

��

A //

;{

S|K|

If the map f was already simplicial (had image in K) on some subcomplex then there exists ahomotopy which is relative to (the subdivision of) this subcomplex.

Proof. Let ι : ∆+ → ∆ denotes the inclusion of the category of all non-empty finite ordinals andtheir injective non-decreasing maps. Then one has an adjunction

ι! : [∆op+ ,Set] � [∆op,Set] : ι∗

The counit ι!ι∗K → K is shown to be a realization homotopy equivalence (to be expanded upon;

the tools are model category theoretic and in Top). On [∆op+ ,Set] (i.e. on the image of ι!) there is a

natural homeomorphism | sdA| → |A| together with a natural homotopy to the realization of thecanonical map sdA→ A. In particular the realization of sdA→ A is a homotopy equivalence. Allfollows at once from the case of the simplex by cocontinuous extension where the homeomorphismshould be obvious and the homotopy is the canonical linear homotopy. Note however that the

24 LUKAS VOKRINEK

homeomorphism (and consequently also the homotopy) is not natural in [n] ∈ ∆ but only in[n] ∈ ∆+. One has a diagram

| sd2A| //

��

| sd2 ι!ι∗K|

∼��

|A| //

9y

|ι!ι∗K|

∼��

|A| //

8x

|K|If f was simplicial on a subcomplex then these homotopies may be chosen to be relative to (thesubdivision of) this subcomplex. Now the top row is a map between simplicial complexes by thenext lemma and thus admits a simplicial approximation by the classical theorem found e.g. in [H]for the absolute case but which works also in the relative case. �

Lemma 5.6. The second subdivision of a simplicial set in the image of ι! is always a simplicialcomplex.

Proof. We leave most of the details to the reader and just state various characterizations usefulfor the proof.

• Simplicial sets in the image of ι! are exactly those satisfying: a face of a non-degeneratesimplex is again non-degenerate (to construct a preimage take all the non-degeneratesimplices).

• Simplicial sets coming from simplicial complexes are precisely those which lie in the imageof ι! and admit at most one non-degenerate simplex with a given set of vertices.

• The first subdvision of a simplicial set in the image of ι! satisfies the following condition:the vertices of each non-degenerate simplex are all different.

• If a simplicial set satisfies the condition from the previous point then its subdivision is asimplicial complex.

�

Theorem 5.7. For each simplicial set K the unit of the adjunction K → S|K| is an anodyneextension.

Proof. We need to prove the following extension property where X → Y is an arbitrary Kanfibration.

K //

��

X

����

S|K| // Y

Essentially we construct a deformation of S|K| onto K. If K was a Kan complex this would bean actual deformation but for a general simplicial set K we need to use subdivision to make thisprecise. We describe now a subdivided version of a cylinder that we will be using in this proof.

We denote the cylinder by cylk A and it will have two ends sdk A∼−→ cylk A and A

∼−→ cylk A. Itis defined inductively starting from cyl0A = A with the next step constructed from the previousone as the pushout

sdk+1Aε

∼ //��

∼��

sdk A∼ // cylk A

��

sdk+1A∼ // sdk+1A×∆1 // cylk+1A

The remaining maps are thus also weak homotopy equivalences. Also important is the fact thatthere is a projection cylk+1A → cylk A that squashes the newly added part. We assume thatfor all simplices ∆d → S|K| with d < n there are given maps cylk ∆d → S|K| together with

SIMPLICIAL SETS 25

lifts cylk ∆d → X whose restriction to sdk ∆d lies in K and whose restriction to ∆d is the givensimplex. To get an actual deformation we have to assume some form of compatibility with faceand degeneracy operators. Note however that k might vary so that this is not as straightforwardas one might hope. Using the projections we see that every cylk ∆d → S|K| yields also a map

cylk+1 ∆d → S|K|. We may thus express the compatibility by passing to a common subdivisionexponent and ask for strict equality. We assume that this is satisfied inductively.

For a simplex ∆n → S|K| we then have by induction a map cylk ∂∆n → S|K| and one

can extend these two maps to cylk ∆n → S|K| using the fact that (cylk ∆n,∆n ∪ cylk ∂∆n)is anodyne. By the previous lemma one may subdivide further and make a homotopy fromsdk ∆n → cylk ∆n → S|K| to a simplex in K that is relative to sdk ∂∆n. This can be expressed

by increasing k and extending to a map cylk ∆n → S|K| whose restriction to sdk ∆n lies in K.We therefore obtain a diagram

sdk ∂∆n //��

∼��

sdk ∆n //��

∼

��

K

!!

��

cylk ∂∆n //

%%

• //��

∼��

X

����

cylk ∆n //

55

S|K| // Y

(the map sdk ∆n → cylk ∆n is a weak homotopy equivalence; by Lemma 5.4 so is • → cylk ∆n).The indicated lift then provides the induction step. In the case that the original simplex liedin K one could choose k = 0 and the extension is then forced equal to the value of K → X atthat simplex. A lift in the original diagram is given by taking for each simplex ∆n → S|K| thecomposition ∆n →Mk∆n → X. �

We define now a weak homotopy equivalence of topological spaces. The most direct definitionis that of a map which becomes a homotopy equivalence upon the application of the functor Sof the singular simplicial set. One can show that this may be equivalently expressed8 using thehomotopy right lifting property against all injective maps, i.e.

A //��

��

SU

��

B //

77

ST

g'

where the homotopy is required to be constant on B. We will relate this to the more usualdefinition using the adjunction between the geometric realization and the singular simplicial set.It gives an equivalent lifting problem

|A| //

��

��

U

��

|B| //

88

T

f&

We may interpret this property in the generating case of A → B being the boundary inclusion∂∆n → ∆n as follows. Assume for simplicity that U → T is injective. The solid diagram thenrepresents an element of πn(T,U) and the existence of the lift says that this element should betrivial. Therefore an (injective) continuous map U → T is a weak homotopy equivalence if andonly if all the relative homotopy groups πn(T,U) are trivial or equivalently if U → T induces anisomorphism on all homotopy groups. This translation can be easily extended to the case of anon-injective map (say by replacing the map by the inclusion into the mapping cylinder).

8Hint: use the factorization of SU → ST into a homotopy equivalence followed by a fibration as above andobserve that the homotopy equivalence admits a retraction that is a trivial fibration.

26 LUKAS VOKRINEK

Corollary 5.8. The counit |ST | → T is a weak homotopy equivalence.

Proof. The composition ST → S|ST | → ST is the identity while the first map is a weak homotopyequivalence by the previous theorem. Since all simplicial sets are Kan complexes, it is a homotopyequivalence and thus S|ST | → ST is its homotopy inverse. �

Corollary 5.9. Denoting the classes of weak homotopy equivalences by W ⊆ sSet and W ⊆ Topthe adjunction | − | a S induces an equivalence of categories

| − | :W−1sSet oo' // W−1Top : S

Proof. The localization sSet → W−1sSet is universal among functors out of sSet that send theelements of W (i.e. the weak homotopy equivalences) to isomorphisms. The functor | − | of geo-metric realization therefore induces | − | :W−1sSet→W−1Top because the geometric realizationof a weak homotopy equivalence is a homotopy equivalence, | − |(W) ⊆ W, which becomes an iso-morphism in W−1Top. As also S(W) ⊆ W by definition, the other functor passes to one betweenlocalizations. Since the unit and the counit are natural transformations whose all components areweak homotopy equivalences they give natural isomorphisms on the localizations. �

At this point it is worth to describe the localizations more concretely. In the localizationW−1sSet homotopic maps f0, f1 become the same for they fit into the diagram

fk : Kik // K ×∆1 h // X

where both i0 and i1 are right inverses of a weak homotopy equivalence pr : K ×∆1 → K, i.e. anisomorphism inW−1sSet, and therefore i0 = i1 there. For a Kan complex X and a weak homotopyequivalence f : K → L the induced map f∗ : [L,X] → [K,X] is a bijection. This implies that[−, X] is in fact a functor (W−1sSet)op → Set and thus two maps K → X equal in the localizationW−1sSet have to be homotopic. Together we have

W−1sSet(K,X) = [K,X]

In particular restricting to Kan complexes the localization is the same as the homotopy category.To describe abstractly what happens for simplicial sets that are not Kan complexes we simplyrecall that every map has a factorization as a trivial cofibration followed by a Kan fibration.Applied to K → ∗ this yields an (injective) weak homotopy equivalence K → X from K toa Kan complex X. In the localization every object is therefore isomorphic to a Kan complexand we may describe W−1sSet up to equivalence of categories as the homotopy category of Kancomplexes (i.e. morphisms are the homotopy classes of maps). Similarly W−1Top is equivalent tothe homotopy category of CW-complexes (or realizations of simplicial sets say).

6. The saturation of weak homotopy equivalences

There is one more important property of the localization. The class W is so called satu-rated. This means that it consists of all morphisms that become isomorphisms in the localizationW−1sSet. This is reduced to homotopy equivalences by passing to topological spaces and usingthe following theorem.

Theorem 6.1. A realization homotopy equivalence is a weak homotopy equivalence.

