Cohomology of exact categories and (non-)additive sheaves

COHOMOLOGY OF EXA CT CA TEGORIES AND (NON-)ADDITIVE SHEA VES DMITR Y KALEDIN AND WENDY LOWEN Abstract. W e use (non-)additiv e shea ves to introduce an (absolute) notion of Hochsc hild cohomology for exact categ ories as Ext’s in a sui table bisheaf category . W e compare our app r oac h to v arious deﬁnitions presen t in the liter- ature. 1. Introduction Given an asso ciative algebr a A ov er a ﬁeld k , one can deﬁne its Ho chschild homology gr oups H H q ( A ) a nd its Hochsc hild cohomolog y groups H H q ( A ). Non- commutativ e geometry , in its homolo gical v ers ion, starts w ith the observ ation that Ho chsc hild ho mology classes b ehav e “ as diﬀerential forms”, while Ho chschild co- homology classes are similar to vector ﬁelds. When A is co mm uta tive and Spec A is a smo oth algebra ic v ariety ov er k , this observ atio n b ecomes a precise theorem, namely , the famous theorem of Ho chsc hild, K ostant and Rosenber g [7]. In the general case, b oth H H q ( A ) and H H q ( A ) still carr y some additiona l str uc tur es analogo us to what one ﬁnds for a comm uta tive algebra. F or H H q ( A ), the r elev ant structure is the Connes -Tsygan diﬀerential B which gives rise to cy clic homolo g y – this is analog ous to the de Rham diﬀerential. F or H H q ( A ), the structur e is the so-called Ger stenhab er bra ck et which turns H H q ( A ) into a Lie algebr a – this is analogo us to the Lie brack et of vector ﬁelds. There are certain natural co mpatibili- ties b etw een the bra ck et and the diﬀerential, axiomatized by Tsygan and T amarkin under the na me of “ non-commutativ e calculus ” [30]. If o ne thinks of an algebra A as a simple example of a “non-co mm uta tive alge- braic v ariety”, then Ho chschild homo lo gy usually gives rise to ho mologica l inv ari- ants o f the v ar iety , suc h as e.g . de Rham or cristalline cohomology . Hochschild cohomolog y , o n the other hand, is intimately related to a utomorphisms and defor- mations of A . F o r re a l-life applications, it is highly desir able to extend the basic theory of Ho chsc hild homolog y and cohomolog y to “more general” non-commutativ e v a ri- eties. This can mean diﬀerent things in diﬀerent cont e xts; but at the very leas t, o ne should b e able to dev elo p the theory for an ab elian c ategory C (a motiv ating obser- v a tion here is that if tw o algebras A , B ha ve e q uiv a lent catego r ies A -mo d ∼ = B - mo d of left mo dules, then their Ho chsc hild homolo gy and cohomolo gy are c a nonically ident iﬁe d). F or Ho chsc hild homology , this has been accomplis he d in a more-o r-less exhaustive fashion by B. Keller [15] back in the 1990 ies. F or Ho chschild cohomol- ogy , the story should b e simpler: morally sp eaking, the Hochschild cohomolo gy algebra H H q ( C ) should just be the algebra o f Ext’s from the identit y endofunctor The ﬁrst author has b een partially supp orted by AG Laboratory SU-H SE, RF go vernmen t gran t, ag. 11.G34.31.0023, the RFBR gr an t 09-01-00242 and the Science F oundation of the SU- HSE aw ard No. 10-09-0015. The second author is postdo ctoral fellow with the Researc h F oun- dation - Fl anders (FW O) and ac knowledge s the support of the European Union, ERC grant No 257004-HHNcdMir. 1 COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 2 of C to itself. Ho wever, ﬁnding an a ppropriate categ ory where these Ext’s can b e computed is a delicate matter. Perhaps becaus e of this, the rigorous co ho mologica l theor y app eared later than the homological one; essentially , it was sta r ted in [21], [20], where a Ho chsc hild cohomolog y theor y for ab elian ca tegories is cons tr ucted, and its relation to defor - mations of the categor y is discussed. But unfortunately , the theo r y that exis ts so far is clo sely mo deled on the theory for asso cia tive algebr as. As a r esult, it lacks some essential featur es which should in fact b ecome automatic in the consistently categoric al appr oach. This b ecomes quite o bvious when one tries to apply the the- ory to co ncrete problems ; for one ex ample of this, we refer the r eader to [1 0], wher e the applica tio n intended is to Ga bb er ’s inv olutivity theorem. The presen t paper arose as an a ttempt to at least ﬁll the g aps no ted in [10], and at most, to sketc h a more- or-less comprehensive theory of Ho chsc hild coho mology of a belia n catego ries and its r elation to deforma tions. As it happ ens, alrea dy the deﬁnitions of Ho chschild coho mology , when do ne accurately , take up quite a lot of space. This is a s far as we get in this pap er, relegating b oth the Gerstenhab er brack et and the deformation theor y story to subsequent w o r k. One additional thing that emerg e s c learly in the catego rical approach is the ability to work “absolutely ”, not over a ﬁxed ﬁeld k . The motiv ating example here is v er y bas ic: the categ o ry of v ec tor spaces over Z /p Z has a na tural “ﬁrs t- order deformation” to the category of mo dules ov er Z /p 2 Z . A truly comprehensive Ho chsc hild coho mology theo ry for ab elia n categor ies should include this example, and assig n to it a non-triv ial deformation cla ss. Some of the theories we construct in the pre s ent paper should b e a ble to do this. In order to a chiev e this, we hav e to sp end quite a lot of time on foundations , but w e b elieve that ultimately , this is time well spent. The ge ne r al outline o f the pap er is as follows. As we hav e already noted, the deﬁnition of Ho chsc hild coho mology should be ob vio us once o ne ha s an appro priate category o f e ndo functors of o ur ab elian catego ry C . If C is the categ ory o f mo d- ules over an algebra A , then a natural ca ndidate for its endofunctor catego ry is the categ ory of bimo dules over the same a lgebra – this is what g ives the classica l Ho chsc hild co homology . An “absolute” v er sion of this stor y also exists, and it has bee n known for quite some time no w , starting from [8]. Howev er, the situation for a gener al ab elian categor y C turns o ut to b e somewha t delicate. In Section 2, we discuss in some detail v ar ious em b edding theo rems which a llow o ne to r epresent a n ab elian category C as a categ ory o f s heav es on itself, then deﬁne its endo functor category as a catego ry of sheaves o n C op × C , a nd so on. In the mo dule categor y case, everything is known, but we repro duce the results for the c o nv enience of the reader; the general case req uires using a ppropriate Grothendieck top ologies , and this seems to b e new. As an unexp ected bo nus, w e discov er along the way that a very natural relaxation of so me conditions pro duces ex actly the exact categor ies in the sense of Q uillen, so that the whole story gener alizes to exa ct categ ories without any changes at all. In Section 3, w e discuss the derived versions of the she a f cate- gories and v arious exactness pro pe r ties of natural functors b etw een them. Then in Section 4, we a re ﬁnally able to in tr o duce Hochsc hild cohomology . W e also discuss other deﬁnitions present in the literatur e and prov e v ario us co mparison theorems betw een them. Of co ur se, to b e useful, such a list o f comparis on theo r ems should be exhaustiv e; this we hav e strived to a chiev e, to the best of o ur knowledge. In particular, we do treat the a bsolute case – the relev ant no tion here is Mac Lane homology and c o homology . Finally , in the la st section, we discuss infor mally what do es n ot work in our approach, esp ecially in the abs olute ca se, and what is the COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 3 relation betw ee n our w o rk and more a bstract theory based on v a r ious triangulated category enha nc e men ts. A cknow le dgement. Ov e r the years, w e ha ve had an opportunity to discuss Hochsc hild homology and cohomolog y with many p e ople, and b eneﬁted a lot from these dis- cussions. The ﬁrst author is par ticularly grateful for the things he has lear ned from M. Kontsevic h, L. Hesselho lt, D. T amar kin, and B. Tsygan. The second a uthor is grateful to Ragnar B uch weitz for bringing the deﬁnition of Hochsc hild cohomology in § 4.5 to her attention and for e x plaining the pro of of Theor em 4.6 for mo dule categorie s. She is als o gr a teful to Michel V an den Bergh for numerous things she learned fro m him, some da ting ba ck to her master thesis - under his sup er vision - on Gro thendieck ca tegories and additive sheaves. After the ﬁrs t version o f the pap er w a s po sted to the w eb, we hav e r eceived several v a luable comments and explanations from T. Pira s hvili, for whic h we are very grateful. 2. Sheaf ca tegories This section contains some basic notions a nd fa c ts concerning sheav es taking v a lues in the categor y Ab of abelian groups. The setting in which we will w o rk is tha t of single morphism to p o logies, i.e. topolo gies for which covers a r e deter mined b y the morphisms in a cer tain collection Λ. Our main a pplication is to exa ct categor ies C , for which C comes naturally equipp ed with the s ingle deﬂa tion top olo gy , and C op with the single inﬂation top ology . In this context, w e intro duce a num b er of bifunctor categories consisting of bifunctor s that are a dditive in some o f the v a riables and sheav es in some o f the v ar iables. 