Lawvere, Taking Categories Seriously

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Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 124.TAKING CATEGORIES SERIOUSLYF. WILLIAM LAWVEREAbstract. The relation between teaching and research is partly embodied in simplegeneral concepts which can guide the elaboration of examples in both. Notions and constructions, such as the spectral analysis of dynamical systems, have important aspectsthat can be understood and pursued without the complication of limiting the models tospecic classical categories. Pursuing that idea leads to a dynamical objectication ofDedekind reals which is particularly suited to the simple identication of metric spacesas enriched categories over a special closed category. Rejecting the complacent description of that identication as a mere analogy or amusement, its relentless pursuit [8] iscontinued, revealing convexity and geodesics as concepts having a denite meaning overany closed category. Along the way various hopefully enlightening exercises for students(and possible directions for research) are inevitably encountered: (1) an explicit treatment of the contrast between multiplication and divisibility that, in inexorable functorialfashion, mutates into the adjoint relation between autonomous and nonautonomous dynamical systems; (2) the role of commutation relations in the contrast between equilibriaand orbits, as well as in qualitative distinctions between extensions of Heyting logic; (3)the functorial contrast between translations and rotations (as appropriately dened) inan arbitrary non symmetric metric space.The theory of categories originated [1] with the need to guide complicated calculationsinvolving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects. Since then it is still actively used for those problems but alsoin algebraic geometry [2], logic and set theory [3], model theory [4], functional analysis[5], continuum physics [6], combinatorics [7], etc. In all these the categorical concept ofadjoint functor has come to play a key role.Such a universal instrument for guiding the learning, development, and use of advancedmathematics does not fail to have its indications also in areas of school and college mathematics, in the most basic relationships of space and quantity and the calculations based onthose relationships. In saying take categories seriously, I advocate noticing, cultivating,After the original manuscript was stolen, I was able to reconstruct, in time for inclusion in theproceedings of the 1983 Bogota Workshop, thissketch (as Saunders Mac Lanes review was to describeit). Interested teachers can elaborate this and similar material into texts for beginning students. Forthe present opportunity, I am grateful to Xavier Caicedo who gave permission to reprint, to ChristophSchubert for his expert transcription into TEX, and to the editors of TAC.Received by the editors 20041102.Transmitted by M. Barr, R. Rosebrugh and R. F. C. Walters. Reprint published on 20050323.2000 Mathematics Subject Classication: 06D20, 18D23, 37B55, 34L05, 52A99, 53C22, 54E25, 54E40.Key words and phrases: General spectral theory, Nonautonomous dynamical systems, Symmetricmonoidal categories, Semimetric spaces, General convexity, Geodesics, Heyting algebras, Special mapson metric spaces.The article originally appeared in Revista Colombiana de Matem aticas, XX (1986) 147178. c _ Revista Colombiana de Matematicas, 1986, used by permission.12 F. WILLIAM LAWVEREand teaching of helpful examples of an elementary nature.1. Elementary mutability of dynamical systemsAlready in [1] it was pointed out that a preordered set is just a category with at most onemorphism between any given pair of objects, and that functors between two such categories are just orderpreserving maps; at the opposite extreme, a monoid is just a categorywith exactly one object, and functors between two such categories are just homomorphismsof monoids. But category theory does not rest content with mere classication in the spiritof Wolan metaphysics (although a few of its practitioners may do so); rather it is themutability of mathematically precise structures (by morphisms) which is the essentialcontent of category theory. If the structures are themselves categories, this mutability isexpressed by functors, while if the structures are functors, the mutability is expressed bynatural transformations. Thus if is a preordered set and A is any category (for example the category of sets and mappings, the category of topological spaces and continuousmappings, the category of linear spaces and linear transformations, or the category ofbornological linear spaces and bounded linear transformations) then there are functors Asometimes called direct systems in A, and the natural transformations
1 11 1 Abetween two such functors are the appropriate morphisms for the study of such directsystems as objects.An important special case is that where = 0 1 , the ordinal number 2, then thefunctors 2 A may be identied with the morphisms in the category A itself; likewiseif = 0 1 2 is the ordinal number , functors X Aare just sequences of objects and morphismsX0 X1 X2 X3 . . .in A, and a natural transformation X f Y between two such is a sequence Xnfn Ynof morphisms in A for which all squaresXnfn
Yn
Xn+1fn+1
Yn+1TAKING CATEGORIES SERIOUSLY 3commute in A (here the vertical maps are the ones given as part of the structure of Xand Y ).Similarly, if M is a monoid then the functors M A are extremely important mathematical objects often known as actions of M on objects of A (or representations of Mby Aendomorphisms, or . . . ) and the natural transformations between such actions areknown variously as Mequivariant maps, intertwining operators, homogeneous functions,etc. depending on the traditions of various contexts. Historically the notion of monoid (orof group in particular) was abstracted from the actions, a pivotally important abstractionsince as soon as a particular action is constructed or noticed, the demands of learning,development, and use mutate it into: 1) other actions on the same object, 2) actions onother related objects, and 3) actions of related monoids. For if M X A is an action andM
h M is a homomorphism, then (composition of functors!) Xh is an action of M
,while if A C is a functor, then CX is an action of M on objects of . To exemplify,if M is the additive group of timetranslations, then a functor M A is often calleda dynamical system (continuoustime and autonomous) in A, but if we are interested inobserving the system only on a daily basis we could consider a homomorphism M
