Categories of Actions and Morita Equivalence - Elkins, Zilber

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    R O C K Y M O U N T A I NJ O U R N A L O F M A T H E M A T I C SVolume 6 , Number 2 , Spr ing 1976

    CATEG O RIES O F ACTIO NS AND MO RITA EQ UIVALENCEB . ELKIN S A N D J . A . ZILB ER

    0. Introduction. L et A b e a small categ ory. By a (right) A-action ismeant a (covariant) functor A Ens, where Ens is the category of sets,and by a morphism of A-act ions a morphism of the correspondingfunctors. T he category of A-actions will be den ote d by Ens A , and Awil l be cal led the ope rator category. An abstract character iza t ion ofcategories of actions is given in [3]. Two (small) categories A and Bare cal led Mori ta-equivalent (AMB) if their categ ories of actions areequivalent in the usual sense (Ens A E n s5 ); i.e., if there exist functorsF and G be tween EnsA a n d E n sB such that FG an d GF are isomorphicto the co rresp on din g identity functors. T he following question s arise:I . How can we character ize Mori ta equivalence intr insical ly?I I . How can we construct al l categories Mori ta-equivalent to a givenone?Answers are given to I and II in [5] , when the operator categoriesare (finite) groups (in which case Morita equivalence implies isomorphism ), an d in [1] a nd [7 ] , wh en they are arbi t rary mo noids. Th esituation is analogous to the Morita theory for modules over a ring,descr ibed, e .g. , in [2] , [4] .In the present paper we provide answers to I and II for the case ofarbi t rary operator categories . One form of the answer can be s tatedin terms of the notion of weak equivalence w between categories . Aweak functor f :A* B is l ike an ordinary functor, but without therequirement that the image of an ident i ty morphism be an ident i tymo rphism ( it is then perforce an idem poten t) . Th ere is a correspond ing notion of weak fi inctor morphism, and the result can be stated(Theorem 4.4):

    Two (small) categories are Mo rita-equivalent if and only ifthey are weakly equivalent.An alternative formulation (Theorem 3.6') is in terms of the idem-potent completion A constructed in 3 for any category A:

    Two (small) categories are Morita-equivalent if and only iftheir idempotent completions are equivalent (in the usualsense).

    Received by the editors on September 3, 1974.Copyright 1976 Rocky Mounta in Mathemat ics Consor t ium199

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    200 B. ELKINS AND J. A. ZILBERThe idempotent complet ion A also provides a means of answeringquestion II (Theorem 3.9).

    Our ini t ia l approach to the problem made use of an Ei lenberg-Gabriel -Watts theorem for act ions, descr ibing an arbi t rary cocont inu-ous functor from Ens A to En sBin term s of a tensor pro du ct con struction.The final version circumvents this idea, but an account of the construction is included, both because of i ts relevance and because of i tsind epe nd en t interest . After this pa per had be en wri t ten, we discove redin [8] that P. Freyd had t reated quest ion I with Ens replaced by Ab,the category of abel ian groups, using the ideas of idempotent complet ion an d am enab le category. This does not seem to diminish th edesirab il i ty of the prese nt exposit ion.1. Prel iminaries . Fu nct ion al operat ion will general ly be wri t ten onthe right; in particular, composition of functions (and morphisms) isfrom left to righ t. An ac tion F:A-+ Ens is equivalent, in the categoryE nsA , to one in which the setsaF, a G |A|, are disjoint. Th us, an actionam ounts to a grad ed set X = {Xfl},a G |A|, with operations Xa :x>xafor every morphism a :a a' in A, taking x G X f l into xa E Xa, andsatisfying the axioms

    (i) xla = x (x G Xa) and(ii) (xd)a' = x(aa').Viewed in this manner, the action will be called a (right) A-set. Thecategory Ens A can therefore be regarded as the category of right A-sets,and w hen i t is so rega rde d a morphism in Ens Awill be called an A -map.An A-map / : X > Y is the n a ma p of grad ed sets such th at (xa)f= (xf)a. Sim ilarly, a left A-set m eans a gra ded set X = {Xa}, a G |A|,with operat ions X a taking x G Xa' into ax G Xa for any morphisma: a a ', and satisfying the axioms

    (i) lax = x (x G Xa) and(ii) a(afx) = (aa')x.

    A left A-set corresponds to a left A-action i .e. , a contr av arian t functorfrom A to En s an d a m oun ts to a right A-set, w he re A is the catego ryopposite to A. "A-set", unmodified, will mean "right A-set". Note thatbecause of our convent ion concerning order of composi t ion, the notation is counter to the common one for semisimplicial complexes; in thepre sen t langu age , thes e are left A-sets, w he re A is the category offinite ordinals.

