Andre spectral sequences for Baues-Wirsching cohomology of categories

ANDR ´ E SPECT R AL SEQUE NCES FOR BA UES-WIRSCHING COHOMOLO GY OF CA TEGORIES IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONKS A B S T R A C T . W e construct spectral sequences in the framework o f Baues-W irsching cohomology and h omology for functors between small categories an d analyze particular cases including Grothen dieck fibrations. W e also give applications to more c lassical cohomolo gy and homology theories including Hochschild-Mitchell cohomology and those studied before by W atts, Roos, Quillen and others. I N T R O D U C T I O N In a fundamental paper [4], Baues and W irsching introduced a ve r y general version of coho mo log y for small categories . This Baues-W irsching cohomology us es general coefficients g iven by natural systems. A natural syste m o n a small category C is a functo r from the factorization category F ( C ) into the category Ab of abelian groups. Baues-W irsching cohomology generalizes at once many of t he previ ously const ructed versions of cohomology for categories , inc luding H ochschild-Mitchell cohomology [20], the various versions of cohomology of small categories i ntroduced by W atts [29], Gr othend ieck [14], Roos [28] and Quillen [26], MacLane cohomology of rings [17], [19 ], [25 ] and the coho mo log y o f algebraic theories [17]. In this articl e we analyze the functoriali ty of Baues-W irsching cohomology with respect to functors between small categories. Namely , for a functo r u : E → B , w e construct natural spectral seque nces relating the Baues-W irsching cohomology of the categor y E with that of the category B and associated local fiber data. W e derive a first quadrant cohomology spectral sequence of t he form E p,q 2 ∼ = H p B W ( B , H q ( − /F u, D ◦ Q ( − ) )) ⇒ H p + q B W ( E , D ) which is functorial with respect to n atu r al transformations and whe re H q ( − /F u, D ◦ Q ( − ) ) = lim q − /F u ( D ◦ Q ( − ) ) : F B → A is a particular natural system kee ping track of the fiber d ata. W e also introduce the dual concept of Baues-W irsching homology of small categ ories and d e rive a homology spectral sequen ce dual to the Key words and phrases. Andr ´ e spectral sequence, Baues- Wirsching cohomology and homology , Grothendieck fibrations . 1 2 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S cohomology s pectral s equence above. Specializing to particular coeffici ent syste ms, for e xample given by bimodules, modules, local o r trivial systems, then induces s pectral sequences for Hochschild-Mitchell cohomology and homology and for the other cl assical cohomology and homology theories of small categories. An interpretation of the cohomology of algebraic theories in terms of B aues-W irsching cohomology due to Ji bladze and Pirashvil i [17] allows us also t o stud y morphisms of algebraic the ories cohomologically . The main ingr edient in the const ruction of t hese spectr al se quences is the Andr ´ e spectral sequence [1]. Given a functor between small categories, Andr ´ e constructed a spe ctral sequence for the cohomology and homolog y o f the categories involved. Using an interpretation of the Baues-W irsching cohomology and homolog y groups as particular Ext and T or gr oups in appropriate functor categories we exte nd the Andr ´ e spectral seq u ences to the context of Baues-W irsching theo ries and un ify the existing spectral se quence const ructions for coh o mology and homology of categories stu died before. Spe cial cases of these spectral seque nces were also constructed by Cegarra [7] and recently by Robinson [27] to stu dy the functoriality of cohomology o f diagrams of groups and algebras. In t he particular ca se that the functor u : E → B is a Gr othend ieck fibration of small categories, the local categorical d ata can be ident ified more explicitly in te rms of fiber data. The coho mo log y spectral sequ ence for a Grothendieck fibration turns out to be equ ivalent t o the one constructed by P irashvili and Redondo [23], which converges to the cohomology of the Grothendieck cons truction of a pseud ofunctor . W e again analyze several special cases, including a general Cartan-Leray type spectral s e quence for categorical group actions. In the p articular case of bimodule coefficients