h Mwhere M
= N is the additive monoid of natural numbers, and concentrate attention onthe predictions of the discretetime, autonomous, futuredirected dynamical system Xh.In other applications we might have M
= M = the multiplicative monoid of real numbers, but consider the homomorphism M ( )p M of raising to the pth power; then if weare given two actions M
A on objects of A, a natural transformation X f (Y )pisjust a morphism of the underlying Aobjects which satisesf(x) = pf(x)for all in M and all T x X in A, i.e. a function homogeneous of degree p. Anextremely important example of the second mutation of action mentioned above is thatin which is the opposite of an appropriate category of algebras and the functor Cassigns to each object (domain of variation) of A an algebra of functions (= intensivequantities) on it. Then the induced action CX of M describes the evolution of intensivequantities which results from the evolution of states as described by the action X.A frequentlyoccurring example of the third type of mutation of action arises from thesurjective homomorphisms M
M from the additive monoid of timetranslations M
to the circle group M. Then a dynamical system M
X A is said to be periodic ofperiod h if there exists a commutative diagram of functors as follows:M
X
h
gggggggg XM4 F. WILLIAM LAWVERE2. Kan quantiers in spectral analysisMost dynamical systems are only partly periodic, and such an analysis can convenientlybe expressed by Kanextensions as follows (we do not assume that M, M
are monoids):For a functor M
h M and a category A, the induced functor AM AM
willoften have a left adjoint X h
X and a right adjoint X h
X. These Kanadjoints vastly generalize the existential and universal quantiers of logic, special casesarising when all the objects of A are truthvalues. (Usually this just means that theobjects of A are canonically idempotent with respect to cartesian product or coproduct.)Kan adjoints also generalize the induced representations frequently considered when A is alinear category (i.e. nite coproduct is canonically isomorphic to nite product) and whenM
M is of nite index, in which case there is a strong tendency for h
( ) andh
( ) to coincide. The dening property of these as adjoints are the natural bijectionsh
X YX Y hT h
XTh Xbetween M
natural, respectively Mnatural transformations, where Y , T are functorsM A (that is objects of the category AMwhose morphisms are the Mnaturaltransformations) and X is a functor M
A (that is an object of the category AM
whose morphisms are M
natural transformations). Since these rened rules of inference uniquely characterize the adjoints up to unique natural isomorphism, if M
kM
h M are two functors for each of which the two Kan adjoints exist, then from theassociativity of substitution, Y (hk) = (Y h)k, follow the two rules(hk)
Z = h
(k
Z)(hk)
Z = h
(k
Z).If M
is a discrete category 1 with I objects (and no morphisms except the identitymorphisms) and if M is the single morphism category 1, then there is a unique functor1 1, often also called I and the Kan adjoints are just the coproduct and productfunctors respectively:I
X =
iI XiI
X =
iI Xiwhere a functor I X A is just a family of objects. It is chiey in regard to theexistence of Kan extensions that questions of largeness and smallness enter categorytheory. The class of all categories A for which h
( ) and h
( ) exist in A can becalled the smallness of M
h M, while dually (in the sense of Galois connections)the class of all functors h for which these exist over a given A can be called the degree of(bi) completeness of A, with obvious renements for left completeness where only
is considered and for right completeness where only
is considered. Informally we mayTAKING CATEGORIES SERIOUSLY 5just say that M
h M is suciently small for A or that A is suciently complete forh when these constructions can be carried out.Returning to the example of a period, i.e. a surjective homomorphism M
h Mfrom the additive group of timetranslations to the circle group, the induced functorAM AM
is just the full inclusion, into the category of all Adynamical systems (continuous, autonomous), of the subcategory of those that happen to have period h. Then the construction h
X just gives the part of X consisting of the hperiodic states. More preciselythe following adjunction morphism (derived from the rule of inference for
)(h
X)h Xwill typically be the inclusion of the hperiodic part.[Of course there is alsoX (h
X)hobtained by forcing the arbitrary dynamical system X into the hperiodic mold, withan accompanying collapse of states, whose detailed understanding depends on a detailedunderstanding of the collapsing or quotient process in A. The quotient process is just ingeneral op1
( ). Here op1 is the nite category E
V in which the two compositesat V are both the identity (implying that the two composites at E are idempotents whichabsorb one another in a noncommutative way) and functors op1 A are often referredto as (reexive) graphs in A; 1 itself can be concretely represented as the full categoryof the category of categories consisting of the two objects V = 1 and E = 2 = 0 1 inwhich representation the two arrows V
E in 1 are the two adjoints of the uniqueE V . The reexive graph in A arising from a period h (homomorphism of monoids)and a particular dynamical system X is just X given byEh X
X_where Eh is the set of all pairs m
1, m
2 for which h(m
1) = h(m
2) and Eh X is thecoproduct of Eh copies of X. The detailed properties of op1
X depend sensitively onthe nature of the category A, usually in concrete examples more so than do the detailedproperties of the dual construction 1
X, where X is the 1 X given byX
XEh_(where XEhdenotes the product of Eh copies of X, and where, as throughout this bracket,we have followed the usual abuse of notation of using the same letter X to denote alsothe object X(0) of A that underlies the action of the monoid M
); thus 1
X is the(h
X)h outside this bracket.]6 F. WILLIAM LAWVEREThe full period spectrum of a dynamical system M
X A can be regarded as asingle functor as follows. Say that a period h2 divides a period h1 if there exists anendomorphism q of the circle group M such thatM
h1{{{{{{{{ h2
ggggggggM q
Mh2 = qh1(then q is unique because h1 is assumed surjective and q itself is surjective because h2 isassumed to be). Denote by Q the category (actually a preordered set) whose objects arethe periods and whose morphisms are the q as indicated. By the denitions, if h2 dividesh1 and if X is (part of) a dynamical system of period h2, then it is (part of) a dynamicalsystem of period h1 as well:M
h1
