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C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M ICHAEL BARR Building closed categories Cahiers de topologie et géométrie différentielle catégoriques, tome 19, no 2 (1978), p. 115-129. <http://www.numdam.org/item?id=CTGDC_1978__19_2_115_0> © Andrée C. Ehresmann et les auteurs, 1978, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Vol. XIX - 2 (1978 ) CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE BUILDING CLOSED CATEGORIES by Michael BARR * examining and generalizing the construction of the cartesian closed category of compactly generated spaces of GabrielZisman [1967 . Suppose we are given a category 21 and a full subcategory 5 . Suppose E is a symmetric monoidal category in the sense of EilenbergKelly [1966], Chapter II, section 1 and Chapter III, section 1. Suppose there I am interested here in is, in addition, «Hom» functor lil°P X 5--+ 6 which satisfies the axioms of a closed monoidal category insofar Eilenberg-Kelly for Then that, granted certain this we can of these show a be extended to a closed monoidal objects of 21 which have thesis suffices to One original product in 5 is the cartesian structure on Z-presentation. category is cartesian closed. As a result, only of compactly generated simplicially generated, sequentially generated, etc... spaces derive the cartesian closedness but also of the full subcategory additional hypo- a show that when the product, then the resultant we they make sense. reasonable hypotheses on lbl and 21, as 1. STRUCTURE ON even not 5. (1.1) We suppose that T is equipped with symmetric monoidal structure. This consists of a tensor product functor - O - : 5 x 5 --+ 5, an object I and coherent commutative , associative and unitary isomorphisms. See EilenbergKelly for details. We suppose also a functor ( - , - ) : 5oP X 21 --+21 and natural equivala ences * I would like to thank the National Research Council of Canada and the Canada Council for their support. 115 C, Here A E 21 . The C 1 , C 2 E 5, first and third combine give also to ( 1.2 ) We make one further hypothesis about the relationship between T and U . Namely that for every object A e 1 there be a set Cw --+ A} of maps with domain in T such that every C - A with C E 5 factors through at least one effect, demanding that each category of 5-objects over 21 weak initial family. Such a family is called a terminal 5-sieve for 21. am, in Cw --+ A . I have a ( 1.3 ) We suppose, finally, that I has regular epimorphism/ monomorphism factorizations of its 5-PRESENTED 2. morphisms. OBJECTS. ( 2.1 ) I define A c a be to 5-generated if there is a regular epimorphism : Because of the cancellation that if there is I GY --+ A} some is Cw--+ a properties of regular epimorphisms, it follows any regular epimorphism of that kind we may suppose the A factors through terminal 5-sieve for A . For each CY--+ A and so regular epimorphism factors : 5-presented provided there is the ( 2.2 ) WOe say that A is Assuming that A is 5-generated by Xi CY--+ of that map it is sufficient that there be if there is such case a terminal ( 2.3 ) that ever, as be happens that 6 - there is will be (2.4) One diagram and that B is the kernel A epimorphism I Cç it is clear that it suffices 4B. seen, word of 5-presented no in I alone - is or even distinction between being Z-presented Z-generated is the a pair In any take for Cç} generator for 21. In to Z-sieve for the kernel pair B . It often case, a an coequalizer diagram a and 5-presented.How- important thing. warning is in order. It is entirely possible that an object a full subcategory of 1 which contains Z but not in 21. 116 This is because the coequalizer condition is less exigent in a subcategory. ( 2.5 ) Suppose {Cw--+ A is a terminal 6-sieve for A , Ao = Z Cw, A1 = I and IC6-+A, XA Ao 1 is a terminal 5-sieve. Then we have a diagram: Ctfr This is diagram call it a a coequalizer 6-presentation iff A is 5-presentable and in that case we of A . ( 2.6 ) Let f: A --+ B . Whether or not A or B is 5-presentable we can find diagrams of the above type where For all hence we we have commutes. are equal through get that the map For an CO) 4 A 4 fo : L Cw--+ any Vi the two ZCç Cç through some Cç--+ B and such that maps and hence determine some B factors and results in a a map CY --+ Bo XBBo commutative Ao - Bo is another choice for fo, map Ao 4 Bo X B Bo which similarly has is that the map f induces a map, «unique If go : 117 the a