Morita Theory For Derived Categories: A Bicategorical Perspective

MORIT A THEOR Y F OR DERIVED CA TEGORIES: A BICA TEGORICAL PERSPECTIVE NILES JOHNSON Abstract. W e presen t a bicateg orical p ersp ective on deriv ed Morita t heory for rings, DG algeb ras, and spectra. This pers pective draws a co nnection b et w een Mori ta theory and the bicateg orical Y oneda Lemma, yiel di ng a conceptual uniﬁcation of Mori ta theory in derived and bicategorical con texts. This is m otiv ated b y study of Rick ard’s theorem for derived equiv alences of rings and of Mori ta theory for ring sp ectra, whic h we present i n Sections 2 and 4. Along the w a y , we gain an understanding of the barriers to Morita theory for DG algebras and give a conce ptual explanation f or the coun terexample of Dugger and Shipl ey . 1. Introduction A bicategoric a l per sp e ctive on Mor ita theo ry is r o oted in the obse r v ation that Morita theor y is a theo ry of bi mo dules, not simply left mo dules or right mo dules. T o give an incomplete survey , this per sp ective has yielded extensions to the theory of distributors ov er enric hed categor ies b y Fisher- Palmquist and Palmquist [FPP 75], to s ubfactor theory by M ¨ uger [M ¨ ug03], to bialg e broids by Szlach´ a nyi [Szl04], a nd to von Neuma nn a lg ebras by B rouw e r [Bro03]. A largely disjoint b o dy of work has studied Mor ita theory in derived c ontexts. This beg an with the work o f Ric k ard, studying derived categ ories of rings in [Ric89] and [Ric9 1]. Rick ard’s results were re-treated by Sch wede in [Sc h04] following w ork of Keller in [Kel9 4]; cf. [KZ9 8, Ch. 8] for a very readable ov er view. Dugger, Sc hw ede and Shipley give par tial extensio ns to ring sp ectra and diﬀerential g raded algebras in [SS03], [DS07], and rela ted w ork. Derived Mor ita theory for diﬀerential graded categories has b een s tudied by Keller in [Kel94] and T o¨ en in [T o¨ e07]. The dev elopment of derived Morita theory has r equired mo r e delicac y than its bicategoric al counterpart, with the counterexample of [DS07] b eing a barrier to ex p ected gener alizations of Rick a rd’s theorem. The work of Dugger , Shipley , and T o¨ en is motiv ated in par t by this unsa tisfactory situa tio n. W e present a co nceptual uniﬁcation of bicatego rical Morita theor y with Mo rita theory for derived categorie s by developing Morita theory in triangulated bicategor ies. In Section 3 , we in tr o duce bica t- egorica l language for those to whom it is unfamiliar, and in Subsection 3.6 we describ e what is meant by a tria ngulated bicategor y . F rom this v a ntage, we a re able to g ive a conceptual expla nation of de- rived Mor ita theory for w hich the r esults (and coun terexample) of [DS07] and [T o¨ e07] b ecome v erifying examples. This is a chiev ed in three stages. First, in Section 5 w e remind the r eader of the bicategorical Y oneda Lemma (5.3) and explain that what is often called Morita theory is a corollary ( 5.4) of this Y oneda Lemma. W e encoura ge the intuition that bicatego rical Morita theor y is as ele men tary as the bicategor ical Y oneda Lemma; th is gives one pos sible reason tha t bicategorical persp ectives have yielded such an abundance of genera lizations for classica l Mor ita theor y . Second, w e mo dify a standard o bs erv ation from the context of enriched categor y theory to explain that, in bicateg ories with left and rig h t int ernal homs, Mor ita theory must necessarily fo cus on equiv alences which are enriched o ver the internal homs (5.5). This g ives a r eason for the re sults a nd exa mples mentioned a bove. Finally , in Section 6 w e apply our understanding o f the Y oneda Lemma. Our per sp ective allows us to re- frame the issue o f derived Morita theory and sheds some new light on the subtleties there. In Subsection 6.2 w e discuss the re lationship of Morita theory to ambien t enrichmen ts . In classica l Mo rita theory the ambien t ab elian enrichmen t is automatically preserved (Remark 2.3), but this is not the case in all other contexts. This provides, for example, a reaso n why the development o f Morita theor y has met unexpected barriers in the DG case. W e fo r eshadow the bica tegorica l p ers pec tive by o utlining a pr o of of Rick ar d’s theo rem in Section 2 . After establishing bicatego r ical terminology in Section 3, Section 4 g ives the details of this pr o of. In Subsection 4.1 we g ive a genera lization o f this theorem to r ing s pec tra. The las t t wo sections c over some basic mo del-theore tic results for our bicategory of DG-alge bras and their bimo dules; they ar e veriﬁcations that exp ected results from the theory of mono idal stable model categories genera lize to our co n text in straightforward wa ys. Section 7 g ives a bica tegorical development of the standard mo del structure for DG alg ebras, and Section 8 describ es the formal structure a rising from c hange of base a lgebra. 1 2. Outline In this section, we demonstrate our p ersp ective by giving Rick ar d’s theo r em for der ived Morita theory of rings , together with an outline of its proo f. F or r eference, we give a statemen t of the classical Morita theo rem together with its proo f, and we follow these with some remark s ab out derived Morita theory for DGAs. The counterexample of [DS07 ] is g iven as Example 2.8, and shows that Rick ar d’s theorem do es not g eneralize to DGAs a s stated. W e g ive some hints a bo ut wher e this break down o ccur s , to b e describ ed more fully after we hav e developed the appr opriate bicateg orical p ersp ective. Note. In the following, w e implicitly understand “mo dule” to mean “rig ht -mo dule”, unless it is otherwise qualiﬁed. The most frequent instance of this opp osite q ua liﬁcation will b e that the endo morphism r ing of a right-module a cts on its left , and vice-versa. Theorem 2.1 (Ric k a rd) . L et k b e a c ommu tative ring, and let R and S b e k -algebr as. The derive d c ate gories D k ( R ) and D k ( S ) ar e e quivalent as triangulate d c ate gories if and only if ther e is an obje ct T of D k ( S ) with the fol lowing thr e e pr op erties. ( i ) T is (quasi-isomorp hic to) a b ounde d c omplex of ﬁnitely-gener ate d pr oje ct ive S -mo dules. ( ii ) T gener ates the triangulate d c ate gory D k ( S ) . ( iii ) T he gr ade d endomorphism algebr a D k ( S )[ T , T ] ∗ is c onc entr ate d in de gr e e zer o and isomorphic to R as a k - algebr a. Theorem 2.2 (Mor ita) . L et R and S b e rings. The c ate gories M od R and M od S ar e e quivalent if and only if ther e is an obje ct P of M od S with the fol lowing thr e e pr op erties. ( i ) P is a ﬁnitely-gener ate d pr oje ctive S -mo dule. ( ii ) P gener ates the ab elian c ate gory M od S . ( iii ) T he endomorphism ring H om S ( P, P ) is isomorphic t o R . Pr o of. F or the class ic al theorem of Morita, we make use of the bicateg ory M of rings and their bimo dules. The dual basis lemma gives that condition ( i ) is e q uiv alent to the canonical co ev a luation map ν : P ⊗ S Hom S ( P, S ) → Ho m S ( P, P ) being an iso mo rphism. With co nditio n ( iii ), this can b e phra sed in the bica teg orical co nt ext b y saying that ( P, Hom S ( P, S )) form a dual pair over S and R , which means that the functors − ⊗ R P and − ⊗ S Hom( P, S ) a re an adjoin t pair . The generating condition, ( ii ), is equiv a le n t to the canonica l ev aluation map ε : Ho m S ( P, S ) ⊗ R P → S b eing an iso morphism, and hence this dual pair is an inv ertible pair, g iving an adjoint eq uiv alence of categor ies. The conv er se is also easy to see class ically , since an eq uiv alence of ca tegories F : M od R → M od S induces an iso morphism on the morphisms b etw een mo dules, and therefore a ring isomorphism R ∼ = Hom R ( R, R ) F − → Hom S ( F R, F R ) . Moreov er, the other tw o pr o pe rties a r e enjoy ed by R and preser ved by equiv alences , so tak ing P = F R gives the conv e r se.  R emark 2.3 . F or our future discussion, it is worth noting that this argument takes adv antage of the elementary fact that the a belia n g roup s tructure on Hom R ( R, R ) is necessa rily pr eserved by F . I n fact any left adjoint functor b etw een ab e lian categor ie s is automa tically enr iched ov er ab elian groups . This is neither exp ected nor true of mor e gener al enrichmen ts. This p oint of view on the c la ssical theo rem is re a dily g eneralized to the pro of of Rick ard’s the- orem. In order to clarify the pro of, we separate Ric k ard’s theore m into a w ell-known lemma and tw o prop ositions. Lemma 2.4 . L et E b e a DG k -algebr a whose homo lo gy is c onc en tr ate d in de gr e e zer o. Th en E is quasi-isomorphi c to its homolo gy, and henc e t her e is a t riangulate d e quivalenc e D k ( E ) ≃ D k ( H ∗ E ) . Pr o of. Let ( E + ) n =      E n , n > 0 Z 0 ( E ) = k er ( d 0 ) , n = 0 0 , n < 0 Then the pro jection a nd inclus io n deﬁne a zig- zag of quasi-isomorphis ms H 0 ( E ) ≃ ← − E + ≃ − → E , and base change along these maps g ives equiv a lences of derived categor ies.  Deﬁnition 2.5 (formality) . DG algebr a s which are quasi-isomo rphic to their homology are called for mal. 2 Prop ositio n 2.6. L et R and S b e k - algebr as. If F : D k ( R ) ≃ D k ( S ) is an e qu ivalenc e of t riangulate d c ate gories, t hen ther e is an obje ct T ∈ D k ( S ) with the fol lowing two pr op erties: ( i ) T is (quasi-isomorp hic to) a b ounde d c omplex of ﬁnitely-gener ate d pr oje ct ive S -mo dules. ( ii ) T gener ates the triangulate d c ate gory D k ( S ) . Mor e over, the DG endomorphi sm algebr a E nd S ( T ) is quasi-isomorphi c to R as a DG k -algebr a. Pr o of. A co mmon pro of of this pro po sition (see [Sch04], for ex a mple) is to remar k that the tw o co nditions are pr eserved b y exact equiv alences a nd are enjoy ed by R regarded a s a module ov er itse lf, hence also T = F R has these prop erties. Equiv alences induce isomor phis ms o n homolo gy of endomor phis m DG k -algebr as, and so E nd S ( T ) has ho mology which is c oncentrated in degree 0 a nd isomo rphic to the homolo gy of R (that is, R itself ). Lemma 2.4 shows that E nd S ( T ) is therefore for mal, a nd hence E nd S ( T ) and R are quasi-is omorphic DG k - algebra s .  