If we had the full 2-out-of-3 property of Theorem 6.3 we could prove the theorem easily fromthe square

K // ∼ //

��

S|K|

∼��

L //∼ // S|L|

(the map on the right is a homotopy equivalence between Kan complexes as the functor S preserveshomotopy equivalences). In fact we prove Theorem 6.3 as a corollary to the recognition theoremfor weak homotopy equivalences.

SIMPLICIAL SETS 27

Proof. By factoring through the mapping cylinder we reduce to

A //��

��

X

����

B //

>>

Y

where we produce the solution inductively. Suppose first that B is obtained from A by attachinga single simplex. Then by the simplicial approximation theorem there exists a homotopy

sdk(∆n ×∆1) −→ B

which maps sdk(∆n × 1) to A and restricts to the standard map

sdk(∆n ∪ (∂∆n ×∆1)) −→ ∆n ∪ (∂∆n ×∆1) −→ ∆n −→ B.

Therefore we have a lift in

sdk ∂∆n //

��

sdk((∂∆n ×∆1) ∪∆n) //

��

∼��

A //

��

X

����

sdk ∆n // sdk(∆n ×∆1) //

44

B // Y

We can then solve the following problem with sdk(∆n, ∂∆n) = sdk ∆n∪sdk ∂∆n ∂∆n that admits by

Proposition 5.3 and the left properness Theorem 4.2 a weak homotopy equivalence from sdk ∆n andhence also from ∆n. In particular the map to ∆n is a weak homotopy equivalence by Lemma 5.4.

sdk(∆n, ∂∆n) //

∼��

%eX

����

∆n //

99

Y

One can make the homotopy relative to the boundary ∂∆n according to Lemma 6.4. This makesthe lift compatible with the map A→ X and one thus obtains a lift in the original diagram.

To prove the general case we invoke the maximality principle – there exists a maximal extension.It therefore must be defined on the whole of B as otherwise it could be further extended by theabove argument. �

Theorem 6.2 (saturation). The class of weak homotopy equivalences of simplicial sets is satu-rated. A map is a weak homotopy equivalence if and only if it becomes an isomorphism in thelocalization W−1sSet. �

In particular (and this will be used a lot) it satisfies the following 2-out-of-3 property.

Theorem 6.3 (2-out-of-3 property). If in a commutative triangle two out of the three maps areweak homotopy equivalences then so is the third. �

Another important corollary is that we may compute the homotopy groups of geometric real-izations of Kan complexes simplicially. If X is a pointed Kan complex then

πn|X| = [|∆n/∂∆n|, |X|]∗ ∼= [∆n/∂∆n, S|X|]∗ ∼= [∆n/∂∆n, X]∗

as all the sets of homotopy classes of maps are equal to the morphisms sets in the localizations. Thiswas proved by Milnor. Note that if X is not Kan the last isomorphism does not hold but one mayinstead approximate this by colimk[sdk(∆n/∂∆n), X]∗. There is however no bound on the numberof iterations needed to compute all the elements. As we saw for the circle ∆1/∂∆1 after subdividing

k-times, there is still only a finite number of homotopy classes of maps [sdk(∆1/∂∆1),∆1/∂∆1]∗whereas π1|∆1/∂∆1| = Z is infinite. On the other hand we may conclude from this that πnS

m = 0

when n < m simply because there is no non-trivial map sdk(∆n/∂∆n)→ ∆m/∂∆m.

28 LUKAS VOKRINEK

Lemma 6.4. If the homotopy in the diagram

K //

f

��

`X

����

L //

>>

Y

is fibrewise and if A ⊆ K is a simplicial subset on which f is injective then one may find a lift upto homotopy that is relative to A.

Proof. Start with (K×∆1)∪L = (K×∂1∆2)∪ (L×0) and attach A× (∆2/∂0∆2) along anodyneA× ∂1∆2. Next attach L× ∂2∆2 along anodyne (L× 0) ∪ (A× ∂2∆2) and finally attach K ×∆2

along anodyne (K × Λ0∆2) ∪ (A×∆2) to obtain the anodyne extension in

(K × ∂1∆2) ∪ (L× 0) //

��

∼��

X

����

(K × ∂0∆2) ∪ (L× 1) // (K ×∆2) ∪ (L× ∂2∆2)/ ∼ //

44

L // Y

where the quotient identifies A× ∂0∆2 to A. Therefore the composition of the diagonal with thehorizontal map on the left produces a lift with a fibrewise homotopy that is also relative to A. �

We would like to characterize the existence of the lift in a very conceptual way which also givesmuch stronger conclusions than just the mere existence of the lift. We express the existence of alift in the square

Af

//��

i

��

X

p

����

Bg

//

88

Y

where the map on the left is injective and the one on the riight is a Kan fibration. A lift in thisdiagram is a 0-simplex of hom(A,X) and the fact that the two squares commute is expressed inthat it maps to a specified 0-simplex (f, g) of

hom(A,X)×hom(B,X) hom(B, Y ).

Theorem 6.5. The map hom(B,X) −→ hom(A,X)×hom(B,X) hom(B, Y ). is a Kan fibration thatis trivial (and in particular surjective) if one of i, p is trivial.

Proof. By adjointness all claims the first claim is equivalent to a product (L,K)× (B,A) of anyanodyne pair with (B,A) being again anodyne, one of the “trivial” cases is just symmetric andthe remaining one reduces to a product of pairs being a pair via the characterization of trivialfibrations as maps having the right lifting property against injective maps. �

This yields a stronger version of the lifting properties. If one of i, p is trivial then one canconclude for example that any two lifts are fibrewise homotopic relative to A as the fibres of themap from the statement (the spaces of lifts in the original square) are contractible. Suppose nowthat none of i, p is trivial. Then the theorem still claims that the map is a Kan fibration. Usingthe left lifting property against (∆1, e) this means that a solution exists for a pair (f, g) if andonly if it exists for a homotopic pair (f ′, g′). This is sometimes useful.

The case when i is not injective and we are asking for solutions up to a fibrewise homotopy couldbe also expressed via this theorem. Here we will describe the dual situation under the conditionthat both X and Y are Kan complex. In that case we ask for a lift up to a homotopy of mapB → Y that is constant on A (we say that the homotopy is relative to A). The solution is then a

SIMPLICIAL SETS 29

strict lift in

Af//

��

i

��

X // hom(∆1, Y )×Y X

����

Bg

//

55

Y

Again a solution exists for (f, g) if and only if it exists for a homotopic one (f ′, g′).

Remark. The space of lifting problems for hom(∆1, Y ) ×Y X → Y above is in fact the space ofhomotopy commutative squares (with the homotopy as part of the data). A solution for such alifting problem is then a map B → X together with a homotopy of maps B → Y that restrictsto the given homotopy on A (in the classical case the homotopy in the problem is constant andthat is why we require the homotopy in the solution to be relative to A). As every homotopycommutative square is homotopic to a commutative one we actually see that the existence of thesemore general lifting problems (those for homotopy commutative square) follows from the existenceof the classical lifting problems.

There is a yet more general version with i not injective and in that case the problem consistsof a homotopy commutative square, the solution of a map B → X and two homotopies, one ofmaps A→ X and one of maps B → Y together with two homotopies of homotopies between theirimages in hom(A, Y ) and the given homotopy from the problem.

There is a dual theorem to the above claiming that sSet is right proper. It depends on adual characterization of weak homotopy equivalences. A map f : X → Y is a weak homotopyequivalence if and only if |f | : |X| → |Y | is a homotopy equivalence. One may express thisequivalently by the requirement that for every commutative solid square

A //��

��

S|X|

��

B //

88

S|Y |f&

there exists a lift up to homotopy relative to A. If A and B are finite simplicial complexes one mayuse the simplicial approximation theorem to replace this problem (after passing to subdivisions)by a homotopic one with S|X| replaced by X and S|Y | by Y . A solution of such a problem shouldbe sought however in S|X| which we can express by passing to further subdivisions. Reducing atthis point to boundary inclusions we proved

Theorem 6.6. A simplicial map f : X → Y is a weak homotopy equivalence if and only if foreach commutative square

∂∆n //��

��

X

��

∆n //

77

Y

g'

there exists a lift up to a “homotopy” relative to ∂∆n after taking a sufficient number of subdivisionsof ∆n ×∆1 (the domain of the homotopy). �

The subdivisions are necessary unless Y (and hence also X) is a Kan complex. One mayextend this characterization to injective maps of finite simplicial sets word by word but to extendto arbitrary injective maps one must allow an “infinite iteration” of the subdivision process – thisonly makes sense by considering the right adjoint to sd called Ex, applying it on the right, anditerating it there. One obtain the so called Kan’s Ex∞-functor. Therefore a map f : X → Y isa weak homotopy equivalence if and only if Ex∞ f : Ex∞X → Ex∞ Y has the homotopy rightlifting property against all injective maps, i.e. if and only if Ex∞ f is a homotopy equivalence.

30 LUKAS VOKRINEK

Theorem 6.7. The category of simplicial sets is right proper. Given a pullback square

K //

��

L

∼��

X // // Y

with L → Y a weak homotopy equivalence its pullback K → X is a weak homotopy equivalenceprovided that X → Y is a Kan fibration.