up which has diagram maps homotopy » from lifting = together lifting h : Ao - Bi . two to a determine The a upshot A1==++ Ao to B1 ==++Bo; is induced functor now the on let TT A we unique a ( 2.7 ) It is if map, and 7TB be the naturally denoted standard argument a category U respective coequalizers, rf: to see TT A 4 that 7T which lands in the full It is clear also that there is 77 B. is a there well defined endo- subcategory 5-present- of natural TT A 4 A which is an isoobjects. morphism iff A is 5-presented and, finally, that 7T determines a right adjoint to the inclusion of that subcategory. We let r21 denote the full subcategory and let 7T denote also the retraction 21 --+ r 21 . ed (2.8) PROPOSITION. Let is isomorphism for all Cc 5. Then an PROOF. imply are { Cw --+ A} be {Cw A 4 B} is Let that and hence rr f is (2.9) an for pairs they are have the same map such that a 77f is 5-sieve for B . Let maps, for A . Then the Ao = Z Cw . hypotheses Now both namely isomorphic for all C e 6 . Hence, Ao XA Ao terminal T-sieves and it follows immediately an Under the same hypotheses, if isomorphism. 3. INDUCED STRUCTURE ON r 21 . ( 3 .1 ) We define There is isomorphism. an and that isomorphism. COROLLARY. ted, f is one isomorphic also be terminal a --+ the kernel A0 XB A0 f: A- B a a 1-1 correspondence between maps With A E 21 , 118 A and B are Z-presen- Finally, which we have gives, upon We prove it is application of r , a map isomorphism by using ( 2.9 ). To corresponds uniquely, using right adjointness of rr ( 3.2 ) Since a an regular image of a ular factorization is inherited 21-sieves is evident. Hence PROPOSITION. by r 4. INDUCED STRUCTURE ON (4.1 ) We will noidal now poses of this In view of The (3.2) ( A, B ) same 21 21 . able C --+ [ C1 O C2 , A ] is (0..presented, the reg- The continued existence of terminal to conclude : satisfies all the hypotheses of Section 1. 7T U. given 1 . In order to tensor and internal hom simplify notation we to a closed mo- will, for the pur- Section, suppose that this is ( 4.2 ) Now choose and let extend the structure on 7T 7r map several times : 5-presented object we are The category a a permissible. T-presentation ( see ( 2.5 ) ) be defined argument (A ’, B) --+ (A, B) as in as the equalizer ( 2.6 ) shows that of any map A 4 and this is well defined and functorial. (4.3) Similarly define A OB by choosing 119 5-presentations A’ induces a map and then A OB is the ( 4.4 ) P ROP OSITION. coequalizer For of fixed CE5, the functor (C,-) commutes with projective limits. PROO F . V’e use ( 2.9 ) . Suppose We get and conversely. (4.5)PROPOSITION. Let CE5. Then PROOF. Let be is a 5-presentation. Then an equalizer. But Since is an (4.6) equalizer, so are PROPOSITION. PROOF. We use For A (2.9). Let fixed, ( A, - ) commutes B = projlim Bw . 120 If with projective limits. conversely. and ( 4.7 ) PROPOSITION. For any A1, A2, BE 21, P ROO F. Let be a T-presentation from which the (4.8) with of Al . Then we have equalizers isomorphism follows. PROPOSITION. For B fixed, the functor (-, B : 21°p --+21 commutes projective limits. PROOF. Let A = ind lim AO) . and the result f ollow s from (4.9) P ROPOSITION. For we have for any C a 6-presentation. is a coequalizer so E, C 4 proj lim( AO) , B ) , ( 2.9 ). C E 5 , (C O A, B) = (C, (A, B)). P ROOF. Let be c Then that 121 are equalizers. (4.10) PROPOSITION. L et A, BE 21 and : sentation. Then is a coequalizer. PROOF. The identity and hence lift to a e is a reflexive Now the are rows single coequalizers, diagonal equalizers ( 4.11 ) of . same a is true a C 0) --+ A Thus of standard a 5-presentation of B, diagram chase to see that the fixed object and consider the resultant diagram sets. ) PROPOSITION. coequalized by I of from which it is column is. ( Hom it into of map coequalizer.The and the are maps For P ROOF. Let 122 be is a of Al. Then coequalizer. From ( 4.8 ) a are 6-presentation equalizers, (4.12) and ( 4.9 ), from which the PROPOSITION. we have that isomorphism follows. There is 1-1 a correspondence between : PROOF. Let 6-presentation. be a is an equalizer. equalizer. (4.13) Since it is a coequalizer, it That the third is follows from THEOREM. equipped with - 0 - Given the and a applying hom (1, -) to Section 1, the category closed monoidal category. hypotheses of (-, -) is follows that the first line of an r 21, 5. CARTESIAN CLOSED CATEGORIES. ( 5.1 ) Suppose the functor -O - :5 x --+ E is the cartesian product, denoted as usual