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    CATEGORIES OF ACTIONS AND MORITA EQUIVALENCE 2 0 1If B is another small category, we mean by an A-B-biset a bigradedset X = {Xba}> a G \A\, b GE |B|, which is simultaneously a left A-setand a right B-set, such that a(x) = (ax) whenever either side ofthis equ ality is defined. Th is am ounts to a right (A0 X B)-set, or a left(A X B)-set. It also amo unts to a contravariant functor X * from A tothe category EnsB of right B-sets, with X*: a I- X, where Xa is theB-set Xa = U {Xba\b G |B|}; and, of course, equally well to a co-variant functor X* from B to the category Ens A of left A-sets, withX*:b h-Xb = U {Xba\a E |A|}. To indicate that X is an A-B-bisetwe shall sometimes write it as XBA. In particu lar, the usua l hornfunctor on A, being a covariant functor from A0 X A to Ens, is

    equivalent to an A-A-biset AAA

    , which we will denote by A all overagain. Its com ponen tsA are the hom -sets A(a, a').Just as two-sided operation can always be regarded as one-sided,so can one-sided operation be regarded as two-sided: the operatorcategory on the other side is the unit category, U, consisting of oneobject and its identity morphism. W e return for the time being to adescr iption of the right theory.Among the covariant functors A > Ens are the representable ones,viz., the functors A(a, ). In the abo ve notation, these are the rightA-sets A, which we call themodels in the category EnsA . The Yonedalemma asserts that for any A-set X, the collection of A-maps EnsA(A a, X)is in 1-1 correspondence with the set X a; namely, to each elementx Xacorresponds the unique A-map< px: Afl X such that(1.1) lrf) , =XIf a : aa' is a morphism in A, so that a is an element of the A-setA a, then(1-2) < P x = < P a < P x \andiff : X Yis an A-map, then(1-3)

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    202 B. ELKINS AND J. A. ZILBERWe shall often write [x] for Aa if x G Xa. For indexing purpose s i tis useful to regard [x] as the set of pa irs (x, a) , a G Aa, with (x, a)a'

    = (x,aa ) and (x,a)7 x a V-LV x) = (xaa v v x ) .The d iagram Dx assigns to each object x the model [x], and to eacharrow (x, a) the A-map [x, a] : [x a] -> [x ]. O ne sees, using (1.2),that X is the colimit of D x , the cone from Dx to X being given by thefamily of maps {D' , whereD and D ' are over index categories / and / ' , is a functor F :J >J'together with a functor morphism from D to FD '. In th e case of Df,the functor mo rphism co nsists objectwise of isomorphisms.) T he assignment Xh->D X , ft-*Df con sti tutes a functor from E nsA to thecategory of diagrams in M(A).The index category (Yon0, X) for the diagram Dx will be called thefundamental category of the A-set X.By the cocompletion of a catego ry A is m ea nt a catego ry Z2 Asuch that

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    C A T E G O R I E S O F A C T I O N S A N D M O R I T A E Q U I V A L E N C E 2 0 3

    (i) A is cocomplete, and(ii) any functor C, where C is an arbi t rary cocomplete category, extends to a cocont inuous functor C, the extension being unique up to an isomorphism(which is the identity on A).

    These propert ies determine the cocomplet ion, i f i t exis ts , up to equivalence; and, indeed, the category Ens A is the cocompletion of A, thela t te r be ing imbedded in Ens A as M(A). (Fo r (i i) , see [9 ] , p . 108;the extension Y, the image of any orbitofX is contained in some o rbit ofY.We shal l be deal ing with epi-mono factor izat ions of idempotents in

    an arbi t rary category. Co ncern ing these, w e ma ke the following easi lyverified observations.1.2. L E M M A . For any morphisms

    (1.5) a^b-^a,the following two conditions are equivalent:(a ) IT is epic, \imoniCyand e = n /LI idempotent.(b) /IST = 1.

    The equivalent conditions of Lemma 1.2 are referred to by sayingtha t the idempotent splits and the factorization (1.5) is called asplitting of the idem po tent e un de r these condi t ions.

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    204 B. ELKINS AND J. A. ZILBER1.3. L E M M A . For any idemp otent :a -+ a, denote by D(e) thediagram consisting of the single object a and the morph ism e. Then

    has an epi-mono factorization (1.5) if and only if b is the colimit ofthe diagram D(e), with n as corresponding cone ; or, equally well, ifand only if b is the limit of D(e), with JJL as corresponding cone .Hence, also, the imag e b of e is unique up to isomorphism.1.4. L E M M A . The image, under any functor p : AB, of a splittingof an idem potent ein A is a splitting of an idempotent e

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    C A T E G O R I E S O F A C T I O N S A N D M O R I T A E Q U I V A L E N C E 2 0 5

    In particular, every m odel Aa is projective, since it obviously satisfiesthe cond ition of the lemm a.2.2. PROPOSITION. L et X = U* Xjfeea representation of X as a co-product in EnsA . Then X is projective if and on ly if each X* is projective. In particular, X is projective if and only if each of its orbits isprojective.PROOF. If each Xj is projective , then so is X = I IiXi9since this is thecase in an arbitrary categor y. Co nverse ly, supp ose X is projective , andle t m:X*Px be an in Lemm a 2.1. S ince clearly Px = \l{ PXi, and

    x = JIipx{>the map m must be of the form JJ_imif with m{pXi = 1for eachi. By Lem ma 2.1, this imp lies that eachX*is projective.Thus, in order to describe all projectives in EnsA , it remains tocharacterize the indecomposable ones.