Since H ∗ E nd S ( T ) = D k ( S )[ T , T ] ∗ , this pr ov es one implication in Rick ar d’s theorem. The other implication is proved b y aga in applying Lemma 2.4 in the case E = E nd s ( T ). If R is iso morphic to H ∗ E , then formality ensures that R a nd E are quasi-isomor phic and hence D k ( E ) ≃ D k ( R ). The follo wing prop osition then pr ov es this dir e ction of Rick a r d’s theorem, by spe c ializing to the cas e that S is a DG k -algebr a co ncentrated in deg ree 0. Note. Dualizable mo dules ov er a DG k -a lgebra are deﬁned in Subs e c tion 3 .5, but for the cur rent ar- gument it is enoug h to o bserve that when S is a DG k - algebra concentrated in deg r ee 0, then a rig ht- dualizable S -module is simply a bounded complex o f ﬁnitely generated a nd pro jective S -mo dules. Prop ositio n 2. 7. L et S b e a DG k -algebr a, and let T b e a DG S -mo dule. If T has the fol lowing two pr op erties, t hen D k ( S ) and D k ( End S ( T )) ar e e quivalent as triangulate d c ate gories. ( i ) T is a right-dualizab le S -mo dule. ( ii ) T gener ates the triangulate d c ate gory D k ( S ) . The pro of is g iven in Section 4 be low. It shows that T has a dual and the des ired equiv alence is given b y the der ived tensor pro duct with T ; its inv er se is derived tensor with the dual of T . Such equiv alences are ca lled standar d derive d e quivalenc es . Since DG k -algebra s a r e, in gener al, not for mal, we do not exp ect Ric k ard’s theor em to generalize to DG k -algebra s as stated. How ever, if the third condition for T is strengthened to a requirement that E nd S ( T ) b e qua s i-isomorphic to R , then Pro po sition 2.7 is a pro of for one direction. Prop osition 2 .6 is not ge ne r ally true when R and S are ta ken to b e DG k -algebra s, a nd this is the ma in ba rrier to generalizing Ric k ard’s theorem. The diﬃculty is that one do es not hav e formality for DG k -algebras in general. Mo re precisely , forma lit y is use d in the pro of of P rop osition 2 .6 to show that an equiv alenc e of derived catego ries (o f rings) is suﬃcien t to guarantee a quasi- isomorphism of DG k -algebra s b etw een a ring and the endomorphism DG k - algebra of its image under the equiv a lence. Such a q ua si-isomor phism is ne ither ex pec ted nor present in greater g enerality . W e investigate this in Section 5, but for now w e give an exa mple to illus trate how Pr o p o sition 2.6 ca n fail in the DG s itua tion. Example 2.8. In [DS07 ], an exa mple of t wo DG r ings is given: C = Z [ e ] / ( e 4 ) with | e | = 1 and d ( e ) = 2, and A = H ∗ C . The mo del ca tegories of C -modules a nd A -mo dules are Quillen equiv alent, but there is no po ssible bimo dule with the prop erties listed in Rick a r d’s theor em. That there can be no such bimo dule is prov e n b y noting that A is a DGA ov er Z / 2, but C is not quasi-isomor phic to any DGA ov er Z / 2. Since the endomorphism alg ebra of any A -mo dule would also b e a DGA ov er Z / 2 , there cannot be a C - A -bimo dule with endomo rphism DGA quasi-is o morphic to C . The argument tha t these DGAs do have Quillen equiv a lent categ ories of mo dules in volv es a THH (top ologica l Hochschild homolog y) calculation which pr o duces an equiv alence of S -a lgebras betw een their Eilenberg - Mac Lane spectra . More details can b e found in [Shi06]. T o understand the fo r ce of this example b etter, we note that the equiv alence s arising in Lemma 2.4 and Prop ositio n 2.7 are standar d derive d e quivalenc es ; they ar e given by derived tensor with a DG- bimo dule. These a r e manifestly induced by Quillen equiv alences of mo del ca teg ories, na mely the un- derived tensor on the categ ories of DG-mo dules. The ex a mple ab ov e s hows, how ever, that the prope rty of being induced b y a Quillen equiv alence is not suﬃcien t to c haracter ize the standard derived e q uiv- alences. T o reiterate, the DGAs in the example do ha ve Q uille n equiv alent module categories, but the induced e q uiv alence o f der ived ca tegories c annot be a standard derived equiv ale nce. A k ey to fully characterizing standard derived equiv alences is an o bserv ation ab o ut the organized wa y in which standard derived equiv alences preserve bi module structures. If M is an R - S DG-bimodule, then − ⊗ R M pr eserves left -mo dule structure for all right R -mo dules. By neg lect, this can be regar ded 3 as a functor from right DG R -mo dules to right DG S -mo dules , but to do so forgets to o muc h. In the example a b ove, the Quillen equiv alence of right-module categ ories do es not preserve left -mo dule structure, s o it cannot induce a s tandard de r ived eq uiv alence. An alter na tive p ersp ective might point out that the sta nda rd derived equiv alences a ls o preserve categoric al enric hment. That is, with M as ab ov e, − ⊗ R M induces morphisms o f hom obje cts , and is compatible with the enriched compositio n in the e xpe c ted wa y . The Quillen pair of functors pro duced in the example of [DS07] is not a pair of DG-enric hed functors. The p oint of Prop osition 5.5 is that these tw o p er s pec tiv es are in fact eq uiv alent. Moreov er, Corollar y 5 .4 in ter prets the Y o neda Lemma (5.3) as a statement that these (equiv alent) prop er ties do characterize the standard deriv ed e q uiv alences. Thes e obse r v ations are unlikely to b e surprising to an enriched category theorist, as they a r e the a pparent g e neralizations (or sp ecializ ations) of standa rd results to o ur context, but they have b een included for the algebr aist or top olo gist who may be unfamilia r with this p ersp ective. In [KZ98, Ch. 8], Keller remarks tha t there a re no known examples of non- standar d der ived equiv alences for rings. Our characterization of Morita theo ry via the Y oneda Lemma yields the following prop osition. The notation D k ( X, Y ) deno tes derived catego ries of bimodules, descr ibed in more detail below. Also no te that End k ( A ) is taken to mean the deriv ed endomorphism ring. Prop ositio n (See 6.7) . L et k b e a c ommutative ring, let A b e a D G k -algebr a and let f : D k ( A ) → D k (End k ( A )) b e an e quivalenc e of triangulate d c ate gories. Then f is a standar d derive d e quivalenc e if and only if the following c onditions hol d. ( i ) The e qu ivalenc e given by f is an enriche d e quivalenc e. ( ii ) Ther e is an enriche d e quivalenc e f ′ : D k ( A, A ) → D k (End k ( A ) , A ) . ( iii ) The t wo e quivalenc es, f and f ′ ar e c omp atible in the fol lowing sense: If T ′ , U ′ ∈ D k ( A, A ) and T , U ∈ D k ( A ) = D k ( A, k ) , t hen ther e ar e natur al maps Ext A ( T , U ′ ) → Ext End k ( A ) ( f T , f ′ U ′ ) in D k ( k , A ) Ext A ( T ′ , U ) → Ext End k ( A ) ( f ′ T ′ , f U ) in D k ( A, k ) which c ommute with the p airing induc e d by c omp osition. (That is, the squar es in Remark 6.3 c ommute.) Spec ia lizing to the cas e that A and Ex t k ( A, A ) a re concentrated in deg r ee 0, this pro po sition implies that a derived equiv a lence of ring s is standard if and only if it pr e s erves bimo dule structur e as descr ibe d ab ov e; s ee Prop osition 5.5. T o make the c haracteriza tion of standard derived eq uiv alences clear, we cannot avoid in tro ducing bicategoric a l language. In par ticular, we mu st describe the notio n of pseudof unctor , esp ecially r ep- r esente d pseudofunctor , a nd st r ong tr ansformation of (represented) pse udofunctor. This lang uage is relev ant b ecause a c omp onent o f a transfo rmation b etw een pseudofunctors is a functor b e tw een certain categorie s, and the que s tion of whether a given derived equiv a lence is a standard derived equiv alence is pr ecisely the same a s whe ther the given functor is a comp onent of a stro ng transfo rmation b etw een t wo sp eciﬁc pseudofunctors. W e addres s this fully in Section 5, but we b egin in Section 3 by intro- ducing our bicategorical cont ext. In Section 4 w e illustrate the bica tegorical language with a pro of o f Prop ositio n 2.7, and in Section 7 and Sectio n 8 w e g ive a further development of the structure present in our bica tegorical fra mework. T his is the foundation for our applica tions of the Y oneda Lemma in Section 5. 3. Bica tegorical Context W e make us e of a bica tegorical context to o rganize and clar ify our understanding of Mor ita theory . In this section, we introduce this or g anizational to ol for tho se to whom it is unfamiliar. F or the classical Morita theor e m, we consider M , the bicategor y o f r ings, bimo dules, a nd bimo dule maps . F or Rick ard’s theorem, we c onsider D G k , the bicategor y o f DG k -algebr as, D G -bimo dules and their maps. Ass o ciated to this bic a tegory , we have a derived bicatego ry , D k . W e deﬁne these bicatego ries b elow, and in the remainder of this sec tion we discuss bicateg ories with a tria ngulated str ucture, ta king D k as a motiv ating example. Pr e cise and concise deﬁnitions ca n b e found in [Lei98], while [La c07] provides a mor e ex panded guide. 3.1. Rin g s and Mo dules. The 0 -cells of M are rings, and for any rings A and B , M ( A, B ) is the category of ( B , A )-bimo dules . So a ( B , A )-bimo dule B M A is a 1-cell M : A → B . The 2-cells b etw een t wo 1-cells M : A → B and N : A → B a re the bimo dule maps f : B M A → B N A . Given three 0-cells, A , B , and C , and tw o 1-cells, M : A → B and L : B → C , the ho rizontal comp osite of L with M is 4 written L ⊙ M : A → C . Since M is a ( B , A )-bimo dule, and L is a ( C , B )-bimo dule, L ⊙ M is deﬁned using the tensor pro duct ov er B ; this pr o duces a ( C, A )-bimo dule, as desired: L ⊙ M = L ⊗ B M . A bicatego ry has, for ea ch 0-cell, A , a unit 1 -cell A → A satisfying usua l unit conditions. W e denote this 1-cell also by A ; in the case of rings, this is A regar ded as a n ( A, A )-bimo dule. 3.2. Clos ed structure for M . A close d structu r e for a bica tegory deﬁnes right adjoints for ⊙ . F or the bicategory M the right adjoints for − ⊙ M and M ⊙ − are well known. The r ight a djoint to − ⊗ B M is Hom A ( M A , − ), homomor phisms of right A -modules, while the r ight adjo int to M ⊗ A − is Hom B ( B M , − ), homomorphisms of left B -mo dules. T o extend these notions to more general bic a tegories, the adjoint to − ⊗ B M is called “ target-ho m” , or “ right-hom”, and denoted M ⊲ − . The adjoint to M ⊗ A − is ca lled “source- hom”, or “ left-hom”, and denoted − ⊳ M . The adjunctions a r e written as M ( V ⊙ M , W ) ∼ = M ( V , M ⊲ W ) M ( M ⊙ T , U ) ∼ = M ( T , U ⊳ M ) The existence of these a djo ints is a closed structure for a g eneral bicateg ory , and we will us e ⊲ and ⊳ to denote the r ight-hom and left-hom functors in genera l. The or ientation of the triang les is intended to help the rea der remember the source and target of the 1-cells M ⊲ W and U ⊳ M . Here , W and M hav e common sourc e, A , and if C denotes the ta r get of W , then M ⊲ W is a 1-cell B → C . Likewise, M and U have common target, B , and if D denotes the so urce o f U , then U ⊳ M is a 1-cell D → A . A techn ically complete desc r iption of closed structur es can be found in [MS06]. 