2.1. Additi v e top ologies. In this section we mainly ﬁx so me notations and ter- minology . F or categorie s C , D with C small we denote by Fun ( C , D ) the categor y o f functors from C to D , and we put Fun ( C ) = Fun ( C op , Ab ). F o r Z -linear ca tegories C , D with C sma ll we denote by Add ( C , D ) the c ategory of additive functors from C to D , and w e put Mo d ( C ) = Add ( C op , Ab ). Ob jects of F un ( C ) are called functor s while o b jects of M o d ( C ) a re calle d mo dules. By a top olo gy on a small categor y C we mean a Gro thendieck top olog y . O n a small Z -line ar catego ry we will als o use the parallel enriched notion of an additive top olog y (see [4], [26], [19]). This is obtained from the usual no tion of a Grothendieck topolog y b y re pla cing Set by Ab and Fun ( C op , Set ) b y Mo d ( C ). More precisely: Deﬁnition 2.1. An additive top olo gy T on a small Z -line a r catego ry C is given by specifying for each ob ject C ∈ C a collection T ( C ) of submo dules of C ( − , C ) ∈ Mo d ( C ) satisfying the following axioms: (1) C ( − , C ) ∈ T ( C ). (2) F or R ∈ T ( C ) and f : D − → C in C the pullback f − 1 R in Mo d ( C ) of R along f ◦ − : C ( − , D ) − → C ( − , C ) is in T ( D ). (3) Co ns ider S ∈ T ( C ) a nd an arbitrary submo dule R ⊆ C ( − , C ). If for every D ∈ C and f ∈ S ( D ) the pullback f − 1 R is in T ( D ), then it fo llows that R ∈ T ( C ). An additiv e top olog y on a one-ob ject Z -linear category corresp onds precisely to a Gabriel topolo gy on a ring [6]. As usual, a s ubmo dule R ⊆ C ( − , C ) is identiﬁed with the set ` D ∈C R ( D ) ⊆ ` D ∈C C ( D , C ), i.e. R is considered as an “additiv e sieve”. A submodule R ∈ T ( C ) is called a c over (of C ) . An additive topolo gy T on C deter mines a Gr othendieck category Sh add ( C , T ) ⊆ Mo d ( C ) of additive she aves , i.e. mo dules F ∈ Mo d ( C ) such that every cov er R ⊆ C ( − , C ) in T ( C ) induces a bijection F ( C ) ∼ = Mo d ( C )( C ( − C ) , F ) − → Mo d ( C )( R, F ) . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 4 Conv erse ly a ny Grothendieck categor y A can be represented as an additive shea f category for suitable c ho ic es of C (see [1 9]), the easiest choice for C b eing a full generating sub categ ory as in the Gabr iel-Popescu theorem [27]. 2.2. Sing le mo rphi sm top ologies. Let C b e a small (res p. sma ll Z -linear) cate- gory and Λ a collection of C -mor phisms. W e deﬁne a subfunctor (r esp. a s ubmo d- ule) R ⊆ C ( − , C ) to be a Λ - c over if R (consider e d as a s ieve) co ntains a morphism λ ∈ Λ. If the Λ-cov e r s de ﬁne a top ology T Λ (resp. an additive top olo gy T add Λ ) on C , then this top olo gy is c alled the single Λ - top olo gy (res p. the additive single Λ -top olo gy ). Let us now sp ell out what it means fo r F ∈ Fun ( C op , Set ) to b e a s heaf fo r T Λ . F or λ : D − → C in Λ , a co mpa tible family of elements with respe ct to the cov er h λ i g enerated b y λ corr esp onds to an element x ∈ F ( D ) such that for every commutativ e diag ram D λ / / C E α 1 O O α 2 / / D λ O O we hav e F ( α 1 )( x ) = F ( α 2 )( x ). Hence, the shea f prop erty with r esp ect to λ says that for s uch an elemen t x ∈ F ( D ) there is a unique e lement y ∈ F ( C ) with F ( λ )( y ) = x . Recall tha t a ﬁltered colimit colim i F i is called monoﬁlter e d if all the transition morphisms F i − → F j are monomor phis ms. Lemma 2.2. A monoﬁlter e d c olimit of she aves (in Fun ( C op , Set ) ) r emains a s he af. Pr o of. Consider a monoﬁlter e d colimit co lim i F i of sheaves F i and λ : D − → C in Λ. Suppo se x ∈ c olim i F i ( D ) is compatible and let x i ∈ F i ( D ) be a representative o f x . Consider α 1 , α 2 : E − → D with λα 1 = λα 2 . Now F i ( α 1 )( x i ) and F i ( α 2 )( x i ) b ecome equal in colim i F i ( E ), but since this colimit is mo noﬁltered, w e obtain F i ( α 1 )( x i ) = F i ( α 2 )( x i )in F i ( E ). Hence, x i is compatible and there exists y i ∈ F i ( C ) with F ( λ )( y i ) = x i . F urthermo re, if y , z ∈ colim i F i ( C ) b ecome equal in colim i F i ( D ), appropria te representativ e s y i , z i ∈ F i ( C ) b eco me equal in F i ( D ), a nd hence y i = z i and y = z .  If C is small Z -linear , it makes sense to consider b oth T Λ and T add Λ on C . The subfunctors R = h λ i ⊆ C ( − , C ) of morphisms factoring through a given λ ∈ Λ ar e additive (whence submo dules) a nd constitute a basis for both T Λ and T add Λ . W e are ma inly interested in s he aves taking v alues in the category Ab of ab elian groups. Consider Fun ( C ) = Fun ( C op , Ab ), the ca tegory Sh Λ ( C ) = Sh ( C , T Λ ) ⊆ F u n ( C ) of (non-additive) sheav e s of ab elian gro ups o n C , Mo d ( C ) = Add ( C op , Ab ) and the catego r y Sh add Λ ( C ) = Sh add ( C , T add Λ ) ⊆ M o d ( C ) of additive sheaves o n C . By the pr evious o bs erv ations, we ha ve Sh add Λ ( C ) = Sh Λ ( C ) ∩ Mo d ( C ) . Recall that an ob ject A in a categor y A is ﬁnitely gener ate d if A ( A, − ) : A − → Set commutes with monoﬁltered colimits. W e ha ve the following natural s ource of single morphis m top ologies: Prop ositi o n 2.3. L et A b e a Gr othendie ck c ate gory and C ⊆ A a s m al l fu l l additive sub c ate gory. The fol lowing ar e e qu ivalent: (1) The obje cts of C ar e ﬁn itely gener ate d gener ators of A . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 5 (2) Λ = { λ ∈ C | λ is an epimorphism in A} deﬁnes an additive single Λ - top olo gy T add Λ on C with Sh ( C , T add Λ ) ∼ = A . Pr o of. If (1) holds, there is an additive top ology T on C with Sh ( C , T ) ∼ = C and for this top o logy R ⊆ C ( − , C ) is a cov e r if and only if ⊕ f ∈ R C f − → C is an epimorphisms in A . Since C is ﬁnitely genera ted, there are ﬁnitely many mor phisms f i : C i − → C in R , i = 1 , . . . n , for whic h f = P n i =1 f i : ⊕ n i =1 C i − → C is a n epimorphism. But since R is an additive subfunctor, in fact f ∈ R . Con versely , suppo se (2) holds. Ob viously C genera tes A , so we ar e to show tha t C ∈ C is ﬁnitely gener ated in Sh ( C , T add Λ ). This easily follo ws fr o m the fact that C is ﬁnitely generated in Mo d ( C ) and Le mma 2.2.  R emark 2.4 . If all the mor phisms λ ∈ Λ of Prop os ition 2.3 b ecome s plit epimor- phisms in A , the top ology T add Λ is reduced to the trivial top ology with Sh ( C , T add Λ ) = Mo d ( C ). This situation is e q uiv a lent to the ob jects in C ⊆ A b eing ﬁnitely gener a ted pr oje ctive g enerator s in A . Often a collection Λ ca n b e directly seen to deﬁne single Λ-to po logies: Prop ositi o n 2. 5. L et C b e a smal l c ate gory (re s p. s m al l Z -line ar c ate gory) and Λ a c ol le ction of morphisms su ch that: (1) Λ c ontains isomorphi sms; (2) F or λ : D − → C in Λ and f : C ′ − → C arbitr ary, the pul lb ack λ ′ : D ′ − → C ′ exists and is in Λ ; (3) Λ is st able under c omp osition. Then Λ deﬁnes a single Λ - top olo gy (r esp. an additive single Λ -top olo gy) on C . Suppo se Λ determines sing le mor phism top ologie s T Λ and T add Λ . The inclusions i ′ : Sh Λ ( C ) ⊆ F un ( C ) and i : Sh add Λ ( C ) ⊆ Mo d ( C ) hav e exact left adjoint sheaﬁﬁca tion functors a ′ : Fun ( C ) − → Sh Λ ( C ) and a : Mo d ( C ) − → Sh Λ ( C ) res p ectively . Deﬁnition 2 . 6. A functor F ∈ Fun ( C ) is we akly Λ -eﬀac e able if and only if for every C ∈ C a nd ev e r y x ∈ F ( C ), there exists a morphism λ : C ′ − → C in Λ with F ( λ )( x ) = 0 . Let W Λ ⊆ Fun ( C ) b e the full sub catgo ry of weakly Λ-eﬀac e able functor s, and W add Λ ⊆ Mo d ( C ) the full sub catego ry of weakly Λ -eﬀaceable mo dules. Clearly W add Λ = W Λ ∩ Mo d ( C ). F rom the concrete formulae for shea ﬁﬁca tion a nd the fact that the h λ i constitute a basis for T Λ and T add Λ , it follows that: W Λ = Ker( a ) W add Λ = Ker( a ′ ) . In par ticular, W Λ and W add Λ are lo ca lizing Serre sub categ ories o f Fun ( C ) and Mo d ( C ) resp ectively , a nd Sh Λ ( C ) = W ⊥ Λ Sh add Λ ( C ) = ( W add Λ ) ⊥ where F ∈ W ⊥ ⇐ ⇒ [ ∀ W ∈ W : Hom( W, F ) = 0 = E xt 1 ( W , F )] (se e for example [17]). W e obtain commutative diagrams: (1) Mo d ( C ) j / / a   F u n ( C ) a ′   Sh add Λ ( C ) i O O j ′ / / Sh Λ ( C ) . i ′ O O COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 6 Lemma 2 .7. In t he ab ove diagr am, j ′ is an exact funct or. Pr o of. Consider a n exa ct sequence 0 − → A − → B − → C − → 0 in Sh add Λ ( C ). In particular, 0 − → i ( A ) − → i ( B ) − → i ( C ) is exa ct in Mo d ( C ) and we can complete it int o a n ex a ct se quence 0 − → i ( A ) − → i ( B ) − → i ( C ) − → M − → 0 in M o d ( C ). Since a is exact, this implies a ( M ) = 0 , or in other words M ∈ W add Λ . But j , being o bviously exact, maps this seq ue nc e to the exact se quence 0 − → j i ( A ) − → j i ( B ) − → j i ( C ) − → j ( M ) − → 0 in F un ( C ). Since a ′ ( j ( M )) = j ′ ( a ( M )) = 0 we hav e an exa ct seq uenc e 0 − → j ′ ( A ) − → j ′ ( B ) − → j ′ ( C ) − → 0 in Sh Λ ( C ) as desired.  R emark 2.8 . No te that the inclusion j : Mo d ( C ) − → Fun ( C ) has a left adjoint “ad- ditivization” functor which is not exact. C o nsequently , it is imp ossible to expr ess additivity of functors by means o f a top olog y on C . 2.3. Additi v e shea ves insi de non-additive shea ves. Let C be a small additive category . It is well known that the inclusio n j : Mo d ( C ) ⊆ Fun ( C ) is an exac t e mbedding and a Serr e sub categor y (see e.g. [2 5] and the references therein). In this section we extend the result to the inclusion j ′ : Sh add ( C ) − → Sh ( C ) in cas e Λ determines single morphism top olog ies T and T add on C (we suppress Λ in a ll nota tio ns). The ingredients of the pro of are well known, but w e include them for completeness . W e start with the following observ ation: Lemma 2.9. The inclusion j : Mo d ( C ) − → Fun ( C ) is an ex act emb e dding which is close d under extensions. Pr o of. Since C is an additive category , j is fully faithful. That Mo d ( C ) is closed in F u n ( C ) under e xtensions eas ily follows from the 5-lemma .  Next we extend Lemma 2 .9 to she aves: Prop ositi o n 2.1 0. The inclusion j ′ : Sh add ( C ) − → Sh ( C ) is an ex act emb e dding which is close d under ext ensions. Pr o of. Consider a n exa ct sequence 0 − → F ′ − → F − → F ′′ − → 0 in Sh ( C ) with F ′ , F ′′ ∈ Sh add ( C ). This means that we ha ve an exa ct sequence 0 − → F ′ − → F − → F ′′ − → W − → 0 in Fun ( C ) in which F , F ′′ are additive and W is w ea kly eﬀaceable. W e are to show that F is additiv e. By Lemma 2.11, W ( 0) = 0 and hence also F (0) = 0. It r emains to show that for A, B ∈ C , the canonical map η : F ( A ⊕ B ) − → F ( A ) ⊕ F ( B ) is an isomor phism. By Lemma 2.12, η is an epimorphism. F urthermo re, from the diagram 0 / / F ′ ( A ⊕ B ) / / ∼ =   F ( A ⊕ B ) / / η   F ′′ ( A ⊕ B ) ∼ =   0 / / F ′ ( A ) ⊕ F ′ ( B ) / / F ( A ) ⊕ F ( B ) / / F ′′ ( A ) ⊕ F ′′ ( B ) we deduce that η is also a monomor phism.  