    2.3. PROPOSITION. An A-set X is an indecomp osable projective ifand only ifX is the idemp otent image of a model, i.e., the image of amodel under an idempotent A-ma p.PROOF. Since an anti-isomorphism preserves idem potents, thisamounts to the condition that X be the image ofanA-mapAe: Aa-* Aafor some idempotent e: a - * a in A. Suppose X is an indecomposableprojective, and letm :X Px be a map such thatm px = 1 (Lemma2.1). By L emm a 1.1, there is an index x in the decomposition (2.1),and a map j :X >[x] = Aa, such that j * ix m, where ix is as in(2.2). Hencej -

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    206 B . E L K I N S A N D J . A . Z I L B E RFor an arbitrary category A, we denote by I(A) the (full) subcategory of EnsA whose objects are the images of models under idem-potent maps; by Proposition 2.3, this is the same as the subcategory ofindecomposable projectives. Thus I(A) contains the category ofmodels M(A). W e now state the condition for equality.2.4. DEFINITION. Idem potents split in the category A if each idem-potent in A has a splitting in A. Such categories will be called propercategories.2.5. PROPOSITION. L et Abe a small category. Then M(A) = I(A)if and on ly if idempotents split in the category A.PROOF. It suffices to observ e that, bec aus e of the anti-isomorp hismbetween A and M(A), and in view of Lemma 1.2, the image of anidempotent A\ Aa^> Aa is of the form Ab for some b G A if and onlyif has an ep i-mon o factorization

    a -=>b -rj> in A, and then the epi-mono factorization o fAeis

    Aa -+ Ab -* Aa-A* A3. Idempotent completion and Mrita equivalence. W e begin thissection by obtaining an invariant characterization of the categoryI(A) for arbitrary A. This has indee d already be en d on e from thepoint of view of regarding I(A) as a subcategory of EnsA: by Prop.2.3, I(A) is characterized as the subcategory of indecomposable projectives. This implies that if E nsA and EnsB are equivalent; so areI(A)and I(B). In fact, w e can already state:3.1. PROPOSITION. Sup pose the small categories A and B are proper.Then

    EnsA EnsBA B.PROOF. AS just observed, EnsAE nsB =>I(A)=*I(B), and, byProp 2.5, this implies AB . On the other hand, an equivalence between A and B implies an equivalence between the functor categoriesEnsAand EnsB .Our object now is to characterize I(A) from the opposite point ofview ; nam ely, as a supercategory ofA.3.2. DEFINITION. A category D A is called the idempotent completion ofAif

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    C A T E G O R I E S OF A C T I O N S ANDM O R I T A E Q U I V A L EN C E 207

    (i) idempotents spli tin,and(ii) anyfunctor C,wh ere ide mp otents sp li tin C,extendstoa functor

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    208 B . ELKIN S A N D J . A . ZILB ERfollows from the commutativity of the squares

    (3.1) a and

    in A, and therefore of the diagram

    Pa

    PaV

    T T a >,

    ~ ~ ?

    a

    TTa

    in C. Fin ally, this is an isomorph ism of functors, since the m orphisma7Ta ' has as inverse the morphism a ;jfa.3.4. COROLLARY. Suppose

    (i) Ais afilli subca tegory of A ;(ii) every object in is the image of an idempotent in A;(iii) every idempotent in A has an image (i.e., has an epi-monofactorization) in A.Then A is the idempotent completion of A .P R O O F . All tha t has to be sho wn is tha t idem po tents spli t in A; i .e. ,that every idem pote nt in A has an image. Let :a > abe an arbi t raryidempotent in A. The object a is the image of an idempotent in A,with corresponding epi-mono factor izat ion rrfi. T h e n neix is also anidempotent in A, and therefore has an epi-mono factorization T T /LL ' .But then has the factorization e fin ' JLIV, which is epi-mono becau se X T T - e i m = I X T T H T T = 1 .Since the category I(A) obviously satisfies the conditions of thecorollary except, of course, that it is A0 ~ M(A) which is the subcategory it follows that I(A) is a realization of the idempotent completion of A. But then any other realization must satisfy the condit ions of the corollary (in terms of A), since under an equivalence between supercategories of A0 these condi t ions are preserved. Thus theco nd ition s of the corollary a re nec essary as we ll as sufficient. In th e"left" theory, the corresponding category 1(A) is precisely the idem-potent complet ion of A

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    CATEGORIES OF ACTIONS AND MORITA EQUIVALENCE 2 0 9It is convenient to give a more explicit construction of the idem-potent complet ion (cf . [6] , p. 61) . We cal l the new version A. Anobject in A is an id em po ten t :a>a in A. A morphism in A, withdomain e : a-* a and codomain ' :a' >a', is a tri p le (e, a,e ') suchtha t a = a ' = a . Th us, a m orphism is given by a com m utat ivediagram

    Co m pos ition is given by t he ru le (e, a,' ) ( * ' , a ' , " ) = ( , a a ' , " ) . N o t ethat the ident i ty morphism l e at the object is (e, e, e). Note also thatfor a given pa ir , ' , the m orph ism s a such that the above diagram iscommutat ive are precisely those of the form ea'e', for an arb itrarymorphism a' :a > a\The functor ^ A : A A given by ah-> la , ( a :a * a ')H> ( la , a, l a ) ,im bed s A as a full su bca tego ry ofA .

