Relative functor categories and categories of algebras

Relative Functor Categories and Categories of Algebras ‘l’he starting point ot’ our considerations is the following situation, to in this paper. f)enote by .Y” the categorl 1~ abstracted and generalized denote the binary product functor, and let ; CJf sets ar1d mappings. Let natural isomorphisms be the terminal object of .Y. ‘There are canonical NAB<. : (A x B) x c----f *-I ;. (B % C), r,4 : =3 x I ---+--I, lA : 1 < .-1-*.+J. and k,, : -4 ,q B --+ B ‘* .-I, colzerml in the sense of Mac I,anc [28]. Also, there exists a natural isomorphism w,,~(- : Hom&A x B, C) ---f Homy(J. Homy(B, C)) in CY, by virtue of which the functor ~ x B : .Y’ -, .Y is ad&zf (or Zeft adjobzt) to the functor Homy(B, -) : :Y’ + Y’, for each object B of .Y’. These properties make .Y’ into a closed calegouy. in a sense to bc made precise later (I .9). (A n Lq uivalent notion is that of a symmetric ~~zo~roirlaI closed catgory (Eilenberg and Kelly- [S]).) There exists an equivalence of categories @ : .‘/’ + Ad(.‘/‘, .Y), where b>~ Ad(.Y’. ,Y) we mean the catcgor) whose objects are the endofunctors of ./ exists, taking as morphisms natural for which a coadjoht (or Y&/?t adjo&t) transformations. More precisei!, define the functor d, by the rules On the other hand, H -+ x.’ B; f--p - x .f. C‘lcarly, @ is full and faithful. let ‘/’ : .Y --F .Y be given, together with a functor G : .‘f + .Y and an rrcQunctir,rz in : 7’~ G, i.c., a natural isomorphism aAc‘ : Honl,,(T(J), C’) + Y ‘T(I) -1 G is then defined IIon?,(.-l, C(C)) in -‘I’. An aci~junction B : of the following mappings: b!- letting Pnc be the composition Horn,, (. I A’ T( I ), C) 2” -t Hon~,~,(- I, HomJ,( ---‘? -L Hom,y(.-l, IIom~P(.4 Ir’( I ), C’)) Hom:/( i: I, G(C)) I . Cg(C))) From the esistencc of 3: and ,L?follows the existence of a canonically d&cd ,; T(l) --+ 7’. Thus, @ is an ccluiv-alence of natural isomorphism y : categories. A peculiarity of the functor @ : .‘/’ --f Xd(.Y, -Y) just defined, is that it carries monoids (i.c., semigroups with identity) into standard constructions in .‘/. Let us examine more closelv this fact. Tl’e recall first that a 7??onoill (in U) is any set I,1 together \\-ith mappings e : I - A and nl : .1 ._ ‘1 f .I, satisfying the following asioms: AlIon 3. e X A f m = I., . If [A, r, vz] is any monoid, then the functor 7’ structure of 3 standard ronstuutior~ in .‘I/‘ (C;odement also called a triple in .Y’ by Eilcnberg and LIoore natural transformations 7j : I,y -+ 7’ and I/- : TT ---f .d x E and ~1,~ CI,~.!.,’ .-1 Y ~1. Indeed, the triple Iw virtue of the monoid axioms, the following: COMfY. I Tp~~=pT.I~. Constr. 2. Tri . !-L= T. Constr. 3. @(A) ~ ‘/ A has the [13]; dually Huber 11141; [9]) together with the T, defined b!; vA z ~,,l. T = [7’, r/, p] satisfies, 77‘.p=T. On the other hand, if T [T, 7, /L] is any adjoint standard construction in .‘f with n : 7’ --I G and a canonically defined natural isomorphism y : - : T( 1) - +7 (as above), then the set ‘1 m=T(1) has the structure of a monoid with E ~~ q1 and ?I? = yTcl) . p1 , as it is easily checked. Therefore, CDsets up a hijection lxtzuen monoids (in ,‘Y) and adjoint standard constructions in .Y. Let % be any category (locally small) with only one object, denoted by o. \\‘ith ?? is associated the monoid [Ax , F, m] with A, Hom,(o, o), P : 1-->A, the mapping Lvhich selects I, , and m : A, x Av ---+ A, , the composition law of ‘6. Denote tn. TX the adjoint standard construction in .Y associated with [,!,( , (I, 77/I. 011 the OIIC hand \ve may form the functor ratqorv .Y”“, htrsrd o/l .‘/‘. its objects are the functors .Y : ‘g + .Y’. Its morphisms are natural transformations x : S + -Y’ : % --t .‘f. There exists a functor (; . ‘/” *-If given by the rules .Y mm% -Y(o); x + x,, . On the other hand ‘6 . we ma!- const&t the mtqoq) of 011 T% -algebras .Y “. Its objects arc pairs IA4, [] \I-here *-I is a set and 6 : :I /‘, A ,(; --> .4 is a mapping which is an “action of -I,( on A-1” in the. sense that the follo\ting conditions hold: 66 UUS(:I; T% u/g 2. c(u, nz(X, A’) f([(u, A), A’), for all o E -4, A, A’ E AL . ‘I’hc morphisms of -‘F” are called T~~-/ronzomo~p-ph~~~~~~. A mapping ,/: ..I + B is a T’r: -homomorphism .f : [.A, [] L [B, 01, if the foilon-ing condition is satisfied: ‘l’hc abo\c example is liable to a t\vofold gc-ncraiization. I;irst. ?\c shall requirement on % and ;11low an;\- opt of objects. wai\ c the “one-object” SecondI!-. \~c shall replacc .(/’ by an arbitrar!- c!osed categor! 9 \vith certain completeness properties (examples of \I-hich are the cattgories of ahclian grollps. modules over a commutative ring, 5nM!l categoric~5. 13anach spacvs3 stc.). ad th assurr1< shea\x3 i:f sets over a topological space, Kvllev _ spaces;. that ‘8 is not onl>- a categor!- but also a category based on .F, or a -Y-categor!,. l’llc corresponding functor categories (also ca!lcd ~‘+tr~c.for. crn~q~~i~s) -P” arc- f(,rmcd I;!- ietting the objects he the .,P-iunctors .\- : ‘6 t .P, ~1~1 the morphisms .iP-natural transformations v: .\’ f .I-’ : % ~+-9. Also in this case there is associated with any small .P-categor!. ‘4 dn adjoint standard constrllction (relative to -9) T% in the categor!- obtained I,!, taking the product of -4 with itself owr th,c‘ set of objects of %, such that an isomorphism exists Ixt\vczn the functor catcgor! Y” dnd the categor!. of all TX-algebras. This generalization forms the core of Section 3. \T’e dc\ote Section 1 tn an and to the exposition of categorv notions rclatiw tn a monoidal categq definition of closed categories. I-‘or this \vc follo\v I3&ahou ([4], [5], [h]) to a certain extent. \Ye also incorporate results due to 1Iac Lane [ZS], Kelly (j/h], [/7]), Linton [22], and Eilcnberg and Kc!l~ [a]. In Section 2 \yc dcvclop a theory of standard constructions in .?-catqorics, where .Y Essentially, \vc sho\v that if the definition of is an\’ monoidal category. a standard construction is modiiicd I,\ aIio\ving onI!- rc!ative notions tc, enter, then the main constructions of the theor;; of standard constructions (as developed b:; Huber [14], Eilenberg and Moore [‘i], Beck [2, 31, and 1,inton [27]) can be made relative as xvell. Adjoint (relative to .Y) standard constructions are stressed for the solv purpose of their direct applicabilit) to the study of .Y-functor categories. Some of these applications, made RELATIVE I‘;:NC‘TOR CATEGORIES 67 possible by the results of Section 3, are the subject matter of Section 3. For example, we deduce in this \vay that functor categories 9’ inherit from the base category .P certain properties (such as completeness); that functor .P : .Y” ---f .4”’ induced bk- .d-functors 5 : %’ - V, have adjoints, and that a generalized ~rrsion of the Yoneda lemma [36] for -P-functors (also shown directly by Linton [22] ant! Eilenberg and Kelly [S]) holds. .Is a final application of the theor\- of relative standard constructions IT-Ccharacterize abstractlv those categories \vhich are representable as functor categories based on some closed category. U\, introducing the notion of c~orz in a @-categorv (a generalization of the Ation of small projective) an alternative form of this theorem is ohtaincd in the case where the closed category .Y has certain additional properties, besides completeness. iVe require that the underlying set functor with which .Y comes equipped be faithful and that it reflects invcrsc limits. and also that a cogenerator exist for .4. From this formuiation it is easier to recover a theorem of Frcyd [/O] for functor categories based on abelian groups, and a theorem of the author [7] for functor categories based on sets. ‘rhe aim of this section is twofold. First, we wish to describe the type of categories envisioned in this paper to he bases for the formation of functor categories. Secondly, we wish to lay down the background required in order to develop a theory of standard constructions in relative categories. We start by defining monoidal categories; then we define category notions relative to a monoidal category, and finally WC say when is a monoidal category cal1ed closed. ( I. I ). A category .P is said to be ,x1: .P ‘< .Y --t :d; (ii) an object aPQR : (P 13 ,O) i-j K - P ,\$ (0 $1 yP : P ,b3 Z --f P and lP : j! 2; P + is commutative. a nzomidul category if there is (i) a functor % of 9; (iii) a natural isomorphism I-?), in 2; (iv) natural isomorphisms P. in 9, satisfying the following axioms: BI’SCE 68 AI(’ 2. The diagram ‘VJ’ /\ QP is commutative. In Rbnabou [4], monoidal categories (called ccrte,pr.i~5 7Liti! frrzchplicatior~) are defined 13~the above data (i)-(iv) and a general rohr,i,iice corrdition for u, y, 1. in\n!Ang an infinite number of statements. In Llac Lane [_78,291 this is replaced by a finite number of statements. the tinal form of u hich is given 1,~.ICell! [/6] and reproduced aho\-e. is a monoidal ‘l’he category .Y (with \, , I , (I, K and I as in the Introduction) catcgor!.. The category .