by - X - . We would like to know when the induced tensor on ;7 21 is 123 the cartesian Of product. course we may well be different from those purposes of know by example in 21 . As above, we products in 17 21 replace 17 a by U for that exposition. ( 5 .2 ) Let A E 21 and let be 6 -pre s en tation . Th en for a C E5 , coequalizer.The projections Cfu X C -+ C give a map Z ( Cw X C) --+ C, which coequalizes the maps from I ( CY XC) and hence induces a map from is a A O C also to C. The map coequalizes AO C to the A . This C’c T , the map factors through whose projections map have two gives any on A e C --+ A X C. If C’ -+ A X C is and determines A and C a a map from a map with map that are the 5-presentation is the map and thus induces CY X C given ones. This implies that the AO C --+ A x C is a regular epimorphism since otherwise there would to be a map C’ -+ A X C which doesn’t factor through the image. ( 5.3 ) Thus for so a Cw --+ A some from I maps a regular epimorphism. By adjointness Z(CwXC ) --+ (ZCw ) X C is a regular epimorphism. Let us now add following H YPOTHESIS. For any set of onical objects map Z(Cw X A ) --+ (ZCw ) X A I C w} is 124 a of Z and any AE 21 , monomorphism. the can- ( 5.4 ) This hypothesis allows Let Ao = Y- C60 5-sieve terminal a hence a AoXC --+ A X C coequalizer, ( 5 .5 ) that so XA Ao is a that {CY --+ Ao XA Ao} are regular epimorphism and The sieve presentation. ZCY--+ Ao immediately is so Since also is in the above conclude us to Now ing (4.10) a it follows that regular epimorphism, from which A O C = A X C is immediate. simply to is repeat the argument with object B an in place of C, us- initiate the argument used in ( 5.2 ). ( 5 .6 ) One easy way in which the hypothesis of ( 5.3 ) may be satisfied is if 5 has finite sums, if any map from an object of Z to a sum in 5 factors through a finite and sum in 6 the injections to a sum are monomorphic. First consider the map If this fails with the w = to same be monomorphism, composite in (ZCw ) a 1, ... , n , suffice so there XC. are two maps C’ A finite number of are injection into the sum a monomorphism, equal. An analysis of the argument used in (5.4) show A 0 C = A gument ( 5.7 ) to derive the THEOREM. C for C E 5 which X indices, say that the maps from C’’ factor and with the to ==++Z ( Cw X C) hypothesis Given the for can right This hypotheses of Section two maps shows that this suffices be fed arbitrary A . 125 it follows the can back into this be summarized 1 with the tensor on aras (Z product, then the resultant category hypothesis of (5.3) is satisfied. the cartesian provided the 77 U is cartesian closed 6.EXAMPLES. (6.1 ) The first example leads to the category of compactly generated spaces. Let 21 be the category of Hausdorff spaces and 5 the full subcategory of compact ones.The tensor is the cartesian product and the hom equips the function space with the uniform convergence topology. That the hypotheses of Section 1 are satisfied is standard ( Kelley [1955] ), and that these of (5.6) are is trivial. The construction gives the category of compactly generated spaces with internal hom given by the compact/open topo- logy. (6.2) With I as above, any full subcategory of compact spaces may be taken as 6 provided it is closed under finite products. In fact, even that restriction may be in Z are dropped provided 5-generated. we can show that finite For then the classes of spaces products of objects generated by T and generated by its finite product closure are the same. Since the full subcategory generated by the one is cartesian closed - the hypothesis of (5.3) being satisfied in any case - it follows that the full subcategory ge- these nerated buy 6 is also. (6.3) For example, take for 5 the category consisting of the space with the usual spaces topology and its sequentially generated. endomorphisms. We call the 6-generated It is clear that first countable spaces - in particular all Cn - are sequentially generated and so are their quotients (see Kelley, problem 3-R where it is shown that the euclidean plane with one axis shrunk to a point is a quotient of a first countable space that isn’t first countable ) . Conversely, any sequentially generated space is a quotient of a union of copies of C and hence a quotient of a first countable space. 