    3.5. PROPOSITION. Ais the idempotent completion of A.P R O O F . Any idempotent in ais an ide m po ten t in A. In A it ha s the factorization

    ( l f l , > l f l ) = = ( l p , ) ( , , lf l ) ,an d this is epi-mon o, be caus e

    ( , , la ) ( l a ? , ) = (e , , e ) = le .The same factorization also shows that any object in A is the imageof an idempotent in

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    210 B . E L K I N SAND J. A. Z I L B E R3.6. T H E O R E M . Let A and B be arbitrary small categories. Then anequivalence E n sAE n sB induces an equivalence I(A) I(B).

    Conversely, an equivalence I(A) I(B) determines an equivalenceE n sA E nsB ,unique up to isomorphism.P R O O F . It remains to prove the second half. Let

    l(B)an d \\t;I(B)I(A)be an equivalence pair , and deno te by

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    CATEGORIES OF ACTIONS AND MORITA EQUIVALENCE 2 1 13.8. L E M M A . TWO categories A and B have equivalent idempotentcompletions if and only if there exists a category C such that(i) Aand B are full subcategories ofC, and(ii) every object in C is the image of an idempotent in A, and alsothe image of an idempotent in B.

    P R O O F . Let C be such a category , and C i ts idem po tent com plet ion.Then C is also the idempotent completion of both A and B, as followsfrom the characterization in Cor. 3.4 of the idempotent completion.Converse ly , suppose A and B hav e equiva lent idemp otent completions D A an d D B. There exists a category C containing both and S, such that the inclusions A C C and Bd C are equivalences.(For example, if the functor f :A is one of an equivalence pair ,le t C be the "mapping cyl inder" Cf obtained by ident i fying a morphism(a, 1) in the base A X 1 of the cylinder X I, where I is the categoryconsis t ing of two objects 0 and 1 and one isomorphism between them,wi th the morphism f(a) in S.) Th en C is an idem pote nt com plet ionof both A and B, and therefore satisfies conditions (i) and (ii).Final ly, we can answer quest ion II of the Introduct ion in thefol lowing way. Given a category A, consider the expl ici t idempotentcompletion A. If B is any category such that B is equivalent to A, thenthe e qu ivale nc e takes B into a full sub categ ory B ' of A such th at Ais the idempotent complet ion of B'. Th is m ean s tha t every object inA is the image of an idem pote nt in B \ In par t icula r , every object in

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    212 B. ELKINS AND J. A. ZILBERA weak functor differs from an ordinary one, which for emphasiswe shall also call a strong functor, only in that identity morphismsneed not be preserved; but since an identity morphism satisfies theequal i ty l a l a = la, under a weak functor i t goes into an idempotent.As for a morphism t :/ > g between weak functors, i t is readily seento am ou nt to a family {ta \a G A} of mo rphism s in B such tha t(i) for any morphism a :a a ' in A,af * ta, = ta ag , and(ii) for any object a in A,\af ta = ta l a g = ta.T h e first eq ua lity in (ii) is alre ad y co nta ine d in (i). If / an d g arestrong functors, the notion of weak functor morphism reduces to theordinary one.Composition of weak functor morphisms is as in the ordinary case:com pose objectwise. This yields a category F u n c t^ A , B) of weakfunctors from A to B. Observe that for a weak functor f : A B, theident i ty morphism 1^:// is given by (l/)a = la / ; the assignm e n t aI-laf is in general not a morphism from f to f Th e mul t ipl icat ion

    F u n c t J A , B ) X F u n c tU )(B, C) -> FunctJA, C) ,which on functors is composition, is defined in the same way as in theordinary case. One has the obvious extension of the not ion of equivalence betw een categories to that of weak equivalen ce w.For any category A, let A be a category satisfying the hypothesisof Lemma 3.3, and denote by iA : A C A the inclu sion functor. As inthe proof of Lemma 3.3, select for each object a G A an idempotenta of which i t is the image, and a corresponding epi-mono factorizationa = ^oMa? W 1th a = 7Ta= /uta= la if a E. A. Define rA: ^> A asfollows: for any object a G , arA = dom ain ( = co dom ain) of e0 ; andfor any morphism a :a a ', ocrA = iraafiat (called pa in the proof ofLemma 3.3) . Then rA is a weak functor; it is not in general a (strong)functor, since i t takes an identity morphism la into the morphism fl.

    4.2. L E M M A . Let A D A be any category satisfying the hypothesisof Lemma 3.3; in particular, can be the idempotent completion ofA. Then the inclusion iA : A C determines a weak equivalenceA w A.P R O O F . The weak functor in the other direction is rA. SinceiA ' rA = 1 , i t remains only to show that rA iA is isomorphic to theidentity functor 1^. We assert that the isomorphism is given byM : 1 -> rA - iAandTT :rA iA - 1,wheren = {fia},^ = KJ , t f ^ .That /x and TT are weak functor morphisms follows from the commuta-

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    C A TEGOR IES OFA C T I O N S ANDM O R I T A E Q U I V A L E N C E 213

    t ivity of the d iagra m s (3.1). (T he fact that on e of th e functors inv olved namely, 1^is a strong functor automatically implies the extraequal i ty requi red for thecasea = la.) For thecom posi te \m : 1 -1> ( p w ) a = /Va = la- F o r t h e C o m p o s i t e TT[Ly (irfl)a = 7Tafla = f l= lfl(fA ' *A)>a nd asal ready no ted, this m eans tha t wfxis the ident i tymorphism atrA iA.