-I/ of ahelian groups is monoidai tvith the tensor product and the group of integers. i\Iorc generally. for an! commutative ,, and A-. ring K, the category CHP/~ of all K-modules is monoid,il lvith These last two categories are also monoidal vith thr direct s:un and the .z<w) Ajcct. as i5 any category with finite products. I;or ~.aniple, ‘A I,/, the “categorv” of all categories is monoidal \\ ith the product a~x! th<, catcgor\denoted 1, having only one morphism. 1~ ali thaw es;unpie~. the nntur;;l isomnrphisms (1, r, and I, arc detinrd c~u~itnicall!-. Other ~samplc5 shall 1~ given later on. \2.ith an!- monoidal categor! -9 are associated (I%nahou [6]: I*Gienhcrg and Kelly [b’]) the notions of categor!. and functor rclativc to y1 as follo:~.s. (1.2). .A .r/‘-rutqoYy .4 consists of: (ii a class of objects tlenotcci ()I,(.-/~); (ii) for any ‘4, B E Oh(.~/), .lii ol>ject --/(. 1, B) of -P; (iii) for cvcr!’ 1, II, C’= f)h(.r/) a morphism cABc. : .:/(.I, B) .d(B. C’j f .-/(.d. C‘) in -f; (i\) for cvcr!- .-I E Ob(.r/) a morphism .i4 : % f-c/(.1. -I) iv .j, s:!tisf!ing: P-mf 1 ‘I’hc diagram JA(A.B) 9 A(B, C,] ,r A(C. D!- .i b--:.x Lil - [;‘(i3. C’ , -:‘c..D: , 4.4. 1st ,’ c 3 AK, a\;( &A, ‘.I!) @ d(C. is commutative. Y-cut 2. The diagram is commutative. D) <L4. Bl ‘Z il(B, D’ REL.%'I‘IVE -Y-rat FUN(‘TOR 69 ('ATIXORIES ‘I’he diagram 3. is commutative. Let .d and .%’ be any two .f-categories. A .Y-fun&or 7’ : .c/ -+ .r/’ consists of: (i) a function, denoted also by T, 7’ : Ob(.pY) --f Ob(.r/‘); (ii) for every BT) of .d (we shall _1, B E Ob(.d), a morphism 7tdB : .4(.-l, B) l .d’(AT, cl-aluate functions on the left from now on), satisfying: P+?“Ct I . ‘The diagram is commutative. 4-fuuct 2. ‘The-diagram A(A,B)c~,~(B.c) 1 ~~-~-- +.&AT.BT) c &A? C) 0 A'(BT.cT) / C’ T,ZC --- ---~~-~~ + t +?(AT.CT) is commutative. Fl’hrn .d = .Y these notions reduce to the ordinary notions of categor! and functor. However, this is the only example of a monoidai category for which this happens. .-/J-categories and functors are usually called additive, \\hile categories and functors relative to VCI/ are l7~~pucategouies and Ilyperfzozctors ([S]). (I .3). Given a monoidal category Y, the notions of monoid and monoid homomorphism make sense (BCnabou [.5]; 1Iac Lane [29]). A ~nonoin in :Y is any object A of .Y together with :Y-morphisms e : % - A, wz : A ::3 A -+ A, satisfying the usual axioms for an ordinary monoid (cf. the Introduction) where x is replaced by @ and 1 by Z. A monoid homomorphism f : [A, e, m] + [A’, e’, M’] is given by any .Y-morphism f : A --f A’ satisfying the following conditions: Monhomom I. e.f Man 2. f @f * m‘ = m . f. homom =e’. 70 BUNGE Monoids in .Y are ordinary monoids. In .c/&, a monoid is a ring with unit; in .&~~/K it is an associative K-algebra. An example of particular interest to us is the following, considered by BCnabou [5]. Let I’ be any category. Then, the category (3, X) of all endofunctors of A”’ and natural transformations has a monoidal structure for which the standard constructions in .i/’ are precisely its monoids. This monoidal structure on (.‘I’, .‘?“) is given hy, composition and the identity. Composition 7’, 7” l 7’. 7” may he regarded as a functor if the rule 7, CJ -r 7~ : T, T3 --f 7;’ . T.J for natural transformations 7 : 7; + T, and cr : T, + T,’ is given by letting 70 be either side of the equation rT2 . ‘T;o 7;~ . 77’,‘, which is Codement’s fifth rule [13]. 1,X . 7’ provide The identities (7’ . T’) . 7”’ II’ . (‘1” . II’“); ‘1’ . I:* ~~ 7’ (.P’, /“) v ith the structure of a monoidal category. A monoid in (,?‘, ./‘) is [T, 71,~1 \v h crc 1’ : .+’ -+ .Y’ is a functor and thus given by, a triple T q : 1, -f 7’ and [L : 7’ 7’+ 7’ arc natural transformations satisfying C’WLQ~. I 3, the axioms for a standard construction ill .6‘ (given in the Introduction for .‘I’ =- .Y). I-1 morphism 7 : T .b T’ of standard cwutuuctions should then be any monoid homomorphism, i.e., any natural transformation 7 : 7’ + 7” such that the following hold: (1.4). A functor @ : 9 --•f g (between monoidal categories .y and 2) is said to be ;I monoidalfunctor (1141;[S]) if th ere is (i) a natural transformation a g-morphism 4” : Z ---+Z@, + -~~po:P@~Q@+(P@Q)@; (ii) satisfying : MF I. aPO,QO,RO ‘p@ @+QR MF 2. P@@~“.c$pz.~p@ ‘$P,(Q&:R) -= 4P,Q 8 R@ ‘d(Pf$iQ),R ‘aPQR@- -Fp. Rlonoidal functors are of interest by virtue of their effect on relative categories and relative functors. The following is shown by BCnabou [6]. With a .?-category & a *p-category .ri =: AXI@ is associated as follows. Let Ob(.rf) ~~~Ob(&); d(A, B) -= &‘(A, B)@p; cAABC=:m4J/(A,B,,,,ti(B,C) . c,,,@; j, :- 4” . j,@. If T : .rl + .d’ is a .Y-functor, aL?-functor T -= TO : .c-t.d’ is induced by [@, c$,+O], where T has the same object function as 7’ and TAB =: TAB@. (I .5). For every monoidal category W, the functor Horn&Z, -) : .y mm*.Y is monoidal with aPQ : Hom9(Z, P) x Hom9(Z, Q) --+ HomS(Z, P p, 0) and U” : 1 + Horn&Z, Z) given by (Y, y) oppo = ril . s @ y, and the mapping which picks up the identity Iz . Denote by .d ! the ordinary category which RELATIVE FUNCTOR 71 C‘.iTEGORIES the monoidal functor [Hom?(Z, -), 0, $1 associates with a given .Y-categoq .%. It follows from the above that HomI.,lI(zA, B) z Hom.F(Z, -‘i(-g, B)). Composition ,4 d’ B -% C in i ,d : is given by the morphism z 2 Z 5; Z -“ol/, .,d(d, B) (ii .d(B, C) & .rY(.4, C) in 9. i’l’ l.4 in j .4 is k-or Ah .I t Cfb(l .d I) = (X(d), tl ~c identity morphism given by the morphism jA : Z---t .4(-‘1, -4) in 9. II!- the .b-category structure on *“I, the following structure exists for the ordinary category ~.-/ . The family ./(=I, B) of objects of .Y may be made into a functor ,?I((, -) : ~.d iPi >: .4 + .4 such that jtiren followd by the functor Hom,,(%, --) : .P + .!I II()m -, (-, --) : .P/ 01’ x I .c/ ~ -+ .Y. For J E Horn; C) bag letting -4(--l,?) : HomI,d,(,4, B) ---f HomI,/,(9, .p/( -4, by the composite morphism Z -2 ,r/(,-l, II) r is nat. equi\-. to d (B, C’), \vc define (,Y) .c/(=~,JJ) be given B) \,j,; z :‘!K!‘S:’ f For x t HomI,,i(N, C’) \vt’ define .q:r, H) L%‘.-1(B, C) -L .d(;I, C). (A, C) by letting (z) -c/(x, C) be given -J(.Y, (I’) : Hom~rlI(B, C) + Homi~-l/l Z & .d(B, C) 11 % ‘i! .d(B, C’j r2EB*~+ 4(/l, C). Conversely, a Y-category .-1 may also . be gi\.en as a pair [ .c/ 1, .d(-, -)] where ~.d ~ is a category and .-/(-, -) is a “lifted Horn-functor”, i.e., a functor such that Horn,,(Z, -,P’(, -)) E Horn ,, (-, -). Similarly, it can be shown that it is equivalent to give a -Y-functor 7’ : .-I --> .d’ by an ordinary functor I 7’ ~ : .Y + .rJ’ ~ together TAB : .?/(A, B) + .d’(_-17’, BT), such that with a natural transformation Hom,9(Z, TAI,) is the mapping which I T induces on the Hom-sets. This point of view is taken up by Kelly [/7], and is ahead! implicit in the work of Iian [/5]. by the composite morphism d(:-I. R) xzj d(B, C) -L (I .6). Let T, , T2 be any two Y-functors .d --f .-1’. A -Y-nutural trans,formution y : Tl --•f T2 is given ([S]) by a family ya t Horn rii(A,ITI , ,4T,) indexed by Ob(.d), satisfying .~Y-?r.t. The diagram &A,B) (T,)AB .-#(AT,.BT,) WAB I A’(AT,.BT,) A’(Y,, BT,t A'(ATl,~B) I ~~(AT,.BT,) is commutative. From Y-n.t. follows (by first applying the basic functor Hotn,,(Z, then evaluating at a given x E Horn, r1,(A, B)) the commutativit); diagram -) and of the In other words, every .b-natural transformation is a natural transformation, also denoted bj. y : 7; + Tz , without bars around. In general, an ordinarv natural transformation between .d-functors riced not be a .Y-natural transformation; however, this is the cast if the functor Honl,9(%, -) : .i/’ + :i is .f&i/&/ (Kelly [/7]). The following arc the rules of composition for .b-natural transformations and .d-functors, the result being always a .4-natural transformation: (i) foi y : 7; + 7:L : .c/ --+ .