Proof. With the above characterization the proof is dual: one lifts the subdivided homotopysdk(∆n×∆1)→ Y to X to yield a compatible map sdk ∆n → K at its end. The lifting is possible

because sdk ∆n → sdk(∆n ×∆1) is an anodyne extension. �

A word of motivation: in homotopy theory problems are quite often expressed in terms of theexistence of a ceratin map (which typically has some interpretation – a bundle, a cohomology classetc.). Another common feature is the use of an inductive construction of such a map which it thenexpressed as an extension problem

Af

//

i

��

X

B

88

In general one may expect such an extension to exist only if the map i : A → B is injective.For non-injective maps one could still ask that the triangle should commute up to homotopy. Wewill see now that for an injective i this amounts to the same: if there exists an extension up tohomotopy then there exists a strict extension (and in fact one that is homotopic to the original “upto homotopy” extension). This is best formalized in the so-called homotopy extension property.In the above extension problem the “up to homotopy” extension together with the homotopyconstitute exactly the horizontal map in the following diagram.

(A×∆1) ∪A B //

��

X

B ×∆1

66

A simplicial set X is called a Kan complex if for every injective i : A→ B the above left extensionexists. This extension is a homotopy starting at the “up to homotopy” extension in the originaldiagram and ending at a strict extension that we were looking for.

At this point we may say what a weak homotopy equivalence is: it is a map i : A → B forwhich i∗ : [B,X] → [A,X] is a bijection for all Kan complexes X. It is however hard to proveanything with this definition.

More generally we speak of a lifting problem and its relative version where a partial lift hasbeen already specified and which we want to extend

Af

//

i

��

X

p

��

Bg

//

88

Y

Again such problems have typically some interpretation such as a reduction of a bundle, a reductionof coefficients, a nullhomotopy etc. If this extension is again only up to homotopy (which fortechnical reasons should be fibrewise) then we may ask whether a strict extension could be found.

SIMPLICIAL SETS 31

This again happens to be

(A×∆1) ∪A B //

��

X

��

B ×∆1 //

55

B // Y

A map X → Y which has the right lifting property against all maps on the left is called a Kanfibration. We therefore see that if in the square as above A→ B is injective and X → Y is a Kanfibration the problem of finding a lift up to homotopy is quite the same as that of finding a strictlift (if in general we do not want to insist on fibrewise homotopies there is a problem of coherence.If the two triangles commute up to a homotopy then these should be constant when composed togive a homotopy of maps A→ X or at least homotopic to one another).

This is very useful because certain lifts up to homotopy should be expected to exist when oneof the maps in question, say p : X → Y is moreover a homotopy equivalence with a homotopyinverse s. In that case the diagonal up to homotopy could be chosen to be sg. By the above astrict lift should exist but it is somewhat technical to make one of the homotopies constant and theother fibrewise. In any case the claim is correct and in the case that p is a homotopy equivalencea lift exists always. Let us consider on the other hand a Kan fibration p for which a lift existsagainst all i : A → B injective. If we were able to apply the lifting property against p itself wewould get that p is in fact an isomorphism. As p is not injective and we have to “replace” it byan injective map. This could be always achieved up to a homotopy equivalence via the so calledmapping cylinder (X ×∆1) ∪X Y which we have already met above. It is true that this mappingcylinder deforms onto its base Y . We have a diagram

X

p

uu��

X

p

��

Yin //

id

22(X ×∆1) ∪X Y //

55

i)

Y

where the composition of the inclusion of Y into the mapping cylinder with the dashed lift yieldsa homotopy inverse for p. In this case the homotopy equivalences are characterized by the liftingproperty. Now we turn to the dual situation. Does every injective map having a left liftingproperty against all Kan fibrations have to be a homotopy equivalence? This time the answer isno as not every map can be replaced by a Kan fibration up to a homotopy equivalence. Thosemaps having the left lifting property against all Kan fibrations are called anodyne extensions ortrivial cofibrations. They will be exactly the injective weak homotopy equivalences. We may nowdefine weak homotopy equivalences as those simplicial maps that have the following homotopy leftlifting property

Af

//

i

��

&fX

p

��

Bg

//

88

Y

For each strictly commutative square there exists a lift making the lower triangle commute strictlyand the upper one up to a homotopy. We saw that it is technically advantageous to require thehomotopy to be “fibrewise”, i.e. such that its composition with p is a constant homotopy. In thiscase when i is injective we may always replace the lift by a homotopic one for which both trianglescommute strictly and therefore an injective homotopy equivalence is indeed a trivial cofibration.

7. Summary

Important to the theory is that it is equivalent for a map to be a weak homotopy equivalenceand to be a homotopy equivalence upon applying the geometric realization functor. Only from

32 LUKAS VOKRINEK

this it is clear that every homotopy equivalence is in fact a weak homotopy equivalence (since thefunctor of geometric realization preserves homotopies and consequently homotopy equivalences).

The important properties that we proved are encoded in the notion of a model category. It is acategory with three subcategories: C of cofibrations, F of fibrations and W of weak equivalences.In our case these are the injective maps, Kan fibrations and weak homotopy equivalences. Themaps that are at the same time cofibrations and weak equivalences are called trivial cofibrations,similarly for trivial fibrations. The important axioms are:

• lifting: the following lifting properties hold

(C ∩W) � F, C � (F ∩W),

namely trivial cofibrations have the left lifting property against fibration and similarlycofibrations against trivial fibrations,

• factorization: every map has a factorization as a trivial cofibration followed by a fibrationand also a factorization into a cofibration followed by a trivial fibration,

• 2-out-of-3: if in a commutative triangle two out of three maps are weak equivalences, so isthe third (in our case this follows from the characterization of weak homotopy equivalencesas maps whose geometric realization is a homotopy equivalence),

• retracts: the classes C, F and W are closed under retracts.

Most of the homotopy theory can be done just with these axioms as we will see shortly in someexamples.

The importance of model categories lies also in comparison of the “homotopy categories”, moreprecisely the localizations. In our case we prove that the adjunction | − | a S is a “Quillenequivalence”, most importantly it induces an equivalence

W−1sSet'←−→W−1Top

Now if f : K → L is a weak homotopy equivalence and X is a Kan complex the induced map onohomotopy classes of maps

f∗ : [L,X]→ [K,X]

is bijective. This is easily seen in the case that f is also injective, i.e. an anodyne extension. Thenany map K → X extends to L by

K //��

∼��

X

����L //

>>

∗and if two maps g0, g1 : L→ X are given whose restrictions g0f, g1f are homotopic then the mapsg0, g1 together with the homotopy can be organized into a map (K×∆1)∪ (L× ∂∆1)→ X whichcan be extended to a map L ×∆1 → X, i.e. to a homotopy between g0 and g1. This is because(L,K)× (∆1, ∂∆1) is an anodyne extension.

This is our third point of view on weak homotopy equivalences – maps which induce a bijectionf∗ : [L,X]→ [K,X] for every Kan complex X. This implies quite easily that

W−1sSet(K,X) = [K,X]

whenever X is a Kan complex. On one hand we need to show that homotopic maps become thesame in the localization. This is a very nice argument regarding the cylinder K ×∆1. The twoinclusions of its ends fit into

K

i0��

id

{{

g0

((K K ×∆1poo h // X

K

i1

OO

id

cc

g1

66

SIMPLICIAL SETS 33

showing that i0, i1 are both right inverses to p which is a weak homotopy equivalence, hence anisomorphism in W−1sSet. This forces i0 to be equal to i1 in this localization and consequentlyg0 = hi0 = hi1 = g1 in W−1sSet.

Thus the localization functor sSet→W−1sSet factors through the homotopy category,

sSet //

((

Ho(sSet)

��

W−1sSet

and W−1sSet is in fact a localization of Ho(sSet) (at the images of weak homotopy equivalences).Since every Kan complex X already “sees weak homotopy equivalences as isomorphisms”, thereis nothing to localize and this yields

W−1sSet(K,X) = [K,X].

Similarly W−1Top(U, T ) = [U, T ] whenever U is a CW-complex (or a geometric realization of asimplicial set). This gives a way of “computing” the maps in the localization W−1sSet also forsimplicial sets that are not Kan complexes:

W−1sSet(K,L) ∼=W−1sSet(K,S|L|) ∼=W−1sSet(|K|, |L|) = [|K|, |L|].

In particular one may “compute” homotopy groups in this way – but without the requirementthat the target is a Kan complex this does not help much in the actual computation.