    4.3. LEMMA. Le Afo ant/category, B a proper category, and faweak functor from A to B. Then f is isomorphic to a (strong)functor.PROOF. Select for each object a E:A an epi-mono factorization

    7Tafiaof the i dempoten t laf and deno te the cor responding imagebyag:af -f>ag --of.

    F o r any morphism a:a->a' in A, let ag= fxa*af iral . Theng : A* Bis astrong functor; for, firstly, ifa ' :a' >a", then(g,and the m a p s pa, aG A, an isomorphism /x,:g /. Thisfollows fromthecom muta t iv i tyofthe d iagram

    for any a: a >a ' ; again, the extra equal i ty needed for the casea= 10is automatically satisfied, becauseg is astrong functor. Fin ally,(pir)a = /vra= lag= lag, whi l e (TTp)a= rrapa = \af so that bothfin an d7TLare ident i ty m orphisms.

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    214 B . E L K I N S AND J . A. Z I L B E RIt follows from Lemma 4.3 t ha t if B is proper , the categoryF u n c t ^ A ,B) is equiva lent to its subcategory Fu nc t (A,B). This isbecause the lat teris a full subcategory.Now cons ider any categories A and B together with their idem-potent comple t ions ~DAandBZ) B. On the oneh a n d , the equivalences Aw A, B wB imply FunctMJ(A ,B) F u n c t ^ A ,). On theother hand, by L e m m a 4.3, Functu?(A ,B) Funct( , B). Th ere isthus an e q u i v a l e n c e F u n c t ^ A ,B)Funct (, B). Fur the rm ore , ifCis a third category, with idempotent complet ion CZ) C,the n clearlythe rvlant equivalences inducean equiva lence be twee n them ult ipl ications

    F u n c t J A ,B) X F u n c t J B ,C) - F u n c t J A ,C)andFunct (A,B) XFunc t (B ,C) -Funct (A,C);

    in par t icular , the m onoid al categories Functu )(A ,A) and Funct (,)are equivalen t . Asaconsequence ,wehave :4.4. THEOREM. A M B*=> A ^B.F o rtheexplicit com pletionsA and B theequ ivalen ce pa ir of functorsw :Func t (A ,B) -F u n c t J A , B) and s: F u n c t J A ,B) - Funct (,B)are found to be as follows. If F :A - B, its image Fw u n d e r u; isde te rmined by the equa li ti e s (1^ 0 , 1^ )F= \aF,aFw> 1 F)for every m or-phism a :aa ' in A. (Theobject laF in A is then the sameas themorphism laFw in A.) If T:F >G is a morphism in Funct (A,B)(so that T is a family {T} , wh eree runs through thei dem poten ts inA), its image Tw :Fw -> Gw is de t e rmined by the equalit ies Tia =( l a F , Taw, laG). If / :A -> B, its image / ' u n d e r s is given by

    (e , a, e')fs = (ef af, e' / ) ; and if : /g is a morphism inF u n c t ^ A , B ) , its image F is given by t= (ef ta eg = ef *a, eg),wh ere e runs through idem potentsin A.5. Tensor products and the Ei lenberg-Gabr ie l -W at t s theorem.F o r any A-B-biset X = XBA and B-C-biset Y= Yc, we define anA-C-biset X Y= XBA YCBas follows. The set (X Y) consistsofall pairs (x, j/) such thatxG Xfoaand t/ G Ycbfor somefo,subject tothe equivalence relat ion~ genera ted by the r equ i r ement (x, y) ~

    (x,y), w h e r e is a morphism in B. We shal l make no notat ionaldis t inct ion between a pair (x,t/) and the cor responding equiva lenceclass. Left andr ight op erat ionsby A and C aredef ined unam biguously by the formulas a(x,y)= (ax , t/), (x,t/)y = (x,yy). If Z is aC-D-biset , then clearly(X Y) Z X (Y Z).

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    C A T E G O R I E S O F A C T I O N S A N D M O R I T A E Q U I V A L E N C E 2 1 5

    Th e A -C-biset X Y can b e reg arde d as a col imit in the categoryE n s A c of A-C-bisets in the following way. Consider the fundamentalcateg ory ( 1) of the o bject X in th e categ ory En s A B . In left-rightlanguage, i t can be described as follows: an object is an elementx G X; an a rrow is a tr iple (a,x, ) :otx x; and the composi teofa'ff'( '^>y~rfi x

    is, (a'a,x,

    f)a oa > x.

    The diagram Dx assigns to the object x the A-B-biset A^X Bh (wherex EXba), which we can regard as the set [x] of triples (a, x, ),a G 4 , ] 8 B b , with a'(a,x,)'= (a'a,x,'); an d to the arrow(a, x, ) the A-B-map \X B:(a',cc,')\-+(a'a,x,').