%’ and 6 : Tz , T:, : .r/ + .c/‘, y6 : T, + T,, : .r/ r .-/I is defined by (yS),, ~~6,~; (C) for y : ‘/‘, F T,, : .d --f .d’ and T, : ~1’ + ;d” , let yTT3 : TIT, -+ T,Z’, he given b!- (~7’:~)~ d ya7’,, ; (iii) for T, : .T/ -+ .rJ’ and 6 : ‘/‘, + 71, : .d’ + .c/“, let T,S : .rJ -+ .r/” he given b!; (7;6), &47-, Al Y-natural transformation y is a .//‘-natural isomorphism iff each ;ln is an isomorphism. (1.7). Let -v’, A bc an!’ .Y-categories, and 7’ : ~.e/ t .& r, G : .~&1 b .rJ i ordinary functors. \Ve say (Kelly [/7]) that T is -r/5-an’joill/ to G if there exists a natural isomorphism (or :Y-n~“unction) 01: .&(-4 T, R) --t &(A, BG) in 9. If this is the case, I[-c write cy : .Y(T + G). The follo\\ing results are taken from Kelly [17] and I,inton [22]. (I .7.1). IJet Y : .Y( 7’ i G). ‘I’hcn, ,X : T-3 G is also an adjunction. Clearly, eAB : Horn ,#,(=3 T, B) + Horni,,, (i-1, BG) is a natural isomorphism. An equivalent formulation for adjointness n : T+ G (Hubcr [14]) is expressed by the existence of natural transformation 7 : I ,d, + 7’G and E : GT -~~+! ,#; which satisfy 7T . TE 7’ and Gy . tG G. ‘The correspondence :Y 1 (7, c) is given by the r&s ‘I 4 ~-=( 1AT) “.I,AT ; ClJ (~id%IG,B)Y~ and (.4 T ~~I’%B) n :-- q4 . yG; (-3 -fL BG) cc1 = XT * cn. If IYb (~1,E), t!lcn q is said to be the unit and E the counit for the adjointness relation n : T + G. (1.7.2). Let 7’ : i .Q’ I + .%’ 1 and G : 9 i -F .c/ ~ he functors, and let in ‘v (?, <) : .Y(T - G) be a Y-adjunction. Then, (i) 7’ and G arc also .Y-functors; (ii) ~7and E arc also .4-natural transformations. This can easill, bc seen b\- defining -Y-functor structures on 7’ and G as follows. I,ct II .q+ ( H’). NBB,(;’ . I ‘44 c/CA, -q,qt) . (N,~,~,#,.)-‘, and G,,, (1.7.3). A&umc no\~ that ‘I’ : .r/ r -8 and G : .ti -* .r/ are .Y-functors, and that there is an ordinary adjunction Cl : 7’ + G, such that if ~1x (~7.c), then 77and t are ./P-natural transformations. (In&r these conditions it can he shop n that T is also .b-adjoint trj G. Define T~,~ : .;l1(<4T, R) + -c/(. 1, BG) and PdB : .-/(.4, UG) t .‘A(.AT, B) 1,) o(,,~~ 6.,r,n ‘/(TV . HG) an<] ,T I_ j7.B 714.m . fl(A4 T, F,~). Clearly, /? In ii \\ay analogous to Eilcnbcrg and AZoore [‘i], \!c shall extend t!lc notioii of .P-adjointncss to include -9-natural transformations. REL.lTIVE FUNCTOR 73 CATEGORIES (1.8). Let 2 : .Y(T+ G) and CY’: Y( T’ --I G’). Also, let 4 : T+ T’ and G’ be .‘Y-natural transformations. \Te say that y5 is -Y-adjoint to I/,:Gin lvhich case we write .4($ + 4; 3;, ,Y’), whenever 4’1(relatii-e to Y and ol’), the diagram is commutative. As in [9] the usual properties of composition may be shown. It is of interest to point out that given $ (and Y, x’), the existence and uniqueness of a .bnatural transformation 4 : G + G’ with .Y($ - 6; 01,a’) follo\t-s. Moreover, $ is iso iff 4 is iso. The proof given in [9] of this fact can easily be adapted to the -d-situation. (1.9). A monoidal category .f is said to be a clos~tl cntqory conditions are verified: CC 1. if the follo\ling For each P, R E Ob(.Y), the functor Hom,(P is representable (with representing wpQR : Homy(P (3 pi R) : .Y + .‘/ objects Zom(P, R) of .Y, and isomorphisms (8 Q, R) + Hom,F(fJ, S*/w(P, R))). It follon-s that Hom&Z, CC 2. There K ~~ “PQ : P $0 synr I. Xo,,(P, R)) g Homy(P, R) exists a natural isomorphism (or symnetry ---f 0 0 P in Y’, satisfying the following: ‘Cpg . Kop l,,,o for 13) . Sym 2. aPoR . K~,~,:]~ . aQRP m-z~~~ (2 R . anpR .Q (Q ~~~ . (These conditions insure, according to AIac Lane [28] and Kellv _ [16], the coherence of a, I’, I, K.) The abo\c definition of closcn catqory corresponds to the definition of svmmet~ic mmidal closed category in Eilenberg and Kelly [Y]. A justification for our choice of terminology is of a practical nature: it is shorter, and all the real esamples of closed categories ([a]) 1inown are also monoidul closed, and in most cases of interest also symmetric monoidal closed. ilnother remark is that a closed category (in our sense) whose underlying set functor is /aithfd is the same as an autonomous category (in the sense of Linton [22]) whose unticrlying set functor is representable. (1 .iO). \\‘e shall now examine categories. some of the salient features of closed ‘i’he natural isomorphism w sets up an ordinary ad.junction, for each -)) 2 P c Ob(.Y), wp : I’ :) + .FcM(P, -). Since Horn,@(%, XC+, Hom,,,(~ , ), it follon-s that there exists a natural isomorphism Q, 2’ ~m(l’, Ii)) for each P i- Oh(Y). Thus, LIPOh : XC.M(P pj Q, K) --> Yea@, -(c’p: Y(P cc, -) + Z/,,,(P, --)). The second assertion is clear b\- virtue of the symmetry. It also follows the txistence of a natural isomorphism j//;,hw(K, P)) -r .F c~m(K, 3 o,@, P)), for each op HoXp : -fo,t(Q, I’ :- C‘b(Y). therefore the last assertion holds also. ( I i 1). Representable -P-.functors. Let .Y be a closed categor!; .r/ any .f-category. For each .-1 t Ob(.d), the functor .-/(A, -) : .-I -+ .Y is a Yfunctor. ‘I’his can be seen by defining, for any two objects B, C of C%, a morphism (-r/(-4, -))Bc : .d(R, C’) + ~I.,~~(.~(~I, B), .&‘(.4, C)) of .4 as the morphism corresponding to cADC : -I(.-!, B) 5:) .-1(R, C) l .+(-,I, C) under O. The naturalit,- of the morphisms c,IBc’ yields the fact that the (.-/(d, -))DC. pro\-ide .~I(il, -) with the structure of a -4-functor. Rv cl representable .f-valued 9’-functor OIL .-/ WC mean any .P-functor _V : :c/’ + :Y for which there exists an A4 t C)b(.d) and a .P-natural isomorphism a : .4(&g, -) -F S. (1.12). Let us conclude section 1 1~~.listing some examples of closed categories. In the references included with each example, these categories mentioned are exhibited qua closed categories. A common feature to (i)-(vii) is the fact that the monoidal structure is given by the binary product and the terminal object (Cartesian closed categories, [S]). These examples are the following: RELATIvE FUNCTOR CATEGORIES (9 (ii) (iii) (iv) (4 (i-i) (\-ii) (l-iii) (ix) 75 the category of sets and mappings (Lawvere [ZO]); the category of all (small) categories (Lawvere [21]); the category of set-valued functors on a small category (Bunge [7]); the category of sheaves of sets over a topological space (Godement [/3]; Verdier [35]); the category of quasi-topological spaces and quasi-continuous mappings (Spanier [34]); the category of Kelley spaces and continuous mappings (GabriclZisman [/2]); the category of sets with base points and mappings which preserve base points (Eilenberg and Kelly [a]); the category of abelian groups and group homomorphisms (Freyd [ZO]; Mitchell [31]); of k’-modules and module homomorphisms the categor! ([W ; [34); (4 (xi) (xii) the category of sheaves of X-modules over a topological space, with .X a sheaf of commutative rings (Godement [13]; \-erdier [35]); the category of real (or complex) Banach spaces with linear transformations of norm less than or equal to 1 as morphisms (hIityagin-!&arc [32]); any algebraic (Lawvere [ZY]) or equational (Linton (231) categoryby the characterization given whose theory is “commutative”, by Linton [24] (see also Frcyd [11] in connexion with this). 2. KELATIVE ST.PNDARD CONSTRUCTIO~V Throughout this section B shall be a monoidal category. For any P-category <d, the category 9(.d, &) of all .Y-functors S : .c’l --f .d and Y-natural transformations is monoidal with composition and the identity .‘Y-functor on .d. By analogy with (1.3) we define the following. construction in .d is any monoid in .9(-r/, &‘). Thus, (2.1). A -Y-standard it is given by a triple T =~=[T, 7, ~1 where T : 21 + .d is a .Y-functor and 7 : l,d - T, p : TT + T are 3-natural transformations satisfying Constr. 1-3 (cf. the Introduction). A morphism 7 : T - T’ is a monoid homomorphism, i.e., a Y-natural transformation 7 : T --f T’ satisfying MConstr. 1 and 2 (cf. (1.3)). Let us denote by .dY-s,C. the category determined by all the Y-standard constructions in the 8-category .d and the above morphisms of -Y-standard constructions. of morphisms of 9. First lvc sho\x that the ahow definition makes -r/T( , ) into a fuuctor and that the morphisms Iy~A,*IIH,,,, \-ield a natural transformation 1 ‘~I : r/‘(- , ) -F .d( , --). Let [--1, (1 1~ >I T-al&t-a ‘I’hc diagram t/j + [C’, U] a T-llonlomorphisrn. and ,f : [II. A(.