h2
cccccccccX
AM Y1
MY2Y2 = Y1Similar reasoning shows that for any dynamical system X, q induces in a functorial wayan inclusion(h2
X)h2 (h1
X)h1of the h2periodic part of X into the h1periodic part of X, whenever q is the reason forh2 dividing h1. Thus we get a functor X : Qop A (where X(h) = (h
X)h) thatin turn depends functorially on X so that X X denes the periodic prespectrumfunctorAM () AQopfrom dynamical systems in any suciently
complete category A into the category ofdirect systems in A indexed by the poset Q of periods.More closely corresponding to the usual notion of spectrum is the following: the weightattached to a given period h is not so much the space (h
X)h of states having that periodas it is the smaller space of orbits of such states, where in general the notion of orbit spaceis the left adjointAM M
( ) Ato the functor induced by the unique M
1. Since for any h, the space (h
X)h ofhperiodic points is itself a dynamical system in its own right, hence combining these ash varies through Q we get a lifted prespectrum functor indicated by the dotted arrowbelow; the latter can be composed with the functor induced by the orbit space functorTAKING CATEGORIES SERIOUSLY 7(upon parameterizing by Q its inputs and outputs):AM
()
()
tttttttttt (AM
)Qop
(M
( ))Q
AQopAQopto yield what could be called the periodic spectrum X X. The periodic spectrum Xof a dynamical system X can in many cases be pictured asQh h
where the darkness of the line at a period h Q is proportional to the size of thespace M
((h
X)h) of equivalence classes of states of period h, where two states areequivalent if the dynamical action moves one through the other.There is one other point (not in Q) which may also be considered part of the spectrum,namely the map M
1 whose corresponding M
X is the space of xed statesof the dynamical system X. If M
is a group, then the xed point space is usuallya subspace of the orbit space, for example if A is the category of sets and mappings.The same conclusion follows if M
is any commutative monoid. However, for the threeelement monoid (essentially equivalent, insofar as actions are concerned, to the category1 mentioned above) consisting of the morphisms 1, 0, 1 with the multiplication tableij = i one can nd examples of (right) actions op1X o on sets (i.e. reexive graphs)having any given number of xed points (i.e. vertices) but only one orbit so that the mapop1
X op1
X is not at all a monomorphism.3. Enhanced algebraic structure in dynamics and logicIncidentally, the above remark that both groups and commutative monoids share a property not true for all monoids can be made more explicitly algebraic by the followingexercise. If ( is a category that is either a group, i.e. every morphism in ( has a twosidedinverse, or a commutative monoid, then ( acts on its endomorphisms in the followingway: for any morphism X
a X in (, and for any endomorphism x of X, we can denean endomorphism xaof X
such thatxa = axaX
xa
a
Xx
8 F. WILLIAM LAWVEREand moreover this is an action in the sense that 1a= 1 andxab= (xa)bX
b
X
a
Xx
and even an action by monoid homomorphisms in that(xy)a= xaya1a= 1for any two endomorphisms x, y of the codomain of a. Of course if ( is itself a monoidthen all its morphisms are endomorphisms, and if all morphisms in ( are monomorphisms(a cancellation law) then there could be at most one operation x, a xawith the crucialproperty xa = axa. In the intersection of the two claimed cases, (i.e. for abelian groups)the two formulas for xareduce to the same (trivial) thing. If we restrict consideration tothe full subcategory of monoids determined by groups and commutative monoids (i.e. theunion of the two kinds of objects but containing all four types of homomorphisms betweenthem) we get an example of a natural operation on the underlyingset functor that doesnot extend (from the full subcategory) to all monoids; note that the algebraic structureof a full subcategory of an equationally dened algebraic category may have additionaloperations as well as additional identities between the given operations. In our examplethe subcategory is the union (made full) of two full subcategories which are themselvesequationally dened (in the sense that each consists of all algebras satisfying all theidentities on all its natural operations). If we take the algebraic category equationallydened by the identities listed above, we getMonoids with commutation rule
Gr Comm
full
underlyingset functor
Monoids.hhhhhhhhhhhhhhhhhhhhSetswhere the descending dotted arrow is a faithful functor which however does not reectisomorphisms. That is, there exists a monoid (necessarily not satisfying the monomorphiccancellation rule) on which there exist two dierent self actions satisfying the commutationrule xa = axa. Of course, one interest for operations of this sort on a monoid ( is the strongproperties it implies for the category oCopof right actions on sets (or any Boolean toposo) in particular with regard to the properties of the intrinsically dened intuitionisticnegation operator dened on the subactions A of any action X byA = x X[ r ([xr / A].TAKING CATEGORIES SERIOUSLY 9Namely, if the monoid ( admits a selfaction with a commutation rule as above, then anynonempty A contained in X = T (= the action of ( on itself by right multiplication)satises A = T. By contrast, if ( is so nongrouplike and noncommutative as the freemonoid on two generators, then every principal A = w( T satises A = A.4. Functors from school mathematics mutate monoids into ordered sets,and backBefore passing to the discussion of nonautonomous dynamical systems let us point out acrucial example of a functor which occurs in school mathematics: suppose x y z arenonnegative integers or nonnegative reals, then the dierences y x, z y, z x arealso nonnegative and satisfyz x = (z y) + (y x)0 = x xEven though this theorem is a very familiar and useful identity, it cannot be explainedby either the monotonicity property nor as a homomorphism property in the usual sense,for in fact it is a structurepreserving property of a process (namely dierence) that goesfrom a preordered set to a monoid. As we have already pointed out, posets and monoidsare on the face of it very opposite kinds of categories, thus it appears that once wehave recognized the necessity for giving a rational status to something as basic as thedierence operation discussed above, we are nearly compelled to accept the category ofcategories, since it is the only reasonable category broad enough to include objects asdisparate as posets and monoids and hence to include the above dierence operator asone of the concomitant structural mutations. To be perfectly explicit, let us denote therelation x y (in the poset of quantities in question) by f, and similarly y z by g.Dene(f) = y xand similarly (g) = z y and (1x) = 10 for any x where 1x denotes x x. 0 may beidentied with (the identity morphism of) the unique object of the monoid of quantitieswhere composition is addition. Then satises(gf) = (g)(f)(1x) = 10and hence is precisely a functor from a poset to a monoid. We can be still more explicit.In our example, what does f : x y mean? We could identify f with the proof thatx y holds, that is with the nonnegative quantity f such that x + f = y, or in other10 F. WILLIAM LAWVEREwords with the morphism f in the monoid for which0y