8. Resolutions

Let us start with an example of a resolution that everybody should know, the bar resolution ofZG. It is given by the chain complex

· · · −→ ZG⊗Z ZG⊗Z ZG −→ ZG⊗Z ZG −→ ZG

where on the group elements the maps are given g ⊗ h ⊗ k 7→ gh ⊗ k − g ⊗ hk and g ⊗ h 7→ gh.When tensored (over ZG) with Z it gives the complex

· · · −→ ZG⊗Z ZG −→ ZG −→ Z

whose homology groups are the homology groups of G. Similar resolutions can be constructedin various other contexts via simplicial methods. The resolution is constructed by iterated ap-plications of the functor ZG ⊗Z − : ZG−mod → ZG−mod. Firstly it does not depend on theZG-module structure and factors through Z-modules, namely in the free-forgetful adjunction

L : Z−mod � ZG−mod : R

it is the composition LR. Its most significant properties are summarized in: “LR is a comonad”(a short definition is: a composition of a right and left adjoint). We will first talk about monadsas these are closer to algebraic way of thinking, i.e. we will summarize the important properties ofthe composition RL. As part of the adjunction there are given natural transformations: the unitη : Id→ RL and the counit ε : LR→ Id out of which one constructs µ = RεL : RLRL→ RL.

Definition 8.1. A monad on a category C is an endofunctor M : C → C together with naturaltransformations η : Id→M and µ : M2 = M ◦M →M for which the diagrams

MηM//

id!!

M2

µ

��

MMηoo

id}}

M3 µM//

Mµ

��

M2

µ

��

M M2µ// M

are commutative. One can express this as M being a monoid in the (monoidal) category (End C, ◦)of endofunctors together with their composition.

34 LUKAS VOKRINEK

From this it follows that there is a unique natural transformation µ : Mk →M for each k ≥ 0(unique among transformations formed out of M , η and µ). It is given e.g. as a composition

Mk Mk−2µ−−−−−−→Mk−1 −→ · · · −→M2 µ−−→M

(or η if k = 0). The axioms say that any other way of forming a natural transformation results inthe same.

Example 8.2. A lot of (universal) algebra can be expressed in terms of monads. Speakingconcretely of groups say we consider the free-forgetful adjunction

L : Set � Gp : R

where R associates to a group its underlying set and L associates to a set S the free group LSgenerated by this set. Concretely LS is a set of certain (equivalence classes9 of) terms whoseindeterminates are the elements of the set S. The natural transformation η is simply the inclusionof the generators in the free group. The natural transformation µ is more complicated. It isclosely related to the way a free group is a group. One may evaluate every term in elements ofMS = RLS and obtain again an element of MS. This describes precisely µS : M2S → MS. Asan example

(aba−1)−1(ab)(ba)−1 7−→ b−1

where the expressions inside the brackets are elements of MS (i.e. terms in elements of S) and thewhole expression on the left is thus an element of M2S, the right hand side is just the “evaluation”of the term on the left in the free group LS.

It is of course essential for the translation of universal algebra into the language of monads thatalgebras (groups in our example) can be expressed in a very simple way in terms of the monad.Concretely to specify a group structure on a set A is to associate to each term, i.e. an element ofMA, its evaluation in A. This is simply a map MA → A. There are only two constraints, theelements of A viewed as terms have to evaluate to the elements themselves and there is a furtheraxiom about terms of terms: it amounts to the same to compose terms and then evaluate theresult or to evaluate the inside terms (those in brackets) and evaluate the resulting term again.This is formalized in the definition of an algebra for a monad.

Definition 8.3. An M -algebra structure on A ∈ C is a morphism a : MA→ A for which

AηA//

id!!

MA

a

��

M2AMaoo

µA

��

A MAa

oo

is commutative.

As an example µ : M2A → MA is an M -algebra structure on MA – it is the free M -algebraon A. The square on the right then says that a : MA → A is an M -algebra morphism from thefree M -algebra MA to A (expressing A as a “quotient” of the free algebra MA).

Again there is a “unique” morphism a : MkA→ A for each k ≥ 0 defined e.g. as

MkAMk−1a−−−−−−→Mk−1A→ · · · →MA

a−−→ A

(or id for k = 0). Again one may use any other composition made out of a, M , η and µ; theaxioms imply that one gets the same.

We will relate now monads to (augmented) simplicial objects. We consider the category ∆ε ofall finite ordinals, denoting the empty one by [−1] = ∅. The main idea (which we will not useexplicitly) is that ∆ε is a free strict monoidal category on one object [0]. Since (End C, ◦) is a

9The relation is by identities that hold in all groups. Equivalently we may agree on choosing representatives for

the equivalence classes say by terms written in some normalized way – for groups this could be words composed ofelements of S and their formal inverses where it does not happen that an element appears right next to its inverse.

SIMPLICIAL SETS 35

monoidal category with an object M there has to be a unique strict monoidal functor ∆→ End Csending [0] to M . We will describe this functor explicitly

[−1] 7−→ Id

[0] 7−→M

[n] 7−→Mn+1

(as the monoidal structure on ∆ is that of an ordinal sum and [n] is the (n+ 1)-fold ordinal sumof [0] the value has to be Mn+1). More interesting are the values on morphisms. Let θ : [m]→ [n]be a morphism in ∆ε. Then the natural transformation associated to θ is given by decomposing[m] into a disjoint union of the preimages θ−1(i) and applying µ on each of them

Mm+1 = M |θ−1(0)| ◦ · · · ◦M |θ

−1(n)| µ◦···◦µ−−−−−−→M ◦ · · · ◦M = Mn+1

As an example there is a unique morphism [1] → [0] whose value is µ : M2 → M and there areexactly two morphisms [0] → [1] whose values are Mη, ηM : M → M2. The fact that this reallydescribes a functor BM : ∆ε → End C follows from the uniqueness of µ : Mk →M . By evaluatingat c ∈ C we obtain BMc : ∆ε → C.

Definition 8.4. A functor ∆ → C is called a cosimplicial object in C. A simplicial object is afunctor ∆op → C (so that a simplicial set is a simplicial object in Set). An augmented simplicialobject is a functor X : ∆op

ε → C. By restriction it has an underlying simplicial object. The imageof [−1]→ [0] is a morphism X0 → X−1 called an augmentation.

Let A ∈ C and consider the composition

BMA : ∆εBM−−−−→ End C evA−−−→ C

If A is an M -algebra, there is an extension of the diagram BMA to a diagram ∆> −→ C where∆> is the category of all non-empty finite ordinals and all the non-decreasing maps between themwhich preserve the bottom element. We denote [n]∗ = {0 < · · · < n <∞}. When η : [m]∗ → [n]∗we have a decomposition of [m]∗ into the disjoint union of η−1(i) where η−1(∞) contains ∞.Therefore we may define the value of BMA on η by

Mm+1A = M |η−1(0)| ◦ · · · ◦M |η

−1(n)| ◦M |η−1(∞)|−1A

µ◦···◦µ◦a−−−−−−−→M ◦ · · · ◦MA = Mn+1A

We say that BMA is an augmented cosimplicial object with extra codegeneracies. There is a“dual” version where the base point is the bottom element. We denote the category by ∆⊥ andwrite [n]∗ = {−∞ < 0 < · · · < n}. There is a situation in which cosimplicial objects augmentedin this way appear naturally. This provides a generalization of the case of algebras. We say that afunctor X : D → C is a left M -module is there are given natural transformations x : MkX → X,k ≥ 0, satisfying the axioms of an algebra. Dually Y : C → E is a right M -module if naturaltransformations y : YMk → Y , k ≥ 0, are provided that satisfy the usual axioms of a rightmodule. Prime examples are the adjoint functors L,R when M = RL; namely L is a left RL-module via εL : LRL → L and R is a right RL-module via Rε : RLR → R. We define theone-sided bar constructions

B(Id,M,X)n = Mn+1X, B(Y,M, Id) = YMn+1

The first case is essentially the same as the algebra case so we only write the second in more detail

YMm+1 = YM |η−1(−∞)|−1 ◦M |η

−1(0)| ◦ · · · ◦M |η−1(n)| y◦µ◦···◦µ−−−−−−−→ YM ◦ · · · ◦M = YMn+1

As the notation suggests there is a zero-sided bar construction B(Id,M, Id) which is the one westarted with and also a two-sided bar construction

B(Y,M,X)n = YMn+1X

which has two kinds of extra degeneracies and as such is a functor ∆⊥,> → [D, E ]. The point of∆⊥,> is that it is isomorphic to ∆op. We have similarly ∆op

>∼= ∆⊥ and ∆op

⊥∼= ∆>.

Lemma 8.5. There is an isomorphism ∆op>

∼=−−→ ∆⊥.