~T,~T) ’ &AT. A(AT.CT) is commutati\-e. (2.2.3) is commutatiw. ---A(AT'n) Since? 7’ is a .b-functor. &(A, II) _TI’“_- On the other hand, &A;. it +L,(AT. fOllO\VS IIT, K! C) hy (1.5) that the diagram (2.2.4). d([, B) . d(A’1’, 6) ,~/((,g) _ .-/(.-I,R) . .-/([. C’). From(2.2. i), (2.2.2), (2.2.3) and (2.2.4) follow the equations By (2.2. I), the equalizer of the pair 1’,,,. . .=/(,-32’, u), -,J(E, C’) is Cy,f4,c,jC,i,, there exists a unique morphism -r/‘( [;3, (1, g) : .~/~([.-l. [], Therefore, [B, ii]) + .eIT([.l, (1, [C, u]) such that .dT([.3, 4],g) . ($J![c,i,, (2.2.5). q4,E,~B,s, . .d(.l,,?) Similar arguments can be produced in order to show that a T-hornomorphism h : [C, U] + [Lq, [] induces a morphism -rIT(lr, [B, H]) : .C/T([-Aq t], LB,01)+ .P/‘([C, u], [B, P]) with the property that (2.2.6). lT,&B,u, . .“/(/I, B) dT(/z? [B, i’]) . I.;c.t,,,fl,H, .‘I’hus, notonl\l iS /A’(~-, -) : ( c/ l‘y _I .-/ ;1 + .Y a functor, but also, by (2.2.5) and (2.2.6), ITT : A”(-, -) - --/(-, ) 15 ‘. d natural transformation. il;c show next that the above definitions provide .<I 9 with the structure of a -Y-category for which 1 .T is a .i/-functor. The functor Hom,p(Z, -) : .d -+ .‘I preserves equalizers. At the level of the underlying sets the equalizer of the pair T,, ’ Horn, ml,(=2T,(I), Horni C/ ([, B) is the mapping Horn! ./ T([.-l. f]. [B, 01) + Hom,,,/,(A, B) induced by the ordinary functor lTT. (A morphism .f : :I + B is a T-homomorphism ,f : F.-l,(1 + [B, H] if‘f ,fT . G -: 4 ..f.) Therefore, Horn, -/~,([3!, [I, [B, H]) G Hom,9(Z, ,~/‘([-a, [I, [B, Cl])). In order to show that FT is -@-adjoint to I:‘, by (1.7.3) it is enough to show that FT is a .Y-functor, and that the natural transformations 7 and E by virtue of which FT is adjoint to CT are also -f-natural transformations. IVe proceed as follows. Since !L : 7’7’ + T is a -Y-natural transformation, the diagram (2.2.7) r$(A.B) ~ T.\:i~---+?"(AT,BTI T.IT-'--+I?(ATT.BTT) UT I 'r Iii i T IIQAT.BT) .&jrA,BTT ffhTT,u~) i : A(.~TT.BT) 78 BUNGIS is commutative. Since, b!- definition, Ufa7.,PA,LBT.tiB,is the equalizer of the pair l’,,7.,BI. . .-/(,ilTT, 1~~). ‘/(pA , RT), it follows from (2.2.7) that there exists a unique morphism FTB : ~/(~-1,B) --+ .-JT([Arl’, /la], [B7’, ,ug]) such thatPf,. . I$,.,GAltnr,,i~, II’,, Since FT : .cJ --F .-f’ ‘T has this property at the level of the underlying sets, the morphisms P;,‘, defined above provide a -i”-functor structure for Fr. Since 7 is .4-natural, it remains to verify that E : kTTFT --* 1 r,,~ is also Y-natural, which amounts to the verification that the diagram is commutative. Since C-TATirA,,B til is an equalizer and therefore, manic, the commutativity of (2.2.8) foil0 ws ’ from the easily justifiable equations: This completes the proof. 1 Let .c/ and & be an!- .P-categories. Let F : .d -+ .a and 1: : .a --z .M;’ be .Y-functors such that cx x (71,l ) : Y(F 4 I). From a theorem of Huber [/4] (cf. also [9]) we can conclude the existence of a standard construction T -_ [7‘. 7, ~1 in .r/ , induced by cs - (~1,c) : F --I fi : i A? i + / .d /. Also, there exists a functor @ : .# -+ .G/ IT with the property that @ * UT = c’. ‘l’he definition of the fuuctor @ is given by the rules I3 E+ [BU, EBC~]; .f+ fr’. T .= [7‘, 7, ,L] is given by the triple (FC, 7, FGU]. (2.3) PROPOSITION. T [‘I’, 7, ~1 = [FI ‘, 7, FCC] is rr .Y-standuvd construction in .d. The functor CD: .9?---f .dT is a .V-functor, unique with the property @ . fTT (7 : :9!?--f .-/. Proof. From (1.6) it follows that T = [T, 7, p] = [FU, 7, FeU] a S-S.C. in .%. Since E is ./P-natural, the following equation holds: is RELATIVE I'UNCTOR CATEGORIES (2.3.1). ‘Therefore (X.2) 79 1 .BII, . FB,,,,, . .&(Rl P, Ed,) = .H(tn , B’), for all B, 8’ E Oh(.d). also the following equations hold: I;,,, . T,,,,,, . .-/(B1-T, c,yU) :- U,,, . F,,.,,, . I’BuF,B’UF . .v’(Bt’T, c~‘U) .- C’BB,. FBLIJFU- :W(BI:F, Ed,) . L-HuF,n, -= ‘A(Q ) B’) * f’yBLrF,B, ~~--- ri,,, * d(t,I:, B’ C). By (2.2.1) there exists a unique morphism DBB, : .W(B, B’) -+ -dT(B@, B’@) such that QBu, . C’zO,B,Q =-~ I-,,, . 1Ve leave to the reader the verification (by arguments similar to the ones employed in the proof of (2.2)) of the fact that the morphisms DD,,, provide @ ?x:ith the structure of a :Y-functor. 0 (2.4) DEFINITION. In the situation of (2.3) wc say that I. : .H -+ .?/ is .7-constructionable whenever the .?‘-functor @ : .H - .dT is a Y-equivalence of categories. (When .4 =mm .‘I, this definition reduces to the definition of a tr~pleahle functor, as in Beck [2].) (2.5). In order to !X able to define adjoint standard constructions we need to recall some dual definitions and statements. A\ d-standard coconstruction (called cotripk in [Y]) in ~1 is a triple G -= [G, E, I,] where G : .-/ - t +’ i5I ‘7 .‘Y-functor ‘rnd E : G + 1 , , 11: G -+ GG are -b-natural transfrumations satisfying axioms which are dual to the axioms for a standard construction (cf. [9]). With a given :Y’-s.cc. G m= [G. E, v] is associated a category of G-coalgebras, ‘1 .d ~ and a functor ‘CT : o .c/ + .r/ / which has a coadjoint GJ/ : / .c+/ +Gi .d Also assuming that 4 has equakzers, it can he shown that the above ma\; be lifted to a .Y-category G.r~ and a -d-functor “I- : o.4 -+ .n/ with a .Y-coadjoint ‘I-: .c/ -P G.d, such that G .~ GJ7GIF. .\s in the case of T-algebras, the key definition is that of the morphism G c,-rA,,]lB.s] : “4-43 71,[BY61)+ .-/(=2, B), for any tw-o given G-coalgebras L--1, rl, LB,61. v-5.1). G~.Ca.,][B*s]is defined to he the epwzlizer of the pair of -Y-morphisms. If Y-categories LY, g and !d-functors U : ~4 - .m/, I’ : ,d --• .%’ are given, together with Y-natural transformations ij : la + UT, < : J-G --) l,,/ by virtue of which V is coadjoint to U, then there exists a .Y-s.cc. G = [G, E, v], 80 BUNGE said to bc coinduced b\- I - I,., and a :f-functor Y : .iA - f “.5! unique with the property Y . G 1. - I -_ Define G [J-IT, C, J,‘$jr’], and let W be given by the rules H - [HI-,ijAIy;.f-~+fl’. In these circumstances, the -Y-functor I ’ : .ti + .v’ is called .d-cororrstr~urtiomhle whenever !F’ is a Y-equivalence of categories. (2.6). Let T [T, 7, p] bc a -f-standard construction, and G [G, t, V] a .4-standard coconstruction both in the Y-category .w’. il.c say that T is A : .P(T --i G) such that :P-at/joint to G if there esists a .iP-adjunction -Yy?j -_ t; a, I) and .U(p -1 I’; 1, xx), with notation as in (1.8). In this case standuvd comtruction in .V. \vt: u:ritc :x : -4(T - G), and call T a -Y-adjoint The following are the rclativc versions of theorems of Eilenberg and AIoorc [9] for adjoint triples. Proqf. Define l and v via the following commutative diagrams: From the remarks of (I .8) follo~vs that E and 11are uniquely determined and that 77- c and p -I jl as required. It also follows (as in [9]) that G [G, E, v] is a Y-standard coconstruction in .-1. B Pwqf. lt is shown in [Y] that L : T + G. Clearly, also <a : .4(T + G). Also in [9] a functor 1, \\ith the prop-ties claimed above is constructed. Tiephrasing the definition of [Y], the definition of I, : ~.c/ F z ‘1 -4 is as REL4TIVE IXNCTOR C4TEGORIES XI follows. Denote bv 7’ : l,Q --* 7’G and C’ : GT -+ I,# the unit and the- counit for .b-adjointness N’ =: ia: : .4( T ---I G), Lvhere 7’ =: FCY and G ~~ i -C’. The rules [-I, [] -f [=-I, S,] z [.3, 7; . fG]; ft-+.f, define 1,. An inverse for I, has f. \Ve must now show that f, the rules [--I, S] - [&-I, t6] [A3, 67’ . ~61; fb can he gi\-cn the structure of a .Y-functor. I,c‘t -4, ,!I e Oh(.:/), and let S : -3 t A G and H : 137’ + B lx any .r/morphis~~s. 13~ the .Y-naturality of 7’ and the Y-functor structure on G, the diagram (2.8.1) belo\\- is commutative. B!- the Y-naturalitv of C and , the Also, .&functor structure on ‘f’, the diagram (2.8.2) 1>eIom is commutatiw. remark that G,47,B . .-/(q;, BG) I’.~~, and that 7:,,,,, . .,/(.17’, l ,) ~~~(i:,,l) ‘. .-ksume now that [A, E] and [B, Ci] are T-algebras. and (2.8.2) follow the equations: On the other hand, by (24, the equalizer .d(A, So) is given by GO~,n.,y~I~B,6 , . Thus, From (2.2.1), (2.8.1) of the pair G,, . -d(S, , BG), there is a unique morphism x2 RUSGE &A,~J~LWI dh the property thatLLA,SJIB,Bl . CC~a,pil~s.a81 = Cr&ILB,Bl. These morphisms provide a .Y-functor means of arguments as in (2.2). structure 1 for L as it is easy to verify by In a situation as in (2.8) there are, by (2.3) and (2.5), 9-functors CD: .z?