3.3. Diﬀe rential Graded k -alg ebras. The bicategor y D G k is similar to M , but here the 0-cells are diﬀerential gra ded k -algebr as, the 1 -cells ar e DG bimodules, and the 2-cells are maps of DG bimodules. Like M , D G k also has a clos e d str ucture. F or tw o 1-cells with co mmon s ource, P and Q , the tar get- hom P ⊲ Q denotes the diﬀerential g raded hom o ver their common source, and likewise ⊳ denotes the diﬀerential g raded ho m ov er common targets. 3.4. The derived bicategory , D k . F or ea ch pair o f DG k -algebras , A and B , ther e is a mo de l structure for D G k ( A, B ) which is a direct generaliz ation of the standa rd mo del structur e for chain co mplexes ov er a ring, a nd which Section 7 desc rib es in mor e deta il. W e use this mo del structure to under stand and work with the derived categor y o f ( B , A )-bimo dules, which we deno te by D k ( A, B ). There is a canonical functor from D G k ( A, B ) to D k ( A, B ) and where it a dds clarity to o ur expo sition we let γ : D G k ( A, B ) → D k ( A, B ) denote this functor. Note that, for a k -alg ebra S , D k ( S, k ) = D k ( S ) is the usual derived categ ory of (right) S -mo dules. The mo del str ucture on each 1-cell categor y satisﬁes the pushout pro duct conditio n for ⊙ -comp osition (Prop ositio n 8.2), so D G k is a mo del bicatego ry . The derived tensor and hom g ive a closed bicategory structure for the categories D k ( A, B ), so we rega rd D k as the derived bicategor y of D G k . 3.5. Duality in bicategories. Thr oughout this subsection we cons ider ﬁxed 1-cells X : B → A a nd Y : A → B in a closed bicategory B . Deﬁnition 3 . 1 (Dual pair ) . W e say ( X, Y ) is a dual pair, or ‘ X is left-dual to Y ’ (‘ Y is right-dual to X ’), or ‘ X is right-dualizable’ (‘ Y is left-dualizable’) to mean that we hav e 2-cells η : A → X ⊙ Y and ε : Y ⊙ X → B such that the following comp osites a re the resp ective identit y 2-cells. X ∼ = A ⊙ X η ⊙ i d − − − → X ⊙ Y ⊙ X id ⊙ ε − − − → X ⊙ B ∼ = X Y ∼ = Y ⊙ A id ⊙ η − − − → Y ⊙ X ⊙ Y ε ⊙ id − − − → B ⊙ Y ∼ = Y Deﬁnition 3.2 (Ba se and cobase for a dua l pair) . When ( X, Y ) is a dua l pair in a bicategory B , we term the sour c e of X (the target of Y ) the b ase o f the dual pair, and we term the so ur ce of Y (the targ et of X ) the c ob ase of the dual pair . Thus, the ev aluation map of the dual pair is a tw o- cell from Y ⊙ X to the base 1 -cell, and the coev aluation (unit) is a tw o- cell from the coba se 1-cell to X ⊙ Y . Deﬁnition 3.3 (Inv e r tible pair ) . A dual pair ( X , Y ) is ca lled inv ertible if the maps η and ε are isomor - phisms. Equiv alently , the adjoint pair s descr ib ed above are adjoint equiv alences. Dualit y for monoidal categ ories has b een studied at length, and dualit y in a bicategor ical context has b een intro duced in [MS06, § 1 6.4]. The deﬁnition of duality do es not r equire B to b e closed, but we will make us e of the following ba sic facts a b o ut duality , some of which do require a clo sed structure on B . 5 Prop ositio n 3.4. A 1-c el l X ∈ B ( A, B ) is right-dualizabl e if and only if the c o evaluation ν : X ⊙ ( X ⊲ A ) → X ⊲ X is an isomorphism. Mor e over, this is the c ase if and only if the map ν Z : X ⊙ ( X ⊲ Z ) → X ⊲ ( X ⊙ Z ) is an isomorphism for al l 1-c el ls Z with tar get A . Prop ositio n 3.5. L et ( X , Y ) b e a dual p air in B , with X : B → A and Y : A → B . (1) F or any 0-c el l C , we have two adjoint p airs of functors, with left adj oints written on top: B ( A, C ) −⊙ X / / B ( B , C ) −⊙ Y o o B ( C, A ) Y ⊙− / / B ( C, B ) X ⊙− o o The st ructur e maps for the dual p air give the triangle identities ne c essary to show t hat the dis- playe d functors ar e adjo int p airs. (2) If B is close d, t hen Y is c anonic al ly isomorphic t o X ⊲ B , and for any 1-c el l Z : B → D , the natur al map Z ⊙ ( X ⊲ B ) → X ⊲ Z is an isomorph ism. The right-dualizable 1- cells in the bicatego ry M a re the ﬁnitely-g enerated pro jective bimo dules. More precisely , they are ﬁnitely-generated pro jective as righ t-mo dules ov er their so urce (the base of the duality). Lemma 8.9 shows that the retracts of ﬁn ite c el l bimo dules (Deﬁnition 7.1) are r ig ht -dualizable in D k , a nd Lemma 8.10 shows that the converse is also tr ue . 3.6. T riangulated bicategories. W e reca ll ﬁrst the deﬁnitions of lo ca lizing sub catego ry and ge nerator for a triangula ted category , and then give a deﬁnition (3.9) o f tria ng ulated bicateg o ry suitable for our purp oses. In pa rticular, under this deﬁnition D k is a triangula ted bicatego r y . Deﬁnition 3.6 (Loca lizing sub categor y) . If T is a tria ngulated ca tegory with inﬁnite co pr o ducts, a lo c alizing subc ategory , S , is a full triangulated sub c ategory o f T which is clo sed under copro ducts fr om T . R emark 3 .7 . This is equiv alent to the deﬁnition for arbitrar y tria ng ulated ca tegories of [Hov99], (which requires that a lo calizing sub catego ry be thick) because a tr iangulated sub c ategory automatically satisﬁes the 2-out- o f-3 proper t y and b ecause in a ny tr iangulated catego ry with count able copro ducts , idemp otents hav e s plittings . See [Nee0 1, 1.5 .2, 1.6.8, and 3.2.7 ] for details. Deﬁnition 3.8 (T riangulated gener ator) . A set, P , of ob jects in T (tria ngulated category with inﬁnite copro ducts, as ab ov e) is a s et of triangulate d gener ators (or simply gener ators ) if the only loca lizing sub ca teg ory containing P is T itself. Deﬁnition 3.9 (T riangulated bicateg ory [MS06 , § 16.7]) . A closed bicategor y B will be called a triangulate d bic ate gory if for each pair of 0-cells, A and B , B ( A, B ) is a triangulated catego ry with inﬁnite copro ducts, a nd if the susp ension, Σ, is a pseudofunctor (Subsection 5.1) on B , and further more the lo cal tr iangulations on B are compatible as describ ed in the following tw o axio ms . (TC1) F o r a 1 -cell X : A → B , there is a natural isomor phism α : X ⊙ Σ A → Σ X such that the compos ite b elow is m ultiplica tion by − 1 . Σ 2 A = Σ(Σ A ) α − 1 − − → Σ A ⊙ Σ A γ − → Σ A ⊙ Σ A α − → Σ(Σ A ) = Σ 2 A (TC2) F o r any 1-cell, W , the functors W ⊙ − , − ⊙ W , W ⊲ − , and − ⊲ W are exact. If B is a triangulated bicategor y and P , Q are 1-cells in B ( A, B ), we emphasize tha t B is tr ia n- gulated by writing the ab elian g r oup of 2-cells P → Q a s B [ P , Q ] a nd by writing the gra de d ab elian group obtained b y taking shifts of Q as B [ P , Q ] ∗ . T o emphasize the source and target of P and Q , w e may a lso write B ( A, B )[ P, Q ] ∗ , a s in Theorem 2.1 (where we write D k ( S ) instea d of D k ( S, k )). 6 4. Proof of Proposition 2.7 In this section we prove Prop osition 2 .7, which genera lizes one direction of Rick ard’s theorem to the case of DG k -alg ebras. W e work in the closed triang ulated bica tegory D k , beg inning with a few general sta temen ts. Deﬁnition 4.1 ( ⊙ -detecting 1- cells) . In any lo cally additive bicategory , B , a 1-c e ll W : A → B is called ⊙ -dete cting if triviality for any 1-cell Z : C → A is detec ted b y triviality o f the comp osite W ⊙ Z . Tha t is, Z : C → A is zero if and o nly if W ⊙ Z = 0 . A colle c tio n of 1 - cells, E , in B ( A, B ) is called jointly ⊙ -dete cting if the ob jects ha ve this prop erty jointly; that is, Z = 0 if a nd only if W ⊙ Z = 0 for all W ∈ E . R emark 4.2 . If B is a monoidal additive ca tegory with mo noidal pro duct ⊙ , the unit ob ject is ⊙ - detecting. In arbitrar y lo cally additive bicateg ories, if A 6 = B then B ( A, B ) may not hav e a single ob ject with this prop erty , but in relev ant examples the collection of all 1-cells, o b B ( A, B ), do e s have this pr op erty jointly . As a coun ter-p oint to this remark, we hav e the following lemma. Lemma 4.3 . L et B b e a triangulate d bic ate gory, and let P : A → B b e a gener ator for B ( A, B ) . If the c ol le ction of al l 1-c el ls, B ( A, B ) , is jointly ⊙ -dete cting, then P is ⊙ -dete cting. Pr o of. Given a n y 1-c e ll Z : C → A with P ⊙ Z = 0 , let S b e the full sub catego ry of 1-cells, W : A → B for which W ⊙ Z = 0 . This is a lo calizing sub categor y o f B ( A, B ), and by ass umption P ∈ S , so S = B ( A, B ), and hence Z = 0.  R emark 4.4 . Since the functors P ⊙ − are exact, the pro per t y o f P ⊙ − detecting trivial ob jects is equiv alent to P ⊙ − detecting isomorphisms (meaning that a 2-cell f is an is omorphism if and only if P ⊙ f is so). Now we turn to the pr o of. Supp ose T is a chain complex of (righ t) S -mo dules sa tisfying the dual- izability and genera tor conditions of Prop osition 2 .7, and let E deno te the DG k - algebra Hom S ( T , T ). One mig h t ca ll our ﬁrs t step ‘cobase e x tension’, a s we describ e how to extend the dualizable o b ject T to a dual pair with base S and cobase E . The c hain complex T is a r ight DG-mo dule ov er S , a nd ca n be considered as a left mo dule over the DG k -algebr a E . W e let e T denote T regar ded a s a 1- cell S → E , and let T denote the 1-cell S → k . These 1-cells a re r elated by base c hange a long the unit map (of DG k -algebr as) k → E . Restr icting scalars on either the left or right of the DG k -a lgebra E gives r is e to a dual pair ( k E , E k ), and the 1-ce ll T : S → k is re cov er ed from e T : S → E as the 1 -cell comp osite k E ⊙ e T . Moreov er, T ⊲ S is rec overed as the comp os ite ( e T ⊲ S ) ⊙ E k , and the co ev a luation map of ( T , T ⊲ S ) is r ecov e r ed from that of ( e T , e T ⊲ S ). In mo r e common language, one might say that T ⊲ S is a right E -mo dule, and the coev a luation map of ( T , T ⊲ S ) is a map of E - E bimo dules. Because T is (right-)dualizable in D G k ( S, k ), it follows that e T is (right-)dualizable in DG k ( S, E ). Lemma 4.5 b elow shows that, therefore, e T is dualizable in D k ( S, E ). T o ﬁnish the pro of, we us e this fac t to s how that, b eca use T is a genera tor, e T is in vertible. Then the in vertible pair ( e T , e T ⊲ S ) establishes an equiv a lence of categor ies, as describe d in Deﬁnition 3.3. Lemma 4.5. If X is dualizable in D G k ( A, B ) , then γ ( X ) is dualizable in D k ( A, B ) . Pr o of. The dualizable o b jects in D G k ( A, B ) are retracts of ﬁnitely-g enerated pro jective (rig ht-)mo dules ov er A ⊗ k B op , and hence coﬁbra nt. This is shown b y a sta nda rd dual-bas is-type argument, and the int erested reader will ﬁnd the deta ils in Le mma 8.8. Beca use X is co ﬁbrant, the functor X ⊙ − preserves weak equiv alences. This a ls o is a s tandard result, and a pro of ca n b e found in [KM95, II I.4.1]. Recall that γ denotes the canonica l functor D G k ( A, B ) → D k ( A, B ). Let Q ( X ⊲ A ) b e a coﬁbrant replacement for X ⊲ A . Then γ ( X ) ⊙ γ ( X ⊲ A ) = X ⊙ Q ( X ⊲ A ), and γ ( X ⊙ ( X ⊲ A )) = X ⊙ ( X ⊲ A ), and even though γ is not a stro ng mono idal functor, we nevertheless hav e an isomo rphism in D k ( B , B ): γ ( X ) ⊙ γ ( X ⊲ A ) ≃ − → γ ( X ⊙ ( X ⊲ A )). It is a formality now to chec k that the duality r elations hold for γ ( X ) a nd γ ( X ⊲ A ), and therefore γ ( X ) is dua lizable in D k ( A, B ). F or those who wish to see it, this formal a rgument is given explicitly for monoidal categories in [LMS86, I I I.1.9 ].  