such a Cartan-Leray type sp ectral se quence was also studied by Ciblis and R edondo [10], [23] for Galois coverings of k - linear categ ories, w h ich p roves to be an important calculational tool for the repr esentation theory o f finite-d imens ional algebras. In a sequel to this paper we aim to s tudy cohomology and ho mo log y theories with even more g eneral coe f ficient systems, d ue to Thomason [30], given by functo rs from t he comma category ∆ / C into abelian g roups or more generally into a model category , where C is the given small category and ∆ t he s implex category . These w ill allow for the construction of spectral sequences gen e ralizi ng the classical Bousfie ld-Kan homotopy limit and colimit spectral sequences . 1. C O N S T R U C T I N G S P E C T R A L S E Q U E N C E S F O R F U N C T O R S 1.1. Con structing Andr ´ e spectral sequences. In this s ection we review and unify t h e constructions of various spe ctr al se quences for functors between small categories. For a given small ca tego ry C and an abelian ca teg ory A , for example if A = Ab is the category of abelia n groups, we can do homological algebra ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 3 in the functor categ o ry Fun ( C , A ) . It turns o ut that Fun ( C , A ) is agai n an abelian category with enoug h projective o bjects and by a the orem of Gr othe n d ieck [14] also with enough injective objects and so t he classical functorial cons t ructions of homological algebra li ke Ext and T or ca n be performed in Fun ( C , A ) (see [12], [14]). Let us first r ecall t he definitions of the coho mology and homology of a small category via derived limits and colimits, and t he construction of the Andr ´ e spectral seq u ences for general functo rs betwe en small catego ries (see [1], [16], and [22]). Definition 1 .1. Le t C be a smal l category and F : C → A a functor t o an abelian category A . (i) Assume A is complete and has exact products. The n -th cohomology of C with coefficients in F is define d as: H n ( C , F ) = lim n C F . (ii) Assu me A is cocomplet e and has exact coproducts. T he n -th homology of C with coefficients in F is d efined as: H n ( C , F ) = colim C n F . Now let u : E → B be a functor between tw o small categ ories E and B and F : B → A a functor fr om B to any category A . W e get an induced functor betwe en functor catego r ies u ∗ : Fun ( B , A ) → Fun ( E , A ) , u ∗ ( F ) = F ◦ u. If the catego ry A is complete then the functor u ∗ has a r igh t adjoint functor , the right Kan e xtension, Ran u : Fun ( E , A ) → Fun ( B , A ) . If A is mor eover an abelian category w ith exact products, the n Ran u has right satellites (see [8]), R q u ∗ := Ran q u : Fun ( E , A ) → Fun ( B , A ) . In this g e neral s ituation we have the following cohomology s pectral sequence du e to And r ´ e [1] (s e e also [16], [22]). Theorem 1.2. Let u : E → B be a fu nctor between small categories and F : E → A a fun ctor to a complete abelian category A with exact pr oducts. Then ther e exists a first quadrant spectral sequence of the form E p,q 2 ∼ = H p ( B , ( R q u ∗ )( F )) ⇒ H p + q ( E , F ) which is functorial with r espe ct to natural transformations, wher e R q u ∗ = Ran q u is the q -th right satellite of Ran u , the right Kan extension along the functor u . For every functor u : E → B and object b ∈ B let Q b : b/u → E be the forgetful functor from t he comma category b/u to E . W e can identify the 4 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S q -th righ t s atellite R q u ∗ = Ran q u in te r ms of the comma categor y as follows (this is the d ual state me nt of [18, Application 2]): ( Ran q u F )( b ) ∼ = lim q b/u ( F ◦ Q b ) . These isomorphisms ar e natural in objects b and functors F of Fun ( E , A ) . Therefor e we can ide ntify the E 2 -term o f the A ndr ´ e sp ectral sequence and obtain the following: Corollary 1.3. Let u : E → B be a functor between small categories and F : E → A a fun ctor to a complete abelian category A with exact pr oducts. Then ther e exists a first quadrant spectral sequence of the form E p,q 2 ∼ = H p ( B , H q ( − /u, F ◦ Q ( − ) )) ⇒ H p + q ( E , F ) which is functorial with res pect to natural transformations and where H q ( − /u, F ◦ Q ( − ) ) = lim q − /u ( F ◦ Q ( − ) ) : B → A . W