aaaaaaax0f
0commutes. That is f is a morphism in the so called comma category 0/( where 0 is theunique object of the monoid (. Of course, f is just the dierence, so that is identiedas the forgetful functor0/(
(welldened for any comma category. Note that the comma category will be a poset onlyin case ( satises a cancellation law.This construction can also be applied to T = multiplicative monoid whose morphismsare positive whole numbers. Then 1/T is isomorphic to the poset whose objects are positive whole numbers ordered by divisibility, and under that identication, the functorialityof the forgetful functor back to the monoid is expressed byn[m & m[r = rn = rm mn .There are nontrivial consequences of these observations, for any forgetful functor (such as1/T T) on a comma category satises the unique lifting of factorizations property:(f) = vu implies there are unique v, u such that f = v u, ( v) = v, ( u) = u. If T isa category satisfying a suitable local niteness condition then we can dene an algebrastructure on the set a(T) of all complexvalued functions on the set of morphisms of Tby the convolution formula( )(f) =
ba=f(b)(a).Then the unique lifting of factorizations property of a functor T
T is just whatis needed to induce a convolutionpreserving homomorphism a(T) a(T
). In case Thas cancellation, we thus get an inclusion of the Dirichlet algebra a(T) into the algebraa(1/T) associated to a poset; in particular the function (dened as the inverse, whenit exists, of the constantly 1 function) of P becomes the function of the poset 1/T.Since the ordering in 1/T is by divisibility, one thus sees how the functions , 2, etc. ina(T) can be related to counting primes.TAKING CATEGORIES SERIOUSLY 115. Nonautonomous systems, monoidal categories, Dedekind completeness,and the construction of the real numbers as an abstraction of dynamicalwaitingtimesBecause right actions (= contravariant functors) of ( are more directly related to theanalysis of ( itself (for by the CayleyDedekindGrothendieckYoneda lemma, there is acanonical full embedding ( oCop) we are led to repeat the above discussion also for thecomma categories (/0 ( whose morphisms are commutative triangles x
bbbbbbb y0In case ( is an additive monoid with cancellation, we would thus naturally write x y todenote the existence of a morphism x y in (/0. Since for any object X of a categoryof the form oCop(where o is the category of sets and ( is any small category) there is anequivalence of categoriesoCop/X = o(C/X)opwhere (/X is the category of elements of X, we get in particular for a commutativeand cancellative monoid ( thatoCop/T o(C/0)opwhere T is ( acting on itself on the right and (/0 is the poset described above. Howis the equation in the box to be interpreted in terms of dynamical systems? Well, T isthe simple autonomous dynamical system whose states reduce to just the instants oftime themselves, and an object of the left hand category is just an arbitrary autonomousdynamical system X equipped with an equivariant morphism X T; (of course manyX, for example any periodic one, will not admit any such further structure X T).The nature of the functor from left to right is just to consider the family of bers (anotherinstance of a
construction) Xt of the given map X T as t varies through T, andwhenever t
t the global dynamics of X induces a map Xt Xt , which completesthe specication of a functor ((/0)op o corresponding to X. A natural interpretationof the objects of o(C/0)opis that they are nonautonomous dynamical systems, such asarise from the solution of ordinary dierential equations which contain forcing termsor whose inertial or frictional terms depend on time in some manner (such as usury orheating) external to the selfinteraction modeled by the dierential equation itself. Ingeneral for a nonautonomous system the space of states Xt available at time t may itselfdepend on t. The (left) adjoint functor is, as already remarked, actually an equivalenceof categories; it assigns to any nonautonomous system the single state space
tTXt12 F. WILLIAM LAWVEREwith the single autonomous dynamics naturally induced by the nonautonomous dynamicsgiven as Xt Xt for t
t. In case all the instantaneous state spaces are identiableas a single YXt = Y all tthen we see that the associated autonomous system is identiable withT Yand this is just the universallyused construction for making a system autonomous: augment the state space by adding one dimension.In case ( is a commutative monoid, the category (/0 becomes a symmetric monoidalcategory in the sense that there is a tensor functor(/0 (/0 (/0induced by the composition (which we will also write as +) and having the terminalobject00of (/0 as unit object. In many cases this tensor has a right adjoint Hom((/0)op((/0) (/0 that can be naturally denoted by subtraction; it is characterized(assuming ( has cancellation) by the logical equivalencet + a st s a .In the fundamental example where ( is the monoid of nonnegative real (or rational)numbers, the meaning of subtraction is forced by this adjointness to be truncated subtraction. This adjointness persists after completing, as discussed below.For a poset (using for ) to be complete means that for any functor M
h M,h
( ) exists in the poset (these are essentially inma) and also that h