Lemma 2 .11. Supp ose W ∈ F un ( C ) is we akly eﬀac e able. Then W (0) = 0 . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 7 Pr o of. Consider a n element x ∈ W (0). Ther e e x ists a Λ -morphism C − → 0 such that W (0) − → W ( C ) maps x to 0 . But the map W ( C ) − → W (0) induced by 0 − → C , b eing a mor phism of ab elia n gr oups, maps 0 to 0. Since W (0) − → W ( C ) − → W (0) is the identit y , this proves tha t x = 0 and consequently W (0) = 0 .  Lemma 2. 12. If F ∈ Fun ( C ) satisﬁes F (0) = 0 , t hen for A, B ∈ C the c anonic al morphism F ( A ) ⊕ F ( B ) − → F ( A ⊕ B ) − → F ( A ) ⊕ F ( B ) is e qual to the identity. Pr o of. Let s A , s B , p A , p B denote the canonical injections a nd pro jections asso ci- ated to A ⊕ B . Then w e ar e now dealing with their images under F . W e hav e F ( p A ) F ( s A ) = F ( p A s A ) = F (1 A ) = 1 F ( A ) and lik ewis e for B . Moreov er, since F (0) = 0, w e a lso ha ve F ( p A ) F ( s B ) = F ( p A s B ) = F ( 0) = 0 and s imila rly for F ( p B ) F ( s A ). This ﬁnishes the pro of.  Theorem 2.13. L et C b e a smal l additive c ate gory. The inclusions j : Mo d ( C ) ⊆ Fun ( C ) and j ′ : Sh add ( C ) ⊆ Sh ( C ) ar e Serr e sub c ate gories. Pr o of. W e already show ed in Lemma 2.9 and Prop ositio n 2.10 that bo th inclus ions are ab e lian subc ategories that are close d under extensions. W e need to show that they ar e closed under sub q uotients. Firs t, consider a n exact se q uence 0 − → F ′ − → F − → F ′′ − → 0 in Fun ( C ) in which F is additive. First of a ll, F ′ (0) and F ′′ (0) ar e zer o as a sub o b ject and a quotient ob ject of F (0) = 0 . N ow consider morphisms a, b : C − → C ′ in C and consider f ′ = F ′ ( a + b ) − F ′ ( a ) − F ′ ( b ), f = F ( a + b ) − F ( a ) − F ( b ) and f ′′ = F ′′ ( a + b ) − F ′′ ( a ) − F ′′ ( b ). Then the comm utative diag ram 0 / / F ′ ( C ′ ) f ′   / / F ( C ′ ) f   / / F ′′ ( C ′ ) f ′′   / / 0 0 / / F ′ ( C ) / / F ( C ) / / F ′′ ( C ) / / 0 immediately yie lds that f = 0 implies that bo th f ′ = 0 and f ′′ = 0. F o r the second claim, consider a n exa ct seq ue nce 0 − → F ′ − → F − → F ′′ − → 0 in Sh ( C ) and supp os e that F is additive. Then F ′′ = a ( Q ) where 0 − → F ′ − → F − → Q − → 0 is ex act in Fun ( C ) and a is sheaﬁﬁcation. W e just obtained that bo th F ′ and Q are additive. Hence also F ′′ = a ( Q ) is additive.  2.4. Sing le morphism top ologi es with kernels. Let C b e a sma ll categ ory and suppo se Λ determines a single Λ-top ology . Supp ose more ov er that the mo rphisms in Λ hav e kernel pair s. In this situation, the notion o f sheaf b ecomes mor e tangible . F o r λ ∈ Λ, consider the kernel pa ir D λ / / C P κ 1 O O κ 2 / / D λ O O COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 8 A pres heaf F ∈ F un ( C op , Set ) is a s heaf if and only if for every λ ∈ Λ with kernel pair ( κ 1 , κ 2 ), (2) F ( C ) F ( λ ) / / F ( D ) F ( κ 1 ) , , F ( κ 2 ) 2 2 F ( P ) is an equalizer diagram. W e immediately deduce the follo wing streng thening of Lemma 2.2: Lemma 2 .14. A ﬁ lter e d c olimit of she aves (in Fun ( C op , Set ) ) r emains a she af. Example 2 .15 . If C is a r egular category [1 ], then Λ = { λ | λ is a co equa lizer } satis- ﬁes the conditions of Pro po sition 2.5. Since a coeq ua lizer is alwa ys the coe qualizer of its kernel pair, F ∈ Fun ( C op , Set ) is a sheaf for T Λ if and only if F maps coequa l- izers of kernel pair s to equaliz er diagr ams. Now we return to the s etting of a small Z -linear category C on which Λ determines single Λ-top olog ies. W e supp ose moreover that the mo rphisms in Λ have kernels. Let F : C op − → Ab b e a (p ossibly non-a dditiv e) functor. Let us write down the sheaf prop erty as concretely a s p oss ible . F o r λ : D − → C in Λ, we obtain a dia gram (3) 0 / / K κ / / i 1 # # G G G G G G G G G D λ / / / / C K ⊕ D κ + p 2 O O p 2 / / D λ O O in which the squar e is a kernel pa ir. The sheaf prop erty for F with resp ect to λ requires that the sequence (4) 0 / / F ( C ) F ( λ ) / / F ( D ) F ( κ + p 2 ) − F ( p 2 ) / / F ( K ⊕ D ) is exact. In the situation where F : C op − → Ab is additive, exa c tnes s of (4) clearly reduces to exactness of (5) 0 / / F ( C ) F ( λ ) / / F ( D ) F ( κ ) / / F ( K ) . Deﬁnition 2.16. An additive functor F : C op − → Ab is called Λ -left exact if for every exact sequence 0 / / K κ / / D λ / / C with λ ∈ Λ the sequence (5) is exact in Ab . Let Lex Λ ( C ) ⊆ Mod ( C ) denote the full sub categor y of Λ-left exac t mo dules. W e th us hav e: Sh add Λ ( C ) = Lex Λ ( C ) . In a sense, the non- additive sheaf category Sh Λ ( C ) captures a kind of Λ-left exa ct- ness with additivit y “removed”. 2.5. Exact categories. Let C b e an exact categor y in the sense of Quillen [2 8, 12]. The exact str ucture on the additive category C is g iven b y a collection of so called c onﬂations (6) K κ / / D λ / / C, exact in the sense that κ is a kernel of λ and λ is a cokernel o f κ , satisfying some further axio ms. Let Λ be the co llection of deﬂations , i.e. mor phisms λ turning up in a conﬂation (6), a nd let Ω b e the collection o f inﬂations , i.e. morphisms κ turning COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 9 up in a co nﬂation (6 ). The further axioms of an exact category can be summarized as follows: (1) Λ satisﬁes the conditions of Pr op osition 2.5. (2) Ω op satisﬁes the conditions of P rop osition 2.5 in C op . Note that since κ is re q uired to b e a co kernel of λ , the entire exact structure is in fact determined by the collection Λ. F rom now on, the exact structure of C being sp eciﬁed, we will drop the mention of Λ from our notations and terminology . In this wa y we na turally recover the standard notions of weakly eﬀaceable functors and left exact functors. It is well kno wn (see [12]) that the cano nic a l embedding C − → Lex ( C ) : C 7− → C ( − , C ) is such that (6) is a co nﬂation in C if and o nly if 0 / / K κ / / D λ / / C / / 0 is an exact sequence in Lex ( C ). Let Ind ( C ) ⊆ Mo d ( C ) denote the full s ubca tegory of ﬁltered colimits of C -ob jects. F o r a Grothendieck ca tegory D , let fp ( D ) denote the full sub categ ory of ﬁnitely presented ob jects. Prop ositi o n 2.17. We hav e C ⊆ fp ( Lex ( C )) and Ind ( C ) ⊆ Lex ( C ) . The c ate gory Lex ( C ) is lo c al ly ﬁnitely pr esente d with C as a c ol le ction of ﬁn itely pr esente d gener a- tors. In p articular, fp ( Lex ( C )) is the closur e of C in Lex ( C ) under ﬁ nite c olimits and every obje ct in Lex ( C ) is a ﬁlter e d c olimit of ob je cts in fp ( Lex ( C )) . If C ∼ = fp ( Lex ( C )) , then Ind ( C ) = Lex ( C ) . Pr o of. The ob jects C ( − , C ) are ﬁnitely presented in Mo d ( C ), s o by Lemma 2.1 4 they are ﬁnitely pr esented in Le x ( C ) a s well. By the same lemma, ﬁltered colimits of C -ob jects in M o d ( C ) re ma in left exa c t. The statements concerning lo cal ﬁnite presentation a re standard, in particular F in Lex ( C ) can b e written as ﬁltere d colimit of fp ( Lex ( C )) /F − → Lex ( C ) : ( X → F ) 7− → X . If C ∼ = fp ( C ), then ag ain by Lemma 2.14, this colimit can b e computed in Mo d ( C ).  Examples 2.18 . (1) Let R b e a ring. Let C 1 = free ( R ) b e the c ategory of ﬁnitely gener ated free modules and C 2 = proj ( R ) the category of ﬁnitely generated pro jective mo dules. Both sub categor ie s of Mo d ( R ) are closed under ex tensions (which ar e automatica lly split) whence inherit a n exa ct structure fro m Mo d ( R ). By Remark 2.4, the top ologies T Λ and T add Λ are trivial whence Lex ( C i ) = Sh add ( C i ) = Mo d ( C i ) ∼ = Mo d ( R ) and Sh ( C i ) ∼ = F u n ( C i ) . (2) If C is a small a belia n categ ory with the canonical e x act str ucture, then C is close d under ﬁnite colimits in Lex ( C ) whence by Pr op osition 2.17, C ∼ = fp ( Lex ( C )) a nd Ind ( C ) = Lex ( C ). No w let A be a loca lly cohere n t Grothendieck catego ry , i.e. A is lo cally ﬁnitely presented and fp ( A ) ⊆ A is an abelia n subca tegory . Then b y Pro po sition 2 .3, A ∼ = Lex ( fp ( A )) ∼ = Ind ( fp ( A )). These facts are well known (see for example [26]). (3) F or a general Grothendieck ca tegory A the kernel of an epimor phism b e- t ween ﬁnitely presented ob jects is not itself ﬁnitely presented, so fp ( A ) do es not inherit a n exact struc tur e from A . By Prop osition 2.3, it do es how ever alwa ys inherit the single A -epimor phism top ology T for whic h A ∼ = Sh add ( fp ( A ) , T ) . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 10 (4) Clea rly , the oppo site category C op of an exact category b eco mes exact with Ω op playing the role of Λ. Th us, we obtain a canonical em b edding C op − → Lex ( C op ) = Lex Ω op ( C op ) . The deﬁnition of derived categ ories of abelian categor ies can b e extended to exact categ ories (see [24], [14]). Prop ositi o n 2 . 19. Consider the c anonic al emb e dding C − → Lex ( C ) . The c anonic al functor D − ( C ) − → D − ( Lex ( C )) is ful ly faithful. Pr o of. By [1 4, Theor em 12.1], this immediately follows from Lemma 2.20.  Lemma 2.20 . Consider an epimorphism F − → C in Lex ( C ) with C ∈ C . Ther e is a map C ′ − → F with C ′ ∈ C such that t he c omp osition C ′ − → F − → C r emains an epimorphism. Pr o of. By P rop ositio n 2.1 7, C is ﬁnitely presented in Lex ( C ), a nd Lex ( C ) is a lo cally ﬁnitely presented ca tegory . Consider f : F − → C a s stated. W r iting F = colim i M i as a monoﬁlter ed c o limit o f its ﬁnitely g enerated s ubo b jects, we hav e C = colim i f ( M i ). Since C is ﬁnitely pres ent e d, the identit y 1 C : C − → co lim i f ( M i ) factors through some f ( M j ) − → colim i f ( M i ) = C which is then necessarily an iso- morphism. Thus, we o btain an epimorphism M = M j − → F − → C with M ﬁnitely generated. Now ther e is an epimorphism ⊕ i C i − → M and since M is ﬁnitely gener - ated, an epimorphism ⊕ n i =1 C i − → M . Finally , since C is additive, C ′ = ⊕ n i =1 C i ∈ C and we obtain the desired epimorphism C ′ − → M − → F − → C .  