Applying the lemma to e T w e have, for any C , the a djoint pair of functors induced by a dual pair (Prop osition 3 .5) shown b elow. Because E = e T ⊲ e T , the unit of this adjunction is a n iso morphism–the inv ers e to the co ev aluation map. D k ( E , C ) −⊙ e T / / D k ( S, C ) . −⊙ ( e T ⊲S ) o o 7 W e ﬁnish the pro of of Pr op osition 2.7 by showing that the c ounit ev al : ( e T ⊲ S ) ⊙ e T → S is an isomorphism in D k ( S, S ). Since k is the ground r ing for our bicategory D k , the 1-c e lls of D k ( S, k ) are joint ly ⊙ -detecting (Deﬁnition 4.1), a nd s o the g enerator co ndition o f P rop osition 2.7 means that T itself is ⊙ -detecting (Lemma 4.3). Thus, ev aluation ( e T ⊲ S ) ⊙ e T → S is an isomor phism in D k ( S, S ) if and only if the map ( k E ⊙ e T ) ⊙ ( e T ⊲ S ) ⊙ e T 1 ⊙ ev al − − − − → k E ⊙ e T is so (Remark 4.4). The duality o f e T a nd e T ⊲ S implies that the comp osite below is the identit y and the ﬁrst map, induced b y the unit of the adjunction, is a n isomorphism so the s econd must b e also. k E ⊙ e T ∼ = − → k E ⊙ e T ⊙ ( e T ⊲ S ) ⊙ e T 1 ⊙ ev al − − − − → k E ⊙ e T Hence the second map in this comp osite is an isomor phism, and so the counit fo r the dual pa ir ( e T , e T ⊲ S ) is an isomor phis m in D k ( S, S ), giving a n equiv alence of triang ulated categ ories, as we wished to show. D k ( E , C ) −⊙ e T / / D k ( S, C ) −⊙ ( e T ⊲S ) ≃ o o This equiv a lence is suitably na tural in C , making it a stro ng transforma tion of the represented pseudo- functors D k ( E , − ) and D k ( S, − ). A more complete picture of str ong transformations and their co nnection to the Y oneda Lemma for bicategories is describ ed in Section 5. 4.1. Rick ard’s theorem for sp ectra. In this subsection we pro ve a result ana lo gous to Prop osition 2 .7, but working instead with a commutativ e S -a lgebra, k , a nd the bica tegory S k of k -alg ebras and their bimo dules. W e extend o ur previous notation to let D k denote the bicateg o ry o f derived ca tegories for sp ectra. One ma jor diﬀerence is that ins tead of working with dualiza bility on the le vel o f mo del categorie s, as w e ha ve for algebraic derived Mor ita theory , w e shift to the notion of dua lizability in the bicategory of derived ca tegories. The principles and gene r al appr oach a re the same, but the details mu st be mo diﬁed slightly . F or exa mple, the ‘cobase extensio n’ step in the algebraic case is near ly transparent, but req uires a lemma in the context of sp ectra. One can pr ov e r e sults ab out sp ectr a which a re dualiza ble on the mo del-catego rical level but, unlike the DG cas e , it is diﬃcult to ﬁnd examples of such spec tra. With a r elative a bundance of sp ectra which ar e dualizable in the derived bicategory , we shift our focus in that direction. F or the remainder of this section, w e use the term ‘dualizable’ to mean dualizable in the bica tegory D k . Prop ositio n 4.6 . Le t A b e a k -algebr a, and let T b e a ﬁbr ant and c oﬁbr ant A -mo dule, with endo- morphism k - algebr a E = F A ( T , T ) . If T has the fol lowing two pr op erties, then D k ( A ) and D k ( E ) ar e e quivalent c ate gories. ( i ) T is (right-)dualizab le as an A -mo dule. ( ii ) T gener ates the triangulate d c ate gory D k ( A ) . As with the algebra ic version, our pro of pro ceeds in tw o pa rts. Fir st (‘cobas e extension’) we show that a dual pair in D k betw een tw o k -algebras ca n b e extended to the endo mo rphism a lgebra of the left dual, and that in so doing w e pro duce a new dual pair who se unit is an isomorphism. This ro ugh description is made precise in the sta temen t of Lemma 4.8, after introducing notation for the restriction of scalars functors. In the seco nd part of our pro of w e use the unit isomorphism of this new dual pair , together with a genera ting condition, to detect that the ev a lua tion map is also an isomor phism (in D k ). Hence the new dua l pair is an inv ertible pa ir , giving an equiv alence o f catego r ies. Notation 4.7. Given a map of k - algebras ι : B → E , we hav e t wo restr iction-of-sca lars functors: o ne for restriction of le ft mo dules , and ano ther fo r r estriction of rig ht mo dules. F or a ny k -algebr a A , W e let ι ∗ L : S k ( A, E ) → S k ( A, B ) denote restr iction o n the left (target), and ι ∗ R : S k ( E , A ) → S k ( B , A ) denote restriction on the right (source). Both functors cre a te weak-equiv alences and ﬁbrations. Lemma 4.8 . L et A and B b e k -algebr as, and let T b e ﬁbr ant and c oﬁbr ant in S k ( A, B ) , with endo- morphism k -algebr a E = F A ( T , T ) . If T is (right-)dualizable in D k , then ther e is a homotopy dual p air ( e T , e D ) with b ase A and c ob ase E whose unit is an isomorphism. This dual p air extends T in the sense that ι ∗ L e T ≃ T , wher e ι : B → E is t he k -algebr a map adj oint t o the action of B on T . Pr o of. Because T is co ﬁbrant and ﬁbrant in S k ( A, B ), no replacements ar e nec e ssary and E is the derived endomo rphism mono id of T . The unit map k → E is o bta ined as the co mpo site of algebra maps k → B → E . Let e T b e a coﬁbr ant replace men t for T in S k ( A, E ). Recall that T is coﬁbra n t in S k ( A, B ), and hence has the LLP with res pect to acyclic ﬁbrations. W e co ns truct e T by the usual factorization of the map from the initial o b ject, and the forgetful functor ι ∗ L creates weak equiv a lences and ﬁbra tions, so the lifting prop erty for T gives a weak eq uiv alence T ≃ − → ι ∗ L e T . 8 The canonical dual of T is F A ( T , A ) = T ⊲ A ∈ S k ( B , A ), and we let D denote a coﬁbrant replacement for F A ( T , A ) in S k ( B , A ), so that we ha ve a weak eq uiv alence D ≃ − → F A ( T , A ). The canonical dual of T has a right-action of the endomorphism k -algebra, E , and we let e D be a coﬁbra n t replacement for F A ( T , A ) in S k ( E , A ), co nstructed aga in by the usual factorization. Since the forgetful functor ι ∗ R creates weak equiv alence s a nd ﬁbra tions, we ha ve an acyclic ﬁbration ι ∗ R e D ≃ − → F A ( T , A ) in S k ( B , A ). Because D is coﬁbrant, the w eak equiv alence D ≃ − → F A ( T , A ) lifts w ith r esp ect to acyclic ﬁbrations and hence w e have a w eak equiv a lence D ≃ − → ι ∗ R e D . Now we s how that ( e T , e D ) is a dual pair in D k . The w eak equiv a lences e T → T and e D → F A ( T , A ) in S k ( A, E ) a nd S k ( E , A ), r esp ectively , give maps e T ⊙ e D → T ⊙ F A ( T , A ) → E and e D ⊙ e T → F A ( T , A ) ⊙ T → A in S k ( E , E ) a nd S k ( A, A ), resp ectively . Moreover, the ﬁr st map is an isomo rphism in D k ( E , E ) b ecaus e its ima ge under ι ∗ L ι ∗ R is a compo site of tw o isomorphisms in D k ( B , B ): ι ∗ L e T ⊙ ι ∗ R e D ∼ = T ⊙ D ∼ = ι ∗ L ι ∗ R E . The inv erse to this map g ives the unit for the dual pair , and the duality diagr ams commute because the corre spo nding diagrams for T and F A ( T , A ) do. Hence the functors − ⊙ e T and − ⊙ e D induce an adjunction D k ( A, C ) −⊙ e T / / D k ( E , C ) −⊙ e D o o and the unit of this adjunction is an is omorphism.  Lemma 4.9. L et T , e T , e D b e as in L emma 4.8, with B = k . If T gener ates D k ( A, k ) , t hen e T is ⊙ -dete cting. Pr o of. As in the algebra ic case , this follows b ecause k is the ground o b ject, and hence the collection of all 1- cells is jointly ⊙ -detecting (Deﬁnition 4.1). The g enerator condition therefore implies that T itself is ⊙ -detecting (Lemma 4 .3 ). Now ι ∗ L creates weak equiv alence s , and ι ∗ L ( − ) = ι ∗ L ( E ) ⊙ − , so e T is also ⊙ -detecting.  Using Lemmas 4.8 a nd 4.9, we ﬁnish the pro of of P rop osition 4 .6 as in the algebra ic case. Both the compo site and the ﬁr st ma p display ed below are is omorphisms, and hence the second map is also an isomorphism. But the second map is e T ⊙ − applied to the counit, and since e T is ⊙ - detecting, the counit of the dua l pa ir is therefore a n isomo r phism in D k ( A, A ). E ⊙ e T → e T ⊙ e D ⊙ e T → e T ⊙ A 5. The Bica tegorical Yoneda Lemma This section describ es the Y oneda Lemma for bica tegories. F ollowing [Str80], we avoid giving the detailed deﬁnitions, and instead give some general description follow ed by exa mples, which w ill be our main in ter est. As in Section 3, we suggest [Lac07] o r [Lei9 8] for further background. 5.1. Pseudofunctors. If A and B are bica tegories, a pseudofunctor P : A → B (also called a m or- phism ) is the bicategor ical v ersion of a functor. It is a function o n 0-cells and for each pa ir of 0-cells a functor A ( A, B ) P AB − − − → B ( P A, P B ) . These functors ar e compatible with ⊙ -comp ositio n in that there are 2-cell isomo r phisms P B C X ′ ⊙ P AB X ∼ = − → P AC ( X ′ ⊙ X ) satisfying the natural asso ciativity a nd unit compatibilit y conditions. Our focus is on the r epresented pseudofunctors. These are a bicategorical version of repre s ent ed functors for categor ies, and they take v alues in the bicateg ory C at . In this bicategory , the 0-cells are categorie s, the 1-cells a r e functor s , a nd the 2-cells a r e natural transforma tio ns of functors . F or any bicategory B with 0-cell A , we hav e the repr esented pseudofunctor B ( A, − ) : B → C at . F o r a 0-cell E ∈ B , this pseudofunctor gives a categor y , B ( A, E ). F o r a 1-ce ll M : E → E ′ , we hav e the functor M ⊙ − : B ( A, E ) → B ( A, E ′ ) , and 2 -cells M → M ′ give natural tr ansformations of such functors. The compa tibilit y isomor phisms which make B ( A, − ) a pseudofunctor a r e precisely the asso ciativity isomorphisms ( M 2 ⊙ ( M 1 ⊙ − )) ∼ = ( M 2 ⊙ M 1 ) ⊙ − . In this con text, our intro ductory rema r k ‘Morita theory is about bi mo dules’ can b e rephrased as the comment tha t Morita theor y is ab out repr e s ent ed pse udofunctors. Our remark near the end o f 9 Section 2 that ‘standa r d derived eq uiv alences pres erve bimodule str uc tur e’ can b e understo o d as the observ a tio n that s ta ndard der ived equiv alence s are transformations of represented pseudofunctors. 