( ) exists in theposet (these are essentially arbitrary suprema). To complete the poset of nonnegativereals (or rationals) means roughly to adjoin (reals and) , but since the precise meaningof this in terms of onesided Dedekind cuts is sensitive to the precise nature of the internalcohesiveness/variation of the sets in o, it is fortunate that there is a precise analysisof this process that goes back to the monoid (. Namely, dene Pos T to be theintersection of all the subdynamical systems P of T which are large enough so that, givenany family f(p) T indexed by p P and satisfying f(p) + t = f(p + t) for all p P,t T, there exists a unique s T such that f(p) = s +p for all p P. Then in favorableexamples ( is continuous (not necessarily complete) in the sense thatp Pos = p1, p2 Pos[p = p1 + p2]and Pos itself is the smallest such P. It is then reasonable to consider the subcategory/ i oCopof semicontinuous dynamical systems dened to consist of those X forTAKING CATEGORIES SERIOUSLY 13which every possible future Pos f X comes from a unique present state, or in otherwords for which the inclusion Pos T induces a bijection (T, X) (Pos, X) betweenthe indicated sets of (equivariant (= natural) morphisms. Then the inclusion i has a leftadjoint i that in turn has a left adjoint / i! oCopincluding the identical category /as an opposite (to i) subcategory of oCop. This implies that / is a topos whose truthvalue object A has as elements all the /subobjects of T; essentially the same is trueof 1 =def//iT, the category of semicontinuous nonautonomous dynamical systems. Butits truth values are also the subobjects of the terminal object 1T because 1 is the toposof sheaves on a topological space [0, ] topologized in such a way that there are as manyopen sets as points, with corresponding to the empty set (or truth value false). Adynamical system X in 1 has a real number [X[ as its support, namely inft[ Xt ,= 0,and this construction can also be viewed as a functor1 1from the topos 1 to the poset 1 = [0, ] (the latter having as arrows) and both 1, 1have tensor and Hom operations related to addition and truncated subtraction, that arecompared by 1 1. Because this discussion can be viewed as a deep objective versionof Dedekinds constructions of 1, it appears that 1 as well as the / that gives rise to itas a comma category, should be taken seriously.6. Metric spaces as enriched categories clarify rotationsas homotopy invariantsIt was extensively discussed in a 1973 seminar in Milan [8] that categories enriched in 1are just metric spaces and hence that a detailed mutual clarication of enriched categorytheory [9] and metric space theory can be exploited. Continuing to take that remarkseriously between 1973 and the Bogot a meeting of 1983 led me to several additionalpoints of mutual clarication, that I will now explain. For a metric space A we have0 = A(a, a)A(a, b) + A(b, c) A(a, c) in 1for any triple a, b, c of objects (points) of /; in general as explained in the cited article,it is better, both for the theory and for the examples, not to insist on further axioms ofniteness or symmetry. 1functors A B turn out to be just distancenonincreasingmaps, and the 1object of 1natural transformations between two such f, g is easilyproved to beBA(f, g) = supaAB(fa, ga).More general than 1functors are the 1modules (= 1relations = profunctors) A Bwhich may be viewed as 1functors BopA 1; the composition of such arising from14 F. WILLIAM LAWVEREthe enriched category notion of Trace (= coend = tensor product of modules) can beshown in this case to reduce to( )(c, a) = infbB[(c, b) + (b, a)]for A B C.Note that if , happened to have only = false and 0 = true as values, then would reduce to the usual , composition of relations as a special case of the aboveleastcost composition that arises when all of 1 = [0, ] is admitted. This relationshipcan be made even more explicit as follows:Let 10 be the twoobject closed category false true with conjunction as tensor andlogical implication as internal Hom. Then the inclusion 10 1 dened in the previousparagraph is actually a closed functor that has a right adjoint 1 10 inducing theposet structure on any metric space; that exemplies the kind of deenrichment processwhich is a universal possibility in enriched category theory, giving the underlying ordinarycategory for categories enriched in any closed category 1. However, in our example thereis moreover also a left adjoint1 0 10to the inclusion that is also a closed functor. [I have called it 0 because of the closeanalogy with other graphs of adjoint functorsccomponentsdiscretepointsc0that occur, such as c = simplicial sets, c0 = sets, and because of the tradition in topologyof calling the components functor 0 because it is sometimes part of a sequence in whichthe next term is the Poincare groupoid 1]. Because 0 is closed, if we consider for anymetric space A the relation dened by0A(a, b)we get another (usually very coarse) preordering. This trivial construction is the key toa pedagogical problem as follows:I wanted to give, for a beginning course in abstract algebra, the basic example of anormal subgroup and a quotient group.Translations Motions Rotations(where each point of the underlying space should give rise to a splitting and hence to aconcrete representation of the abstract rotation group as a subgroup of motions, namelythose motions which are rotations about the point in question). However, it is desirable tobe able to make this basic construction before assuming detailed axioms on the structureTAKING CATEGORIES SERIOUSLY 15of the underlying metric space A. Motions should of course be dened as invertible 1functors f; these will then in particular be distancepreservingA(fa, fb) = A(a, b).But how to dene translations? At rst is seems reasonable to say that they are motionst which moreover satisfy A(ta, a) = constant, for it is not hard to show that the set ofsuch t is normal, i.e.A(ftf1a, a) = A(tf1a, f1a) = same constant.But they may not form a subgroup of the motions, for (theoretically) knowing nothingabout the structure of A, how could we know which constant should result from composingt1, t2 having constants c1, c2? In this case the theoretical worry is substantiated by thepractical fact that we can construct an example of a ve point metric space, indeedembeddable in threedimensional space as three equidistant points on the rim of a wheeland two judiciously placed points on the axis through the center of the wheel, such thatthe translations as dened are not closed under composition. But we shouldnt give up.Dene a translation of an arbitrary metric space A to be an automorphism t such thatsupaAA(ta, a) < .Now it is easy (assuming the metric is symmetric) to prove both parts of the statement thatthe translations form a normal subgroup of the motions, as desired. Even better, thereare many examples of metric spaces A, such as the ordinary Euclidean plane, which havethe property that every translation does in fact move all points through the same distancedue to the searchlight eect: if A(ta, a) ,= A(tb, b) then A(tx, x) can be arbitrarily large,for if there are enough strict translations we can assume that ta = a and constructaxtxbtb??