36 LUKAS VOKRINEK

Proof. The isomorphism is the identity on objects and sends η : [m]∗ → [n]∗ to η∗ : [n]∗ → [m]∗defined by η∗j = min{i | ηi ≥ j}. �

Dually a comonad is a functor T : C → C together with comultiplication maps δ : T → T k whichsatisfy the obvious duals of the defining equations for a monad (monoid). We then define for eachc ∈ C an augmented simplicial object BTc : ∆op

ε → C by (BTc)n = Tn+1C and for θ : [m] → [n]we define θ∗ : (BTc)n → (BTc)m as

Tn+1cδ◦···◦δc−−−−−−→ T |θ

−1(0)| ◦ · · · ◦ T |θ−1(n)|c = Tm+1c

(one could say that a comonad is a monad in the opposite category and then replace ∆ε → Cop

by ∆opε → C). There are again one-sided bar constructions when X is a left T -comodule with

x : X → T kX or Y is a right T -comodule with y : Y → Y T k. We will mostly use

B(R,LR, Id) : ∆op⊥ → [C, E ], [n]∗ 7→ R(LR)n+1

The point of simplicial objects with extra degeneracies is that they are “contractible”, moreprecisely they contract onto their augmentation. We will explain this for augmented objects with

extra degeneracies ∆op⊥ → E . There are two obvious simplicial objects one may obtain from X, the

most obvious one being the “underlying” simplicial object X obtained via the inclusion ∆→ ∆⊥sending [n] to [n]∗ and θ to θ∗ that restricts to θ on finite elements and sends −∞ to −∞ asit should. There is also a constant functor ∆ → ∆⊥ sending everything to the zero (i.e. bothinitial and terminal) object [−1]∗ = {−∞}. We denote the composition cstX−1 as it is a constantsimplicial object at the value X−1 of the augmented simplicial object at [−1]. There are also twoobvious natural transformations (coming from the fact that [−1]∗ is both initial and terminal)that yield

cstX−1 → X → cstX−1

whose composition is the identity. The main point of the theory is that the other compositionX → cstX−1 → X is homotopic to the identity. This has to be defined first. Assume at this pointthat E has all small coproducts and observe that there is a canonical bifunctor

− · − : Set× E → E , S · e =∐s∈S

e,

the coproduct of #S copies of e. We might think of this functor as an “action” of Set on E . Itinduces an obvious action on functor categories

− · − : sSet× sE → sE , (K ·X)n = Kn ·Xn.

In this way we may define a simplicial homotopy between two simplicial maps f0, f1 : X → Ybetween simplicial objects in E to be a simplicial map

∆1 ·X → Y

that restricts to fi on {i} · X. This agrees with our previous definition when E = Set, so thatsE = sSet, as K ·X = K ×X.

Proposition 8.6. Let there be given a simplicial object X : ∆op → E in E together with anaugmentation by X−1 and extra degeneracies. Then the induced canonical maps X → cstX−1 andcstX−1 → X are mutual homotopy inverses.

Proof. The component of the homotopy ∆1 ·X → X in dimension n is to be a morphism

sSet(∆n,∆1) ·Xn → Xn

or as a collection of morphisms Xn → Xn indexed by morphisms [n]→ [1] in ∆. For a morphismthat sends 0, . . . , i to 0 and i+ 1, . . . , n to 1 the morphism Xn → Xn is induced by a morphism in∆⊥ that sends

−∞ <_

��

0 <H

��

· · · < i </

ww

i+ 1 <_

��

· · · < n_

��−∞ < 0 < · · · < i < i+ 1 < · · · < n

SIMPLICIAL SETS 37

�

Remark. There is an equivalent definition of a simplicial homotopy that is given as a specification of a number of morphismshi : Xn → Xn+1 (which should remind you of a chain homotopy). To explain this one defines a functor −⊗− : sSet×E → sEgiven by K ⊗ e = K · cst e or more concretely

(K ⊗ e)n = Kn · e.There is then a generalized version of the canonical presentation, namely the isomorphism∫ [n]

∆n ⊗Xn

∼= X.

Applying this to the domain X of a homotopy ∆1 ·X → Y gives∫ [n]

(∆1 ×∆

n)⊗Xn = ∆

1 ·∫ [n]

∆n ⊗Xn −→ Y.

Therefore to give a simplicial homotopy is the same as to give a (compatible) collection of morhpisms (∆1×∆n)⊗Xn → Y .

Since ∆1×∆n is spanned by a number of (n+1)-simplices this is the same as to give a (compatible) collection of morphisms

∆n+1 ⊗Xn → Y or equivalently Xn → Yn+1. The compatibility relations are written for example in [W] in section 8.3.1.

The last proposition has a number of corollaries. The first is classical homological algebra.

Remark. Let A ∈ sA be a simplicial object in an abelian category A. We define two chaincomplexes associated with X, the first one is called the Moore complex and denoted by the sameletter X. It has the differential

d = d0 − d1 + · · ·+ (−1)ndn.

The second is called the normalized chain complex and is denoted NA. It is given naturally as asubcomplex of A by

(NA)n =

n−1⋂i=0

ker di.

There is a second definition NA = A/DA as a quotient complex of A by its subcomplex generatedby all the degenerate simplices. The composition NA → A → A/DA is an isomorphism. Thefamous Dold-Kan correspondence says that the functor

N : sA → Ch+A

associating to each simplicial object in A its (non-negatively graded) normalized chain complexis an equivalence of categories – one might reconstruct the simplicial object A from NA up to anisomorphism! Secondly the inclusion NA → A is a chain homotopy equivalence so that for thepurpose of say computing homology it does not matter if we work with A or NA, both have theiradvantages – NA is smaller but A has a simplicial structure. Also in the case of an augmentedsimplicial object in an abelian category the augmentation map A→ cstA−1 induces NA→ A−1

(a map to a chain complex concentrated in dimension 0). If A admits extra degeneracies then thesimplicial contraction ∆1 ·A = Z∆1 ⊗A→ A induces

NZ∆1 ⊗NA −→ N(∆1 ·A) −→ NA

which is a chain homotopy (a chain contraction onto A−1) as NZ∆1 = · · · → 0 → Z → Z ⊕ Z.Moreover there is an isomorphism between homotopy groups of A and homology groups of A orNA.

We will spend more time on the so called homotopical algebra, i.e. model categories (in ourcase simplicial sets and various diagram categories). The moral of this whole story is that oneshould think of augmented simplicial objects as chain complexes. In the case BT c with T = LRthe values (BT c)n for n ≥ 0 are in the image of L, i.e. they are typically free in some sense.After the application of R the augmented simplicial object admits extra degeneracies making itcontractible (if the augmentation is included) and hence provides a resolution of Rc. Typically Rwould be a functor that reflects exactness (it only forgets parts of the structure but the exactnessremains the same) and in this situation the original simplicial object is a “free resolution” of c. Inhomotopical algebra there is no underlying chain complex to a simplicial object and all the data(or at least all the face operators) is required to set up a resolution. In homotopy theory there ishowever a way to “realize” the simplicial object and make it into an object of the category (rather

38 LUKAS VOKRINEK

than a chain complex). It is often the case that this realization functor respects weak equivalencesand the augmentation in the end induces a weak equivalence |BT c| → | cst(BT c)−1| = c. This isagain some sort of resolution in the sense that |BT c| is better behaved than c (it is cofibrant – wehave not met this notion in sSet as all objects are cofibrant there but it is a dual notion to a Kancomplex; in topological spaces cofibant objects are (mild generalizations of) CW-complexes).

9. The homotopy theory of diagrams

Resolutions in homological algebra are used to defined derived functors which are a homotopycorrection to the original functor. This means that applying F : A → B pointwise induces afunctor ChA → ChB but this functor does not preserve quasi isomorphisms. It preserves homotopyequivalences though and these two classes agree on pointwise projective chain complexes. Thusin order to correct the deficiency of F we define LB which takes a chain complex, replaces it bya quasi isomorphic one that is also pointwise projective and only then applies F . One may thentake homology groups to get the derived functors of F but from the point of view of homotopicalalgebra the main object is the chain complex itself.

We will do the same procedure now with colimits. Let us start with a simple (and too classical)example of the non-homotopy invariance of pushouts.

∗

∼��

∂∆1oo //

∼��

∗

∼��

// ∗

6∼��

∆1 ∂∆1oo // ∆1 // ∆1 ∪∂∆1 ∆1

While all the components on the left are weak homotopy equivalences, the map on the right is not– the pushout of the bottom row has the homotopy type of the circle. The homotopically correctpushout is that on the bottom. This is because the diagram is (projective) cofibrant. One mightalso say that it is the homotopy colimit of the top row. We will describe now what this means.

Let I be a small category and consider the functor category [I, sSet]. We say that a naturaltransformation is a fibration or a weak equivalence if each of its components is a fibration or a weakequivalence. A (projective) cofibration is a natural transformation which has a left lifting propertyagainst all trivial fibrations. We will describe these now more concretely. A map p : X → Y isa (trivial) fibration if and only if each pi : Xi → Y i is a (trivial) fibration. Trivial fibrations aredetected by (non-trivial) cofibrations (boundary inclusions are enough) so that we have

K //��

��

Xi

pi

��≡

FiK //

��

X

p

��

L //

>>

Y i FiL //

==

Y

where Fi is the functor sSet→ [I, sSet] left adjoint to the evaluation functor at i which we denoteby Ui. In particular FiK → FiL is a (trivial) cofibration and it is in fact true that all (trivial)cofibrations are the saturation of these.

Proposition 9.1. The cofibrations are the saturation of {Fi∂∆n → Fi∆n} and the trivial cofi-

brations are the saturation of {FiΛk∆n → Fi∆n}.

Proof. For cofibrations this is the small object argument.The case of trivial cofibrations is more complicated. It is obvious that the maps from the

saturation are cofibrations. From the formula (2) for Fi it is clear that the generating mapsFiΛk∆n → Fi∆

n are pointwise trivial cofibrations. Since the class of pointwise trivial cofibra-tions is saturated the saturation consists of pointwise trivial cofibrations and in particular weakequivalences.