5.2. (Strong) transformations. A transformatio n is a kind o f bicategorica l na tural transfo r mation of functors. A trans formation of t wo represented ps e udo functors, B ( B , − ) and B ( A, − ) is given by (1) A family of functors F C : B ( B , C ) → B ( A, C ). These ar e the c omp onents of F . (2) F or each 1 - cell C K − → C ′ , a natura l tra nsformation whic h, fo r 1-cells X ∈ B ( B , C ), has compo nent 2-cells K ⊙ F C ( X ) → F C ′ ( K ⊙ X ) natural in K and X , with standard asso ciativity and unit compatibilities; namely that the following diag rams commute, with K and X as above, and L ∈ B ( C ′ , C ′′ ). L ⊙ K ⊙ F C ( X ) ) ) S S S S S S S S S S S S S S / / L ⊙ F C ′ ( K ⊙ X )   F C ′′ ( L ⊙ K ⊙ X ) C ⊙ F C ( X ) ∼ =   / / F C ( C ⊙ X ) ∼ = w w o o o o o o o o o o o F C ( X ) Notation 5.1. In the following, we will frequently drop the s ubscripts on the comp onents of our tra ns- formations since they may alwa ys be determined from context and they tend to mak e the text less readable. F or developing Mo rita theory , our in ter est will b e in s t r ong transfo r mations; these are tr ansforma- tions for whic h the compo nen t 2-ce lls sho wn ab ov e are natura l isomorphisms. Restricting atten tion to strong tr ansformations is equiv a lent to res tr icting to tra nsformations which hav e ob ject-wise adjoints (Lemma 5.6). Since the equiv alences Morita theor y seeks to understand a r e, in particular , adjoint pair s, this restriction of s c op e is necessary . Similar ideas are consider ed fo r dis tr ibutors in [FPP 75] a nd [BV02] and for bialgebr oids in [Szl04]. The appropriate morphisms of transforma tions a re called mo diﬁc ations , but we will not make any explicit reference to them b eyond the follo wing deﬁnition. Deﬁnition 5.2. F or tw o bicategorie s A and B , Ψ s [ A , B ] denotes the bicatego ry whose 0-cells are pseudofunctors A → B , 1-cells ar e strong transformations , and 2-cells ar e mo diﬁcations . Lemma 5.3 (Y oneda [Str80]) . F or a pseudofunctor of bic ate gories P : A → C at , evaluation at the unit 1-c el l for e ach 0-c el l, A , of A pr ovides the c omp onents for an e quivalenc e of c ate gories Ψ s [ A , C at ]( A ( A, − ) , P ) ≃ − → P A. Corollary 5.4 (Morita II) . Ψ s [ A , C at ]( A ( A, − ) , A ( B , − )) ≃ − → A ( B , A ) That is, str ong tr ansformations A ( A, − ) → A ( B , − ) ar e given (pr e cisely) by ⊙ - c omp osition with a 1-c el l B → A . In p articular, str ong tra nsformations which induc e e quivalenc es A ( A, C ) ≃ A ( B , C ) for al l 0-c el ls C ar e given by invertible 1-c el ls B → A . The es sential point of the pro of, a s in the 1-categorica l c a se, is the observ ation that for a str ong transformatio n, S , and a 1 -cell Z : A → C , S C ( Z ) ∼ = S C ( Z ⊙ A ) ∼ = Z ⊙ S A ( A ) so that, for a n y C , the functor S C is deter mined by S A ( A ), an o b ject in the categ o ry A ( B , A ). Natural transformatio ns of these functors are determined by mor phisms in A ( B , A ). This equiv alence can be read with v ario us empha s es, yielding v ar io us interpretations. One p ossible int erpretatio n w ould take s trong transformatio ns or strong equiv alences as ob jects o f interest a nd take the equiv alence a s a characteriza tion of these o b jects– they ca n b e only those tr ansformations arising as ⊙ -comp osition with a 1-cell. A co mplement ary interpretation takes the transformatio ns given by ⊙ -comp osition as the basic ob jects of interest, as in the case of the standar d der ived equiv ale nc e s for derived categ ories of DG k -a lgebras. F rom this point of view, the equiv alence is an assurance that functors a rising in this way a re no less, a nd no more, than the strong transformations. In the presence of a c lo sed s tr ucture for our bicateg ory , we hav e a further int erpretatio n. F unctor s given by ⊙ -co mpo sition with a 1 -cell a re naturally enriched ov er an ambien t closed struc tur e; in the following pr o po sition, we formalize what is meant by a family of functors enriched over the internal hom, and show that s uch functors are ne c e ssarily the family of comp onents of a transfor mation. 10 Prop ositio n 5.5. L et A and B b e 0-c el ls of a close d bic ate gory B , and let F b e a famil y of functors F C : B ( A, C ) → B ( B , C ) for 0-c el ls C . The fol lowing ar e e quivalent. (1) F or any C , and any 1-c el ls T ∈ B ( A, C 1 ) , U ∈ B ( A, C 2 ) , V ∈ B ( A, C 3 ) ther e ar e 2-c el ls T ⊲ U → F T ⊲ F U in B ( C 1 , C 2 ) and U ⊲ V → F U ⊲ F V in B ( C 2 , C 3 ) . These 2-c el ls ar e natur al in T , U , and V , pr eserve units , and c ommute with c omp osition, in the sense describ e d by the fol lowing. ( U ⊲ V ) ⊙ ( T ⊲ U ) c omp / /   T ⊲ V   ( F U ⊲ F V ) ⊙ ( F T ⊲ F U ) c omp / / F T ⊲ F V T ⊲ T / / F T ⊲ F T C 1 O O adj. to unit 9 9 r r r r r r r r r r (2) The family F is t he family of c omp onents for a tr ansformation of r epr esente d pseudofunctors. That is, for any 1-c el ls X ∈ B ( A, C 1 ) and K ∈ B ( C 1 , C 2 ) , ther e ar e 2-c el ls K ⊙ F ( X ) → F ( K ⊙ X ) in B ( B , C 2 ) . These 2-c el ls ar e natur al in K and X , and asso ciative and unital. Pr o of. Given maps a s in (1), and 1-cells K and X as in (2), we des crib e the structure 2 -cell K ⊙ F ( X ) → F ( K ⊙ X ) as the following co mpo site: K ⊙ F ( X ) adj. to id K ⊙ X − − − − − − − − − → [ X ⊲ ( K ⊙ X )] ⊙ F ( X ) F − → [ F ( X ) ⊲ F ( K ⊙ X )] ⊙ F ( X ) ev al − − → F ( K ⊙ X ) . Using the deﬁnition of the map, as so ciativity for the structure 2-cell is r educed to the giv en comm utativity with co mpo sition. Unitality follows fro m the unit condition ab ov e. The situation is ex actly reversed for the conv er se. Given maps as in (2 ) a nd 1-cells T and U as in (1), we descr ibe T ⊲ U F − → F T ⊲ F U as a djoin t to the map ( T ⊲ U ) ⊙ F T → F (( T ⊲ U ) ⊙ T ) F (ev al) − − − − − → F U.  Prop ositio n 5.5 shows that a transfor mation of repr esented pseudofunctors can b e interpreted as what o ne might call “a natural family of enriched functors”. The following lemma is proved similar ly , and gives a spe c ia lized interpretation for the strong transfor mations: families of left adjoints. Lemma 5.6. F or F as ab ove, the maps K ⊙ F ( X ) → F ( K ⊙ X ) ar e isomorphisms for al l K and X if and only if F has a family of right adjo ints, B ( A, C ) ← B ( B , C ) : G C and these adj oints have maps K ′ ⊙ G ( X ′ ) → G ( K ′ ⊙ X ′ ) making G into a tr ansformation of r epr esente d pseudofunctors B ( B , − ) → B ( A, − ) . In other wor ds, a tr ansformation F is a st r ong tr ansformation if and only if it has a right adjoint tr ansformation. 6. Practical Interpret a tion In this section, we r eturn our fo cus to Mor ita theory . The question of when a derived equiv alence is a st andar d der ived equiv a lence is r a ised, but not ans wered, b y Rick ar d’s work. E xample 2 .8 shows that a short a nswer to this ques tion is “not always”, and more subtle a ns wers have b een explored in the literature of derived Morita theory (see [KZ9 8], fo r exa mple). Our ins pec tio n of the Y oneda Lemma yields a reinterpretation of this q uestion, and a nother approach to determining when one mig h t give an aﬃrmative answer; we discuss this in Subsec tion 6.1. In Subse c tion 6.2, we turn to the ambien t enrichmen ts which are present in both clas s ical and derived Morita theory . W e again use the ideas of the pr e v ious section, this time to empha s ize the relev ance o f enrichmen ts to Morita theor y . W e follo w this expla na tion with so me exa mples, illustrating how o ne might apply these int erpretatio ns in pr actice. 11 6.1. Comp onen ts of (strong) transformations. In this subsection we let B de no te a clo sed bicate- gory , with the example B = D k as o ur primar y motiv ation. Let A , B , and I b e three ﬁxed 0-cells o f B ; in our motiv a ting e x ample, we take I = k . F urthermore, supp ose f is s imply a functor B ( A, I ) → B ( B , I ). Recalling Co r ollary 5.4 (Mor ita I I), the question o f when f is a st andar d functor is equiv a lent to the question of whether f is a comp onent of a transfo r mation b et ween pseudo functors, that is, whether there is a tra nsformation F : B ( A, − ) → B ( B , − ) with F I = f . In pa rticular, we seek to understand the case when f is a n equiv a lence, and c haracteriz e when f is a co mp onent o f a str ong tr ansformation. As Lemma 5.6 p o int s out, f b eing a comp onent o f a strong transformation implies that its adjoin t is itself a co mp onent o f a tr ansformation. In this situa tion, le t B A denote the full sub-bica tegory o f B whose 0-cells are A and I . There a re four 1-cell catego ries in B A ; these ar e the ca teg ories of 1-ce lls in B from A to A , from A to I , fr om I to A , and from I to I . That is, B A ( x, y ) = B ( x, y ) for x, y ∈ { A, I } . W e ha ve the r epresented pseudofunctor B A ( A, − ) : B A → C at , and we hav e a lso the pseudofunctor B ( B , − ) : B A → C at . Since B is not a 0-cell of B A , B ( B , − ) is un-re presented, but it is nevertheless a pseudo functor on B A and the Y oneda Lemma applies to de s crib e strong tr ansformations B A ( A, − ) → B ( B , − ). Using the Y oneda lemma twice, we hav e the following tw o equiv alences: Ψ s [ B A , C at ]( B A ( A, − ) , B ( B , − )] ≃ − → B ( B , A ) ≃ ← − Ψ s [ B , C at ]( B ( A, − ) , B ( B , − )) . These tw o eq uiv alences say in bica tegorica l la nguage what is apparent to one who considers the pro of of the Y oneda Lemma: tha t stro ng transformatio ns a re determined by their v alues o n the unit 1-cell. One could make this c le a rer by res tricting B A further to a single 0 -cell, A , since for the equiv a lence ab ov e I is ir relev ant. W e hav e chosen to include I so that the equiv alences ab ove provide a pro of for the following. Corollary 6. 