XXXXXXXXXXXXXXXXWhy is the left adjoint to the inclusionfalse, true [0, ]the key to this problem? Because it (in contrast to the right adjoint, which seems toadmit as truly possible only those projects that cost no eort) is given by0(s) = true i s < as is easily veried. Thus it appears we should take seriously the idea that the homotopytheory of metric spaces, that is the 2functor1cat 10catinduced by 0, is in large part the theory of rotations.16 F. WILLIAM LAWVERE7. Convexity, Isbell conjugacy, closed balls, and radii in enriched categoriesThe CayleyDedekindGrothendieckYoneda embeddingA 1Aopfor 1enriched categories A, reduces in the case 1 = [0, ] under discussion to the factthat A(, a) is a distancenonincreasing real function for any point of any metric spaceA, and that the supdistance between two such functions is equal to the distance betweenthe given points or, more generally, that if Aop f 1 is any distancenondecreasingnonnegative real function (not necessarily of the special form indicated) on the oppositeof the metric space A, thenA(a
, a) + fa fa
all a
fa fa
A(a
, a) all a
fa 1(A(a
, a), fa
) all a
fa supa
1(A(a
, a), fa
)fa 1Aop(A(, a), f)but the last inequality is actually an equality because the sup is achieved at a
= a.Now in between we can insert the space of closed subsets of AA T(A) 1Aopby assigning to each F A the functionF(a
) = infaFA(a
, a)which vanishes on (by denition) the closure of F. The sup metric on 1Aop, restricted toT(A) is a rened nonsymmetric version of the Hausdor metric; that is, its symmetrization is the Hausdor metric, but it itself reduces via 1 10 to an ordering which reectsthe inclusion of the closed setsF1 F2 as closed sets i F1 F2 in 1Aop.Since any f 1Aopdoes have a zeroset, therefore a right adjoint to the inclusionT(A)
1AopZ.can be constructed. This leads to the idea that the objects in 1Aopmight be considered asrened closed sets; a point of view already forced upon researchers in constructive analysis and variational calculus by the stringent requirements of their proofs and calculationsthus receives also a conceptual support from enriched category theory.TAKING CATEGORIES SERIOUSLY 17Now an extremely fundamental construction in enriched category theory is the adjointpair known as Isbell conjugation1Aop( )
(1A)op( )#which is dened in both directions by a similar formula(a) = 1Aop(, a)#(a) = 1A(, a)where we have followed the usual practice of letting the Yoneda lemma justify the abusesof notationA(, a) = a in 1AopA(a, ) = a in 1A.The general signicance of this construction is somewhat as follows: if 1 is the category ofsets, or simplicial sets, or (properly construed) topological spaces, or bornological linearspaces, and if the 1category A is construed as a category of basic geometrical gures,then 1Aopis a large category which includes very general geometrical objects that can beprobed with help of A, but that would inevitably come up in a thorough study of A itself.On the other hand, 1Aincludes very general algebras of quantities whose operations (` ala Descartes) mirror the geometric constructions and incidence relations in A itself. Thenthe conjugacies are the rst step toward expressing the duality between space and quantityfundamental to mathematics: ( ) assigns to each general space the algebra of functionson it, whereas ( )#assigns to each algebra its spectrum which is a general space. Ofcourse neither of the conjugacies is usually surjective; the second step in expressing thefundamental duality is to nd subcategories with reasonable properties that still includethe images of the conjugacies:A
qqqqqqqqq.zzzzzzzzA_
/op_
1Aop
(1A)opFor example if 1 is the category of sets and A is a small category with nite products,we could take A to be the topos of canonical sheaves on A, and / to be the algebraiccategory of all niteproductpreserving functors.Our poset 1 = [0, ] appears rather puny compared to the grand examples mentionedin the previous paragraph, but seriousness eventually leads us to try the Isbell conjugacy18 F. WILLIAM LAWVEREon it as well, and in particular to ask which closed sets F T(A) are xed by thecomposed Isbell conjugacies for a metric space A.T(A)
1AopZ.( )# ( )
(1A)op.( )#Note that from the adjointness, #for all in 1Aop, so that the idempotent operation( )#gives a kind of lower envelope for functions and hence a kind of hull Z() Z(#)for the corresponding closed zerosets; the question is what kind of hull?To answer this it is relevant to explicitly introduce the following parameterized familyof special elements of 1Aop:1opA B 1Aopdened byB(r, c)(a
) = 1(r, A(a
, c)),where 1(x, y) = y x is the truncated subtraction in our example 1 = [0, ]; but for anyclosed category 1 denotes its internal Hom, and where the tensor product of 1categories(dened in our case as the metric which is the sum of the coordinate distances) is usedinstead of the cartesian product because it guarantees that B itself is also a 1functor.The letter B stands for closed ball of a given radius and center since0 B(r, c)(a
) i r A(a
, c).An amazing example of the seriouslypursued study of the mutual relationship of akey example with general philosophy is that these closed balls occur and are useful overmany apparently quite diverse closed categories 1, for example in homological algebra.Moreover, let us denote by
the supremum operation on 1Aop
i
(a
) = supii(a
)since that is what it corresponds to under the operation Z of taking zerosets.With the aboveintroduced notation, we see that = #i # , and also that#(a
) = 1A(, a
)= supa
1((a
), A(a
, a
))so that#=
a
B((a
), a
)is the intersection of all the closed balls, centered at all points a
, of specied radius (a
).[Naturally this construction, under the name of end or center, also comes up forTAKING CATEGORIES SERIOUSLY 19general 1]. Now what if is of the form ? The number (a
) is a radius (about anarbitrary center a
) that somehow prefers:(a
) = supa1((a), A(a, a