SIMPLICIAL SETS 39

If on the other hand f is a trivial cofibration we factor it via the small object argument as amap g from the saturation followed by a fibration h. We have a diagram

•g

//��

f ∼��

•

h����

•

77

•

Since g is a trivial cofibration by the previous, h must be a weak equivalence and thus a trivialfibration. As f is a cofibration the diagonal exists by definition. Consequently f is a retract of gand lies in the saturation. �

We say that a diagram D is cofibrant if the unique map ∅ → D is a cofibration.We might apply the theory of resolutions to the adjunctions Fi a Ui but it will be more

convenient to put them together to form a single adjunction

F : [ob I, sSet] � [I, sSet] : U

Here ob I is the category with the same objects as I but only identity morphisms. Thus anobject of [ob I, sSet] is a collection (Ki) of simplicial sets indexed by the objects of I and we haveF (Ki) =

∐i FiKi. Thus it is still true that F maps (trivial) cofibrations to (trivial) cofibrations.

Consider for a diagram D ∈ [I, sSet] its bar resolution

BFUD : ∆op → [I, sSet]

augmented by D. The simplicial object BFUD is called the simplicial replacement of D. ApplyingU we have that UBFUD → UD admits extra degeneracies and thus UBFUD → UD is a simplicialhomotopy equivalence of simplicial objects in [ob I, sSet]. Since ob I is discrete this means preciselythat each (BFUD)i → Di is a simplicial homotopy equivalence of simplicial objects in sSet. Wewant to make this into a (weak) homotopy equivalence of simplicial sets – we have to constructfrom a simplicial simplicial set a simplicial set. There is a very easy way to do this but whichhides much of the geometry

diag : s2Set = [∆op ×∆op,Set]δ∗∆−−−→ [∆op,Set] = sSet,

the precomposition with the diagonal embedding δ∆ : ∆ → ∆ × ∆. Since it is natural it alsoinduces a functor diag : s[I, sSet] ∼= [I, s2sSet]→ [I, sSet].

Theorem 9.2. The augmentation of the bar resolution induces a weak equivalence

diagBFUD → D

in [I, sSet] with a cofibrant domain.

Proof. At this point we will prove that the map is a weak equivalence. First we prove that forK ∈ sSet and D ∈ s[I, sSet] it holds

diag(K ·X) = K × diagX.

This follows directly from the definition10

|K ·X|n = ((K ·X)n)n = (Kn ·Xn)n = Kn × (Xn)n = (K × |X|)n.

Now recall that BFUDi → cstDi admits (for each i ∈ I separately) a simplicial homotopyinverse. A simplicial homotopy ∆1 ·X → Y in s(sSet) induces by the previous observation

∆1 × |X| = |∆1 ·X| → |Y |,

a homotopy on the realizations. Therefore |BFUD| and D = | cstD| are homotopy equivalent. �

10There is a generalization of the “diagonal” to simplicial objects in simplicial model categories where theformula holds only for constant simplicial objects. The theorem still stays true for D pointwise cofibrant.

40 LUKAS VOKRINEK

With this theorem in mind we may define a homotopy colimit of a diagram X ∈ [I, sSet] to be

hocolimI

Xdef= colim

IdiagBFUX

We will need to understand the diagonal functor homotopically in order to prove the theorem.Then we will study the basic properties of the homotopy colimit functor.

Definition 9.3. Let F : Cop×C → D be a bifunctor. Its coend∫ C

F =∫ cF (c, c) is the coequalizer

of the following pair of morphisms∐f F (cod f, dom f) //

// ∐c F (c, c)

where the first arrow maps the component F (c1, c0) corresponding to f : c0 → c1 to the componentF (c0, c0) via F (f, id) while the second maps it to the component F (c1, c1) via F (id, f). It has

the following universal property: morphisms∫ C

F → d are uniquely specified by collection ofmorphisms τd : F (c, c)→ d such that for each f : c0 → c1 the square

F (c1, c0)F (f,id)

//

F (id,f)

��

F (c0, c0)

τc0

��

F (c1, c1)τc1

// d

commutes. In the case that F is some sort of product, say − � − we denote the coend by

−�C −def=∫ C −�−.

Remark. There is a dual notion of an end which is a special limit. The set of natural transformations between two functorshas a nice (and useful) definition as the end

[C,D](F,G) =

∫CD(F,G) =

∫c

D(Fc,Gc).

This is especially useful when computing with “tensor products” and adjunctions where these appear naturally.

Example 9.4. We may view a set Y with a left action of G as a functor X : G→ Set where G isconsidered here as a one-object category. Similarly a set X with a right action of G is a functorY : Gop → Set. The product X × Y is then a functor X × Y : Gop × G → Set. Its coend is the“tensor product” X ×G Y , i.e. the quotient of X × Y by the relation (xg, y) ∼ (x, gy) (these arethe two images of (x, y) under the two parallel maps in the definition of a coend).

Similarly a left R-module N may be viewed as an (additive) functor R → Ab and a rightmodule M as an (additive) functor Rop → Ab. The tensor product in Ab yields a functor M ⊗N :Rop ×R→ Ab whose coend M ⊗R N is again the tensor product over R.

We will now define a “weighted colimit”. For two functors W : Cop → Set and D : C → D wedefine the weighted colimit W ·C D as the coend of the functor W ·D : Cop × C → D that takes(c1, c0) 7→ (Wc1) · (Dc0) the coproduct of #(Wc1) copies of Dc0. It has the following universalproperty (expressing it as a left adjoint)

D(W ·C D, d) ∼= [Cop,Set](W,D(D−, d)).

This is easily seen from the universal property of the coend (the square corresponds to the natu-rality)11. Also note that in the case Wc = ∗ the right hand side is the set of all cones from D to dso that ∗ ·C D = colimC D. It is in fact true that every weighted colimit12 may be replaced by anordinary colimit indexed by the category of elements of W .

11Alternatively one may take the coend out of the hom-set on the left hand side (making it an end), then use

the adjunction and get the end describing the set of natural transformations.12For those that know something about enriched category theory – in the enriched context ordinary colimits do

not generate all weighted colimits. To get a theory of colimits similar to that in the non-enriched case weightedcolimits are preferable.

SIMPLICIAL SETS 41

Example 9.5. The Yoneda lemma has the following interpretation. Let W = C(−, c) : Cop → Setbe the representable functor and D : C → D and arbitrary functor. Then

C(−, c) ·C D ∼= Dc.

This is bacause

D(C(−, c) ·C D, d

) ∼= [Cop,Set](C(−, c),D(D, d)

) ∼= D(Dc, d).

This is related to the canonical presentation for a simplicial set X ∈ sSet

Xk∼= ∆(k,−) ·∆ X =

∫ n

(∆n)k ×Xn

or more compactly

X ∼=∫ n

Xn ·∆n = X ·∆ ∆•.

This is just a rewriting of the canonical colimit – it is the quotient of∐

[n]∈∆

∐x∈Xn

∆n and it

turns out to be the same as that giving the colimit of the canonical diagram. This has to do withrewriting weighted colimits as normal colimits via the category of elements.

Example 9.6. The above reinterpretation of the canonical presentation yields a nice formula forthe geometric realization of a simplicial set X,

|X| ∼=∣∣∣ ∫ n

Xn ·∆n∣∣∣ ∼= ∫ n

Xn · |∆n|

where we only use the fact that | − |, being a left adjoint, preserves colimits.

Example 9.7. We are now able to reinterpret the diagonal as some sort of geometric realization.Let X ∈ s2Set = s(sSet) be a bisimplicial set (i.e. a simplicial object in simplicial sets). Thediagonal is the functor

(diagX)n = (Xn)n ∼=∫ k

∆(n, k)× (Xk)n

or more compactly

diagX =

∫ k

∆k ×Xk

where the product in the integrand is the product of a standard simplex ∆k with the simplicialset Xk – the value of X : ∆op → sSet at k. For our purpouses this is a much better descriptionin that it is given in terms of the values Xk of the simplicial object X rather than its simplices.From now on we will denote |X| = diagX and think of it in this way.

There is another important example, the left Kan extension which we will be using shortly.We have seen already that the value of a diagram D at an object c is “computed” by the coendC(−, c) ·C D. This allows for expressing the “restriction of scalars” as a coend again, namely ifF : C′ → C then the composition DF : C′ → D is given by

DFc′ = D(Fc′) = C(−, F c′) ·C D.

We may now compute the left adjoint of this precomposition functor F ∗ : D → DF (called theleft Kan extension along F and denoted F!) as a coend again

F!D′c = C(F−, c) ·C′ D′

This is because a natural (in c) transformation C(F−, c) ·C′ D′ → Dc is the same as a natural (inboth c and c′) transformation

C(Fc′, c)→ D(D′c′, Dc)

which by Yoneda is the same as a natural (in c′) transformation D′c′ → DFc′ or simply D′ → DF .