2.6. Sheav es in t w o v ariables . If C is an exa ct ca tegory , then b oth C and C op are naturally endow e d with single morphism top ologies : the “ single deﬂation-to p o logy” on C and the “single inﬂa tion-top olog y ” on C op . Hence, it makes sense to consider bimo dules and bifuncors ov er C that are sheav es in either of the tw o v ariables . In fact, we can develop everything for tw o p ossibly diﬀerent sites A op and B , which, for simplicity of exp ositio n, we take to arise from exact categor ies. Consider exact categ ories A and B and the bifunctor ca tegory Fun ( A op × B ). W e will intro duce a list o f sub categor ies F un ∗ ⋆ ( A op × B ), in which we consider functors that are additive in so me of the arguments, a nd sheav es in some of the arguments. W e will indicate additivity by upp er indices ∗ ∈ { ∅ , ⊳, ⊲, ⋄} (wher e ∗ = ∅ means “invisible index” ): Fun ⊳ means additive in the ﬁrs t v a riable (i.e. all the F ( − , A ) are additive), Fun ⊲ means a dditive in the seco nd v ar iable (i.e. all the F ( B , − ) are additive), F un ⋄ means additive in b o th v a r iables (i.e. Fun ⋄ ( A op × B ) = Mo d ( A op ⊗ B )), and F un means additive in none of the v ar iables. In the same way , w e indicate sheav es by lower indices ⋆ ∈ { ∅ , ⊳ , ⊲, ⋄ } . So for example, Fun ⊳ ⊲ ( A op × B ) consists of functors F for which every F ( − , A ) is additive and every F ( B , − ) is a shea f. W e are interested in inclusio ns of the t y p e i : Fun ∗ ⋆ ( A op × B ) − → Fun ∗ ( A op × B ) where the “additivity para meter” is left unchanged, but we hav e inclusions of sheav es in to presheaves in some of the arg ument s. Our ﬁr st aim is to show tha t all these inclusio ns are lo ca liz a tions, just lik e i 1 : Lex ( A ) − → Mo d ( A ) and i 2 : Sh ( A ) − → Fun ( A ) in the one a rgument case. First note that i 1 and i 2 give r ise to a num b er of lo calizations by lo oking a t the induced Fun ( B , i j ) and Mo d ( B , i j ), and dual versions COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 11 of these. Also , it is immediate to write down the corresp onding lo calizing Serre sub c ategories . F or e x ample, F u n ⊲ ⊲ ( A op × B ) − → Fun ⊲ ( A op × B ) is rea lized as Fun ( B , i 1 ), and the co rresp onding lo calizing Se r re sub catego ry consists of functors that are weakly eﬀacea ble in the second arg ument . In general, for ∗ ∈ { ∅ , ⊳, ⊲, ⋄} and ⋆ ∈ { ⊳, ⊲ } , w e put W ∗ ⋆ = W ∗ ⋆ ( A op × B ) ⊆ F un ∗ ⋆ ( A op × B ) the sub catego ry of functors w ea kly eﬀaceable in the argument designa ted by ⋆ . F or example, in the ab ov e exa mple, the r e le v a nt categor y is W ⊲ ⊲ . Next we turn to the cases we hav en’t cov er ed yet , namely the inclusions i : Fun ∗ ⋄ ( A op × B ) − → Fun ∗ ( A op × B ) . F o r lo calizing Serre sub catego ries S 1 and S 2 of a n abelia n catego r y C , we put S 1 ∗ S 2 = { C ∈ C | ∃ S 1 ∈ S 1 , S 2 ∈ S 2 , 0 − → S 1 − → C − → S 2 − → 0 } . The sub c ategories are ca lled c omp atible [5, 31] if S 1 ∗ S 2 = S 2 ∗ S 1 . In this even t S 1 ∗ S 2 is the s ma llest lo ca lizing Serre s ub ca tegory containing S 1 and S 2 and ( S 1 ∗ S 2 ) ⊥ = S ⊥ 1 ∩ S ⊥ 2 . Deﬁnition 2.21. A mo dule W ∈ Fun ∗ ( A op × B ) is ca lled we akly eﬀac e able if for every ξ ∈ W ( B , A ), there exis t a deﬂation B ′ − → B and an inﬂatio n A − → A ′ such that the induced W ( B , A ) − → W ( B ′ , A ′ ) maps ξ to zero. Prop ositi o n 2.22. The inclusion i : Fun ∗ ⋄ ( A op × B ) − → Fun ∗ ( A op × B ) is a lo c alization with c orr esp onding lo c alizing sub c ate gory W ∗ ⋄ + W ∗ ⊳ ∗ W ∗ ⊲ c onsisting of al l we akly eﬀac e able bifunctors. Pr o of. It suﬃces to show that W ∗ ⊳ and W ∗ ⊲ are compatible and tha t W ∗ ⊳ ∗ W ∗ ⊲ consists of the weakly eﬀacea ble bifunctors . Supp ose we have a n exact sequence 0 − → W 1 − → F − → W 2 − → 0 in Fun ∗ ( A op × B ) w ith W 1 ∈ W ∗ ⊳ and W 2 ∈ W ∗ ⊲ . Consider ξ ∈ F ( B , A ). Since W 2 is w eakly eﬀaceable in the second v a riable, there is an inﬂation A − → A ′ such that the image ξ ′ ∈ F ( B , A ′ ) of ξ gets ma pp ed to zero in W 2 ( B , A ′ ). But then ξ ′ is itself the ima g e o f some ξ ′′ ∈ W 1 ( B , A ). Now we can ﬁnd a deﬂatio n B ′ − → B eﬀaceing the image of ξ ′ in F ( B ′ , A ′ ). Clearly , this is indep endent of exchanging the roles of W 1 and W 2 . Conversely , consider a weakly eﬀaceable W . Deﬁne W 1 ⊆ W by letting W 1 ( B , A ) − → W ( B , A ) contain all e lement s ξ that can b e eﬀa c ed in the ﬁrst v ar iable. It is readily seen that the quotient W/W 1 is weakly eﬀaceable in the seco nd v a riable.  3. Derived sheaf ca tegories In this section we inv es tigate the de r ived functors of the v ar ious inclusions of (bi)sheaf categ ories into (bi)functor catego ries of the previous sectio n. 3.1. Mo dels of deriv ed functors. In this subsection we pr ove Lemma 3.1 on the existence o f dg mo dels of ce r tain der ived functors. Let C b e a small exact category . Let ¯ C − → C b e a k -coﬁbr a nt dg resolution of the k -linear category C . Consider ι : ¯ C − → C − → Lex ( C ) − → C ( Lex ( C )) as an ob ject in the model category o f dg functors Dg F un ( ¯ C , C ( Lex ( C op ))) of [20, Prop osition 5.1]. Then a ﬁbra nt replacement ι − → E yields a dg functor E : ¯ C − → Fib ( C ( Lex ( C ))) COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 12 and ﬁbrant replacements C − → E ( C ) natural in C ∈ ¯ C . Now co nsider a left exa c t functor F : Lex ( C ) − → Mo d ( k ). It gives rise to a dg functor F : Fib ( C ( Lex ( C ))) − → C ( k ) . The co mp os ition F E : ¯ C − → C ( k ) induces a functor RF : C ∼ = H 0 ¯ C − → H 0 C ( k ) − → D ( k ) which is a deriv e d functor of F (with restricted domain). F ur ther more the na tur al functor DgFun ( ¯ C , C ( k )) − → Fun ( C , D ( k )) clearly descends to a funcor D ( ¯ C op ) − → Fun ( C , D ( k )) . Next w e will replace F E by an honest dg functor C − → C ( k ). T o this end w e note that ¯ C − → C induces an equiv alence of c a tegories D ( C op ) − → D ( ¯ C op ). Let RF : C − → C ( k ) b e any representativ e in C ( C op ) = C ( Mo d ( C op )) of a pre-image of F E under this equiv alence. Then the induced functor C − → H 0 C ( k ) − → D ( k ) is a derived functor of F (with restricted do main). W e hav e thus prov en: Lemma 3.1. L et C b e a smal l exact c ate gory and F : Lex ( C ) − → M o d ( k ) a left exact functor. Ther e exists a c omplex RF ∈ C ( Mod ( C op )) such that the c orr esp onding dg functor RF : C − → C ( k ) induc es a r estriction C − → H 0 C ( k ) − → D ( k ) of a de rive d functor of F . 3.2. Derived lo calizations. Next we will inv estigate the der ived functors o f the lo calizations of § 2.6. The following general fact will b e useful. Consider a lo caliza- tion i : C − → D of Grothendieck categories with exact left adjoint a : D − → C . W e hav e a derived adjoint pair Ri : D ( C ) − → D ( D ) and La = a : D ( D ) − → D ( C ). Prop ositi o n 3.2. The functor Ri is ful ly faithful. Pr o of. Endow C ( C ) and C ( D ) with the injective mo del structur es for which coﬁbra- tions ar e p oint wise monomor phisms and w e ak equiv alences are qua si-isomor phisms. Since a preserves bo th o f these classes, by adjunction i preserves ﬁbrations a nd ﬁ- brant ob jects. F or ﬁbrant ob jects E a nd F in C ( C ) w e hav e RHom( Ri ( E ) , R i ( F )) = RHom( i ( E ) , i ( F )) = Hom( i ( E ) , i ( F )) = Hom( E , F ) = RHom( E , F ).  3.3. The derived category of left exact m o dule s . Let C b e a s mall exact category . W e will now c har acterize the essential image of Ri : D + ( Lex ( C )) − → D + ( Mo d ( C )). Deﬁnition 3 . 3. Let C b e an exact category a nd T a triangulated category . A functor F : C − → T is called c ohomolo gic al if for every conﬂation A − → B − → C in C , the image under F c a n be completed into a triangle F ( A ) − → F ( B ) − → F ( C ) − → F ( A )[1] in T . A complex K ∈ C ( Fun ( C )) is called c ohomolo gic al if the induced functor C op − → C ( k ) − → D ( k ) is coho mologica l. Examples 3.4 . (1) F or a Gr othendieck categ ory A , the natura l functor A − → D ( A ) is cohomologic a l. (2) If C ′ − → C is an exact functor b etw een exact catego r ies, T − → T ′ is a tria ng ulated functor b etw een triangulated categories, and C − → T is cohomolog ical, then the comp os ition C ′ − → T ′ is cohomolo gical too . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 13 Prop ositi o n 3.5. L et K ∈ C ( Mo d ( C )) b e a b ounde d b elow c omplex. The fol lowing ar e e quivalent: (1) K ∼ = Ri ( L ) for some L ∈ D ( Lex ( C )) ; (2) K ∼ = Ri ( a ( K )) ; (3) RHom( W, K ) = 0 for every we akly eﬀac e able W ; (4) K is c ohomolo gic al. The implic ations fr om ( i ) to ( j ) with i ≤ j hold without the b oun de dness assu mption on K . Pr o of. The equiv alence o f (1) and (2) is obvious since , by P rop ositio n 3.2, aRi ∼ = 1. T o se e that (2) implies (3) we ta ke K a s in (2 ) and W weakly eﬀac eable a nd we write RHom ( W , K ) = RHom( W, R i ( a ( K )) = RHom( a ( W ) , a ( K )) = 0 since a ( W ) = 0. T o s how that (3) implies (4), supp os e that K s atisﬁes (3) a nd consider a con- ﬂation A − → B − → C . There is an asso cia ted exact sequence 0 − → A ( − , A ) − → A ( − , B ) − → A ( − , C ) − → W − → 0 in Mo d ( C ) in which W is weakly eﬀaceable, proving (4). T o show that (4) implies (1), consider the adjunction morphism K − → R i ( a ( K )). W e ar e to s how that the cone L is acy clic in C ( Mo d ( C )). Then L r e ma ins cohomo- logical, a nd since a ( L ) is acyclic in C ( Lex ( C )) the cohomolo g y ob jects H i of L are weakly eﬀace able. Since L is bo unded below, obviously there is a n n with H i = 0 for all i ≤ n . Let us prove that H i = 0 implies H i +1 = 0. Consider ξ ∈ H i +1 ( C ). There is a conﬂation A − → B − → C such that H i +1 ( C ) − → H i +1 ( B ) maps ξ to zer o. The long exact coho mology sequence induced by the triang le L ( A ) − → L ( B ) − → L ( C ) − → L ( A )[1] contains . . . − → H i ( A ) − → H i +1 ( C ) − → H i +1 ( B ) − → . . . so we conclude that ξ = 0. Co nsequently H i +1 = 0.  