))so thatr (a
) i r + (a) A(a, a
) for all ai (a) A(a, a
) r for all ai B(r, a
).Thus (since throwing some larger balls into the family wont change the intersection)#=
B(r, a
)[ B(r, a
)is a geometrical description of the doubleconjugate lower envelope of . In the casewhere is xed under the other idempotent operation on 1Aopcoming from Z, i.e. if isdetermined by its closed zeroset F as explained before(a
) = infaFA(a
, a)then the condition B(r, a
) reduces toa F a
A(a
, a) B(r, a
)(a
)= A(a
, a
) ri.e. toA(a
, a) + r A(a
, a
) for a F, arbitrary a
.In particular this means that F the ball (= zero set of) B(r, a
), so we see thatF F# the intersection of all closed balls that contain F.But in fact that intersection of balls is equal to F#, since the right adjointness of Z saysin particular that for any closed set F and any ball we have the equivalenceF ZB(r, a
)F B(r, a
) .Putting it dierently, since we have in general thatA(a
, a) + A(a, a
) A(a
, a
),if a F implies 0 B(r, a
)(a), i.e. r A(a, a
), then for any a
and any a FA(a
, a) + r A(a
, a) + A(a, a
) A(a
, a
)20 F. WILLIAM LAWVEREso thatA(a
, a) B(r, a
)(a
)meaning thatF(a
) B(r, a
)(a
) as functions of a
,because the right hand side is independent of a and the function F was dened as theinmum (adjointness of
along a diagonal). ThusF#= intersection of all closed balls which contain F= closed convex hull of F,at least in many metric spaces of geometric importance, and for the others the proposeduse of the term convex for those closed sets F satisfying F = F#should be as goodor better than other proposals because of the apparent importance of adjointness in calculations.As noticed above, the numbers (a
) have in many connections the concrete significance of radii for balls about a
. Using the obvious direct limit functor 1A inf 1(which exists because A 1 is a 1functor), we can dene a single radius for itselfby applying the composite1Aop ( )
rad
(1A)op infop
1opIn particular, the radius of a closed set F T(A) 1Aopisrad(F) = infr[ a
[F ZB(r, a
)]where the candidate centers a
are not themselves necessarily in F. (Note that to say wehave a functor T(A) 1opfrom a poset to 1opis to say that the values increase as theobjects in T(A) increase). The radius is a more functorial quantity than the habituallyused diameter; for a symmetric metric space there is the estimate diam 2 radwhereas for certain reasonably well behaved spaces one may also have a converse estimate rad diam. There is a strong tendency for the radius to be realized at a uniquecenter a
. Note that rad(F#) = rad(F).8. Geodesic remetrization as an adequacy comonadNow let us say a few words about the important role of paths in metric spaces. Thecomma categories d/1 = [0, d] with their canonical d/1 1 have a retraction givenagain by double dualization:x 1(1(x, d), d) = d (d x)TAKING CATEGORIES SERIOUSLY 21is always in the interval; moreover single ddualization is an invertible duality(d/1) (d/1)opwhen restricted, provided d < . Denote by 1
1 all those d such that 0(d) = true,i.e. for which d < . Let 1(d) be the symmetric metric space determined by d/1, sothat in particular d ( ) becomes (for d < ) a selfmotion of the interval 1(d), becausesymmetrizing is a functor. Indeed, d 1(d) denes a functor1op 1catusing d
d = 1(d) 1(d
), but also a functor 1
1cat by invoking the retractions.The essential question we want to understand is: to what extent is the structure of anarbitrary metric space A 1cat analyzable in terms of the paths (= 1functors)1(d) Aof duration d, with d variable? Since all constants are paths, such analysis easily maintainsthe points of A. If there is such a path, passing a0 at time 0 and a1 at time d, thend A(a0, a1).This leads to the idea of geodesic distance:(A)(a0, a1) = infd 1
[ : V (d) A with (0) = a0, (d) = a1which can be seen to be a new metric since1(0)
1(d
)
1(d)
1(d + d
)is a pushout in 1cat. That is, if is a path in A of duration d, and
of duration d
,and if(d) =
(0) = a1we must show that if d t, d + d
s d, thenA((t),
(s d))? s t.But we haveA((t),
(s d)) A((t), a1) + A(a1,
(s d))= A((t), (d)) + A(
(0),
(s d)) d t + s d = s t22 F. WILLIAM LAWVEREbecause d t and s d. The other cases of the 1functorality of (d + d
)
A areobvious. Thus (using the logicians symbol for proves) d A(a0, a1)
d
A(a1, a2)
=
d + d
A(a0, a2).Because direct limits in metric spaces are essentially computable in terms of direct limitsof sets and inma of distances, it can be seen thatA = limV
/A1(dom)is an endofunctor of 1cat having a natural distancenonincreasing mapA
Aback to the identity functor; it is in fact the adequacy comonad of 1
1cat, anotion dened for any small subcategory of any category having direct limits. Note forexample that pathconnectedness of A becomes 0connectedness of A, for if two pointsof A are not connectable by a path, then (empty inf) their geodesic distance in A isinnite.9. History still has much to teach, and raises fresh questionsFinally, I believe that we should take seriously the historical precursors of category theory, such as Grassmann, whose works contain much clarity, contrary to his reputation forobscurity. For example, I read there a statement of the sort diversity can be addedwhereas unity can be multiplied together with quite convincing geometrical and algebraic substantiation of these principles. The rst of them suggests the following:If T(A) is a poset of parts of a space and ( is a suitable additive monoid, then theamount by which F must be extended to achieve a diverse G F might be given by(F, G) (; that should again give a functor T(A) ( in thatF G H = (F, H) = (F, G) + (G, H).F