42 LUKAS VOKRINEK

Example 9.8. The adjunction F a U that yields the standard simplicial resolution is induced bythe inclusion I : ob I → I in that U = I∗ and consequently F = I!. Using the above descriptionof I! we might compute at (Ki) ∈ [ob I, sSet]:

F (Ki)j = I(I−, j) ·ob I K• =∐i∈ob I

I(i, j) ·Kj

(there are no non-identity morphisms in ob I so that there is no quotient). Restricting to a singleobject the corresponding formula for Fi is (FiK)j = I(i, j) ·K or simply

FiK = I(i,−) ·K. (2)

The counit of the adjunction F! a F ∗ is given by

C(F−, c) ·C′ DF −→ Dc, [f, x] 7→ (Df)x

and the unit by

D′c′ −→ C(F−, F c′) ·C′ D′, x 7→ [idFc′ , x]

Now if (and only if) F is fully faithful then C(F−, F c′) ∼= C′(−, c′) so that by the Yoneda lemmathe term on the right is C′(−, c′) ·C′ D′ ∼= D′c′ and in fact the unit is an isomorphism.

This should remind one of the restriction of scalars and induction for modules (the inductionis M 7→ S ⊗RM) and indeed if we were speaking about categories enriched over Ab and enrichedcoends we would arive at (a many-object generalization of) the induction-restriction adjunction.

We apply this adjunction to the inclusion ιn : ∆≤n → ∆ of the full subcategory of ∆ on objects[0], . . . , [n]. A functor ∆op

≤n → C is an n-truncated simplicial object. If X is a simplicial object

then ι∗nX is an n-truncated simplicial object (the truncation of X at n) and (ιn)!ι∗nX is a new

simplicial object.

Definition 9.9. The simplicial object sknXdef= (ιn)!ι

∗nX is called the n-skeleton of X. It admits

a canonical map (the counit) sknX → X.

We say that a simplicial object is n-skeletal if it is isomorphic to an object of the form (ιn)!Y .If X is n-skeletal then

X ∼= (ιn)!Y(ιn)!ηY−−−−−−→ (ιn)!ι

∗n(ιn)!Y = sknX

is a section of the canonical sknX → X (by the triangle identity). Moreover since ιn is full andfaithful the unit is an isomorphism and thus so is the canonical map sknX → X.

Lemma 9.10. A simplicial object X is n-skeletal if and only if sknX → X is an isomorphism.

Because ιn−1 factors as ∆≤n−1

ι(n)−−→ ∆≤nιn−→ ∆ one has a canonical map

skn−1X = (ιn)!(ι(n))!ι∗(n)ι∗nX −→ (ιn)!ι

∗nX = sknX

We may thus conclude that skkX is n-skeletal if k ≤ n.We define the n-th latching object LnX of a simplicial object X to be (skn−1X)n. It is equipped

with a canonical map LnX → Xn, the n-th component of skn−1X → X. By the formula for theleft Kan extension one has

LnX = ∆([n], ιn−1−) ·∆≤n−1ι∗n−1X

The (n− 1)-truncated cosimplicial set on the left is generated by s0, . . . , sn : [n]→ [n− 1] subjectto the relations which hold among si in ∆. This implies that LnX is a certain quotient

LnX =

n∐i=0

Xn−1/ ∼

of the coproduct of the (n− 1)-simplices thought of as n-simplices via si where one has to identifythose which are degenerate in more than one way and are thus counted multiple times.

SIMPLICIAL SETS 43

Proposition 9.11. There is a pushout square

(∆n · cstLnX) ∪∂∆n·cstLnX (∂∆n · cstXn) //

��

skn−1X

��

∆n · cstXn// sknX

for each n and an isomorphism between X and the colimit of

sk0X −→ · · · −→ sknX −→ · · · .

Therefore to study the homotopy properties of X we need to understand the maps LnX → Xn.We will do this in detail for the simplicial replacement BFUD of a diagram D : I → C after theproof.

Proof. The square is formed by n-skeletal simplicial objects and as such lies in the image of acolimit preserving functor (ιn)!. It is therefore sufficient to check that the square is a pushoutin dimensions ≤ n. In dimensions < n the vertical arrows are isomorphisms. In dimension n itbecomes

((∆n)n · LnX) ∪(∂∆n)n·LnX ((∂∆n)n ·Xn) //

��

LnX

��

(∆n)n ·Xn// Xn

The vertical map on the left is a coproduct of id : Xn → Xn indexed by elements of (∂∆n)n witha single LnX → Xn corresponding to the unique id ∈ (∆n)n r (∂∆n)n.

The second claim follows from the fact that starting from sknX the sequence stabilizes indimension n. �

Lemma 9.12. If K is an n-skeletal simplicial set and c is an object of C then K · c is alson-skeletal.

Proof. Easy; from the cocontinuity of − · −. �

We say that a simplicial diagram X ∈ s[I, sSet] is Reedy cofibrant if all the canonical mapsLnX → Xn are cofibrations. For the following proof we remind that there is an isomorphism

|K · cstD| = K × | cstD| ∼= K ×D.

(which holds much more generally – in sC when C is a simplicially enriched category).

Theorem 9.13. Let X ∈ s[I, sSet] be Reedy cofibrant. Then |X| is cofibrant.

Proof. The proof is by skeletal induction. For n = 0 we have L0X = ∅ so that | sk0X| = X0 iscofibrant. For the inductive step observe we have a pushout square

(∆n × LnX) ∪∂∆n×LnX (∂∆n ×Xn) //

��

| skn−1X|

��

∆n ×Xn// | sknX|

by the previous proposition and the observation made before the theorem. Now the map on theleft is a cofibration by the following lemma. Hence so is its pushout. The map ∅ → |X| is thetransfinite composition of | skn−1X| → | sknX| and is thus a cofibration too. �

Lemma 9.14. Let K → L be an injective map of simplicial sets and A → B a cofibration in[I, sSet] then the map

(K ×B) ∪K×A (L×A) −→ L×Bis a cofibration too.

44 LUKAS VOKRINEK

Proof. The class of transformations A→ B for which the map is a cofibration is saturated. It alsocontains FiA0 → FiB0 for an injective simplicial map A0 → B0 since the map in question is then

Fi((K ×B0) ∪K×A0 (L×A0)

)−→ Fi(L×B0)

which is again a (generating) cofibration. �

Theorem 9.15. The simplicial diagram BFUD ∈ s[I, sSet] obtained from the free-forgetful ad-junction F a U is Reedy cofibrant. Consequently |BFUD| is cofibrant.

Proof. We need to compute the latching object and the corresponding maps into the simplicialresolution. We have

(BFUD)ni = (FU)n+1Di =∐

i0→···→in

I(in, i) ·Di0

and it is not difficult to convince oneself that the latching object is the sub-coproduct over the“degenerate” chains, i.e. those in which one of i0 → · · · → in is identity. The inclusion is theinclusion of the coproduct over the degenerate chains into the big coproduct. Therefore thisinclusion is a pushout of

∅ −→∐

i0→···→innon−degenerate

I(in, i) ·Di0.

The codomain is cofibrant hence the map is a cofibration and its pushout LnBFUD → (BFUD)nis a cofibration too. �

This finishes the proof that |BFUD| → D is a cofibrant replacement. We will show now (inparticular) that any other choice of a cofibrant replacement will yield weakly equivalent colimitso that we may in fact define the homotopy colimit of a diagram as a colimit of its cofibrantreplacement. This is also one of the classical definitions.

Proposition 9.16. The colimit functor colimI : [I, sSet] → sSet preserves weak equivalencesbetween cofibrant objects.

Proof. This is essentially Ken Brown’s lemma. First colimI preserves trivial cofibrations becauseits right adjoint cstI preserves fibrations (this depends on the characterization of trivial cofibrationsas those that have the left lifting property against fibrations). Then every weak equivalencef : X → Y between cofibrant diagrams may be factored into

f : X // ∼ // (∆1 ×X) ∪X Y∼ // Yzz

∼

ee

where the second map admits a section Y → (∆1 ×X) ∪X Y that is always a trivial cofibrationand the first map is always a cofibration. If f is a weak equivalence then by 2-out-of-3 so is thefirst map in the above factorization and consequently it is a trivial cofibration. By the previoustrivial cofibrations are preserved by colimI so that again by 2-out-of-3 the colimit of the secondmap in the factorization is a weak equivalence and finally so is colimI f . �

Corollary 9.17. The homotopy colimit hocolimI : [I, sSet] → sSet is homotopy invariant – itpreserves all weak equivalences. If D is cofibrant then the canonical map hocolimI D → colimI Dis a weak homotopy equivalence.

The second claim means that it is not necessary to derive the colimit functor on cofibrantdiagrams.

Proof. If D is cofibrant then |BFUD| → D is a weak equivalence between cofibrant diagrams andhence induces a weak homotopy equivalence on colimits. This proves the second claim.

To prove the first claim observe that by the 2-out-of-3 property if D → D′ is a weak equivalencethen so is |BFUD| → |BFUD′|. As it runs between cofibrant objects it again induces a weakequivalence on colimits, i.e. on homotopy colimits of the original diagrams. �

SIMPLICIAL SETS 45

We will now derive a useful formula for the homotopy colimit due to Bousfield and Kan. Wehave already explained that

(BFUD)n =∐

i0→···→in

I(in,−) ·Di0.