126 Thus sequentially generated means a of quotient first countable space. a (6.4) For another example let 5 be the category consisting of the unit interval J and its endomorphisms. Let X be any space such that each point has a countable decreasing basis of first countable.Then the Given such ces. x . By refining and xn connected is determined by { Un} be the sequence, if necessary, we topology sequence {xn }--+ x, a Then map J --+ X xn +1 pathwise let the interval Un . Call it pathwise base neighborhood a may suppose that [ 1 / n+1, 1/ n] by letting entirely inside which lies sets. the convergent sequen- go to a This describes path a at xn E Un . between continuous map J --+ X and it is clear that such maps determine the topology. Since each Jn is pathwise first countable the 5-generated spaces - usually called simplicially generated - form a cartesian closed category.They are the same as the pathwise first countable These spaces. topological examples have also been considered, from ly different points of view - by Day [1972] substantially the same results. (6.5) We can also play and slightWyler [1973] - who obtain this game with the category of separated uniform spaces. A direct limit in the category of uniform spaces of compact Haus- uniformity. For any function which is continuous is continuous, hence uniformly continuous, on every compact subspace, and hence uniformly continuous on the whole space. The same argument shows it is compactly generated in the category of completely regular spa- dorff spaces has the fine ces. The converse pletely regular is also true so spaces which are the resultant category consists of the compactly generated com- in that category. (6.6) We may vary the examples of ( 6.2 ) - ( 6.4 ) by considering pointed Hausdorff spaces for 21 . The is the smash product ( the cartesian product with the naturally embedded sum shrunk to a point) and the internal hom of C to X is the subspace of base point preserving maps with the same compact/open topology. The fact that the m ap tensor product 127 now consists in the identification of proof that the inverse image vation, the equivalence of single a a compact compact set set to a point allows the is compact. From that obser- follows readily from the analogous result for non-pointed spaces. This example may also be varied by considering different possibilities for 5 to get pointed sequentially generated spaces, pointed simplicially generated spaces, etc... topology) is meant a real or complex topological space equipped with an auxiliary norm. Morphisms are required to be linear, continuous in the topology and norm reducing. Various conditions may be imposed on the spaces but the one most useful here is the supposition that the norm and the topology are those induced by mappings to Banach spaces. This means the space is a subspace, both in norm and topology, of a product of Banach spaces. If complete, it is a closed subspace. VUe let 21 be the category of such complete MT spaces. Let B be the full subcategory of the Banach spaces ( i. e. the topology is that of the norm ) and the full subcategory of those whose unit ball is compact. It is known ( see Semadeni [1960] or Barr [1976] ) that B and 6 are dual by functors (6.7) By an MT (for mixed ( describable the space of linear functionals topologized, respectively, topologies). If B E B , C e 6 , ( C , B ) is defined to be the space of morphisms C - B normed by the sup norm. It is a Banach space. If A E 21 , let A C II Bw , then ( C, A ) is the space of morphisms given the topology and norm induced by ( C, A ) C II ( C, Bw ). Finally, for Cj, C2 in 5 , we define as with the weak and the norm which lies in 6 . The completeness Barr [1976 a] . ( which from proof that uses the conditions of section 1 of the spaces, 128 by the way) can are satisfied be worked out REFERENCES M. BARR, M. Duality of Banach spaces, Cahiers B ARR, Closed categories and Banach B. DAY, A reflection theorem for closed 1- 11. Topo. et Géo. Diff. spaces, Idem XVII-4 ( 1976), 15. ( 1976 a), 135. categories, J. Pure Appl. Algebra S. EILENBERG and G.M. KELLY, Closed categories, Proc. P. XVII- 1 Algebra (La Jolla, 1965), Springer, 1966. GABRIEL and M. ZISMAN, Calculus of fractions and Conf. on 2 ( 1972), Categorical homotopy theory, Springer, 1967. J. L. KELLEY, General Topology, Z. SEMADENI, Van Projectivity, injectivity O. WYLER, Convenient Nostrand, 1955. and duality, Rozprawy categories for Topology, General Topology and Applications 3 (1973), 225-242. Forschungsinstitut Mat. 35 ( 1963). fur Mathematik E. T. H. ZURICH and Mathematics Department Mc Gill University MONTREAL, Qu6bec CANADA 129