e can a lso cons ider the du al s itu ation. If t h e abelian category A is cocomplete then the functor u ∗ has a left adjoint functo r , the left Kan extension, Lan u : Fun ( E , A ) → Fun ( B , A ) . If A is moreover an abelian category with exact coproducts the n Lan u has left satellites (see [8]), L q u ∗ := Lan u q : Fun ( E , A ) → Fun ( B , A ) , and we get t he following homo log y Andr ´ e spectral seque nce: Theorem 1.4. Let u : E → B be a functor between small categories and F : E → A a functor to a cocomplete abelian category A with exact copr oducts. Then ther e exists a third quadrant spectral sequence of the form E 2 p,q ∼ = H p ( B , ( L q u ∗ )( F )) ⇒ H p + q ( E , F ) which is functorial with r espect to natural tra nsformations, where L q u ∗ = Lan u q is the q -th left satellite of Lan u , the left K an extension along the functor u . For eve r y functor u : E → B and every g iven object b ∈ B let Q b : u/b → E be now t he forgetful functor fr om the comma categ ory u/b to E . W e can then ide ntify the q -th left s ate llite L q u ∗ = Lan u q as follows (s ee [18, Application 2]): ( Lan u q F )( b ) ∼ = colim u/b q ( F ◦ Q b ) . These isomorphisms ar e again natural in objects b and functors F of Fun ( E , A ) and we can identify the E 2 -term o f the above spe ctral seque nce and obtain: Corollary 1.5. Let u : E → B be a functor be tween small categories, A a cocomplete abelian category with exact copro ducts and F : E → A a functor . Then ther e exists a third qu adrant spectral sequence of the form E p,q 2 ∼ = H p ( B , H q ( u/ − , F ◦ Q ( − ) )) ⇒ H p + q ( E , F ) ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 5 which is functorial with res pect to natural transformations and where H q ( u/ − , F ◦ Q ( − ) ) = coli m u/ − q ( F ◦ Q ( − ) ) : B → A . 1.2. Con structing spectral sequences for Baues-W irsching cohomology and h omology . L et us now r ecall the definition of the B aues-W irsching cohomology of a s mall category and its main p roperties [4]. D ually , we will also introduce the not ion of Baues-W irsching homo log y . Definition 1.6. Let C be a small category . The factor ization catego ry F C of C is the category who se object set is th e s et of mo r p hisms o f C and who se Hom-sets F C ( f , f ′ ) are the sets o f pairs ( α, β ) s uch that f ′ = β f α , b β / / b ′ a f O O a ′ . f ′ O O α o o The composition o f morphisms in the factorization category F C is define d by ( α ′ , β ′ ) ◦ ( α, β ) = ( α ◦ α ′ , β ′ ◦ β ) . W e will introduce the following g e neral coefficient systems: Definition 1.7. Let M be a catego ry . A functor D : F C → M is called a natural system with values in M . For α, f , β as above we write α ∗ = D ( α, 1) : D ( f ) → D ( f ◦ α ) (1) β ∗ = D (1 , β ) : D ( f ) → D ( β ◦ f ) . (2) The factorization ca teg o ry construction is functorial on t he category Cat of small categories. The functor F : Cat → Cat is given on morp hisms as follows: if u : E → B is a functor the n F u : F E → F B is t he functor given o n objects by F u ( f ) = u ( f ) and o n morp hisms by F u ( α, β ) = ( u ( α ) , u ( β )) . There is a functor F C → C op × C given by ( a f − → b ) 7→ ( a, b ) and ( α, β ) 7→ ( α, β ) . This construction is natural in C . W e define now the B aues-W irsching cohomology of a small category as follows: Definition 1.8. Let C be a small category and let D : F C → A be a natural syste m with values i n a complet e abelian category A with exact products. 6 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S W e de fine the B au es-Wirsch ing cochain complex C ∗ B W ( C , D ) as follows: For each integ er n ≥ 0 the n -th cochain object is g iven by C n B W ( C , D ) = Y C 0 f 1 ← C 1 f 2 ←··· f n ← C n D ( f 1 ◦ f 2 ◦ · · · ◦ f n ) . Let N • C be the nerve of C . Considering elements of C n B W ( C , D ) as maps σ : N n C → S f ∈ M or ( C ) D ( f ) with σ ( f 1 , . . . , f n ) ∈ D ( f 1 ◦ · · · ◦ f n ) , where σ (1 C ) ∈ D (1 C ) for n = 0 , the differ ential d : C n B W ( C , D ) → C n +1 B W ( C , D ) is given by ( dσ )( f 1 , · · · , f n +1 ) = ( f 1 ) ∗ σ ( f 2 , · · · , f n +1 ) + P n i =1 ( − 1) i σ ( f 1 , · · · , f i ◦ f i +1 , · · · , f n +1 ) +( − 1) n +1 ( f n +1 ) ∗ σ ( f 1 , · · · , f n ) . The n -th Baues-Wi rsching cohomology is de fined as H n B W ( C , D ) = H