W e ha ve the following partial counterpart for the inclusion Ri ′ : D + ( Sh ( C )) − → D + ( F un ( C )). Prop ositi o n 3.6. L et K ∈ C ( F un ( C )) b e a b ounde d b elow c omplex. If K is c oho- molo gic al, then K ∼ = Ri ′ ( a ′ ( K )) . Pr o of. This is the s a me pro of a s for P rop osition 3.5.  R emark 3 .7 . Co nsider ob jects A, B ∈ Lex ( C ). The fact that a cohomolo gical com- plex K ∈ C ( Lex ( C )) res o lving B yields a coho mologica l complex j ′ ( K ) ∈ Sh ( C ) resolving j ′ ( B ) easily shows that the natural map Ext 1 Lex ( C ) ( A, B ) − → Ext 1 Sh ( C ) ( j ′ ( A ) , j ′ ( B )) is an isomorphism, a fact we already know from Pro po sition 2.10. Corollary 3.8. The fun ct or Ri : D + ( Lex ( C )) − → D + ( Mo d ( C )) induc es an e quiv- alenc e D + ( Lex ( C )) − → g D + ( Lex ( C )) wher e g D + ( Lex ( C )) ⊆ D + ( Mo d ( C )) is t he ful l sub c ate gory of c ohomolo gic al c omplexes. Our main interest in Prop osition 3.5 stems from the following: Prop ositi o n 3.9. Consider a left exact functor F ′ : Lex ( C ) op − → Mo d ( k ) with left exact r estriction F : C op − → Mo d ( k ) in Lex ( C ) . L et Ri ( F ) b e the image of F under Ri : D ( Lex ( C )) − → D ( Mo d ( C )) . Any r epr esentative C op − → C ( k ) of Ri ( F ) induc es a funct or C op − → C ( k ) − → D ( k ) which is a r estr iction t o C op of a derive d functor of F ′ . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 14 Pr o of. Apart from our standard universe U , take a lar ger universe V such that D = Lex ( C ) op is V -small. Since D is ab elia n, w e can extend F ′′ : D − → Mo d ( k ) − → V - Mo d ( k ) to a le ft exact functor ˆ F ′′ : V - Lex ( D ) − → V - Mo d ( k ) : colim i D i 7− → colim i F ′′ ( D i ) . Now we can apply Lemma 3.1 to ˆ F ′′ . W e thu s obtain RF ′′ ∈ C ( V - Mo d ( D op )) induc- ing a res tr iction RF ′′ : D − → V - C ( k ) − → V - D ( k ) of a deriv ed functor of ˆ F ′′ , whic h is itself a de r ived functor of F ′′ . If we restric t R F ′′ to RF ∈ C ( V - Mo d ( C )), then this complex is such that the induced functor C op − → V - C ( k ) − → V - D ( k ) is a restric- tion of a deriv ed functor of F ′′ . By Examples 3.4, R F is cohomological, whence, b y Prop ositio n 3.5, RF ∼ = Ri ( a ( RF )), where we consider i : V - L ex ( C ) − → V - Mo d ( C ) and its left adjoint a . Clearly the n - th cohomo logy ob ject of RF corr esp onds to H n RF : C op − → V - C ( k ) − → V - Mo d ( k ), which is the restriction to C op of the n - th derived functor R n F ′′ : Lex ( C ) op − → V - Mo d ( k ) of F ′′ . Now R n F ′′ is eﬀace a ble, so for C ∈ C there is an epimorphism u : X − → C in Lex ( C ) such that R n F ′′ ( u ) = 0. By Le mma 2 .20 there is a further morphism v : C ′ − → X in Lex ( C ) with C ′ ∈ C suc h that uv : C ′ − → C remains a n epimorphism. In pa rticular, H n ( RF ) ∈ V - Mo d ( C ) is weakly eﬀaceable for n > 0 and H 0 ( RF ) = F : C op − → V - Mo d ( k ). Con- sequently , a ( RF ) = F . Moreov e r , s ince in fact F : C op − → U - Mo d , we ha ve RF ∼ = Ri ( F ) ∈ C ( U - Mo d ( C )).  R emark 3.10 . Any left exact functor F : C op − → Mo d ( k ) in Lex ( C ) has a left exact extension F ′ = Lex ( C )( − , F ) : Lex ( C ) op − → Mo d ( k ) . Since Ri ( F ) is obtained by repla cing F by an injective resolution in Lex ( C ), Prop o - sition 3.9 is a k ind of balancednes s result. 3.4. Lo calization i n one o f severa l v ariables. Let a b e a small k -coﬁbrant dg ca tegory and i : L − → D , a : D − → L a lo calization b etw een Grothendieck categorie s. Cons ider the induced lo c alization i ◦ − : DgFun ( a , C ( L )) − → DgFun ( a , C ( D )) , a ◦ − : DgFun ( a , C ( D )) − → Dg F u n ( a , C ( L )) where the inv olved ca tegories are endow ed with the model structure of [20, Pro p o - sition 5.1]. Lemma 3.11. If F ∈ DgFun ( a , C ( L )) is such that for every A ∈ a , F ( A ) is ﬁbr ant, then F is ( i ◦ − ) -acyclic, i.e R ( i ◦ − )( F ) ∼ = iF in Ho( DgFun ( a , C ( L ))) . Pr o of. T ake a ﬁbr ant res o lution F − → E in DgFun ( a , C ( L )). Since a is k -coﬁbrant, for every A ∈ a , E ( A ) is ﬁbr ant. Consequently , every F ( A ) − → E ( A ) is a weak equiv ale nc e b etw een ﬁbrant ob jects, whence a ho mo topy eq uiv a lence. Since i ( F ( A )) − → i ( E ( A )) remains a homotopy equiv alence, iF − → i E is a weak equiv- alence as desired.  3.5. Sheav es i n one of several v ariable s. As so on a s we want to extend the r e- sults of the previous subsections to bimodules, ﬂatness ov er the ground ring comes int o pla y . The reason fo r this is that in the abs ence of ﬂatness, injectiv e r esolu- tions of bimo dules do not y ield injective resolutions in indiv idua l v ar iables. More precisely , we hav e the following situation. Let C b e a small exa ct categor y and a a small k -linear categor y , a nd consider the catego ry Mo d ( a , Mo d ( C )) ∼ = Mo d ( a op ⊗ C ) . The lo ca lization Lex ( C ) − → Mo d ( C ) gives rise to a lo caliza tio n i C : Mo d ( a , Lex ( C )) − → M o d ( a , Mo d ( C )) . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 15 Lemma 3. 12. F or every A ∈ a , the pr oje ction ev A : Mo d ( a , Lex ( C )) − → L ex ( C ) : F 7− → F ( A ) has a left adjoint given by M 7− → ( A ′ 7− → a ( A, A ′ ) ⊗ k M ) . If a has k -ﬂat homsets, t hen t his adjoint is exact, and ev A pr eserves inje ctives. Pr o of. This is clear.  Let Lex ( Lex ( C )) denote the ca tegory of left exact a dditive functors Lex ( C ) op − → Mo d ( k ). The exa ct inclusion s : C − → Lex ( C ) induces a r estriction π : Lex ( Lex ( C )) − → Lex ( C ) : G 7− → Gs a nd the inclusion functor ι : Lex ( C ) − → Lex ( Lex ( C )) : F 7− → Lex ( C )( − , F ) satisﬁes π ι = 1 Lex ( C ) . Let F ′ : a − → Lex ( Lex ( C )) be an a dditiv e functor with r estriction F = π F ′ : a − → Lex ( C ). The following result extends Prop osition 3.9 to mo dules left exact in one of several v ar ia bles. Prop ositi o n 3.13. L et a , C , F ′ and F b e as ab ove and let Ri C ( F ) b e the image of F under Ri C : D ( Mo d ( a , Le x ( C ))) − → D ( Mo d ( a , Mo d ( C ))) . If a has k -ﬂat homsets, then for any K ∈ C ( Mod ( a , Mo d ( C ))) r epr esenting R i C ( F ) and for any A ∈ a , K ( A ) ∈ C ( Mo d ( C )) induc es a functor C op − → C ( k ) − → D ( k ) which is a r estr iction t o C op of a derive d functor of F ′ ( A ) : Lex ( C ) op − → Mo d ( k ) . Pr o of. Let F − → E b e an injective resolution of F ∈ Mo d ( a , Le x ( C )). Then for ev- ery A ∈ a , F ( A ) − → E ( A ) is an injective resolutio n in Lex ( C ) by Lemma 3.12. Con- sequently , fo r the inclusion i : Lex ( C ) − → Mo d ( C ), we hav e R i ( F ( A )) = i ( E ( A )) = i C ( E )( A ) = R i C ( F )( A ) in D ( Mo d ( C )) hence the result follows from Prop ositio n 3.9.  If, in the ﬁr st ar gument, w e consider functors rather than mo dules, the ﬂatness issue go es awa y . W e a re interested in the following application. Let B and A b e small exact categorie s and let F ′ : A − → Lex ( Lex ( B )) b e a p ossibly no n-additive functor with restriction F = π F ′ : A − → Lex ( B ). Co nsider the inclusion i B : F u n ( A , Lex ( B )) − → Fun ( A , Mo d ( B )) . Corollary 3.14. L et A , B , F ′ and F b e as ab ove and let R i B ( F ) b e the image of F under Ri B : D ( F u n ( A , Lex ( B ))) − → D ( Fun ( A , Mo d ( B ))) . F or any K ∈ C ( F un ( A , Mo d ( B ))) r epr esenting Ri B ( F ) and for any A ∈ A , K ( A ) ∈ C ( Mo d ( B )) induc es a functor B op − → C ( k ) − → D ( k ) which is a r estriction t o B op of a derive d functor of F ′ ( A ) : Lex ( B ) op − → Mo d ( k ) . Pr o of. This immedia tely follows from Pr op osition 3.13 by putting C = B and a = Z A , the free Z -linea r categ o ry on A (having Ob( Z A ) = Ob( A ) and ( Z A )( A, A ′ ) = Z ( A ( A, A ′ )), the free a b elia n group on A ( A, A ′ )), and noting that Fun ( A , Lex ( B )) ∼ = Mo d ( Z A , Lex ( B )).  3.6. Sheav es in t wo v ariables. In this section we consider sheaves in b oth v ari- ables. W e start with a v ers ion of P rop osition 3 .5. F or small exa ct catego ries A and B , co nsider the inclusio ns i : Fun ∗ ⋄ ( A op × B ) − → Fun ∗ ( A op × B ) for ∗ ∈ { ∅ , ⊳, ⊲, ⋄} , along with the der ived functors Ri : D ( Fun ∗ ⋄ ( A op × B )) − → D ( Fun ∗ ( A op × B )) . As usual, the left a djoint s of i and R i ar e denoted by a . F o r mo dules F ∈ Mod ( B ) and G ∈ Mo d ( A op ), F ⊠ G ∈ Mo d ( A op ⊗ B ) deno tes the bimo dule with ( F ⊠ G )( B , A ) = F ( B ) ⊗ G ( A ). COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 16 Prop ositi o n 3.15. F or K ∈ C ( Fun ∗ ( A op × B )) , c onsider the fol lowing pr op erties: (1) K ∼ = Ri ( L ) for some L ∈ D ( Fun ∗ ⋄ ( A op × B )) ; (2) K ∼ = Ri ( a ( K )) ; (3) RHom( W, K ) = 0 for every we akly eﬀac e able W ; (4) K is c ohomolo gic al in b oth variables. (5) K is c ohomolo gic al in the ﬁrst variable (i.e. for every A ∈ A , the c omplex K ( − , A ) ∈ C ( F un ( B )) is c ohomolo gic al). The fol lowing facts hold true: (i) (1) and (2) ar e e quivalent and (1) implies ( 3). (ii) If K is b ounde d b elow, then (4) implies (1). (iii) If k = Z and ∗ = ⊳ , then (3) implies (5). (iv) If k is a ﬁeld and ∗ = ⋄ , t hen (3) implies (4). Pr o of. (i) This is prov en like in Prop os ition 3.5. (ii) Supp ose that K is b ounded below and that (4) holds. T o prov e that (4 ) implies (1), a s in the pr o of of Propo si- tion 3.5 it is suﬃcien t to show that if K is cohomolo gical in both v ariables and has weakly e ﬀa c eable co homology ob jects H i , then H i = 0 implies H i +1 = 0. Consider ξ ∈ H i +1 ( B , A ). T ake conﬂations A − → A ′ − → A ′′ and B ′′ − → B ′ − → B such that H i +1 ( B , A ) − → H i +1 ( B ′ , A ′ ) maps ξ to zero. F rom the diagram H i ( B ′′ , A ′ )   H i ( B , A ′′ ) / / H i +1 ( B , A )   / / H i +1 ( B , A ′ )   H i +1 ( B ′ , A ) / / H i +1 ( B ′ , A ′ ) with exa c t middle row and last column we deduce that ξ = 0. Conseq ue ntly H i +1 = 0. W e now give the pro of of (iii), the proof of (iv) is similar. Let k = Z a nd supp os e K satisﬁes (3). Consider a conﬂation B ′ − → B − → B ′′ in B and the aso ciated exact sequence 0 − → B ( − , B ′ ) − → B ( − , B ) − → B ( − , B ′′ ) − → W − → 0 with W weakly eﬀaceable in Mo d ( B ). F or A ∈ A , the sequence 0 − → B ( − , B ′ ) ⊠ A ( A, − ) − → B ( − , B ) ⊠ A ( A, − ) − → B ( − , B ′′ ) ⊠ A ( A, − ) − → W ⊠ A ( A, − ) − → 0 remains exact in Mo d ( A op ⊗ Z B ). An elemen t P n i =1 w i ⊗ f i ∈ W ( Y ) ⊗ Z A ( A, X ) can b e eﬀaced by comp osing ﬁnitely many B - deﬂations, so W ⊠ A ( A, − ) is weakly eﬀaceable in the ﬁr st v aria ble in F un ⊳ ( A op × B ). Finally , since B ( − , B ) ⊠ Z A ( A, − ) = (( Z A ) op ⊗ B )( − , ( B , A )), we obtain the desir ed triangle b y considering RHom( − , K ).  