-G/~ and Y : .;/) + G.p/, such that @ . C:’ CT and Y . cc’ = 1.. C’learly we must also have @ . I, ~~ Y. ‘Thus, Y is a Y-equivalence of categories iff CD is a -Y-equivalence of categories. ‘I’hus, the following (2.9) I~EFINITIOE. A d-functor 1. : -4 -+ .cl in the situation of (2.8) is called .Y-arrjoint constructionable (or .Y-coadjoiuf coconstructionahle) whenever @ (and thus, also Yj is a .Y-equi\-alence of categories. Before stating our next theorem, let us recall (Linton [Xl; Alanes [3U]j that iwzwsr limits exist in any categor!- of T-algebras provided they esist in the categor! where T is defined. bloreoxrer C’T preseroes and ~~$erts i?r-zerse limits. Dunll~, categories of G-coal~,aebras inherit from the base category all direct &nits which might exist, and Gl~7 pressewes and reflects direct limits. Thus, if T is an adjoint standard construction, from (2.8) follows that .-IT is M ell hchaved with respect to all limits, inverse and direct. (2.10). Given a category .# and a functor C; : :Y - .d, -ti is said to have 1 ‘-coequali,-ers if for any pair of .d-morphisms f, 0” : Y --f I” such that the coequalizcr of the pairfc’, gZ7 exists in ,c/, then the coequalizer off, g exists irj .ti. i repects isomorphisms if given any morphism f : Is + Y’ in M such that ,fl is an isomorphism, thcnf is an isomorphism. Our nest theorem gives ncccssary and suficicnt conditions for a .b-functor A characterization of constructionable to bc .P-adjoint constructionable. (“triplcable”) functors is given by Beck ([2], [.?I) and it carries over easily to the relative case. (2.1 1j THEOREM. Lets/, .39he Y-categories. Let ZJ-: 8 --, .-I he a .Y-functov for zchich there exists a Y-adjoint F ((7, 6) : Y(F + U)) and a .Y-coadjoint I’ ((7, tj : .4( U -I V)). Then, U is .Y-adjoint constructionable {f and only if / 33’ I has C,‘-coequalizers and CTreflects isomorphisms. Proof. From the remark concerning limits in categories of algebras over it is clear that 1& iT (for a .Y-adjoint an adjoint standard construction, S.C. Tj has UT-coequalizers. Also, CT reflects isomorphisms. (Let f : [,4, [] + [B, 81 be a T-homomorphism such that f is an isomorphism in Cd, i.e., there is an &-morphism g : B - A such that gf = B and & := d. Then, g : [B, 01 --* [A, E] is a T-homomorphism, as follows readily from the :.gT.fT.%.g= equations: 0 ag -= IgT .%.gI’(l,jT.%.g-(gf)T.%.g gT . t . f . g = gT . E.) Since these conditions are preserved under an)- RELATIVE FDNCTOR x3 CATEGORIES equivalence of categories which commutes with the underlying .7Y-objcct functors, this settles the necessity part of the proof. Assume the conditions of the theorem are given. Then we must show that the .Y-functor CD: 9 + .c/=, as defined in (2.3), is a :Y-equivalence of categories. Let [--I, [] be a T-algebra. The following is a coequalizer diagram in / .PJ ~: [7’ . cr. (Let (T : -3 + I-: be any .d-morphism wit!) the property that p,,, . o [ . 7)/j . 0 --:- 77.47. (7’ . I.7 ~ Define 12: .-I --+ B by h 71.,,. O. ‘Then, ciz 7ar . p4 . (T == 0. Since 5 is epic by virtue of T-ok 1, it follows that 17 is uniquel!~ determined with this property.) The pair pA , CT is the same as the pair Since .a has C-coequalizers, the pair ‘AF .Wl% has a coequa!izer -=. : .-IF, in ) ilA 1, denoted b\ A-1FAt- + [A, (1 6. S’mce U preserves coequahzers (having a coadjoint) then (/IF A Letf [--I, [] 4) u = z4 7’ -i : [A, 51 - is commutative. diagram z4. [B, 81 be a T-homomorphism. Since E : UI’ ’ - Then, the diagram 1 d is a natural IFUFI t 13FUF transformation, also the 11 t,ii t 'I 13: is commutative. From the equations cAF . fF . k, = fFUF . cBF ’ kg fFUF . t9F . k, == [F * fF . k, , follows the existence of a unique morphism f6 : [-$ ,514 - [B, wk such that kc ..f& = fF . k, . This gives us a functor 4: /.dl=-t~.cL1. In xvhat follows we shou- that & is Y-adjoint to ~5, and therefore, (hy (1.7.2)) 4 is :I .b-functor. Xext we show that the unit X : l,,,T --z di@ and the counit y : @6 r I,, for :V-adjointness, are Y-natural isomorphisms. Thus @ and 6, together with A and y, set up a .f-equivalence between the .iP-categories :?I and .YT, as required. C‘lcarly, ,\:A.C, For J T-algebra [A, [] define Xifr.C-; rja . /z:l, I 1 : [--1. (1 z [.-I, (1 since k,l ’ 6. If B is any object of 3 1 let us remark that the diagram IiUFUF- tnUF -BUF ’ 13 t ill I( c BUF 6 li LB t is commutative, due to the naturality of 6 : 1 Y1 ,9,. Since R@ =: [LILT, cBl :] is a T-algebra, the coequalizer of the pair E~L:J;, eSUF must be the morphism k f8L : Bl.FI;‘- F B@6. Th us, there exists a unique yB : B@$ --f B for lvhich tB (If B == [A, []6 f or some T-algebra [A, 51, then ye is the krRti . YB identity.) The coequalizer of tgl/:~li and tg& is clearly E~C,‘. Since l, E~IJ. On the other hand, then Is(~,~,)( preserves coequalizers, kCCRC.) I . yBI E~C,‘.Since kcCBu)l!_ is a coequalizcr, it is epic and therefore, I RC Since l’T reflects isomorphisms it follows that yR is an isomorYL?l. phism. It remains to verify- that the so defined Y-natural isomorphisms h and 1 satkfq- the adjointness relations. For a T-algebra [L4, f], (X,A,CI)6, . Y,~,;~~; (I,,,,,)& . Y~~,~,& = yra,s16 = I,n,EI . For an object B of / A 1, A,, . (ys)@ 1 (y# ==: I ,y@. This completes the proof. I,et : : T’ --, T be any morphism of -fl-standa;? constructions (cf. (2.1)). ‘rhis induces a functor ; .r/ ~- : .c/ T + -4 1 with the property that / .d ,i . CT.“’ = UT. The functor , .c/ 7 is given by the rules [-A, [] -+ [-il, 7,4 . [I; f -‘.f. (2.12) PROPOSITION. / .d T xith the property Proof. There exists tl wzique .Y-functor 7,‘T : .& --t .o/. that .d~ . CT” For any two T-algebras 4~,m,o1 : d’(V, be the unique morphism structure -di [A, [] and [B, 01, let El, LB,Q - .~“([A 7.~* 51,LB,TB. ‘4) in d for which the following diagram is commutative: on 85 RELATIVE FUNCTOR CATEGORIES The existence and uniqueness of such a morphism is a consequence of the validity of the following equations (which hold since 7 is .Y-natural and the definitions of UT and UT’ (2.1)): U?i,c,[~.s,. TAB. .c/J(JT’, 7~ . 4 = U&,~IIB,~I. TAB. .&‘(A = U&E,,B,S~ ' TAB ' T’, TV) . .&‘(A T’, .Q) .P”/(T~ , BT) . .r/(A T’, 0) = li&,t~fB,g~ . TAB * .ri’(dT, = @4,:][B,S] ' -!('f, = u?&B,O, ’ cd(T.4 B, 0) . J~(TA , B) ' cd(TA ’ &, B)- , B, 0 (2.13). A pair of morphismsf, g : X - X’ in a category .%’ is said to be a reflexive pair if is there exists a morphism d : S’ - X so that df lx, 7. dg. The following theorem, whose proof vve sketch here, is proved by Linton [27]. (2.14). If / %CeIT has coequalizers of rejexive pairs, then any functor 1.c/ ‘7 : 1.cf IT --f 1d IT’, induced by a morphism 7 : T’ - T of standard constructions, has an adjoint. Proof. For any T’-algebra [A, 61, the pair (AT’) FT - 7AFT (AT)FTA @.T > AFT is reflexive with d = qjqFT. Let AFT ---f [a, [I(! JX?IT)” be their coequalizer. It can be shown that (1 ~1 1’)” is a functor, adjoint to / .d IT. 1 3. RELATIVE FUNCTOR CATEGORIES AND CATEGORIES OF ALGEBRAS Let :P be any closed category, as defined in (1.9). (3.1). 11 functor category based on 9 (or a 9-functor category) is any .YV of all the P-valued P-functors on some small Y-category V’, and .Y-natural transformations. For each such % there is a functor U, : .Y”l ---f PobCK) defined bv the rules x:fiy+.y-x: Ob(%‘) ---f Oh(B); y : x - E’ --f (yC : cx ---f c&o,,,, . U, is said to be the underlying object function functor of .P”“. category (3.2). For a given set I, denote by Y-KC/~, the category whose objects are the Y-categories %- for which there exists an isomorphism Ob(C6) -2 1. The morphisms of Y-K///, arc .d-functors u-hose object functions arc isomorphisms. The correspondcncc % l I .,6 : .Yp” > .P’ is contravariantly functoriai ii1 the following sense. ‘I’o each morphism < : %’ --+ % of Y-KU/, is associated a functor .Yl : .P” ---f .Y”’ which commutes vvith the underlying ob,jcct function functors of 9” and 9”. .@ is defined by the rules <Y : ‘6 m~h.P t j.Y : %’ -i 9; y,-z (y. We remark that 9-X l//i is isomorphic to &O/L 9, the categor\ of all monoids in 9. In order to be able to generalize this statement for arbitrar!; I, wc shall assume that the closed category 9 is I-complete, meaning that ./P has equalizers and coequalizcrs as well as products and coproducts of arbitrary families of objects of Y indevcd by any subset of Z. In particular, 9 has always a terminal object 1 and a coterminal object 0. (3.3) PROPOSITION. Let .4 be any /-complete dosed cutegory. Thin, the category .YplyJ is monoida zcith “matrlr nzultiplication”. There exists aI1 isonwphism of catqories K : .,&[‘c’a,fYl,‘~I - f Y-%*1 L, . Proof. Let ,II ==m(M,,) and R- em (i\‘i,) be any tvvo objects of -Fiji. Define JI # .