n ( C ∗ B W ( C , D ) , d ) . Dually , we ca n define the Baues-W irsching ho mology of a s mall category . Definition 1.9. Le t C be a small category and let D : F C → A be a natural sys t em with values in a cocomplete abelian ca tego ry A with e xact coproducts. W e define the Baues-W irsching cha in complex C B W ∗ ( C , D ) as follows: For each intege r n ≥ 0 the n -th chain object is given by C B W n ( C , D ) = M C 0 f 1 ← C 1 f 2 ←··· f n ← C n D ( f 1 ◦ f 2 ◦ · · · ◦ f n ) . Let λ = ( C 0 f 1 ← C 1 f 2 ← · · · f n ← C n ) be a string of n compos able arrows and let in λ : D ( λ ) ֒ → C B W n ( C , D ) be the inclusion. W e first d e fine d i : C B W n ( C , D ) → C B W n − 1 ( C , D ) , 0 ≤ i ≤ n. by set t ing d i ◦ in λ =      in d 0 λ ◦ D ( id C 0 ◦ f n ) , if i = 0 , in d i λ , if 0 < i < n, in d n λ ◦ D ( f 0 ◦ id C n ) , if i = n. Here d i λ is the i -th face of th e string in th e nerve N • C . T he differ ential d : C B W n ( C , D ) → C B W n − 1 ( C , D ) is given by the alternating sum d = n X i =0 ( − 1) i d i . The n -th Baues-Wi rsching homology is d e fined as H B W n ( C , D ) = H n ( C B W ∗ ( C , D ) , d ) . ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 7 Remark 1.1 0. It follows fr om the definition, that even though the coefficient syste ms ar e more g eneral, Baues-W irsching cohomology can also be interpreted in terms of classical cohomolog y of small catego ries over t h e factorization catego ry F C , namely we have an isomorp hism H n B W ( C , D ) ∼ = H n ( F C , D ) . In fact, we can identify Baues-W irsching cohomology as the d erived functors of the limit functor [4 , Theo rem 4.4 and Re mark 8.7], H n B W ( C , D ) ∼ = Ext n F C ( Z , D ) ∼ = lim n F C D , where Z : F C → A is the cons tant natural syste m on C . Dually , we can identify Baue s -W irsching h o mology as th e derived functors of the colimit functor H B W n ( C , D ) ∼ = T or F C n ( Z , D ) ∼ = colim F C n D and therefore get again an isomorp hism H B W n ( C , D ) ∼ = H n ( F C , D ) . The Baues-W irs ching cochain complex can also be seen naturally as a functor from the category Nat of natural syst ems given by all p airs ( C , D ) , where C is a small category and D a natural systems into the categ ory of cocha in complexes. Similarly Baues -W irsching cohomology gives an induced functor from Nat into the categ ory of graded abelian groups Ab . In particular let us note that any equivalence of small catego ries φ : C ′ → C induces an iso mo r p hism of Baues -W irsching cohomolog y groups (see [4, Theorem 1.11] ): φ ∗ : H n B W ( C , D ) ∼ = → H n B W ( C ′ , φ ∗ D ) . The Baues-W irsching cochain complex can further be exte nded to give a 2 -functor fr om an ext ended 2 -category of natural syst ems into the 2 - category of cochain complexes s u ch that the B aues-W irsching coho mology factors through a quotient of Nat [21]. Dually , analogous functoriality properties can be obtained for t h e Baues- W irsching chain complex and the asso ciated Baues-W irsching homology of a small category . Baues-W irsching cohomolog y and homology also generalize many existing coho mo log y and homology t heories whos e coefficient sys tems can be see n as particular cases o f natural syste ms. Let us recall the most important examples here (see also [4 ]). W e have the following general diagram of categories and functors, in which C is a given small category: F C π − → C op × C p − → C q − → π 1 C t − → 1 In t his diagram, π and p are the for getful func tors and q is the localization functor into the fundamental groupoid π 1 C = (Mor C ) − 1 C o f C (see [18]). 8 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S Furthermore, 1 is the the trivial category consisting of one object and one morphism and t is the trivial functor . Now let A be a (co)complete abelian category with exact (c o)products, as above. Pulling back functors fr om Fun ( C ′ , A ) via the functors in the above diagram, where C ′ is any one of the categories in th e diagram, i nduces natural sys tems on the category C and we can de fine various versions of cohomology and ho mology o f small categories , which can all be see n as special cases of Baues -W irsching cohomolog y or homology in this way (see [4, Definition 1.18]). Definition 1.11. Let A be a complete abelian category with