    Take now the same index category, but assign to the object x( G Xba)the A-C-biset Aa X Yh, and to the arrow (a, x,) the A-C-map \ X Y.Thus to x G X is assigned an A-C-biset whose elements can be writ tenas (a, x, y), a G Aa, y G Yb; and to the arrow (a,x, ) is assigned theA-C-map

    (a', otx, y)H-> (a'a, x, y).O ne rea dily sees that X Y is the colim it of this diagra m . It is equallywel l the col imit of ano ther diag ram b ased on the fundam ental categoryof the object Y in the category EnsB c .In pa rticu lar, if X is an A-B-biset and Y is the B-B-biset B, the n thedescription of X Y as a colimit over the diagram Dx coincides withthe desc r ipt ion of X as a col imit over the same diagram . Th us, X B X. Sim ilarly, for any B -C-biset Y, B Y Y.An A-B-map f :X X ' an d a B-C -map g : Y> Y' de termine theA - C-m a p / < 8 > g : X Y - X ' < 8 > Y ' g iv en b y (x, t/)(/ g ) =(xf> lie)- T m s defines a bifunctor , cov arian t in both v ariable s,from Ens A B X E n sB c to En sA c .

    Next, we define for an A-B-biset X = XBA an d a C-B-biset Y = YB

    Ca C-A-biset (X, Y) = (X BA , Y BC). The set (X,Y)ca consists of all (right)B-maps

    Yc. Left and right operations by C and A are definedby the formulas yip =

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    216 B. ELKINS AND J. A. ZILBERgeneral the same thing as the hom-set Ens A B(X, Y); but it is, if A is theu n i t c a t e g o r y .

    An A-B-map / :X ' > X an d a C-B-map g : Y> Y' de term ine theC-A-map (/ , g) : (X, Y) -+ (X' , Y') given by YBc(of C-B-bisets) given by (

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    C A TEGOR IES OF A C T I O N S ANDM O R I T A E Q U I V A L E N C E 217

    Consider now, for a fixed W= WBA, the functor Tw = - WBA:E ns A E nsB . ByL e m m a 5.1,weh a v ethenatu ral isomorphism(XAWB\ YB) (XA,(WB\ YB)),

    in this case an isomorphism between hom-sets . This says that thefunctor Sw= (WBA,): E n s ->E n sA is r ight-adjoint to the functorTw. The un i t 1 -TWSW and couni t SWTW - 1 are given by thesect ion mapsXA->(WBA, XA WBA) and evalua t ion map s (WBA, YB) WBA -+ YB.We show now t ha t all adjunct ions between the two categoriesE ns A and E n sBare of the abov e form. M ore precisely, consider thecategory A d jA B of adjun ctions from E nsA to E nsB . This is equivalentto the category of functors EnsA > E n sB that have right adjoints,which is the same (s ince EnsA has a small set of generatorsthemodels ) as the category of functors EnsA E n s5 t ha t are cocont inu-ous. The notat ion A d jA B will consequently alsobe used to mean thiscategory of functors. We have just descr ibed an assignment WiTwof objects in E n sA B to objects in A djA B . Similarly, an A-B-mapf :W -+W de termines a functor morphism Tf = /; and wehave thereby def ined a covariant functor T :E n sA B > AdjA B . On theother hand,anyfunctor FG A d jA , be ing cocont inuous ,is de terminedu p to isomorphism by its restriction F \ M(A) to the subcategory ofmodels . This rest r ict ion amounts to a contravariant functor from AtoE ns B ,i.e., it amount sto an A-B-biset W;and theisomorphism A W W impli es t ha t the functors Tw andFag reeon M(A). Th us everyfunctor FG A djA B is isomorphic to a functor of theform Tw. Final ly,for any W and W, an arbi t ra ry morphism between the two cocontinuous functors Tw andTw>isde t e rminedby itsvalueson themo dels .This implies that every such morphism is of the form Tf; and therepresenta t ion is unique , because Af~f- We conclude tha t thefunctor T de termines an equiva lence be tween the categories EnsA Ban d AdjA B .Fur the rmore , if C is a third operator category, the relevant equivalences induceanequiva lence be tw eenthem ult ipl icat ions

    A djAs X A djBc AdjAcandE n s A BX E n sB c -* E n sA c .

    This is beca use:(X WBA) Z^B X (WBA ZcB);iff: W-> Wa n d g : Z - Z\ then (x f) g (X f) g; and (/ lz)(lw g) = ( l wg) (/ lZf) = / g. In par t icular , the

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    218 B . E L K I N S A N D J . A . Z I L B E Rmonoidal categories A djA A and EnsA A are equiva lent. As a consequence, the condi t ion for equivalence of the categories Ens A andE n s B is the existence of a pair of mutually inverse bisets WBA and ZAB .We can sum up by the fol lowing Ei lenberg-Gabriel -Watts theoremfor categories of actions.

    5.2. THEOREM. The bicategory whose objects are the categoriesE nsA , and whose hom-sets are the categories AdjA B , is quivalen t tothe bicategory whose hom-sets are the categories E n sA B (with tensorproduct as compo sition). In particular, AMB if and only if thereexist bisets W BA and ZA Bsuch thatW B A Z A B A a n d(5.1) Z A B WBA B.