1. A functor f : B ( A, I ) → B ( B , I ) is a c omp onent of a s t r ong tr ansformation B ( A, − ) → B ( B , − ) if and only if it is a c omp onent of a str ong tr ansformation F : B A ( A, − ) → B ( B , − ) . That is, f is a c omp onen t of a str ong tr ansformation if and only if ther e is a functor f ′ : B ( A, A ) → B ( B , A ) and natur al 2-c el l isomorphisms K ⊙ f ′ ( X ) ∼ = − → f ( K ⊙ X ) , with the app ar ent asso ciativity r e quir ement, for any 1-c el ls X : A → A and K : A → I . In this c ase, the str ong tr ansformation F is determine d by its two c omp onents, F I = f and F A = f ′ . Dropping the condition that the transformation be stro ng, we can use Pro po sition 5.5 in the pre- ceding co n text to achiev e a desc r iption in terms of the in ternal hom. Corollary 6.2 . A functor f : B ( A, I ) → B ( B , I ) is a c omp onent of a tr ansformation B ( A, − ) → B ( B , − ) if and only if t her e is a functor f ′ : B ( A, A ) → B ( B , A ) and, for 1-c el ls T , U ∈ B ( A, I ) and T ′ , U ′ ∈ B ( A, A ) , ther e ar e 2-c el ls T ⊲ U → f T ⊲ f U in B ( I , I ) T ′ ⊲ U ′ → f ′ T ′ ⊲ f ′ U ′ in B ( A, A ) T ⊲ U ′ → f T ⊲ f ′ U ′ in B ( I , A ) T ′ ⊲ U → f ′ T ′ ⊲ f U in B ( A, I ) subje ct t o the c omp atibility with c omp osition and units describ e d in Pr op osition 5.5 (1). R emark 6.3 . In this context, the co mpatibilit y means precisely that a ll pos sible diagrams o f the for m below co mm ute: ( U ⊲ V ) ⊙ ( T ⊲ U )   / / T ⊲ V   ( F U ⊲ F V ) ⊙ ( F T ⊲ F U ) / / F T ⊲ F V T ⊲ T / / F T ⊲ F T unit O O 9 9 r r r r r r r r r r Where ea ch of T , U , and V is taken to be either in B ( A, A ) or B ( A, I ), ‘ unit ’ is taken to b e either I , or A , and F is taken to b e e ither f or f ′ , as a ppropriate. No te tha t if all thre e of the ob jects, T , U , V are taken fro m the same category , this condition is precisely the condition which makes f and f ′ enriched ov er ⊲ . The ﬁrst diagr am sa ys that the enrichm ent m ust commute with the compo sition pairing, and the s e cond diag ram says that the enrichmen t must preserve units. W e now turn to some more practica l interpretations o f Section 5. Combining these results with our prev ious int erpretatio n of Mo rita theor y enables us to give a description of the concepts at work in bo th cla s sical a nd derived Morita theory . W e follow this description with examples of Corolla ry 6 .2, showing that, at least in algebraic co nt exts, the four-par t necessary condition can be veriﬁed fo rmally . 12 6.2. Enriched equiv ale n ces . Pro p o s ition 5.5 shows that the standar d Morita equiv alences in a closed bicategory must b e equiv a le nces which ar e enriched ov er the internal hom, and likewise that families of equiv alences which are enr iched over the internal hom ﬁt toge ther to form standa rd Morita equiv- alences. As no ted in Remar k 2.3, left-adjoint functors (e.g. equiv alences) b et ween ab elian categ o ries are auto ma tically enriched in abe lia n groups, a nd this may b e one reason that enrichmen t ha s b een under-apprecia ted in these contexts. The topo logical Morita theor em o f Sch wede and Shipley [SS03] addresses sp e ctr al Quillen equiv alences– Quillen e q uiv alences enr ich ed in sp e c tra. Likewise, T o¨ en [T o¨ e07] works with DG ca tegories, the morphisms of which a re enric hed functors. P rop osition 5.5 s hows that fo cusing on enriched equiv alences is inevitable. In these pa pe r s the authors also addr ess the imp ortant question of what mo del-theo retic a s sump- tions could be veriﬁed in pra c tice and would gua rantee standard Mo rita equiv a lences on the derived level. O ne po s sible lesson taught by Example 2.8 is that certainly some ass umptions a re necessary in general. The r esults ab ov e, how ever, are indep endent of mo del theory , a pplying to any clos ed bicatego ry . They indicate that Quillen equiv alences which induce enriche d transfo rmations on the derived level are necessarily the appropria te equiv alenc e s fo r the development of Mo r ita theory . This p er s pec tiv e can oﬀer an e xplanation for the results of [DS07] in pa r ticular. There, and in related work, the technical no tion o f additive mo del category is introduced, and it is shown that Q uillen equiv alences b etw een additive mo del categorie s are necessa rily additive functor s, just as in the cla ssical situation. The result, therefore , is that a zig -zag o f Quillen equiv alences for which each intermediate mo del categor y is add itive provides a well- behaved notio n of Mor ita e q uiv alence for additive mo del categor ies. O ur p ersp ective would sugge s t that this can be extended to more g eneral enric hed mo del categ ories, with the appropr iate notio n o f Mor ita equiv alence in thos e settings being enriched Quillen functors . F r om this p oint of view, Exa mple 2.8 is an exp ected example, and others like it will be expected in applications for which Quillen functors ar e not necessarily enriched. Corollar y 6 .2 shows that the pr o p e rty of enrichmen t may b e identiﬁed b y considering only a sp e- ciﬁc spec ial case, arising through our restriction from the bicateg ory B to the full sub-bica tegory , B A generated by tw o 0-cells, A and I . In algebr aic examples, this fo ur-part co ndition can be simpliﬁed even further. W e demonstr ate this b y recalling the following tw o c lassical r esults, with their pr o ofs for reference. T he y show that, in the case of classical Morita theory , the condition in Corollary 6.2 is automatic. Theorem 6.4 (Mo r ita I I [Lam99, 7.18.26]) . L et R and S b e two rings, and let f : M ( R , Z ) / / M ( S, Z ) : g o o b e an e qu ivalenc e b etwe en the c ate gories of right R -mo dules and right S -mo dules. L et Q = f ( R ) and let P = g ( S ) . Then ther e ar e natur al bimo dule st ructur es making P ∈ M ( R, S ) and Q ∈ M ( S, R ) . Using these bimo dule structur es, ther e ar e natur al isomorphisms of functors f ∼ = − ⊙ Q and g ∼ = − ⊙ P. Pr o of. The bimodule structures are recognize d b y the ring isomorphisms R ∼ = Hom R ( R, R ) ∼ = Hom S ( f R, f R ) and S ∼ = Hom S ( S, S ) ∼ = Hom R ( g S, g S ) . The iden tiﬁca tion o f f is obtained by the following computatio n, us ing the fact that g is an adjoint for f a nd that P is dualizable; the identiﬁcation of g is simila r. F o r M ∈ M ( R , Z ), f ( M ) ∼ = Hom S ( S, f M ) ∼ = Hom R ( g S, M ) ∼ = M ⊗ R Hom R ( P, R ) ∼ = M ⊗ R Q = M ⊙ Q.  R emark 6 .5 . The pr o of here implicitly deﬁnes the functor f ′ as the comp osite of the forgetful functor from R - R - bimo dules to right R -mo dules with the functor f . The computation ab ov e shows that the image o f this compo site lies in the sub catego ry of S - R -bimo dules. T o rela te the previous re s ult to the following one, recall that equiv ale nc e s of ab elian ca tegories are, in pa rticular, exa ct and copro duct-pres e rving. Theorem 6 . 6 (W a tts [W at6 0]) . L et R and S b e rings, and let f : M ( R, Z ) → M ( S, Z ) b e a functor fr om the c ate gory of right R - mo dules to t he c ate gory of right S -mo du les. If f is right-exact and pr eserves dir e ct sums, then ther e is a bimo dule C ∈ M ( S, R ) and a natur al isomorphism f ∼ = − ⊙ C . Pr o of. Since f prese rves dire ct sums, it is automa tically enriched o ver Hom R . If T ′ is an ( S, R )-bimo dule, we obs erve that f ( Z T ′ ) ha s a natural S -mo dule structure g iven by the map o f ab elia n gro ups Hom S ( Z S , Z S ) → Ho m R ( Z T ′ , Z T ′ ) → Hom S ( f ( Z T ′ ) , f ( Z T ′ )) 13 and w e deﬁne f ′ ( T ′ ) to be the ( S, S )-bimo dule whose underlying ( Z , S ) bimo dule is f ( Z T ′ ). The cat- egories of bimo dules M ( R , R ) a nd M ( S, R ) are deﬁned to be the sub categ o ries o f left R - mo dules in M ( R, Z ) and M ( S, Z ), resp ectively , and hence the compa tibilit y conditions of Corollar y 6.2 follo w for - mally . This shows that f and f ′ are c omp o nents of a tra ns formation M ( R, − ) → M ( S, − ). The theorem is pr ov en once we show that this is a strong transformatio n. By Lemma 5.6, it suﬃces to show that this tra ns formation has a r ight-adjoin t transforma tio n. This also follows for mally , b y the s pecia l adjoint functor theor e m: f is copro duct-prese r ving, a nd right-exact, and hence has a right-adjoint; f ′ likewise has a right a djoint, and these form a n adjoint trans formation.  In der ived con texts, the co ndition of Corollary 6.2 is no longer a utomatic, but we can s till apply the r esult to obtain the following explicit des c r iption. Prop ositio n 6.7. L et k b e a c ommutative ring, let A b e a D G k -algebr a and let f : D k ( A ) → D k (End k ( A )) b e an e quivalenc e of triangulate d c ate gories. Then f is a standar d derive d e quivalenc e if and only if we have the fol lowing: ( i ) The e qu ivalenc e given by f is an enriche d e quivalenc e. ( ii ) Ther e is an enriche d e quivalenc e f ′ : D k ( A, A ) → D k (End k ( A ) , A ) . ( iii ) The t wo e quivalenc es, f and f ′ ar e c omp atible in the fol lowing sense: If T ′ , U ′ ∈ D k ( A, A ) and T , U ∈ D k ( A ) = D k ( A, k ) , t hen ther e ar e natur al maps Ext A ( T , U ′ ) → Ext End k ( A ) ( f T , f ′ U ′ ) in D k ( k , A ) Ext A ( T ′ , U ) → Ext End k ( A ) ( f ′ T ′ , f U ) in D k ( A, k ) which c ommute with the p airing induc e d by c omp osition. (That is, the squar es in Remark 6.3 c ommute.) 7. Mod el Structure for DG Al gebras W e b egin with so me deﬁnitions and reminders from [KM9 5], and g ive a mo del structure for the category of DG-mo dules ov er a DG algebr a. W e make use o f the ho motopy extension and lifting pr op erty (HELP) to streamline the model- theoretic arguments, and empha size the analogy with topo lo gy . When the DG a lgebra is concentrated in degree 0 (a ring), this is the standa rd mo del structure for c hain complexes over the ring (Remar k 7.8). Let k b e a commutativ e ring and A a ﬁxed DG k -alg ebra. W e let S n = A ⊗ k ( k [ n ]), where k [ n ] is a free DG k -mo dule on a single gener a tor in degr ee n , so S n is a free A -mo dule on a sing le genera tor in degree n . W e let D n be a free A -mo dule with one gener ator in degree n and one in deg ree n − 1; the diﬀerential on D n takes the gene r ator in degree n to that in degr ee n − 1. Finally , w e let I denote a free A -mo dule which has one g enerator, h I i , in deg ree 1 , and t wo generator s, h 0 i and h 1 i in degr ee 0; on generator s, the diﬀeren tial in I is given by h I i 7→ h 0 i − h 1 i . W e let ⊗ denote ⊗ A , and for any A -mo dule M we let i 0 and i 1 denote the inclusions M → M ⊗ I corr esp onding to M ⊗ h 0 i and M ⊗ h 1 i , resp ectively . Deﬁnition 7.1 (Relative cell mo dule) . A map o f A -modules C 0 → C is ca lled a r elative cell A -module if C is the co limit of a sequence of maps C r → C r +1 , with each map obta ine d as a pushout L q i S q i / /   C r   L q i D q i +1 / / C r +1 The maps S q i → C r ab ov e are ca lled the attaching maps for C r . If 0 → C is a r e la tive cell A -module, C is called a cell A -mo dule. If there a re o nly ﬁnitely many cells, then C is called a ﬁnite cell A -mo dule. This is generaliz ed b y the following deﬁnition. Deﬁnition 7.2 ([MS06, 4.5.1 ]) . Let I b e a set of maps in a categ o ry C with copro ducts ⊕ . ( a ) A r elative I -c el l mo dule is a map C 0 → C , with C obtained as a c o limit o f maps, C r → C r +1 , formed by pushouts L q ∈ I X q / /   C r   L q ∈ I Y q / / C r +1 where ea ch X q → Y q is a map in I . 