\; ~~ ((,I4 # N),,) (CIC JI,,, (3 iVizj), and make it into a functor 0 if i -;‘. j. in the obvious way. Let A =- (dij) be given by dji r- % and dij Define natural isomorphisms t&N, : (iv2 # -V) # 0 -> iv1 # (:V # 0); rM : III # A - -11 and l,W : A # M - A4 by the following conditions. Let be given by the requirement that, for each r, k E I, ~~~~~~and (Inl)i, ~~ I,y,, . By (1.10.2), for each P @ 0 E 0 s 0 @ P. Therefore, the above is sufficient to I,. The coherence of a, r, I yields the coherence of a, J’, i. monoidal with the above definitions. Given any monoid [A, e, m] in 4‘NIX’, define a Y-category Let (i) Ob(?Y) = 1; (ii) F(;(;, j) =- Atj ; (iii) c,~~ = m:, : Aij where nzi, is given by the composition Let @Ml, object 1’ of 9, define yM and Thus, :/pzxr is ‘6 as follows. @A,, -+ A,,< , 87 RELATIVE FUNCTOR CATEGORIES (iv) ji == eii : Z--f & . Axioms P-cut l-3 follow directly from Man 1-3 (cf. (1.2); (1.3)). Iff: [A, e, m] --f [A’, e’, m’] is any monoid homomorphism, a ,Y-functor [ : % + V between the P-categories with I objects associated with the above monoids, is given by (i) 5 = II : I + fi (ii) sij =fij : n,j -+ fllj. A functor K defined by the rules [il, e, fn] tt V‘; f++ {, as above, clearly satisfies the requirements of the proposition. 1 Let & be any Y-category. Then, the category (3.4) PROPOSITION. .‘P-ad(.d, ~2) of all 9-adjoint functors -01 - .eZ and Y-natural transformations, is monoidal with composition and the identity. Also, ,420~ .Y-ad(.d, .d) =: -r4op.ad S.C. . Proof. The first statement is clearly true, by virtue of previous remarks (cf. (1.3)). ‘4s for the second statement, a monoid in Y-ad+‘, <9’) is, by definition, a P-standard construction T -mm: [T, 7, ~1 in ~1, such that there exists o( . .Y(T --I G). By (2.7), there is a unique .P-standard cocontruction G : [G, E, V] such that T is .Y-adjoint to G. Thus, T is a B-adjoint standard construction in .J/ (by definition (2.6). Morphisms of .Y-S.C. are also morphisms of P-adjoint S.C. 1 (3.5) PROPOSITION. be given the structure k E I, are Y-functors. Let Y be an I-complete closed category, Then, 8’ may of a Y-category so that the projections n,< : 9 ---) P, Proof. For A, B E Oh(F) let .YI(A, B) = J&,YcN~(A~, Bi). A composition law for 8’ is defined in terms of the composition law of 9’ by the conditions cABC . projri = proj, projk * cAkB,c,~, for each k E I. The morphism (3.5.1) jA : % --+ #(A, A) is given by (3.5.2) j, . proj, == jAk:, for each k E I. The functor Hom,(Z, -) : 9 - Y preserves Hom,,r,(A, B) E Hom,(Z, F(A, products. Thus, B)) = Horn9 This shows that the above is indeed a P-category structure Let the components of =k : PI + B be given by on 9. ‘Then, remark that (3.5.1) and (3.52) at-‘._ kxeciselv required for T,, to be a .f-functor (cf. (1.2)). 1 (3.6) a .Y-djoint ?HOPOslTIoS. A,, : .Y For ench 12 tl, f .P and N .Y-coadjoint the the coherence conditions .f-fu?lctov x,, : 9’ -+ .Y hs Y, : .f ~ f .Y’. Proof: ‘I’hc conditions d/,n, equals the idcntit!. on .Y when i /r 3rl~i the functor constantI!- 0 (the coterminal object) otherwise, d&c a functor dcfinc Ti. . RF (I .iO.2) .X PW(O, I’) 5 I and Ll,, : .P + .Y’. Ihall~ X I /,,(I’, I) = 1 (\5;here I is the terminal object of .4), for each object I’ of :d. .iP-adjunctions B,, : ?(.!I,, ~ xi,) and y,, : -Y/I!“! ~- r),) result from the canonicai isomorphisms given below. (3.7) I’R~POSITIOS. (Jntemal chnmcterization of .b-adjoiilt endo~iu~fors qf Y’). For each Al c Ob(-4’“‘) the functor defined by the rules =1 f .-I # M (zcith (.‘I # AZ). -y!,. At “: M,,); .f --+f# dl (with inj,,. . (f# M), f,, 1x1-II,,) has a .Y-condjoint. ComerseLy, ij’ ‘I’ : .@ + 9’ has o .f-coadJ’oilit then fhew c,vi.rts ,I/,. F Ob(.Y’Y’) and n .b-natural isomorphism ST : T + ~- # ilfT Pw0j: -1 P-coadjoint to the functor #AI : .4’ l 9’ is the functor (AZ; m) : .Y’ em+-4’ such that (M; -4) n,,, A’ CU,( JI,,,. , A,;). A .Y-adjunction # AZ, B) l .4'(L4, (AZ; B)), detincd is gi\it:n b\, the morphisms 31AB: P’(,-1 by requiring that the diagrams be all commutative, for i, k E 1. Let 7’~ Ob(.F-ad(.W, .F)) with 01: .4(7’ -I G). For each k, i F I define Yyki = A, . 7’ . 7r1 and G,,. T, . G . TT,,. The functor Tki is b-adjoint to RELATIVE FUNCTOR 89 CATEGORIES equivalent G,., . It follows then that T,, is Y-naturally - 3 ZT,,j A%more general statement is the following: to the functor (3.7.2). If F : .Y -+ .4 is a functor and E : .4(F + Ff), then F z ~ V, %I;. The following composition of natural isomorphisms yields a -f-adjunction from ~ >J %E’ to H: Both F and ~ 15)%I; are .Y-adjoints alent to ‘\/?,ZF. Denote b!- y,,; : 7‘,., --f 1) ZT,,, (3.72). Define 112, by (L1lT)!i := C,> (Y,;~)~, : (=i’I’), g I,, --I,,7’, , --f ST: 7’+-#MT. 1 (3.8) rules ill to II. Therefore, F is .P-naturally equiv- a natural isomorphism which exists 1~~ ZI,,, . ‘I’hen, the morphisms (STA)i yield the required C,,. -3,, ‘f; ZT,.; clejinen by r/w The f1m-I0I. @ : .P I --f .4-ad&@‘, 9’) # :U; g L+ - #,;I, 1.t (I nmnoidal equi~z~alenceof categories. ‘I’HEOREM. -+ f'wof. uniquely For every 31, A\\‘~ Ob(-4’X’) let (cT,,,,~)~ bc that morphism determined h!- the requirement that the diagram of .Y hc commutative, for every i, .i. k E 1. ‘This defines a natural transformation +M.N : (- # 112)# S --f ~ # (31# 11:), Rhich in fact is an isomorphism. Let (6’) : l,,y,i -> - # A he gi\-cn hy the commutative diagram below, for each i E I. (3.8.2) 90 BUNGE Clearly, the morphisms $, #), d, F, 1, 2~3, I@, I@ are coherent since the morphisms (I, r, I are coherent. Thus, (cf. (1.4)), @ (with 4 and #I) is a monoidal functor. A monoidal functor Y’so that both @Paand YW are naturally equivalent to identity functors is given as follows. For any .Y-adjoint endofunctor 7’of .Y’ there is, by (3.7) a .Y-natural isomorphism P’ : 1’-* -- # MT , with (M,,),, == ZAiTnj = ZT,, . The rules T I+ iWT ; 7 : T --•f T’ c> g, : llZr -+ illr, with (g,Jij ~ (Q+~ dcfinc a functor Y : .4-ad(.W, 9’) -t yprxr Let 4” : d -+ i11(lc9p’,) be such that (#“)ii 1 that Y, 4 and #” are as required. = lz . It can easily be checked l'he functor 47,: 9“” ---f :Y-ad(.YI, 9) tqether aith the (3.9) c OROLLARY. natural transformations $MN : (- # 111)# I$~--f - # (M # V) (for every M, N E Ob(.P, 9I)) and @’ : l(p~, - - # d, induces an equivalence of categories Proof. By (1.4), @ induces a functor @.+: JZo/t WXr-+ &n~ .Y-ad(Y, 9”). Similarly, Y induces a functor Y, in the opposite direction. Also, @*Y, and Y’%cP, are naturally equivalent to identity functors. 1 (3.10). In order to prove our nest theorem it xvi11 be convenient to have available an explicit form of the induced functor @, . Using the definitions of @, 4, (b” and (1.4), we compute this functor in the following way. Let E be a Y-category with I objects. By (1.2), the remaining part of the data for 59 is given by (ii) an object %(Ci , Cj) of .Y for every two objects Ci , C, of 55; (iii) a morphism cczc,c, : ?T(C, , C,) @ U(C, , C,) -P %(C, , C,) of .Y for every three objects C, , C, , C, of CG,and (iv) a morphismjcc!,.: Z-t %(Cj , Ci) of 9 for any object Ci of %?, all these subject to the conditions imposed by axioms P-cat l-3. With 9?, @.+ associates a .Y-adjoint standard construction TF? = [T, 7, ~1 in 8’ as follovvs. The functor T := ~ # %?is given by the rules A++A#FY, with (A#(&), --&Ak@%(Cic,C,); x~Fx#~‘, with inj,: . (x # %)1 = xk 6%%(C,: , Ci). The P-natural transformation 7 : 1cy,) --f -- # ‘G is defined by RELATIVE FUNCTOR The Y-natural transformation requiring that the diagrams 91 C’.iTEGORIES p : (- # %) # %‘ --f -- # 55’ is defined b!, be commutative. ‘l%is defines 0, on the objects of .~Y-%;Q/~. Assume now that 5 : %’ --+ % is a morphism of .+‘-V~dI . Define a morphism of .Y’-s.c. 7i : T%’ --, T% as follows. Let 7i : - # ‘6’ ~ + ~ #Z’ be given by the family (T[)~ , indexed by Ob(.Y’) with (( 7<)A) L uniquely determined b\: the requirement that, for each k t I, Let .Y he an I-complete closed category. (3.11) ~hk.ORIiM. ‘6 E 0b(.Y-‘k?0 /,) there is an isomorphism of categories For 17%: (,y/)T% -+ J/L such that the dia~yram is commutative up to natural equivalence. Atloreovu, if 5 : (6’ - 97 is any morphism is commutative up to natural equivalence. of .Y-‘&ad, , then the diagram each 92 BUNGk Proof. Let [A, [] be any TZ-algebra. Define _Y ~~ X, : % + .Y by the following data: (i) 1Y =- .? : C)b(%‘) z I + Ob(.P); (ii) Xc,c, : K(C, , Cj) + .