Corollary 3.1 6. Supp ose k is a ﬁ eld. The fu nctor R i : D + ( F un ⋄ ⋄ ( A op × B )) − → D + ( F un ⋄ ( A op × B )) induc es an e quivalenc e D + ( F un ⋄ ⋄ ( A op × B )) − → g D + ( F un ⋄ ⋄ ( A op × B )) wher e g D + ( F un ⋄ ⋄ ( A op × B )) ⊆ D + ( F un ⋄ ( A op × B )) is the ful l s u b c ate gory of c omplexes that ar e c ohomolo gic al in b oth variables. F o r small e x act catego ries A and B , consider the inclusions i : Fun ⊳ ⋄ ( A op × B ) − → Fun ⊳ ( A op × B ) and i B : F un ⊳ ⊳ ( A op × B ) − → Fun ⊳ ( A op × B ) which has an equiv alent incarnation: i B : Fun ( A , Lex ( B )) − → F un ( A , Mo d ( B )) . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 17 Consider F ∈ Fun ⊳ ⋄ ( A op × B ) along with its natural extensio n F ′ : A − → Lex ( B ) − → Lex ( Lex ( B ) : A 7− → Lex ( B )( − , F ( A )) . Let F − → E b e an injectiv e r e solution in Fun ( A , Lex ( B )). W e hav e Ri B ( F ) = i B E = E and for every A ∈ A , F ( − , A ) − → E ( − , A ) is an injective resolution in Lex ( B ). According to Corollary 3.14, E ( − , A ) : B op − → C ( k ) induces a res tric- tion of a der ived functor of Lex ( B )( − , F ( A )), a nd E ( − , A ) is itself a restrictio n of Hom Lex ( B ) ( − , E ( − , A )). Prop ositi o n 3 . 17. L et A , B and F b e as ab ove. Supp ose F : A − → Lex ( B ) is exact (i.e. maps c onﬂations t o short exact se quenc es). Then R i B ( F ) is c ohomolo gic al in b oth variables and Ri B ( F ) ∼ = Ri ( F ) . Pr o of. Let Ri B ( F ) = E as ab ov e. By the ab ov e discussion, for A ∈ A , E ( − , A ) : B op − → C ( k ) is cohomolo gical, H 0 E ( − , A ) = F ( − , A ) a nd the other co homology ob ject ar e weakly eﬀacea ble in M o d ( B ). In particular, H 0 E = F and the hig her cohomolog y o b jects are weakly eﬀacea ble, whence a ( E ) = F . Now ﬁx B ∈ B and consider E ( B , − ) : A − → C ( k ). Let us show that this functor is coho mological. Let A ′ − → A − → A ′′ be a conﬂa tion in A . By assumption, 0 − → F ( A ′ ) − → F ( A ) − → F ( A ′′ ) − → 0 is a short e xact sequence in Lex ( B ). No w by natura lity of F − → E we obtain a commutativ e diagram F ( A ′ ) / /   F ( A ) / /   F ( A ′′ )   E ( A ′ ) / / E ( A ) / / E ( A ′′ ) in C ( Lex ( B )) in which the vertical arr ows are quas i- isomorphisms . As a conse- quence, the lower row can b e c o mpleted in to a triangle in Fib ( C ( Lex ( B ))). The func- tor Lex ( B )( B ( − , B ) , − ) : Fib ( C ( Lex ( B ))) − → C ( k ) ma ps this triangle to a triang le in D ( k ) as desir e d. Finally b y Prop ositio n 3.1 5, we conclude that E ∼ = Ri ( F ).  In the remainder of this subsection, let k b e a ﬁeld. F or small exact k -linear categorie s A and B , consider the inclusions i : Fun ⋄ ⋄ ( A op × B ) − → Fun ⋄ ( A op × B ) , i A : F un ⋄ ⊲ ( A op × B ) − → Fun ⋄ ( A op × B ) which has an equiv alent incarnation: i A : Mo d ( B op , Lex ( A op )) − → Mo d ( B op , Mo d ( A op )) and i B : F un ⋄ ⊳ ( A op × B ) − → Fun ⋄ ( A op × B ) which has an equiv alent incarnation: i B : Mo d ( A , Lex ( B )) − → M o d ( A , Mo d ( B )) . Consider F ∈ Fun ⋄ ⋄ ( A op × B ). Prop ositi o n 3.1 8. If F : B op − → Lex ( A op ) is exact, then R i A ( F ) is c ohomolo gic al in b oth variables and Ri A ( F ) ∼ = Ri ( F ) . If F : A − → Lex ( B ) is exact, then Ri B ( F ) is c ohomolo gic al in b oth variables and Ri B ( F ) ∼ = Ri ( F ) . Pr o of. Similar to the pro of of Pr op osition 3.17.  COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 18 4. Cohomol ogy of exact ca tegories In this section we discuss a num b er o f diﬀerent co homology expressions for ex - act catego r ies and more g enerally for linea r sites . W e s tart with ex pressions “of Ho chsc hild type”. Our main results are ov er a ﬁeld. W e r elate the cohomolo gy of a Gr othendieck categor y D of [20] to E xt’s in the large additive functor categ ory Add ( D , D ) (Theo rem 4.6). F or a small exact c ategory C , the co homology of [11] corres p o nds to the co homology of the Grothendieck categ ory Lex ( C ), similar to the situation for abelian catego ries in [20] (Theorem 4.2). W e show that this coho- mology ca n also b e express ed as Ext’s in the ca tegory F un ⋄ ⋄ ( C op × C ) of bimo dules that are sheaves (in other words, left exact) in b oth v a riables (Theorem 4.5). This expression or iginated fro m [10]. F o r mo dule categor ies, some of these Ho chsc hild expressions b ea r re s emblance to an incarnation of Mac Lane cohomology disc ov ered in [9]. Inspir ed by this, w e deﬁne Mac Lane cohomo logy for linea r sites ( § 4.7). Fi- nally , we sho w that for an exact ca tegory C this coho mology can also b e expressed as Ext’s in the catego ry Fun ⊳ ⋄ ( C op × C ) o f bifunctors that are additive in the ﬁrst v a riable and s heav es in both v a riables (Theorem 4 .14). 4.1. Ho c hsc h i ld-Shuk la cohomo logy of dg categories. Let k b e a commuta- tive ring. Let a b e a k -linea r dg categor y and M an a - bimo dule. Rec all tha t the Ho chsc hild complex C ho ch ( a , M ) of a with v alues in M is the pro duct total complex of the do uble complex with p -th co lumn Y A 0 ,...,A p Hom k ( a ( A p − 1 , A p ) ⊗ k · · · ⊗ k a ( A 0 , A 1 ) , M ( A 0 , A p )) and the usual Ho chsch ild diﬀerential. The Ho chsc hild complex of a is C ho ch ( a ) = C ho ch ( a , a ). If a is k -co ﬁbrant, then C ho ch ( a , M ) ∼ = RHom a op ⊗ a ( a , M ) in D ( k ). F o r a arbitra ry , the Sh ukla complex of a is b y deﬁnition the Ho chsc hild complex of a k -coﬁbrant dg resolution ¯ a − → a , i.e. C sh ( a , M ) = C ho ch ( ¯ a , M ) . 4.2. Ho c hsc h i ld-Shuk la cohomolog y of Grothendi ec k categories. In [20], Ho chsc hild-Shukla cohomo logy was deﬁned for ab elian categ o ries. F or a Grothendieck category , a c o nv enient deﬁnition is C gro ( D ) = C sh ( inj ( D )) where inj ( D ) is the linea r c a tegory o f injectiv es in D . Now let ( u , T ) be an additive site with additiv e s heaf categ ory Sh ( u ) and canonica l map u : u − → Sh ( u ). F or every U ∈ u , cho o se an injective r esolution u ( U ) − → E U and let u dg ⊆ C ( Sh ( u )) be the full dg sub ca tegory consisting o f the E U . It is proven in [20] that C gro ( Sh ( u )) ∼ = C sh ( u dg ) . W e ﬁnally re c a ll the following more technical result [20, Lemma 5.4.2 ], whic h will be crucial for us. Let r : ¯ u − → u be a k -coﬁbra nt res olution a nd take a ﬁbr ant replacement ur − → E in the mo del categ o ry DgFun ( ¯ u , C ( Sh ( u ))) of [20, Prop o- sition 5 .1]. Then E naturally de ﬁnes a ¯ u - ¯ u -bimo dule by E ( U, V ) = E ( V )( U ) = Hom Sh ( u ) ( ur ( U ) , E ( V )) and w e hav e (7) C gro ( Sh ( u )) ∼ = C sh ( ¯ u , E ) . In the re mainder of this subsection, let k b e a ﬁeld. Consider the lo calizatio n i ◦ − : Mo d ( u , Sh ( u )) − → Mo d ( u , M o d ( u )) , COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 19 a ◦ − : Mo d ( u , Mo d ( u )) − → Mo d ( u , Sh ( u )) induced by i : Sh ( u ) − → Mo d ( u ), a : Mo d ( u ) − → Sh ( u ). Prop ositi o n 4.1. We have: C gro ( Sh ( u )) ∼ = C ho ch ( u , R ( i ◦ − )( u )) ∼ = RHom Mo d ( u , Sh ( u )) ( u, u ) . F urthermor e, for every natur al tr ansformation u − → F in C ( Mo d ( u , Sh ( u ))) for which every u ( U ) − → F ( U ) is an inje ctive re s olut ion, we have: C gro ( Sh ( u )) ∼ = C ho ch ( u , F ) . Pr o of. Since w e are ov er a ﬁeld, we can ta ke ¯ u = u and u − → E a n injective resolution of u in M o d ( u , Sh ( u )). By construction R ( i ◦ − )( u ) = iE a nd hence C gro ( Sh ( u )) ∼ = C ho ch ( u , iE ) ∼ = RHom u op ⊗ u ( I , R ( i ◦ − )( u )) ∼ = RHom Mo d ( u , Sh ( u )) ( u, u ) . F ur thermore, by Lemma 3.1 1 we have R ( i ◦ − )( u ) ∼ = iF .  4.3. Ho c hsc h i ld-Shuk la cohom o logy of exact categories. Let k be a com- m utative r ing. Let C be a s ma ll exa ct categ ory . In this se ction we discuss some deﬁnitions o f Ho chsc hild-Shukla cohomolog y of C . The ﬁrst deﬁnition is due to Kelle r [11]. Let C b dg ( C ) be the dg categor y of bo unded co mplexes of C -ob jects, a nd Ac b dg ( C ) its full dg sub categ ory of acyclic complexes. Then for the dg quo tien t D b dg ( C ) = C b dg ( C ) / Ac b dg ( C ): C ex ( C ) = C sh ( D b dg ( C )) . In [20], the a uthors deﬁned Hochsc hild-Shu kla cohomo logy o f abelia n categ ories. This deﬁnition ha s the following generalization to exa c t categor ie s: C ex ′ ( C ) = C gro ( Lex ( C )) = C sh ( inj ( Lex ( C )) . Using Pro po sition 2.19 it is ea s ily see n (see [20, Lemma 6.3]) that a concre te mo del for D b dg ( C ) is given by the full subcateg o ry of C dg ( Lex ( C )) of b ounded b elow complexes of injectives with b ounded co homology in C . W e also intro duce the full sub c ategory C dg ⊆ C dg ( Lex ( C )) of p ositively graded complexes o f injectives who se only cohomo logy is in degree zero and in C . The following is prov e n in exactly the same way as [20, Theor e m 6.2]: Theorem 4.2. Ther e ar e qu asi-isomorphi sm s C ex ′ ( C ) ∼ = C sh ( C dg ) ∼ = C ex ( C ) . Sh ukla c o homology o f an exact category in terp olates be tween Shukla co homology of a k - linear categor y and Shukla cohomolo gy of an abelia n category . Of co urse, a n arbitrar y k -linea r categ ory is not exact since it is not additiv e, but this can easily be remedied by adding ﬁnite bipro ducts. Prop ositi o n 4 . 3. L et a b e a k -line ar c ate gory and free ( a ) the exact c ate gory of ﬁnitely gener ate d fr e e a - mo dules with split exact c onﬂations. We have: C ex ( free ( a )) ∼ = C sh ( a ) . Pr o of. W e hav e Lex ( free ( a )) ∼ = Mo d ( free ( a )) ∼ = Mo d ( a ) (see Remark 2.4). Hence it follows from [20] that C ex ′ ( free ( a )) ∼ = C sh ( a ).  COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 20 4.4. Ho c hsc h i ld coho m ology and (bi)sheaf categories . Let k b e a ﬁeld and C a small exact k -linea r ca tegory . Let ι : C − → Lex ( C ) be the canonical embedding. The re s ults of the previous subsectio ns yield: Prop ositi o n 4.4. We have: C ex ( C ) ∼ = RHom Mo d ( C , Lex ( C )) ( ι, ι ) . Pr o of. This is an applica tion of Prop osition 4.1 to u = ι : C − → Lex ( C ).  Let I deno te the iden tity C -bimo dule with I ( C ′ , C ) = C ( C ′ , C ). Using the results of § 3.6, we obtain the following symmetric ab elian expression, whic h also appea red in [10]: Theorem 4.5. We have: C ex ( C ) ∼ = RHom F u n ⋄ ⋄ ( C op ×C ) ( I , I ) . Pr o of. Consider i 1 : Fun ⋄ ⊳ ( C op × C ) − → Fun ⋄ ⋄ ( C op × C ), which is isomo rphic to i − : Mo d ( C , Le x ( C )) − → M o d ( C , Mo d ( C )). Prop osition 4.4 translates into C ex ( C ) ∼ = RHom F u n ⋄ ⊳ ( C op ×C ) ( I , I ). F ro m Pr o p osition 3.18 we further o btain R i 1 ( I ) ∼ = Rj ( I ) for j : F un ⋄ ⋄ ( C op × C ) − → Fun ⋄ ( C op × C ), so C ex ( C ) ∼ = RHom F u n ⋄ ( I , Rj ( I )) ∼ = RHom F u n ⋄ ⋄ ( C op ×C ) ( I , I ) by adjunction.  4.5. Ho c hsc h i ld cohomology and large additive functor categories . Let k be a ﬁeld. The following deﬁnition of Ho chsc hild cohomology of a (po ssibly large) ab elian category D was communicated to the sec ond author by Ragnar Buc hw eitz, who a ttr ibuted it to John Greenlees. One considers the (p ossibly large) ab elian category Add ( D , D ) of additive functors from D to D and puts H H n top ( D ) = Ext n Add ( D , D ) (1 D , 1 D ) . This subsection is devoted to the pro of of the following Theorem 4.6. F or a Gr othendie ck c ate gory D , we have H H n gro ( D ) ∼ = H H n top ( D ) . The theorem is known to hold true for mo dule categ o ries (see [8], [9]), and the pro of of the theorem relies heavily on this ca s e, which we ﬁrst discuss. F o r later use, apart from our standa rd univ erse U , w e intro duce another universe U ⊆ V . As usual, U is s uppr essed in the notations. Let a be a small linear category . Consider the adjoint pair R : Add ( Mo d ( a ) , V - Mo d ( a )) − → Add ( a , V - M o d ( a )) ∼ = V - Mo d ( a op ⊗ a ) : F 7− → F | a and L : V - M o d ( a op ⊗ a ) − → Add ( Mo d ( a ) , V - Mo d ( a )) : M 7− → M ⊗ a − . Let I ∈ V - Mo d ( a op ⊗ a ) b e the iden tity bimo dule and j : M o d ( a ) − → V - Mo d ( a ) the natural inclusio n. Lemma 4 .7 (see [8], [9]) . F or M ∈ C ( Add ( Mo d ( a ) , V - M o d ( a ))) , we have RHom V - Mo d ( a op ⊗ a ) ( I , M | a ) ∼ = RHom Add ( Mo d ( a ) , V - Mo d ( a )) ( j, M ) . Pr o of. Let B ( I ) − → I b e the ba r r e solution of I in V - Mo d ( a op ⊗ a ). C o ncretely , we hav e B n ( I ) = ⊕ A 0 ,...,A n a ( A n , − ) ⊗ k a ( A n − 1 , A n ) ⊗ k · · · ⊗ k a ( − , A 0 ) . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 21 Pro jectivity of L ( B n ( I )) follows automa tically from the adjunction since R is exa ct. T o see that L ( B ( I )) − → L ( I ) = j remains a reso lution, it s uﬃces to chec k its ev a luation at a n arbitra ry X ∈ Mo d ( a ). W e hav e L ( B n ( I ))( X ) = ⊕ A 0 ,...,A n X ( A n ) ⊗ k a ( A n − 1 , A n ) ⊗ k · · · ⊗ k a ( − , A 0 ) , so this is precisely the bar r esolution of X . Finally , we can write RHom V - Mo d ( a op ⊗ a ) ( I , M | a ) = Hom V - Mo d ( a op ⊗ a ) ( B ( I ) , R ( M )) = Hom Add ( Mo d ( a ) , V - Mo d ( a )) ( L ( B ( I )) , M ) = RHom Add ( Mo d ( a ) , V - Mo d ( a )) ( j, M ) .  Obviously , tak ing U = V and M = j = 1 Mo d ( a ) in Lemma 4.7 yields Theorem 4.6 for D = Mo d ( a ). Now let D b e an arbitra ry Gr othendieck category and c ho os e an equiv alence D ∼ = Sh ( u ) = Sh ( u , T ) for an additiv e top olo gy T on a small Z -linea r catego r y u (see § 2 .1). F rom now on, we choose U ⊆ V in such a wa y that Mo d ( u ) and Sh ( u ) are V -small, and we consider the categ ories V - Mo d ( u ), V - Sh ( u ). W e have a commutativ e diag ram: Mo d ( u ) j / / a   V - M o d ( u ) a ′   Sh ( u ) i O O j ′ / / V - Sh ( u ) . i ′ O O . The pro of consis ts o f three steps, and so me remarks on how to get rid o f the additional universe V . First, we take an injectiv e res olution j ′ a − → E in the V -Grothendieck catego ry Add ( Mo d ( u ) , V - Sh ( u )). Then the restriction j ′ aI − → E I for I : u − → Mo d ( u ) yields a functorial choice of injective reso lutions a ′ ( u ( − , U )) − → E ( u ( − , U )) in V - Sh ( u ). By Pro po sition 4.1 and Lemma 4.7, w e hav e (8) C gro ( V - Sh ( u )) ∼ = RHom V - Mo d ( u op ⊗ u ) ( j I , i ′ E I ) ∼ = RHom Add ( Mo d ( u ) , V - Mo d ( u )) ( j, i ′ E ) . F o r the second step, we note that the lo caliz a tion ( a ′ , i ′ ) induces a lo calization a ′ ◦ − : Add ( Mo d ( u ) , V - Mo d ( u )) − → Add ( Mo d ( u ) , V - Sh ( u )) and i ′ ◦ − : Add ( Mo d ( u ) , V - Sh ( u )) − → Add ( Mo d ( u ) , V - Mo d ( u )) . W e thus obtain (9) RHom Add ( Mo d ( u ) , V - Mo d ( u )) ( j, i ′ E ) = RHom Add ( Mo d ( u ) , V - Mo d ( u )) ( j, R ( i ′ − ) E ) ∼ = RHom Add ( Mo d ( u ) , V - Sh ( u )) ( a ′ j, E ) . F o r the third step, we use the following localiza tion induced by ( a, i ): − ◦ a : Add ( Sh ( u ) , V - Sh ( u )) − → Add ( Mod ( U ) , V - Sh ( u )) , − ◦ i : Add ( Mo d ( U ) , V - Sh ( u )) − → Add ( Sh ( u ) , V - Sh ( u )) . Since b oth functors are exact, we obtain: (10) RHom Add ( Mo d ( u ) , V - Sh ( u )) ( a ′ j, E ) ∼ = RHom Add ( Mo d ( u ) , V - Sh ( u )) ( j ′ a, E ) ∼ = RHom Add ( Sh ( u ) , V - Sh ( u )) ( j ′ , E i ) ∼ = RHom Add ( Sh ( u ) , V - Sh ( u )) ( j ′ , j ′ ai ) ∼ = RHom Add ( Sh ( u ) , V - Sh ( u )) ( j ′ , j ′ ) COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 22 Putting (8), (9) a nd (10) together, we now arrive at C gro ( V - Sh ( u )) ∼ = RHom Add ( Sh ( u ) , V - Sh ( u )) ( j ′ , j ′ ) . Finally , w e need s ome remar ks conce r ning the universe V . The functor j ′ : Sh ( u ) − → V - Sh ( u ) a prio ri do es not preser ve injective ob jects, whereas j : Mo d ( u ) − → V - Mo d ( u ) do e s (using the Baer criter ium). Ho wever, Sh ( u ) ha s enough injectives that ar e preverved by j ′ . Indeed, for a sheaf F ∈ Sh ( u ), a n ess e ntial mono mor- phisms iF − → M to an injective M ∈ Mo d ( u ) a c tua lly yields a mono morphism int o an injective shea f, and all inv olved notions ar e preser ved by j . In particula r, j ′ preserves Ext. Thinking of actua l extensions, it is then readily seen that j ′ ◦ − : Add ( Sh ( u ) , Sh ( u )) − → Add ( Sh ( u ) , V - Sh ( u )) also preser ves E xt, whence Ext n Add ( Sh ( u ) , Sh ( u )) (1 Sh ( u ) , 1 Sh ( u ) ) ∼ = Ext n Add ( Sh ( u ) , V - Sh ( u )) ( j ′ , j ′ ) . Let us now lo ok at C gro ( Sh ( u )). If we take for every U ∈ u a “sp ecial” injective resolution a ( u ( − , U )) − → E U , then the dg category u dg ⊆ C ( Sh ( u )) of a ll these resolutions satisﬁe s C gro ( Sh ( u )) ∼ = C ho ch ( u dg ) . T a king the imag e s of the E U under j yields a qua si-equiv alent dg category , whence C gro ( Sh ( u )) ∼ = C gro ( V - Sh ( u )) . This ﬁnishes the pro of o f Theor em 4.6. 4.6. Mac Lane cohomo logy o f Z -line ar categories. Mac Lane cohomolo gy originated in [2 3] as a cohomo logy theory for r ings A taking v alues in bimo dules. In [9], the authors discov er ed a n incar nation allowing for a natural generaliza tion to Mac Lane cohomolog y with v alues in non-additive functors free ( A ) − → Mo d ( A ). W e review the situation for a small Z -linear categor y a . F o r an ab elian gr oup A , denote by Q ( A ) the cub e co nstruction of A [23]. This is a co chain co mplex of ab elian groups in nonp os itive deg rees, together with an augmentation Q 0 ( A ) − → A such that H 0 ( Q ( A )) ∼ = A . F or ab elian groups A and B , there is a natural pairing Q ( A ) ⊗ Q ( B ) − → Q ( A ⊗ B ) . This allows us to deﬁne a diﬀerential g raded Z -linear categ ory Q ( a ) with Q ( a )( A, B ) = Q ( a ( A, B )) for A, B ∈ a and comp os ition morphisms Q ( a ( B , C )) ⊗ Q ( a ( A, B )) − → Q ( a ( B , C ) ⊗ a ( A, C )) − → Q ( a ( A, C )) just like in the ring case. F or M ∈ C ( Mo d ( a op ⊗ a )), w e put C mac ′ ( a , M ) = C ho ch ( Q ( a ) , M ) where the right hand side is Ho chsc hild cohomolo gy of the dg categ o ry Q ( a ) with v a lues in the dg bimo dule M . Now consider the inclusion I : ˜ a − → Mo d ( a ) of a full additive sub categ ory containing a , and a co chain complex M ∈ C ( Add ( ˜ a , Mo d ( a ))). W e denote both the restriction of M to C ( Add ( a , Mo d ( a )) = C ( Mo d ( a op ⊗ a )) a nd the imag e o f M in C ( Fun ( ˜ a , Mo d ( a )) - the category of co c ha in complexes of non-additive functors - by M . W e hav e: Theorem 4.8. [9, Theorem A] Ther e is an isomorphism C mac ′ ( a , M ) ∼ = RHom F u n ( ˜ a , Mod ( a )) ( I , M ) . Consider the following tw o v ar iants o f Mac Lane cohomology: COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 23 Deﬁnition 4.9. (1) F or a Z -linear catego ry a and T ∈ C ( F un ( a , Mo d ( a ))), C mac ( a , T ) = RHom F u n ( a , Mo d ( a )) ( I , T ) . (2) F or a Z -linear ca tegory a and T ∈ C ( Fun ( free ( a ) , Mo d ( a ))), C mac ′ ( a , T ) = RHom F u n ( free ( a ) , Mo d ( a )) ( I , T ) . R emark 4.10 . The tw o notions in Deﬁnition 4.9 ar e related in the following way . It a is a dditive, then a ∼ = free ( a ) and if T and T ′ corres p o nd under the equiv alence C ( Fun ( a , M o d ( a ))) ∼ = C ( Fun ( fr ee ( a ) , M o d ( a ))), then H H n mac ( a , T ) ∼ = H H n mac ′ ( a , T ′ ) . If a is arbitrary , then Mod ( a ) ∼ = Mo d ( free ( a )) and if T a nd T ′ corres p o nd under the equiv a lence C ( Fun ( free ( a ) , Mo d ( a ))) ∼ = C ( Fun ( fr ee ( a ) , M o d ( free ( a )))), then H H n mac ′ ( a , T ) ∼ = H H n mac ( free ( a ) , T ′ ) . By Theor em 4.8, C mac ′ ( a , T ) directly generalizes the earlier deﬁnition for M ∈ C ( Mo d ( a op ⊗ a )) ∼ = C ( Add ( free ( a ) , M o d ( a ))). As proven in [9], Mac La ne cohomolog y a ls o has a natural interpretation in terms of Ho chsc hild-Mitchel cohomolog y . Let a b e a small (non-linea r) category . Rec all from [2] that a na tural system M on a is given b y ab elia n groups M ( λ ) a sso ciated to the mo rphisms λ : A − → B of a , and morphisms M ( λ ) − → M ( λ ′ ) ass o ciated to comp ositions λ ′ = b λa in a with a : A ′ − → A and b : B − → B ′ (satisfying the natural asso cia tivity condition). Ho chsc hild-Mitchel cohomolo gy of a with v alues in M is the cohomolo gy of the natura l “Ho chsc hild type” c o mplex with C n mitch ( a , M ) = Y ( λ 1 ,...,λ n ) ∈ N n ( a ) M ( λ n . . . λ 1 ) where A 0 λ 1 / / A 1 / / . . . λ n / / A n is a sequence of a -morphisms in the nerve of a . W e have C mitch ( a , M ) ∼ = RHom Nat ( a ) ( Z , M ) where Nat ( a ) is the ab elian category of natural systems on a and Z is the constant natural system with Z ( λ : A − → B ) = Z . A bifuncor M : a op × a − → Ab is naturally co nsidered as a natural system with M ( λ : A − → A ′ ) = M ( A, A ′ ). Now w e return to the setting of a Z -linea r categ o ry a . F or T ∈ C ( Fun ( a , Mo d ( a ))), consider T as a co mplex of bifunctors a op × a − → Ab , and hence as a natural system. Prop ositi o n 4.11. [9, Prop ositio n 3.12] We have: C mac ( a , T ) ∼ = C mitch ( a , T ) . 