14 ( b ) The set I is c omp act if, for every map X → Y in I , the source ob ject, X , is small with re spe ct to countable colimits. That is, for ev ery relative I -cell mo dule C 0 → C a s ab ove, the natural map b elow is an isomor phism. colim Hom A ( X, C r ) ∼ = − → Hom A ( X, colim C r ) ( c ) An I -c oﬁbr ation is a map which satisﬁes the LLP with respec t to any ma p satisfying the RLP with r esp ect to all maps in I . Deﬁnition 7.3 (Cell submo dule) . If M = co lim M r and L = colim L r are cell A -mo dules for which each L r is a submo dule of M r and, for each attaching ma p S q → L r , the compos ite S q → L r ⊂ M r is one of the attac hing maps for M r , then L is called a c el l submo dule of M . Theorem 7.4 (HELP [KM95, I I I.2.2]) . L et L b e a c el l submo dule of a c el l A -mo dule, M , and let e : N → P b e a quasi-isomorphism of A -mo dules. Then, given maps which make the solid arr ow diagr am b elow c ommute, ther e ar e dashe d lifts which c ommu te with the r est of the diagr am. L i 0 / /   L ⊗ I   h { { v v v v L i 1 o o g   ~ ~ ~ ~   P N e o o M i 0 / / f ? ?    M ⊗ I c c H H M i 1 o o _ _ @ @ The fo llowing lemma clariﬁes the relatio nship b etw een HELP and quasi- isomorphisms. It is obvious from [KM95, I I I.2.1], although they state and prov e only one dir e c tion. Theor em 7.4 is prov en by using the r elative cell structure L → M to reduce to this lemma. Note. F or one who compares this lemma with [KM9 5], it may be helpful to p oint out that the g rading is coho mological ther e, so they use s a nd s − 1 wher e we use n and n + 1. Lemma 7.5. F or any inte ger n , a map e : N → P of DG-mo dules over A satisﬁes HELP with r esp e ct to t he inclusion S n → D n +1 if and only if e ∗ : H ∗ ( N ) → H ∗ ( P ) is a monomorphi sm in de gr e e n and an epimorphi sm in de gr e e n + 1 . Pr o of. Having HELP with resp ect to S n → D n +1 means having the do tted lifts in any solid-a rrow diagram o f the type shown b elow. S n i 0 / /   S n ⊗ I   θ y y r r r r r S n i 1 o o z { { w w w w   P N e o o D n +1 i 0 / / w ′ < < y y y y D n +1 ⊗ I η e e K K K D n +1 i 1 o o w c c F F In words (using subscripts to denote degree s of elements, and including facto r s of ( − 1 ) n implicitly where appropria te in o ur co rresp ondence b e t ween letters ab ov e and letters b elow), this says that g iven a n y cycle, z n , in N whose imag e in P is homolog ous to a boundar y , z ′ n : z ′ n = dw ′ n +1 and ez n − z ′ n = dθ n +1 , then z n is a boundar y in N of some w n +1 and, moreov er , the imag e of that bounding element in P is homologo us to the diﬀerence b etw een the b ounding element for z ′ n and the b ounding element for ez n − z ′ n : z n = dw n +1 and ew n +1 − w ′ n +1 + θ n +1 = dη n +2 . If e ha s this lifting prop erty , then taking θ n +1 = 0 sho ws that e ∗ is a mono morphism in degree n , and tak ing z n , z ′ n = 0 (so θ n +1 is a cycle in P ) s hows that e ∗ is a n epimorphism in degree n + 1. F or the conv er se, e n being a mono morphism gives the existence of an element, e w n +1 whose b oundary is z n , since ez n is homolo gous to a b oundary in P . The existence of η n +2 follows b ecause e n +1 is an epimor phism and e e w − w ′ n +1 + θ n +1 is a cycle, so ther e is so me c ycle b w in N fo r which ( e e w − w ′ n +1 + θ n +1 ) − e b w = dη n +2 . The elemen t w n +1 is taken to be the diﬀerence e w − b w .  As HELP indicates, we use the inclusions S q → D q +1 and D q → D q ⊗ I to g enerate the mo del structure for DG mo dules o ver A . This is formalized b y any of sev eral s tandard r esults fo r model structures, and we quote one such result here. Theorem 7.6 ([MS06 , 4.5.6 ]) . L et C b e a bic omplete c ate gory with a sub c ate gory of we ak e quivalenc es (that is, a sub c ate gory c ontaining al l isomorphisms in C and close d under r etra cts and the two out of thr e e pr op erty). L et I and J b e c omp act sets of maps in C . If the fol lowing two c onditions hold, 15 then C is a c omp actly gener ate d mo del c ate gory, with gener ating c oﬁbr ations I and gener ating acycli c ﬁbr ations J : (i) (A cyclicity) Every r elative J - c el l c omplex is a we ak e qu ivalenc e. (ii) (Comp atibility) A map has the RLP with r esp e ct to I if and only if it is a we ak e quivalenc e and has the RLP with r esp e ct to J . Note. The ter m c omp actly gener ate d is a sp ecializa tion of the notion of co ﬁbrantly g enerated for the case that I a nd J are co mpact s ets of maps (recall Deﬁnition 7.2). It means that the ﬁbrations are characterized by the RLP with resp ect to J , the acy clic ﬁbrations are characterized b y the RLP with resp ect to I , the co ﬁbrations ar e the retra cts of r e la tive I -cell complexes, and the acyclic co ﬁbrations are the retra cts of rela tive J - c ell complexes . The main adv antages o f compact generation ov er coﬁbrant g e neration ar e that it do es not req uir e one to use the full version of the s mall-ob ject ar gument, but only a sma ll-ob ject argument over count able colimits, a nd that it is suﬃciently ge ne r al for many top olog ical and algebr a ic a pplications, including the one which co ncerns us here. 7.1. Application. In our applica tion, C will b e the categor y o f (DG) A -mo dules, and the weak equiv- alences will b e the quasi-iso mo rphisms. The se t I = { S n − 1 → D n | n ∈ Z } , and the s e t J = { D n i 0 − → D n ⊗ I | n ∈ Z } . By deﬁnition, the relative I -cell mo dules are the relative cell A -mo dules, and since D n and D n ⊗ I are cell A -mo dules , any relative J -cell mo dule is als o a cell A -mo dule. Since S n − 1 and D n are ﬁnite cell A -mo dules , the next lemma shows that I and J ar e compact. Lemma 7.7 (compactness ) . If Z 0 → Z 1 → Z 2 → · · · → Z is a r elative c el l c omplex, and if C is a ﬁnite c el l A -mo du le, t hen the natu ra l map b elow is an isomorph ism. colim Hom A ( C, Z r ) ∼ = − → Hom A ( C, colim Z r ) Pr o of. Assume ﬁrst that C is a b ounded co mplex of ﬁnitely-generated free A -mo dules, with generator s x 1 , . . . , x n . Then an A-mo dule map f : C → colim Z r is uniq ue ly determined by the elements f ( x i ) ∈ colim Z r . Since n is ﬁnite, there is some s such that f ( x i ) ∈ im ( Z s → colim r Z r ) for all i . Hence the lemma holds when C is free; in particular , the lemma holds when C = S n or C = D n . Now if M is a cell c omplex for which the lemma holds, and S n → M is an y map of A -mo dules , then the pushout of this map along S n → D n +1 is an A -mo dule for whic h the lemma holds. Since the lemma holds for the A -mo dule 0 , it holds for every ﬁnite ce ll A -mo dule.  7.2. Acyclicit y. W e note ﬁrst that the inclusio n i 0 : A → I of free A -mo dules given by 1 7→ h 0 i has a deformation re traction, r , given by r h I i = 0 a nd r ( a h 0 i + b h 1 i ) = a + b . An e xplicit ho mo topy h : i o r ≃ id is easily constructed. F or any A - mo dule, M , the inclusion i 0 : M → M ⊗ I has a defor mation retraction induced b y r , and ther efore also any map M → N g iven by pushout along D n → D n ⊗ I has a defor mation retr action. Since S n and S n ⊗ I are ﬁnite cell A -mo dules, w e apply Le mma 7.7 to see that any relative J - cell mo dule is a w eak equiv alenc e . 7.3. Compatibi lity . Let p : X → Y b e a map o f A -modules. Assume ﬁrst that p has the RLP with resp ect to I . Then for maps L S q i − 1 / /   C r / /   X p   L D q i / / C r +1 / / Y where C r +1 is a pushout, w e hav e a lift ⊕ D q i → X and hence a lift C r +1 → X . Therefo re p lifts with resp ect to any relative ( I -)ce ll mo dule, and in particular has the RLP with r e spe ct to J . Moreov er, p has the RLP with resp ect to all maps 0 → S n and S n ⊗ I → S n , and hence p is a weak equiv alence. Now supp ose o nly that p is a weak e q uiv alence and ha s the RLP with resp ect to J . Being a weak equiv alence , p satisﬁes the homotopy extension and lifting prop erty (Theor em 7.4). T o s ee that this implies the r esult, no te ﬁrst that there is an isomorphism of fre e A -mo dules I ∼ = D 1 ⊕ S 0 given by c hanging basis in degr ee 0 , and hence a pro jection I → D 1 which equalizes the tw o inclusio ns i 0 and i 1 : S 0 → I . Moreover, the composite of either inclus ion with this pro jection is the sta ndard inclusion S 0 → D 1 . W e tensor with S n and use the isomor phism D n +1 ∼ = S n ⊗ D 1 to deﬁne a ma p S n ⊗ I → S n ⊗ D 1 ∼ = D n +1 equalizing the inclus ions i 0 and i 1 : S n → S n ⊗ I and such that either 16 comp osite S n → S n ⊗ I → D n +1 is the standard inclusion. In other words, the diagr am b elow commutes. S n i 0 / /   S n ⊗ I x x r r r r S n i 1 o o | | | | | | | | D n +1 S n o o D n +1 s s s s s s s s Now g iven the following commuting s q uare of DG mo dules ov er A , S n / /   X p   D n +1 / / Y we pro duce a lift via the commuting diagram be low, where the map S n ⊗ I → Y is the comp osite S n ⊗ I → D n +1 → Y . S n i 0 / /   S n ⊗ I   y y r r r r r S n i 1 o o { { w w w w   Y X p o o D n +1 i 0 / / < < x x x x D n +1 ⊗ I e e K K K ℓ 9 9 D n +1 i 1 o o c c F F The dashed lifts follow from HELP (Theor em 7.4), and the dotted lift of these, ℓ , exists b ecause p ha s the RLP with r esp ect to J . The compos ite ℓi 0 is the desired lift for the squa re a b ove. 7.4. F urther Structure. Here w e do cument so me facts about the model structure describe d ab ove. R emark 7.8 . This mo del structure is the same as the standard model s tr ucture for chain complexes ov er a r ing. Hov ey’s description [Hov99, 2.3.3] of the standa r d mo del structure for DG-mo dules ov er A when A is a ring (i.e. chain complexes ov er the ring A ) has the sa me weak equiv alences and g enerating coﬁbrations, I , as ab ov e; the generating acyclic coﬁbrations are J ′ = { 0 → D n } . It is clear that the coﬁbrations of these tw o mo del str uctures are the same, and the squares below show that the ﬁbrations of these tw o structures a re also the sa me: RLP with resp ect to J is e q uiv alent to RLP with resp ect to J ′ . 0 / /   D n / /   0   D n +1 / / D n ⊗ I / / D n +1 Prop ositio n 7.9. Al l A -mo dules ar e ﬁbr ant. Pr o of. The inclusion D n → D n ⊗ I has a s ection, hence the map M → 0 ha s the RLP with resp ect to all ge ner ating acyclic coﬁbrations for any M .  