    Co nd ition (5.1) can be ma de mo re prec ise. Since i t asserts tha t thefunctor Tw is an equ ivale nce , i t implies tha t the un it an d coun it forthe adjunct ion Tw H Sw , already specified by the evaluation andsection maps, are isomorphisms:X ( W, X W ),( W , Y ) W Y ,

    na tura l in X an d Y, respectively. Similarly, for Z, w e have isomorphism s Y (Z, Y Z) an d (Z, X) Z X, na tura l in Y an d X. Inpart icular , taking for X a model Aa an d for Y a m ode l Bby observingtha t AaW = W= W a an d Bb Z = Zb, and using natural i ty, weobtain isomorphismsA ( W, W),

    ( W, B)W^B,B - (Z, Z),

    (Z , A) Z A.On the other hand, condition (5.1) asserts not only that Tw an d Tzare separately equivalences, but that they form an equivalence pair;so tha t the two functors

    7V, S z :E n sA - E n sBare isomo rphic, as are also the two functors

    S w , Tz : En sB -> EnsA .

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    CATEGORIES OF ACTIONS AND MORITA EQUIVALENCE 2 1 9Of th e four adjun ctions

    Tw H Sw , S z H Tz , Tw H Tz , S z H Sw ,the unit and counit have been specified only for the first two. Pick adefini te isomorphism between, say, Tw and Sz . This determinesuniquely a conjugate isomorphism between S w an d Tz; it also determines units and counits for the last two adjunctions, together with compat ible isomorphisms between any two of the four adjunct ions, i tbeing understood that the isomorphism between, say, the f i rs t and thethird is th e identity on T\y Th is gives rise, then , to the following(part ly redundant) addi t ional l is t of compat ible isomorphisms, naturalin X an d Y: X W (Z, X),(w,Y) yz,

    X X W Z,Y Z W Y,

    X (W , (Z, X)),(Z ,(W , Y)) Y.

    As before, taking X and Y to be mo dels and u sing natural i ty, we ob tainW ( Z , A ) ,

    (W , B) Z,A W Z,

    Z W B,A (W , (Z, A)),

    (Z , (W , B)) B.Notice also that because we have now selected a uni t and couni tfor the adjunction Tw H Tz , the isomorphisms (5.1) are compatible,in that, e.g., the two isomorphismsW < 8 > Z W ( W < g ) Z ) W ^ A < g > W ^ Wand

    coincide. This is because the two corresponding isomorphisms between the functors Tw Tz Tw a n d Tw coincide. Similarly, there is noambiguity for the isomorphism Z W Z Z.

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    220 B . ELKIN S A N D J . A . ZILB ER6. Some re-examinations. Le t us no w rederive , directly from co nd it ion (5.1), th e result of 3 con cern ing M orita equ ivalen ce. An A-B-biset W determines a category C(W), in the following way. RegardA and B as disjoint; then an object in C(W) is any object in either Aor B; a morphism in C(W) is either a morphism in A or B, or an elem e n t w G W, wi th w : a - * b if w G Wb a; and composition is giveneither by composition in A or B, or by left or right operation by A orB,wh iche ver is releva nt. T he re is an obvious functor from C(W ) ontothe c atego ry $ ; an d in pa rticu lar, for th e A-A-biset A, the catego ryC(A) is isomorphic to the category | X A .O n the othe r han d, let / an d g b e any two functors from A an d B,respectively, to a third category C. Then f and g de termine anA-B-biset [/, g] : an ele m en t of [f,g]l is any triple (a,y,b), wherey : af >fog is a m orp hism in C; an d if a :a a, : b *fo aremorphisms in A and B, then a (a , y , b ) = (a',af ' y g , b ' ) .Next , consider a tensor product W B A Z eB . Each of the categoriesC(W) and C(Z) contains a copy of the category B, and we can thereforeform the amalgamation C(W) *B C(Z), i .e. , the pushout (in the category of categories) of the diagram C(W)

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    CATEGORIES OF ACTIONS AND MORITA EQUIVALENCE 2 2 1morph ism betw een, e .g., a pro du ct W Z W Z and thebiset A, independent of parentheses.

    The compos it ion C ( W Z) C(A) ~ \ X A-> A, where thelast functor is the projection, determines a 1-1 correspondence between morphisms of the form (w, z) : a -* a' in C(W Z) and mor-phisms a :a a ' in A. Write 0(a>,z) = a. T h e function 0 satisfies th eidentit ieseiwtZ) = 6{w,z\(6.1) a (i , z) = 0(aw , s), 0(u>, z )a = 0(w, za).