We may now compute the colimit using the fact that colimI commutes with the diagonal | − |,

hocolimI

D = colimI

∣∣∣ ∐i0→···→in

I(in,−) ·Di0∣∣∣ ∼= ∣∣∣ ∐

i0→···→in

(colimII(in,−)

)︸ ︷︷ ︸

∗

·Di0∣∣∣ ∼= ∣∣∣ ∐

i0→···→in

Di0

∣∣∣.This formula expresses hocolimI D as a diagonal of a bisimplicial set. There is another which isoften very useful where we start with observing that the coproduct is indexed by the n-simplicesof NI. The summand depends on the chain but we will get rid of this dependence via a coend∫ i ∐

i→i0→···→in

Di ∼=∫ i ∐

i0→···→in

I(i, i0) ·Di ∼=∐

i0→···→in

Di0.

The term on the left is∫ iN(i↓I) · Di where N(i↓I) is the nerve of the comma category i↓I of

objects under i (its objects are arrows i→ i0 and its morphisms are arrows i0 → i1 for which theobvious triangle commutes). We may thus write

hocolimI

D ∼=∣∣∣ ∫ i

N(i↓I) ·Di∣∣∣ ∼= ∫ i

N(i↓I)×Di.

This is a particular type of a (simplicial) weighted colimit. For a functor (called the “weight”)W : Iop → sSet and a diagram D : I → sSet the weighted colimit W ×I D is the coend

W ×I D =

∫ IW ×D

of the product W ×D : Iop × I → sSet in simplicial sets. The following observation is crucial.

Proposition 9.18. The weighted colimit W ×I D is homotopy invariant when W is cofibrant13.Concretely if W → W ′ is a weak equivalence between cofibrant diagrams (weights) and D → D′

any weak equivalence then W ×I D →W ′ ×I D′ is a weak equivalence.

Proof. The homotopy invariance in W follows from the adjunction between −×ID : [Iop, sSet]→sSet and map(D,−) : sSet → [Iop, sSet], the second clearly preserves fibrations. The homotopyinvariance in D is similar – W ×I − : [I, sSet] → sSet is adjoint to map(W,−) : sSet → [I, sSet]and the second maps fibrations to “injective fibrations”, i.e. those which have the right liftingproperty against pointwise trivial cofibrations. This follows by induction on the “cell structure”of W . We will not need the homotopy invariance in the D-variable in the proceeding and thusskip details. �

10. Additional topics

For a simplicial set K there is a very special simplicial object in sSet namely

∆K : ∆op K−−→ Setcst−−−→ sSet

where the second functor sends every set S to the constant simplicial set cstS. It is obviousthat its diagonal is |∆K| = K. Moreover the map Ln∆K → (∆K)n = Kn is the inclusion ofdiscrete simplicial sets of the degenerate n-simplices into all n-simplices. In particular ∆K isReedy cofibrant (in fact every simplicial object in sSet is Reedy cofibrant). We may compare thehomotopy colimit with the geometric realization

hocolim∆op

∆K =

∫ [n]

N(∆↓[n])× (∆K)n −→∫ [n]

∆n × (∆K)n = |∆K| = K

It is true that the transformation of weights N(∆↓[•]) → ∆• is a pointwise weak equivalenceof Reedy cofibrant cosimplicial objects (which we have not defined) and as such induces a weak

13. . . and D pointwise cofibrant which is automatic in simplicial sets.

46 LUKAS VOKRINEK

equivalence when the weighted colimit of a Reedy cofibrant diagram is taken. Therefore the abovemap is a weak equivalence yielding a homotopy version of the canonical presentation (there is alsoa version with the original canonical diagram but that is slightly more difficult to explain). Nowthere are further weights that produce interesting functors, namely sd ∆• and ι!ι

∗∆• where in thesecond ι : ∆+ → ∆ is the inclusion of the subcategory of ∆ formed by the injective maps of ordinals.They are both Reedy cofibrant (this is not automatic as with simplicial objects) and pointwiseweakly contractible and thus the canonical transformations sd ∆• → ∆• and ι!ι

∗∆• → ∆• induceweak equivalences

sdK∼−−→ K, ι!ι

∗K∼−−→ K

on the weighted colimits.A functor F : J → I induces a map hocolimJ DF −→ hocolimI D for any diagram D : I →

sSet. This is given essentially by a transformation of weights once we rewrite the first homotopycolimit as

hocolimJ

DF = N(−↓J )×J DF ∼= F!N(−↓J )×I D ∼= N(−↓F )×I D

The first isomorphism comes from the following

W ×J FD = W ×J (I(−, F−)×I D) = (W ×J I(−, F−))︸ ︷︷ ︸F!W

×ID = F!W ×I D

while the second from

(F!N(− ↓ J ))n =

∫ j

I(−, F j)×∐

j0,...,jn

J (j, j0)× J (j0, j1)× · · · × J (jn−1, jn)

=∐

j0,...,jn

I(−, F j0)× J (j0, j1)× · · · × J (jn−1, jn) = N(−↓F )n

where we remind that −↓F is the “comma category” of arrows into F . Its objects are pairs(i → Fj, j) and its morphisms from (i → Fj0, j0) to (i → Fj1, j1) those arrows f : j0 → j1 forwhich the obvious triangle

Fj0

Ff

��

j0

f

��

i

88

&&

Fj1 j1

commutes. Now the map on homotopy colimits is induced by the obvious transformation of weights

N(−↓F ) −→ N(−↓I),

induced by the “forgetful” functor (−↓F )→ (−↓I) sending (i→ Fj, j) to i→ Fj.

Theorem 10.1. A functor F : J → I induces a weak homotopy equivalence

hocolimJ

FD → hocolimI

D

for all diagrams D : I → sSet if and only if each N(−↓F ) ' ∗ is weakly contractible. In this casewe say that F is homotopy right cofinal.

Proof. The point is that the above transformation of weights is between (projective) cofibrantdiagrams. We have seen this for N(−↓I). The second weight is isomorphic to F!N(−↓J ). SinceF ∗ preserves trivial fibrations, its left adjoint F! preserves cofibrations and in particular N(−↓F )is cofibrant.

If N(−↓F ) → N(−↓I) is a weak equivalence, it induces a weak homotopy equivalence onweighted colimits. The opposite direction follows by applying to representable functor D = I(i,−)giving

N(i↓F ) = hocolimJ

FD∼−−→ hocolim

ID = N(i↓I)

and the right hand side is contractible since i↓I has an initial object. �

SIMPLICIAL SETS 47

A typical example of a homotopy right cofinal functor is any right adjoint. The precedingtheorem allows one to compute with homotopy colimits. There is another rule (and all relations canbe deduced from these two in a suitable sense) which allows one to replace an iteration of homotopycolimits to a single homotopy colimit, most classically hocolimI hocolimJ D ' hocolimI×J D forany diagram D : I × J → sSet.

Corollary 10.2 (Quillen’s Theorem A). Suppose that F : J → I is homotopy right cofinal,i.e. such that N(−↓F ) ' ∗. Then NF : NJ → NI is a weak homotopy equivalence.

Proof. The point is that NI (and likewise NJ ) is the homotopy colimit of the terminal diagram∗ : D : I → sSet. To see this compute

hocolimI

∗ = N(− ↓ I)×I ∗ = colimI

N(− ↓ I) = NI.

The induced map on homotopy colimits equals NF and the result follows from the previouscharacterization of homotopy right cofinal functors. �

The line of the proof actually shows that every homotopy colimit hocolimI D admits a canonicalmap to hocolimI ∗ = NI. Quillen’s Theorem B then improves on the previous result: if allthe maps in a diagram D are weak homotopy equivalences the map hocolimI D → NI has thehomotopy fibre over i ∈ NI weakly equivalent to the nerve N(i↓F ), i.e. when replaced by a weaklyhomotopy equivalent fibration the actual fibre is weakly homotopy equivalent to N(i↓F ).

Also of some interest is the following. Combining the canonical homotopy colimit decomposition(that we have not fully proved)

K ' hocolim∆↓K

PK

with the homotopy invariance of the homotopy colimit applied to PK ⇒ ∗ (all values of PK aresimplices that are weakly contractible) we deduce that

K ' hocolim∆↓K

∗ ∼= N(∆↓K),

i.e. K is weakly homotopy equivalent to a nerve of some category. One may improve upon thisand show that up to weak equivalences categories and simplicial sets agree. It is a bit of cheatingthough as the weak equivalences in categories are defined exactly so that this works. In any casethere is some abstract description of what these should be. They are formalized in Grothendieck’snotion of a basic localizer. A basic localizer is a class of functors that are closed under certainconstructions – 2-out-of-3, retracts, functors C → 1 if C has a terminal object, and then someextension property. A theorem of Cisinski states that the class of weak equivalences in Cat is thesmallest basic localizer.

References

[GJ] Goerss, P. G., Jardine, J. F., Simplicial homotopy theory, Birkhuser Verlag, Basel, 1999[H] Hatcher, Allen, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN: 0-521-

79160-X; 0-521-79540-0[W] Weibel, Charles, An introduction to homological algebra, Cambridge University Press, ...