~~~~(d~ , z4,) is the image under w of the morphism Let us now verify that WC have a -f-functor. mutativity of the diagram expressing diagram axiom J-funct 1, is equivalent (AiOlc,) Ai @ C(Ci Notice (by adjointness) By the definitions follo\vs: (ra,)-l to that of the I n. /’ , Ci) LAi On the other hand, since [A, 41 is a T’Z-algebra, (qA);[, first that the com- from T%-alg 1 follows that I :A,-+A,,foreachiEI. given of 17 and of [, the above may also be written * L4, ei,jci . jiL =- (rAi)-’ . Ai (g,jct . inj, . ti 1 I : &4-r as -4,. This shows that .9-funct 1 holds for S. Xext, we observe that the commutativity of the diagram expressing Y-~zMc~ 2, xC,Cj ‘< xCj Ch , ~~~ -~+JfdAi,Aj) Q(Ci,Cj)@e(Cj,CI,) %&(Ai,A I ‘CiCjCk/ is equivalent C.kIAi:\!. (by adjointness) [Ai% ,! I t?(C,,Cj;] to the commutativity 53 C(Cj,Clc) it@ e(r’ 1. .:‘:--,Aj ck) of the diagram @ C(Ci.Ck) RELATIVE From T%-a& 2 follows mutative. FC’XCTOR that, 93 (‘.YI‘EGOKIES for each h ~1, the diagram From the above and the definition belon is com- of p follow the identities: ~A,,~(C,.C,).K(Cj,C~) . -12 c% CC,C,Ck. Eki = . -4 a,~,‘~(C,.C,).I~(Cj.C~) (8 cC,C,Cp ’ “‘Ii -E,; = inji @ %(Cj , C,;) . injj . (pA),,. . Ek = inji @ %(Cj , C,J . inji * C 6, @J%(Cj , C,) . 5,; = inj, .C [,i @ %(Cj , C,.) . fJ: = oi 3 V(Cj , C,) . irrj, . t,, ‘Tbust also P-jknct 2 holds, and therefore X =: S, : %‘ --F 9’ is a :Y-functor. Let X~ : ,Yf ---F & be Let f : [A, 61---f [B, 6’1bc a I’%:-homomorphism. of given by the family (~yf)ci= fi , indexed by the set 1. The commutativity the diagram expressing the fact that ?c,.is a P-natural transformation, i.c., xCiCj C(C,,Cj) is equivalent ----~~-Am(Ai,$) (by adjointness) to the commutativity of the diagram pi A, ‘?I @CC; ,Cj) ---LAj ! I I, r, “CC!.Cj) i t R, f9 ‘?(C,,Cj) I. ,g; -~--~- lfj -I3j From T’G-honmmorphism follows that .f, 3 %(C, , Cj) . iJ,i - fi (3 V(Ci , Cj) . inji f Bj := inji . (fT)j = ing, . 5; . f, -= Q .fj . Th us, 2-f : 1Y, + X8 is a Y-natural transformation. . 19) 94 RUNGE The rules [LJ, [] F+ X-: ; f ‘. t X~ define a functor r,6 : (:P’)T’6 -+ d” and r, 1 -z n c,.“‘. It is also easily seen that & is an isomorphism of categories. An invcrsc to l’<, assigns to a :Y-functor S : %’ m-f .Y the T%-algebra [iI, [] defined by :-1 -V:I? Oh(V) -+ Ob(.Y), and [ : -47’ -+ *4 by letting c, : (--I #‘6); --P z1, be the unique morphism such that, for each h EZ. ini, . t, (aTITc.,c~,) w !, for each j f: I. \l’e finally verify. the last statement of the theorem. Let < : %’ --f $5 be given. 1,c.t us see that i’, . .4i 2 (:Y’)” . f,,, , as required. By-, definition, (cP)i’ assigns, to a TK-algebra [-3, 51, the T%‘-algebra [-;I, (T<)~ . 51. Applying i’,,,’ to this, results the .fl-functor -\1~(7,r,,,.~): %’ -+ 9, which we denote b! -Y, for slrort. ??$ is dcttrmincd by the morphisms ‘i’ra\-cling clockwise along the diagram, we end up with the .P-functor %fm’+% --\;,.f~ , which is determined by the morphisms -4’ & ccrc, . j,~. ‘l’he object functions of 1Ir and < -Xi differ only by the isomorphism < : Ob(%‘) r+ Ob(%). Thus, S, z i . -Y$ . If f : i&4, [] --f [B, 01 is a T’c;-homomorphism then ,f : [--I, (T<)~ [] l [H, (T”)~ . 01 is a TV-homomorphism and to it r,, assigns the Y-natural transformation given by the family (.fj),t,. On the other hand, r,, assigns to ,f the Y-natural transformation ( ,fi) yields lx, with (&x)c! given by (.fj)rEI as well. P’, nhrn applied to .v (fpy) rc6’ . 1 “q. ,x< Thus, (fr,) :P 4. APPLICATIONS In what follows vve shall deal with some applications of the theory of categories of algebras to the study of relative functor categories. A key result throughout these applications is the last theorem of section 3. As in section 3 we shall assume that .Y is an I-complete closed category. Let W be any :F-category with Ob(%) .z I, and form the !Y-functor category :PV. Some direct consequences of theorems of section 2 together with Theorem (3.11) arc listed belolv. The details of their proofs are left to the reader. (4.1) For each CE Ob(%), denote by Ec : 9” -+ .Y the functor obtained by evaluating at C, d-functors ;Y : ‘r;’ - Y and Y-natural transformations y : -Y--f LT. Remark that EC is nothing else than the composite functor UC.. ~yol’(“‘) 25, y’. ,#6 _>(4.2) THEOREM (Completeness of functor categories). Let 2’ be any given class of limits. If .9 is S!‘-complete then 9”O: is 9-complete. In this case, each functor EC : .fcC + .Y ((7 E Ob(%?)) p reserves ..F’-limits and the family collectively reflects Slimits. (&kmcr, 95 RELATIVE FUNCTOR CATEGORIES Proof. First observe that in a product category (such as 9”) limits are defined pointwise. Then apply (3.11) and the remarks just preceeding (2.10). 1 (4.3) THEOREM (Existence of adjoints to induced functors between functor categories). Let 5 : ‘6‘ + K be any morphism of Y-cat, Then, the functor .Yy : :Y” --+ :Ycf defined b-y the rules S -t &Y; y 3 {y, has an adjoint and a roadjoint. Proof. It follows immediately (2.14) and its dual. i from (3.11), (4.2) and (2.8), together with The category -&’ (4.4) THEOREM (P-functor categories are Y-categories). has the structure of a Y-category fey which the functor LTV is a .?-functor, has a .Y-adjoint F, , and a !Y-coadjoint V, . Proof. (2.8). This result follows readily from (3.11) together with (2.2) and I For each C E Oh(V), the evaluationfunctor (4.5) COROLLARY. has a Y-adjoint Fc , and a S’-coadjoint r/, . Proof. It follows from (4.4) and (3.6). EC : .b’ - .Y 1 (4.6) THEOREM (First characterization of functor categories). Let .d be an I-complete closed category. Let .ZYbe a Y-category and Li : .‘/A- Y a .Y-functor such that there is a Y-adjoint F and a .Y-coadj’oint T7,for U. Then, there exists a .‘Y-category %?with Ob(‘%) G I, and a Y-equivalence of categories @ : A’ + Y” such that C = a3 1 U, , rf and only if / CZI/ h as coequalizers and C: rejects isomorphisnu. Proof. This theorem with (2.11). 1 is an immediate consequence (4.7) THEOREM (Representation for -Y-valued C F Ob(%Q there exists a natural isomorphism y = yx : B”(~(C, of (3.11) together Y-functors). -), X) -+ XE, For each . P~oqf. Let OL= olP,X : :Yc(PF, , X) --, Xnm(P, XE,) be the isomorphism which exists by (4.5). Let ip : P -+ Xn~(2, P) be the natural iso1 morphism defined in (1.10.1). Define yX = CS~,~* (ix+1. (4.8) DEFINITION. A category 55”is said to be coregular when it is the case that every epimorphism of 9” is also a coequalizer. L%n alternative form of ‘I’heorcm (4.6) can be given under the further assumption that .Y is a corcguiar cateRor>. ‘I‘his form, gilen helow, will he uscfui when proving our second characterization of functor categories. Proof. Let us see first that tile conditions arc ncccssar\. Assume for the I .rG: .Pm+.Yp,. moment that& -4” for some C’t- Ob(.Y-%(I/,),and that 1’ C.learly i -% is faithful. By assumption .Y has cocqualizyrs. therefore, so does .9”, I:! (4.2). \t’e shov, nc,\t that .4” is coregular. Let p : .Y + 1. be an cpmorphism of .Y”. Ixt the pair (y, S) hc the kernel pair ofp. It follo\n-s from (4.2) that, for each object (‘ of % , the pair (ytr . 8,) is the kcrncl pair of p, It also follows (cf. [3/l) that each pc is an epimorphism of .b. Since h? assrm~ption .Y’ is coregular. each pc must lx! a coequalizcr, in particular, of its kernel pair. Since the family of evaluation functors EC collectiwlv reflects iimits, it follows from the ahwc that p coeq(y, 8) in :Y’/‘“. ‘I’his sho\vs that .d” is coregular. Ohserve no\\ that any equivalence of categories @ : .ti ---f .P’ which commutes with the I:ndcrlying .b’-object functors, must preserve the conditions of the theorem. In order to prove the sufficiency part of the theorem we choose to reduce the conditions to those of (4.6). Since by assumption ~.8 i has coequalizers, we onI\- need to show that the functor IT reflects lsomorphisms. This n-ill require both the condition that .d ~ is a coregular categorv and the fact that ( is faithful, as follo\\s. Assumc,fis a morphisms of .+b’ such thatfl.’ is an isomorphism. Since fr. is both manic and epic and since F is faithful, it folio\\ s that f is both manic and epic. Since ~.ti ! is coregular, f is also a m cocquzlizcr. ‘I’htls (cf. [3/]), .f‘ is an isomorphism. ‘i’he ti,llon+ng notion generalizes categr??~~(cf. [lo]; [3/l). that of a small projecfiu in ilfl additive (4.10) ~)I~FINIl'IOx. An object A of a -i/‘-category .fl is said to lx an uto~ll of .d, whenever it is the case that the 9-functor .ti(K. ) : .H f .P preserves direct limits. (4.1 I j. U’c shall denote hy W each C’ :T Oh(%‘)(cf. (1.1 1)). the object %(C, --) : % -+ .d of .P”, for (4.12) PROPOSITION. Let 9 be a cowzpl~te closed cutegoyy. Then, in 9” twr~ object of the .form ii’ is nn atom of .Y” 97 HELA'lWE FUNCTOK CATEGORIES Proof. proposition By (4.7) and (4.1 I), .b”(@, now follow from (4.2). -) is naturally equivalent to EC . The 1 (4.13) DEFINITION. .A .4-category- .A is said to be atomic if thcrc a generating farnil>- of atoms of .a, indexed b>- a set. is (4.14) PROFO~ITIOS. Let .d be a rotttplete closed category whose uttderlyitzg : ,,,,d(%, mm)is .faitl!ful. Thftt, fhr an-v small .f-cate~~ot7~ %, the ,/iinrtor cateCyory .F is a complete atomic cate,Sory. set flltll-tOl..x I’royf. On the one hand, the composite Y”(W, -) . Hom.,(%, -) is nat. ). On the other hand, .Y”‘(lrc, -) is naturally cquix-. to Homc,,eI(/i”, cquix-aicnt to the evaluation functor Kc. l’he family (I?C)c-iob(C6) is collec‘I‘hus, the family tivel!- t’aithful. The functor Hom,,(%, ) is faithful. (HomCpx)(fjC. -)I is cokctivelv faithful and therefore, the family (I/‘-i is gcncrating for .f” (cf. [3/l). [(4.15) PROP~~TIOY. Jf .Y I\L scellposcered (rowellpowered) and a complete _, % the ,funclor cate,oory 9’” is closed category, then for atiy stttail -f-cafeC~or~ wellpowered (Co~~~ellpoecerPci). Proof. \Ye refer to [/O] t‘or the definitions of w.p. and co-w.p. Remark (epimorphism) if? for each that a morphism 7 of .P ” is a monomorphism c‘ t Oh(%), yc is a monomorphism (epimorphism). 1 In order to get a more internal characterization of functor categories, further assumptions shall bc made of .f. (4.16) 'I‘IIEORIX (,~erottd churaclrr-izatiott of funclor categories). Let 9 be a complete, wellpowered attd roxellpo-wered coregular closed category. rlssume furthermore that the futlrtor Hom,(%, --) : .Y f .<Y is faithful and reflects itteerse limits. Let .Y haze also a coC,anterator. I ‘ttder these conditions, a Y-category .H is -b-eqniaalettt fo a-functor category .b” ,for sotne small .4-category ‘6, if and on/v if .H is a complete weIIpo~wered and cozcellpowered corq&ar atomic CtifegcJYy. Proof. \Ve saw in (-I.?), (4.15), (4.9) and (4.14) that the conditions were wcessarv. Let (k,i,,, he a generating famil!- of atoms of :&. For each j F J then, the .d-functor .8(K) . -) : A + .Y preserves direct limits. Since A is right comptetc, co\vel!powrcd, and has a generating family, the Special .-It/joint Futtclor. Theorem of Fre!-d [IO] (,‘1s : lm p roved, e.g., by Lamb& [/a]), yields the existence of a coadjoint for each filnctor ,/A(Ki , -). Since .9 is closed with a faithful underiying set functor, h!. the considerations of (I .6) follows that this is also a .Y-coadjoint. Denote by (Ki ; ) : .Y --+ .‘A a .P-coadjoint to .Yl(K, , ) : .i/) + .P and let &: 2rC~,~~(.d(A-,) B), P) --•, .d(R, (Kj ; P)) he a -Y-adjunction, for each ; E /. Another way of looking at the above natural isomorphism is that it gives a rcprescntation for the functor NP,,J(A(K, . -), Pj : #J, + .YJ. Since the farnil\- {Kji is generating for -8, ‘i’he functors .#(A-, , ) are collecti\-cl!- faithful. Let ,O he a cogcneratnr for .P. ‘l’hcn the functor .?‘cc,,,,( ~, 0) : .P”’ \ .Y is faithfui. ‘l%ercforc. t!lr fjmil!of functors .?+, (K, ; 9)) p .7? c,~r(.ti(K, , -~),Cl) is collccti~ Cl!. faithful. \\ hi& implies in turn that the farnil? {(k; ; 0)) is cogenerating for ~8. Since 24 is left complete, wellpon-cred and has a cogencrating family, I>! the ahovc mentioned theorem of I;rc>-d ([ii)]. also [IN]), an\ functor ./A + .4 whiclr preserves inverse limits has an adjoint. Ixt us no\\ show that the .!@-functor .M(K, , -) : .# -+ .d Ixeserves inwrsc limits. On the one hand, Homy(%, .:‘/(A, , )) Hom,(K, , -). Or> ttrc otiler Aand, Ilom,A(K, , ) prcscrws inverse limits while, hy one of the assumptions or @, the functor Homa(%, ) reflects inverse limits. ‘i’i,i;s, an adjoint and. as heforc. also a .Y-adjoint K, ‘,>I ~ : .Y --t .a exists for .A(&,-, , -), .ti(he, , ) for Define a functor 1. : .8 f .YJ bv the conditioni I ‘7 each j c -1. \Ye want to show that I- has a -F-adjoint and a Y/‘-coadjoint, and that it is faithful. (‘Iearly I is faithful since tl:e family /-ti(h-, , -)! is collectivel!~ faithful. On the other hand, we have shou II nhorc that. for each j c 1. the functor A(K; , -~) has a .Y-adjoint (denoted K, _~ ~-) and a -fl-coadjoint (dcnotcd (/Y, : -~)). By (3.5) and (3.6) I‘t I\.I_ :It so the case that the projections Ti, : .YJ -+ .:F are .f-functors and have both -P-adjoints and -F-wadjoints, denoted respectively by A, and T,, . r , I;, 1,“; :f, A .b-adjoint to i- is given b!, a functor F defined by --IF I>uall!. a .d-coadjoint is given by a functor I defined by .--II ~--h, (A-, ; Lf). ,f-adjunctions which corroborate these assertions are those giwn below. By (4.9) the proof is now finished. 1 RELATIVE FUNCTOR CATEGORIES 99 ,4 first corollary to this last theorem is a variant of a theorem of Fre1.d [/O] characterizing functor categories based on .dL, the category of ail aheiian groups. (Charncterization of functor categories .dP). (4.17) C’OROLIARY L-1catr,gory :H is equivalent to a functor category of the form .dk” for some small additive category %-, if and only if .& is a complete wellpowered and cozcellporccred coregular additive category with a generating set of small projecfkes. I’r-oaf .r// is an abelian category. In particular, .P// is additive and coregular (ewry epimorphism is a cokernei). It is also complete, wellpowcred and coweiiponcred. It is monoidal with the tensor product and the group of integers, Z. also a generator. Thus, the functor Hom,,e(%. -) is faithful. It also reflects inverse limits. -4 cogcncrator for .d/ is the group of rationals moduio the integers, Q,lZ. On the other hand, by definition, a small (or abstractly finite) projective of an additive category .& is any object K for which the functor ~&(K, -) : .& .+.*/fl preserves direct limits, i.c., an atom of A. Thus 3 is atomic iff .8 has iI generating set of small projecti!-es. b (4. i 8) &NSITIOS. products) is said to f : K --+ x, &YJfactors i tl I), followed by the Also as ;I coroilar!of set \-alued functors A4n object K of a category .Y’ (ivith arbitrary cobc nbstractfy unary, whenever every .F-morphism uniqucl!- into an X-morphism f’ : K --* -I*, (for some corresponding injection into the coproduct. to (4.16) we derive the characterization of categories given b!. us in [7]. (4. i 9) C‘OROLLXRY (~haracteri~ativrl (~‘fUnctOY Categories C”‘). .3 category 2” is equivalent to a functor. categorlt 9” for some small category E, if and ot11y with if :G is u complete, ~zuellpowered and cowellpowered coreCgalar CafqcJTy a generatin: set of abstractly uuarv projecti-ies. Z’roqf. Cleari! .c/ is a complete wellpowered and cowzllpowered coregular catcgorv. It is Cartesian closed and 1 (the terminal object) is a generator. The functor Hoin,y( 1. -) is the identity and thus satisfies tri\-ially all the conditions imposed in (-1.16). A co ~ g enerator for .‘/’ is the object 2 =: I 1. Since coproducts in .Y’ :wc disjoint unions, an object K of a category .F is abstractly unary if and only if X(K, -) : 9’ ---f .Y preserl-es coproducts. Since .;I’ is coregular, an object K of 3” is a projective iff the functor .F(K, ) : .I ---+ Y preserves coequalizers. Thus, K is an abstractly unary projcctiw iff K is an atom. 1 I. .AuI,I?!‘, LI., Categories of functors atid ad!ulnt tuoctors. Nutriif, Rep!.. <;cn&~ i 1964). 2. RIICI~, J, XI., ‘l’riples, algchras and whmnoiogy. l~issert,ltion, C‘olumhi;~ I-ni\crsit! lY67. J’ BECK. J. AI., ‘I’he trlpleahlenesh theorem. ~Iumeogrilphed tlotc’h. C(lrnelI C’ni\ crsit! I Y67. 4. Ulix.wo~ , J.. Cattgories awe multipiicatron. (‘. Ii. .-icrrri. SC;. I’mj.~ 256 ( 1963), 1887-l 8YO. 5. H~SGNOI , J., Xlgtbre CICmentaire dans Ic\ catCgorws ;LIW rn~tltil,lic;~tlor1. (‘. I<. .-lcatl. Ski. 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