    Similarly, for C(Z W ), w rite 0(z,w ) for the corresponding morphism in B.Now reduce the ca tegory K(W, Z) by the equivalence relation ~~ onmorphisms genera ted by put t ing u ; z 6(w, z) and z tv ~ 0(z , w;).Consider the effect of this reduction on the full subcategory [A]gene ra ted by A in K(W, Z) . A mo rphism fi in [A] is either a morphismin A, or of th e form/ut = u ^ Z\ u ;2 * *2 ' u>n * ^;the lat ter rep resentat ion is unique up to replacements w - z++w ' z an d za - w++zaw. to any morphism /x, G [A] ass ign the morphism / l A asfollows: i f / i E A, then fi= J L ;if gi = WI ' zx ' * u> * zn? thenfi (wi, Zi) 6(wn, zn). T he identit ies (6.1) assure th at fi iswell-defined. If now tw o m orph isms //, an d fi' differ by a rep lace m en tof th e form w z *0(ttf, z) (with incorporation of 0(u;, z) into an adjoining factor, if one exists, as an operator), then the identities (6.1) sufficeto ensure fi= fi'. On the othe r ha nd , if fi and /x,' differ by a replacement of the from z w- 0(z, a;) , the equality fi = fi' is ensured bythe consistency con dition . It follows that the effect of the equ ivale ncerelation on the full subcategory [ A] is to reduce i t precisely to A,and therefore A appears as a ful l subcategory of the reduced categoryK= K(W, Z) /~ . The same i s t rue of B.Since th e func tuin 0 is surjective, for each ob ject a G A th er e is apa i r (wa, za) such that 6{wa, za) = la . Then in the category K, wa za= l a . This mean tha t za wa is an idempotent in B, and a is i ts image.Thus , every object in A, and therefore every object in K, is the imageof an idempotent in B. Similarly, every object in K is the image of anidempotent in A. Hence (Lemma 3.8) A and B have equiva lent idem-potent complet ions.Conversely, i f A and B have equiva lent idempotent comple t ions ,then by Lemma 3.8 they are both contained as full subcategories in acategory C where every object is both the image of an idempotent inA and the image of an idempotent in B. Deno t ing by iA an d iB the

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    222 B. ELKINS AND J. A. ZILBERinclusions A C C and B C C, consider the A-B-biset W= [A,B]and the B-A-biset Z = [iB, iA], and form the tensor product W Z.The assignment

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    C A T E G O R I E S O F A C T I O N S A N D M O R I T A E Q U I V A L E N C E 2 2 3

    two distinct A-maps / , g : X X ' , there exists a m em ber Zb of thefamily and an A-map X such that cpf ^ tpg), since thesame is true of the family {Bb} in EnsB . Finally, the full subcategorydetermined by the family {Zb} is equivalent with B, since i t corresponds under Tz to the full subcategory M(B). Th us, if A M B, thenthe ca tegory Ens A contains a generat ing set consis t ing of indecomposable projectives, such that the full subcategory determined by them isequivalent to the category B.Conversely, consider any full subcategory of Ens A de termined bya genera t ing set consis t ing of indeco mp osab le project ives. D en ote byB the opposi te of this subcategory, and by Zb, b G |B| , the members ofthe generat ing set . For the set {Zb} to be a generating set , the following property is necessary (and sufficient): for any object a G A, thereexists an epimorphism Zb > Aa for some b. For , le t4> be the familyof all A-maps

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    224 B . ELKIN S A N D J . A . ZILB ERL L

    I

    i i/I

    i s commutat ive; and the monoid B' ~ B amounts to the monoid eAe(for w hich is the ide ntity m orph ism). This is Prop osit ion 4 of [1]and e quiva lent to Th eorem 6.1 of [7 ] .This example raises the question: when is a category A Morita-equivalent to a monoid? The answer is again provided by Theorem3.9: when A contains an idempotent e such that for every object a G Athere exist morphisms ira an d jxa for which fxa7Ta = 10 and the d iagram

    Va Ma_

    is com m utativ e. T he mon oid in que stion the n consists of all morphisms of the form eae, and these are al l the monoids Mori ta-equivalent to A.As a second example, consider the two well-known parallel versionsof the definit ion of a graph (or better, of the category of graphs): by"arrow s and ob jects," or by "arrows only". Th ey are in essence asfollows. Le t A b e the category who se objects are the category 1and the category fy, and whose morphisms are al l functors betweenthem. Let B be the full subcategory of A generated by the object ^alone. Then an "arrows-and-objects" graph is a left A-set, while an"arrows-only" graph is a left B-set . The fact that the two notions areequivalent is simply the fact that A and B are Mori ta-equivalent .Notice that this is a case of a category which is not equivalent to amo noid bu t is M ori ta-equivalent to a monoid.

    BIBLIOGRAPHY1. B. Banaschew ski, Functors into categories of M-sets, Abh. Math. Sem. Univ.Ham burg 38 (1972), 49- 64.2. H. Bass,Algebraic K -theory, Benjamin, New York, 1968.3. M. Bunge, Relative functor categories and categories of algebras, J.Algebra 11 (1969), 64- 100 .

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    CATEGORIES O F ACTIONS A N D MORITA EQUIVALENCE 2 2 5

    4 . P . M . Co h n , Mo rita equivalences and duality, Q u e e n M a r y C o l l e g e M a t h e ma t ic s No te s , Lo n d o n , 1 9 6 8 .5. A. W. M. Dress , Notes on the theory of representations of finite groups,Pa r t i , B ie le f e ld , 1 9 7 1 .6 . P . J . Fre yd , Abelian categories, H a r p e r a n d Ro w, Ne w Yo rk , 1 9 6 4 .7. U. Knauer, Projectivity of acts and Mo rita equivalence of mono ids, Se migroup Fo r u m 3 ( 1 9 7 1 /7 2 ) , 3 5 9 - 3 7 0 .8 . B. M itche l l , Rings with several objects, Ad vanc es in M ath . 8 (1972) , 1 -161 .9 . B. Pare ig is , Catego ries and junctors, Ac ade mi c Press , Ne w York , 1970 .O H I O S T A T E U N I V E R S I T Y , C O L U M B U S , O H I O 43210

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