Prop ositio n 7.1 0 ([KM9 5, I I I.4.1]) . If X → Y is a we ak e quivalenc e of left A -mo dules and M is a c oﬁbr ant (right) A -mo dule, then M ⊗ A X → M ⊗ A Y is a we ak e quivalenc e. 8. Base Change f o r DG Algebras In this section, we describ e gener al res ults r e garding change of base DG k - algebra. Supp ose that A and B are DG k -alg e bras for a co mm utative r ing , k , a nd supp ose f : A → B is a map o f DG k -algebras . There a re tw o na tural pull-backs of B to the category of DG A -mo dules: let A B B ∈ D G k ( B , A ) denote B with the a ction of A on the left via f , and let B B A ∈ D G k ( A, B ) denote B with the action of A on the right via f . Then the A - A bimo dule obtained from B with A a cting on b oth sides by f is given a s ( A B B ) ⊗ B ( B B A ) = ( A B B ) ⊙ ( B B A ) ∈ D G k ( A, A ). The map f can b e regar ded a s a 2-cell A f − → A B A = ( A B B ) ⊙ ( B B A ) . The m ultiplicatio n for B giv es a 2-cell in D G k ( B , B ) ( B B A ) ⊙ ( A B B ) = ( B B A ) ⊗ A ( A B B ) → B 17 and the duality r elations hold, making ( A B B , B B A ) a dual pair . Hence we have an adjoint pair of str ong transformatio ns extension o f sc a lars: f ! = − ⊙ A B B : D G k ( A, − ) → D G k ( B , − ) and restriction of scalar s: f ∗ = − ⊙ B B A ∼ = A B B ⊲ − : D G k ( B , − ) → D G k ( A, − ) The transforma tion f ∗ is r ight a djo int to f ! , but s ince f ∗ is itself a s trong tra nsformation, it als o ha s its own r ight adjo int, f ∗ = B B A ⊲ − : D G k ( A, − ) → D G k ( B , − ) . 8.1. Lo cal mo del structure. Ea ch 1-cell c ategory D G k ( A, B ) has a mo del structure, described by applying the theory of Section 7 to the DG k -algebra A ⊗ k B op . W e re fer to this a s a lo ca l mo del structure for the bicateg ory D G k , meaning simply a mo del structre on e ach 1-cell category . The gener ating coﬁbrations a nd acyclic c oﬁbrations o f D G k ( A, B ) are denoted by I ( A, B ) a nd J ( A, B ), respectively , and the results below describ e the b ehavior of the base-change tra nsformations ab ov e with resp ect to this lo cal mo del structre. Notation 8.1. In con trast with Section 7, here we let S n , D m , and I denote the c o rresp onding chain complexes ov er k , a nd we le t ⊗ denote ⊗ k . F or any chain complex, M , ov er k , we le t B M A denote the DG ( B , A )-bimo dule B ⊗ k M ⊗ k A . So B M A ∈ D G k ( A, B ). Prop ositio n 8 .2 (Push-out P ro ducts) . The lo c al mo del structur e on e ach 1-c el l c ate gory D G k ( A, B ) is c omp atible with ⊙ - c omp osition of 1-c el ls in t he fol lowing sense: If i and j ar e gener ating c oﬁbr ations, then their pushout- pr o duct is a c oﬁbr ation, and if one of i or j is a gener ating acyclic c oﬁbr ation and the other is a gener ating c oﬁbr ation, then their pushout-pr o duct is an acyclic c oﬁbr ation. Pr o of. Suppo s e ﬁrst that i : S q → D q +1 and j : S r → D r +1 are generating coﬁbra tions in C h ( k ). If we denote by P the pushout b elow, then the pushout pro duct of i a nd j is the induced ma p P → D q +1 ⊗ D r +1 . S q ⊗ S r / /   S q ⊗ D r +1   D q +1 ⊗ S r / / P ( ( R R R R R D q +1 ⊗ D r +1 This map can b e obtained ex plicitly by attaching a cell o f dimens io n q + r + 2 to P , and hence is a coﬁbration. Likewise, if j is taken to b e a g e nerating acyclic coﬁbration D r +1 → D r +1 ⊗ I , the pushout pro duct can b e s e e n explic itly to b e a coﬁbration. Since extension of scalar s to an arbitrar y DG k - algebra preserves coﬁbr ations, the pushout pro ducts ov er ⊙ are still coﬁbrations. When i or j is taken to be a generating acyclic co ﬁbration, w e see tha t the pushout pro duct is still acy clic by r ecalling the defor mation retraction D r +1 i 0 − → D r +1 ⊗ I ≃ − → D r +1 of Subsection 7.2. Extending to another DG k -alg ebra, w e s till hav e these deformatio n retra ctions, so the pushout pro ducts over ⊙ remain weak equiv alences .  Prop ositio n 8.3. If f : A → B is a map of DG k -algebr as, then the adjo int p air ( f ! , f ∗ ) is a Q uil len adjoint p air for e ach C . D G k ( A, C ) f ! / / D G k ( B , C ) f ∗ o o Pr o of. This follo ws from the o bserv ation that, for a chain complex M ∈ C h ( k ), f ! ( C M A ) = C M A ⊙ A B B ∼ = C M B . Hence f ! induces an iso morphism o f sets I ( A, C ) ∼ = I ( B , C ) and J ( A, C ) ∼ = J ( B , C ), wher e I and J are the generating coﬁbrations a nd acyclic coﬁbrations for D G k ( A, C ) and D G k ( B , C ).  R emark 8.4 . A similar statement for ( f ∗ , f ∗ ) is no t true unless A B A is coﬁbrant a s an A -module, since otherwise f ∗ do es not preserve coﬁbratio ns in gener al. Lemma 8.5 ( f ∗ creates w eak eq uiv alences) . If e : X → Y is a map of 1-c el ls in DG k ( B , C ) for which f ∗ e : f ∗ X → f ∗ Y is a we ak e quivalenc e, t hen the original map e : X → Y is a we ak e quivalenc e. Pr o of. If L → M is a coﬁbration in C h ( k ), we hav e no ted a lready that the induced map C L B → C M B is isomo rphic to f !  C L A → C M A  . T o sho w that e : X → Y is a weak equiv alence, it suﬃces to show 18 that e has HELP with re spe ct to maps of this for m (Lemma 7.5). Consider the adjoint lifting dia grams below. f ! C L A / /   f ! C ( L ⊗ I ) A   y y t t t t t t t f ! C L A o o } } | | | | |   Y X e o o f ! C M A / / > > | | | | | f ! C ( M ⊗ I ) A e e J J J J f ! C M A o o a a B B B C L A / /   C ( L ⊗ I ) A   y y s s s s s s C L A o o } } { { { { {   f ∗ Y f ∗ X f ∗ e o o C M A / / = = { { { { { C ( M ⊗ I ) A e e K K K C M A o o a a C C C Lifts ex ist o n the right by hypothesis, and henc e on the left b y adjunction.  Prop ositio n 8. 6. If f ab ove is a we ak e quivalenc e, then the Quil len p air ( f ! , f ∗ ) is a Quil len e quivalenc e for al l C . Pr o of. By [Hov99 , Prop. 1.3.13] it suﬃces to pr ov e tha t, for co ﬁbrant 1-cells M ∈ DG k ( A, C ) a nd N ∈ D G k ( B , C ), the compos ites M → f ∗ f ! M = M ⊙ A B B ⊙ B B A ∼ = M ⊙ A B A and f ! Q ( f ∗ N ) = Q ( N ⊙ B B A ) ⊙ A B B → N ⊙ B B A ⊙ A B B → N are weak equiv a le nces. The functor Q is coﬁbra nt replacement; no ﬁbr ant r eplacement is necessa ry since every o b ject is ﬁbr ant. That the top comp osite is a weak equiv alence follows b ecause M is a coﬁbrant ( C, A )-bimo dule and A → A B A is a weak equiv a lence of ( A, A )-bimo dules (Prop ositio n 7.10). T o see that the bo ttom comp osite is a w eak equiv alence, we consider the comp osite Q ( N ⊙ B B A ) → Q ( N ⊙ B B A ) ⊙ A B B ⊙ B B A → N ⊙ B B A where the ﬁrs t map is the unit of the adjunction, and the se cond is f ∗ applied to the comp osite above. This total co mpo s ite is the coﬁbrant replace ment map Q ( N ⊙ B B A ) → N ⊙ B B A , a nd hence is a w eak equiv alence by deﬁnition. The ﬁrst map in this comp osite is a weak equiv alence b ecaus e Q ( N ⊙ B B A ) is coﬁbr ant (as ab ov e), and hence by the t wo-out-of-three pr op erty , f ∗  Q ( N ⊙ B B A ) ⊙ A B B → N  is a weak equiv alence. By Lemma 8.5 then, the map Q ( N ⊙ B B A ) ⊙ A B B → N is a w eak equiv alenc e .  Corollary 8 . 7. F or f : A ≃ − → B as ab ove, the dual p air ( A B B , B B A ) is invert ible when c onsider e d as a p air of 1-c el ls in the derive d c ate gories D k ( B , A ) and D k ( A, B ) , r esp e ctively. 8.2. Duality in D G k and D k . Lemma 8.8 . If M is right-dualizable in D G k ( A, B ) , then M is a r etr act of a ﬁ n ite fr e e (right-)DG- mo dule over A ⊗ k B op . Pr o of. Since M is rig ht -dualizable, the co ev aluation map ν : M ⊙ ( M ⊲ A ) → M ⊲ M is an isomor phism, and hence there is a map B → M ⊙ ( M ⊲ A ) lifting the unit B → M ⊲ M . W e let Σ i ( m i ⊗ ϕ i ) denote the imag e o f the unit, 1 B , under this map, where m i ∈ M , ϕ i ∈ M ⊲ A = Hom A ( M , A ), and the sum has o nly ﬁnitely many ter ms . W e now show that there is an A ⊗ k B op -mo dule map p , with a s ection ˜ ϕ , making M a retract o f a ﬁnitely-genera ted fre e DG-mo dule, where each e i is a genera tor o f deg ree | e i | = | m i | : L i ( A ⊗ k B op · e i ) p / / M . ˜ ϕ x x The map p is deﬁned by p ( a ⊗ b · e i ) = b · m i · a , and the section ˜ ϕ is deﬁned by ˜ ϕ ( m ) = Σ i ϕ i ( m ) ⊗ 1 B . It is ea sy to see that ˜ ϕ is a sectio n for p by making use o f the fact that id M = ν (Σ i m i ⊗ ϕ i ) = Σ i m i · ϕ i ,  Lemma 8.9. L et M ∈ D k ( A, B ) and su pp ose M is a r etr act of a ﬁn ite c el l ( B , A ) -bimo dule. Then M : A → B is (right-)dualizable in D k and t her efor e t he c o evaluation M ⊙ ( M ⊲ A ) → M ⊲ M is an isomorphi sm in D k . Note. Since w e a r e working in the derived bica tegory , D k , the source - homs, ⊲ , ab ov e a re unders to o d to be the derived ho ms . Since M is coﬁbra n t, the deriv ed and underived homs a re equal on M . Pr o of. One characterizatio n of duality is that the map induced by ev aluation D k [ W , Z ⊙ ( M ⊲ A )] → D k [ W ⊙ M , Z ] b e an isomorphism for all 1- cells W , Z with appropria te sourc e a nd tar get. F rom this po int of vie w, the ﬁve lemma shows that the full sub categor y o f dualizable ob jects in D k ( A, B ) is a thic k subcateg ory (see, fo r example, [MS06 , 16.8.1]). Since the pushouts which build cell mo dules are 19 examples of exact triangles in D K , the result follows by noting that the spheres and disks, S q and D q , are dualizable. The coev a luation map, ν , is deﬁned as the adjoin t to M ⊙ ( M ⊲ A ) ⊙ M → M ⊙ A ∼ = M , induced by the ev aluation ma p, so if M is dualizable in the sense des crib ed ab ov e, then ta king W = M ⊲ M and Z = M pro duces the inverse to co ev a luation.  Lemma 8.10. L et M : A → B b e a 1-c el l in D k , and supp ose the c o evaluation M ⊙ ( M ⊲ A ) → M ⊲ M is an isomorphism. Then M is (qu asi-)isomorphi c to a r etra ct of a ﬁn ite c el l ( B , A ) -mo dule. Pr o of. F ollowing the usual arg ument , we implicitly tak e a coﬁbrant replac emen t for M as a ( B , A )- bimo dule, co lim M r ≃ − → M . T he inverse to coev a luation, comp osed with the monoid map B → M ⊲ M gives η : B → M ⊙ ( M ⊲ A ). Since − ⊙ ( M ⊲ A ) preser ves colimits, and since B is co mpa ct in D k ( B , B ), this ma p factors thro ugh some ﬁnite stage of co lim( M r ⊙ ( M ⊲ A )), and we hav e a lift in the diagr am below for some r . M r ⊙ ( M ⊲ A ) ⊙ M / /   M r   M ∼ = B ⊙ M η ⊙ i d / / 4 4 i i i i i i i i i M ⊙ ( M ⊲ A ) ⊙ M id ⊙ ev al / / M The bottom comp osite is the identit y , and we see that M is a retra ct of M r .  References [Bro03] R.M. Br ou wer, A bica te goric al appr o ach to Morita e quivalenc e for rings and von Neumann algebr as , arXiv:math/0301353v1 [math.OA] (2003). [BV02] F. Bor ceux and E. 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Morita Theory For Derived Categories: A Bicategorical Perspective

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