'&$%'&$%TAKING CATEGORIES SERIOUSLY 23Of course, if 0 T(A) and if ( has cancellation then (G, H) will be determined by H =def(0, H) and F. To further express the quantitative nature of such a measurement ofextension , we can consider the further condition that only depends on the dierence.Since the crucial property of dierence is again that it is a Hom, adjoint this time to unionas ,S H GG S Hwe can express this invariance of the functor using only the union structure on F(A):S[ G S H = S[ G1 S H1 = (G, H) = (G1, H1).It would be interesting to determine for which upper semilattices T, 0, there existsa commutative monoid ( and a functor with this invariance property which moreoverhas the uniqueliftingoffactorizations property previously discussed, i.e., for which theobject X in a process of intermediate expansion F X H is uniquely determined bysuciently many quantitative measurements of its size. For example, area alone is notsucient but ( can be a cartesian product of many dierent kinds of quantities. Againadjointness makes at least an initial contribution to the problem: for each T there is awelldened universal ( and , so that one need only study the lifting question for that.10. Dialectical relation between teaching and research can be exempliedby metrical development of enriched category theoryWe have seen that the application of some simple general concepts from category theoryleads from a clarication of basic constructions on dynamical systems to a construction ofthe real number system with its structure as a closed category; applied to that particularclosed category, the general enriched category theory leads inexorably to embedding theorems and to notions of Cauchy completeness, rotation, convex hull, radius, and geodesicdistance for arbitrary metric spaces. In fact, the latter notions present themselves in sucha form that the calculations in elementary analysis and geometry can be explicitly guidedby the experience that is concentrated in adjointness. It seems certain that this approach,combined with a sober appreciation of the historical origin of all notions, will apply tomany more examples, thus unifying our eorts in the teaching, research, and applicationof mathematics.References[1] Eilenberg, S. and Mac Lane, S., General Theory of Natural Equivalences. Trans.Amer. Math. Soc. 58(1945), 231294[2] Grothendieck, et. al., Theorie des Topos et Cohomologie Etale des Schemas. SpringerLecture Notes in Mathematics 269, 270, 1972.24 F. WILLIAM LAWVERE[3] Lawvere, F.W. (editor), Toposes, Algebraic Geometry and Logic. Springer LectureNotes in Mathematics 274, 1972.[4] Lawvere, Maurer, and Wraith (editors), Model Theory and Topoi. Springer LectureNotes in Mathematics 445, 1975.[5] Banaschewski (editor), Categorical Aspects of Topology and Analysis. Springer Lecture Notes in Mathematics 915, 1982.[6] Lawvere and Schanuel (editors), Categories in Continuum Physics. Springer LectureNotes in Mathematics 1174, 1986.[7] Joyal, A., Une theorie combinatoire de series formelles, Advances in Mathematics,42(1981), 181.[8] Lawvere, F.W., Metric Spaces, Generalized Logic, and Closed Categories, SeminarioMatematico e Fisico di Milano 43(1973), 135166, and Reprints in Theory and Applications of Categories, no. 1(2002), 137.[9] Kelly, G.M., Basic Concepts of Enriched Category Theory. London MathematicalSociety Lecture Notes Series 64, Cambridge University Press, 1982 and Reprints inTheory and Applications of Categories, (2005).University at Bualo, Department of Mathematics244 Mathematics Building, Bualo, New York 142602900Email: [email protected] article may be accessed from http://www.tac.mta.ca/tac/reprints or by anonymous ftp atftp://ftp.tac.mta.ca/pub/tac/html/tac/reprints/articles/8/tr8.dvi,psREPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles from thebody of important literature in Category Theory and closely related subjects which have never beenpublished in journal form, or which have been published in journals whose narrow circulation makesaccess very dicult. Publication in Reprints in Theory and Applications of Categories will permitfree and full dissemination of such documents over the Internet. Articles appearing have been criticallyreviewed by the Editorial Board of Theory and Applications of Categories. 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After a work is accepted, the author or proposer must provide to TACeither a usable TeX source or a PDF document acceptable to the Managing Editor that reproduces atypeset version. Up to ve pages of corrections, commentary and forward pointers may be appended bythe author. When submitting commentary, authors should read and follow the Format for submission ofTheory and Applications of Categories at http://www.tac.mta.ca/tac/.Managing editor. Robert Rosebrugh, Mount Allison University: [email protected] editor. Michael Barr, McGill University: [email protected] editors.Richard Blute, Universite d Ottawa: [email protected] Breen, Paris 13: [email protected] Brown, University of North Wales: [email protected] Brylinski, Pennsylvania State University: [email protected] Carboni, Universit` a dell Insubria: [email protected] de Paiva, Xerox Palo Alto Research Center: [email protected] Getzler, Northwestern University: getzler(at)math(dot)northwestern(dot)eduMartin Hyland, University of Cambridge: [email protected] T. Johnstone, University of Cambridge: [email protected] Max Kelly, University of Sydney: [email protected] Kock, University of Aarhus: [email protected] Lack, University of Western Sydney: [email protected] William Lawvere, State University of New York at Bualo: [email protected] Loday, Universite de Strasbourg: [email protected] Moerdijk, University of Utrecht: [email protected] Nieeld, Union College: [email protected] Pare, Dalhousie University: [email protected] Rosicky, Masaryk University: [email protected] Shipley, University of Illinois at Chicago: [email protected]c.eduJames Stashe, University of North Carolina: [email protected] Street, Macquarie University: [email protected] Tholen, York University: [email protected] Tierney, Rutgers University: [email protected] F. C. Walters, University of Insubria: [email protected] J. Wood, Dalhousie University: [email protected]