4.7. Mac Lane cohomolo g y of additiv e si tes. In this subsection, w e a dapt the notions o f the pr evious subsectio n to the situation o f a linear site. Let ( u , T ) b e a Z - linear site with additive sheaf catego ry Sh ( u ) and ca nonical functor u : u − → Sh ( u ). W e start with an analo g ue of Pr op osition 4.1. Consider the loca lization i ◦ − : Fun ( u , Sh ( u )) − → Fun ( u , Mo d ( u )) , a ◦ − : Fun ( u , Mo d ( u )) − → Fun ( u , Sh ( u )) induced by i : Sh ( u ) − → Mo d ( u ), a : Mo d ( u ) − → Sh ( u ). Prop ositi o n 4.12. W e have: C mac ( u , R ( i ◦ − ) u ) ∼ = RHom F u n ( u , Sh ( u )) ( u, u ) . F urthermor e, for every natur al tr ansformation u − → F in C ( Fun ( u , Sh ( u ))) for which every u ( U ) − → F ( U ) is an inje ctive re s olut ion, we have: C mac ( u , F ) ∼ = RHom F u n ( u , Sh ( u )) ( u, u ) . COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 24 Pr o of. The ﬁrst line immediately follo ws from the adjunction. F urthermore, by Lemma 3.11 (with a = Z u ) w e hav e R ( i ◦ − )( u ) ∼ = iF whence the seco nd statemen t follows.  W e deﬁne C mac ( u , T ) = RHom F u n ( u , Sh ( u )) ( u, u ). 4.8. Mac Lane cohomolo gy of exact categories and (bi)sheaf cat egories. Let C be an ex a ct Z -linear catego ry with canonical embedding ι : C − → Lex ( C ). This subsec tio n is parallel to § 4 .4. W e deﬁne Mac La ne cohomology of C to be Mac Lane cohomo lo gy o f the natural site ( C , T ) where T is the sing le deﬂation top olo gy . Concretely , (11) C mac , ex ( C ) = RHom F u n ( C , Lex ( C )) ( ι, ι ) . W e hav e the fo llowing a nalogue of Prop ositio n 4.3: Prop ositi o n 4. 13. L et a b e a Z -line ar c ate gory and free ( a ) the ex act c ate gory of ﬁnitely gener ate d fr e e a - mo dules with split exact c onﬂations. We have: C mac , ex ( free ( a )) ∼ = C mac ′ ( a , I ) . Pr o of. W e have Lex ( free ( a )) ∼ = Mo d ( free ( a )) ∼ = Mo d ( a ) (s e e Remark 2.4). Hence the res ult immediately follows from the deﬁnitions.  Let I deno te the iden tity C -bimo dule with I ( C ′ , C ) = C ( C ′ , C ). Using the results of § 3.6, we obtain the following expression in ter ms o f s heav es in t wo v ariables: Theorem 4.14. We have: C mac , ex ( C ) ∼ = RHom F u n ⊳ ⋄ ( C op ×C ) ( I , I ) . Pr o of. Consider i 1 : Fun ⊳ ⊳ ( C op × C ) − → Fun ⊳ ( C op × C ), which is isomo rphic to i ◦ − : Fun ( C , Lex ( C )) − → Fun ( C , Mo d ( C )). The deﬁnition (11) tr anslates into C mac , ex ( C ) ∼ = RHom F u n ⊳ ⊳ ( C op ×C ) ( I , I ). F ro m Prop o s ition 3.17 w e further obtain Ri 1 ( I ) ∼ = Rj ( I ) for j : F u n ⊳ ⋄ ( C op × C ) − → F un ⊳ ( C op × C ), so C mac , ex ( C ) ∼ = RHom F u n ⊳ ( I , Rj ( I )) ∼ = RHom F u n ⊳ ⋄ ( C op ×C ) ( I , I ) by adjunction.  5. Discussion T o ﬁnish the pap er, le t us now explain informally and without pro ofs the moti- v a tions b ehind o ur v arious deﬁnitions and c o nstructions. First of all, our emphasis on a b elian and exact catego ries seems dis tinctly old- fashioned; these days, it is muc h more common to s ta rt with a triangulated categ ory (for example, the der ived category D ( C ) o f an ab elian categor y C instead of the cat- egory C itself ). The problem with this approach is that of course just a triangula ted category is not enough – the categ ory o f e x act functor s from a triangulated cat- egory to itself is not triangulated. T o get the correct endo functor ca tegory , one needs some enha ncement, see e.g. [3]. When working ov e r a ﬁeld, a DG enha ncement (see [13], [16]) would do the job, but at the cost o f technical c omplications which o bscure the ess e n tial conten t of the theory . Th us a purely ab elian treatment is also useful. Moreov er, there is one po int where the ab elia n tr eatment should b e co ns iderably simpler. Namely , as sume given an a be lia n c a tegory C and another ab elian categor y C ′ which is a “s quare-zer o extension” of C in some sense (for exa mple, C could be mo dules ov e r some algebra , and C ′ could b e mo dules over a s quare-zer o extension of this algebr a). Then we hav e a pair of a djoint functors i ∗ : C − → C ′ , i ∗ : C ′ − → C , with i ∗ being exact and fully faithful, and the total der ived functor L q i ∗ . It turns o ut that in a r ather COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 25 general situation, it is the co mpo sition L 1 i ∗ ◦ i ∗ : C − → C of the ﬁrst derived functor L 1 i ∗ with the embedding i ∗ which serves as tangent space to C inside C ′ . Mor eov er, taking the appropriate ca nonical truncation of the total derived functor, w e obtain a complex of functors with 0-th homolo g y isomorphic to the identit y id and the ﬁrst homology iso mo rphic to L 1 i ∗ ◦ i ∗ . By Y oneda, this complex r epresents a class in Ext 2 ( id , L 1 i ∗ ◦ i ∗ ) , and it is this class that should b e the Hochschild cohomolo gy class of the square-ze ro extension. Of course, even with the v a rious functor categories in tr o duced in the present pap er, making the ab ov e sketc h precise requires s ome work, and we re le gate it to a subsequent pap er. Nev er theles s, it is alrea dy obvious that the ab elian context is essential: if one works with enhanced tr iangulated categories , one canno t separate L 1 i ∗ from the total derived functor L q i ∗ . When working abs o lutely , the situation b ecomes muc h more complica ted from the technical p o int of view. DG enhancement is no longer suﬃcient; among the theories existing in the litera ture, the ones which would a pply a re either sp ectral categorie s, see e.g. [29], or ∞ -catego ries in the sense of Lurie [22]. Both req uir e quite a lot of pre liminary work. How ever, surprising as it may b e, a t lea st in the simple ca se mentioned in the int r o duction, — namely that of C b eing the categor y Z /p Z -vector spac e s, — the correct “a bs olute” endofunctor categ ory of C is very easy to describ e. Namely , let Fun ( C , C ) be the categor y o f all functors from C to itself that comm ute with ﬁltered direct limits. It is an ab elian category , so that we can tak e its derived category D ( C , C ). Then the tr iangulated catego ry of “ absolute” endofunctors of C should b e the full triangulated sub categor y D add ( C , C ) ⊂ D ( C , C ) spanned by functors whic h are additive. W e note that this is diﬀer ent from the derived categor y of the ab elian categ o ry of additive functor s – indeed, since every additive functor in Fun ( C , C ) is given b y tensor pr o duct with a ﬁxed vector spac e V ∈ C , the latter is just the deriv ed catego ry D ( C ). How ever, there a re higher Ext’s betw een a dditive endofunctors in D ( C , C ) which do not o ccur in D ( C ). F or exa mple, for any v e c tor spac e V , co nsider the tensor p ow er V ⊗ p , and let σ : V ⊗ p − → V ⊗ p be the lo ngest cycle p er mu tation. Then one can consider the complex ( V ⊗ p ) σ id + σ + · ·· + σ p − 1 / / ( V ⊗ p ) σ and it is easy to show that the ho mo logy o f this co mplex is naturally isomo rphic to V b oth in deg ree 1 and in degree 0. The complex is functor ia l in V , th us deﬁnes by Y oneda an element in Ext 2 ( id , id ) in the category D add ( C , C ). This element is in fact non-trivia l, and cor resp onds to the squa r e-zero extension Z /p 2 Z of the ﬁeld Z /p Z . The categor y D add ( C , C ) is the simplest ex ample of a tria ngulated categ o ry of “non-additive bimodules” whose imp or tance for Mac Lane homology a nd top olog- ical Ho chschild homology has b een known since the pio neering w or k of Jibladze and P irashvili in the 198 0ies, see e.g. [8], [9 ], [18]. What we would like to do is to obtain a s imilar categor y for an a b elian catego ry C which is not the category of mo dules over a n algebra (and for example do es not have eno ugh pro jectives). Our best approximation to the corr ect catego ry is Fun ⊳ ⋄ ( C op × C ). W e b elieve that it do es give the cor rect a bsolute Ho chsc hild cohomolo gy . How ever, one s igniﬁcant problem with this category is that it do es not hav e a natural tensor str ucture – COHOMOLOGY OF EXACT CA TEGORIES AND (NON-)ADDITIVE SHE A VES 26 this is not surprising, since its very deﬁnition is asymmetric . When C is the cate- gory of Z /p Z -vector s pa ces, the triangulated category D add ( C , C ) do es hav e a tensor structure given by the comp o s ition of functors; howev er, our Fun ⊳ ⋄ ( C op × C ) gives something like a DG enhancement for D add ( C , C ), and the tensor pro duct app ea r s to b e inco mpatible with this DG enhancement. P e r haps this is unavoidable, and one should exp ect D add ( C , C ) to be a gen uinely “ top ologica l” triang ulated catego r y , with a sp ectral enhancement instead o f a DG one. 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Cohomology of exact categories and (non-)additive sheaves

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