exact products for coh o mology , or a cocomplete abelian category with exact coproducts for homology . Then: (i) M is called a C -bimodule if M is a functor of Fun ( C op × C , A ) . Define the coho mology H ∗ H M ( C , M ) = H ∗ B W ( C , π ∗ M ) and dually the homo log y H H M ∗ ( C , M ) = H B W ∗ ( C , π ∗ M ) . (ii) F is called a C -module if F is a functo r o f Fun ( C , A ) . Define the cohomology H ∗ ( C , F ) = H ∗ B W ( C , π ∗ p ∗ F ) and du ally the homolog y H ∗ ( C , F ) = H B W ∗ ( C , π ∗ p ∗ F ) . (iii) L is called a local system on C if L is a functor of Fun ( π 1 C , A ) . Define the coho mology H ∗ ( C , L ) = H ∗ B W ( C , π ∗ p ∗ q ∗ L ) and dually the homo log y H ∗ ( C , L ) = H B W ∗ ( C , π ∗ p ∗ q ∗ L ) . (iv) A is a trivial system on C if A is an abelian group (resp. an object in A ), i. e. a functor of Fun ( 1 , Ab ) (resp. a functor of Fun ( 1 , A ) ). Define the cohomology H ∗ ( C , A ) = H ∗ B W ( C , π ∗ p ∗ q ∗ t ∗ A ) and dually the homo log y H ∗ ( C , A ) = H B W ∗ ( C , π ∗ p ∗ q ∗ t ∗ A ) . These various cohomology and homology theories can be identified with the ones known previously in th e literature. W e have the following result (see [4, S ection 8]). Proposition 1.12. Let A be a complete abelian categ ory with exact p rod ucts. Then: (i) For any C -bimodule M of Fun ( C op × C , A ) we have: H n H M ( C , M ) ∼ = Ext n F C ( Z , π ∗ M ) ∼ = Ext n C op × C ( Z C , M ) . (ii) For any C -mod ule F of Fun ( C , A ) w e have: H n ( C , F ) ∼ = Ext n F C ( Z , π ∗ p ∗ F ) ∼ = Ext n C ( Z , F ) ∼ = lim n C F . (iii) For any local system L on C of Fun ( π 1 C , A ) we have: H n ( C , L ) ∼ = Ext n F C ( Z , π ∗ p ∗ q ∗ L ) ∼ = Ext n π 1 C ( Z , L ) ∼ = lim n π 1 C L ∼ = H n ( B C , L ) wher e B C is the classify ing space of the small category C . Pro of. Th e statements of (i) and (ii) follow immediately fr om [4, Pr opos ition 8.5]. Th e canonical C -bi module Z C : C op × C → A used in (i) carries any object ( A, B ) to the fr ee abelian gr oup g enerated by the morphism set Hom C ( A, B ) . The state ment of (iii) is due t o Quillen [26, § 1, p. 83, (1)].  ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 9 Dually , we have a similar characterization for the various ho mology theories of a small category . Proposition 1.13. Let A be a cocomplete abelian category with exact copro ducts. Then: (i) For any C -bimodule M of Fun ( C op × C , A ) we have: H H M n ( C , M ) ∼ = T or F C n ( Z , π ∗ M ) ∼ = T or C op × C n ( Z C , M ) . (ii) For any C -mod ule F of Fun ( C , A ) w e have: H n ( C , F ) ∼ = T or F C n ( Z , π ∗ p ∗ F ) ∼ = T or C n ( Z , F ) ∼ = colim C n F . (iii) For any local system L on C of Fun ( π 1 C , A ) we have: H n ( C , L ) ∼ = T or F C n ( Z , π ∗ p ∗ q ∗ L ) ∼ = T or π 1 C n ( Z , L ) ∼ = colim π 1 C n L ∼ = H n ( B C , L ) wher e B C is the classify ing space of the small category C . Pro of. Th e proofs of (i) and (ii) are just dual t o t hose o f t he preceding theorem and th e s tatement of (iii) was pr oven by Quillen [26, § 1, p.83, (2)].  The cohomology and homo log y theories in (i) for C -bimodule coeffic ients are the o nes introduced by Hochschild and Mitchell, and which ar e also referr ed to as Hochschild-Mitch ell cohomology [9, 20]. Ho chschild- Mitchell h o mology o f catego ries was also introduced and s tudied before by Pirashvi li and W aldhausen [2 5 ]. Special cases of (i i) have been s tudied by W atts [2 9], Roos [28] and Quillen [26]. These in tu rn al so naturally generalize g r oup cohomology and homology , where a g roup G is simply seen as a categ ory with one o bject and morphism set G . W e will now de rive s everal spectral sequences for Baues-W irsching cohomology and homology fr om the respective Andr ´ e spectral sequences as constructed in the first paragraph. Let us con s ider a functor u : E → B betw e en small categories and the following diagram of categories and functor s as defined be fo re: F E π / / F u   E op × E u op × u   p / / E u   q / / π 1 E π 1 ( u )   t / / 1 F B ˜ π / / B op × B ˜ p / / B ˜ q / / π 1 B ˜ t / / 1 In order to de al with all t he dif ferent cases of coe f ficient sys tems at once, let us write C for any of t he categories in the lower row of the above diagram and denote by u ′ : F E → C the associated composition of functors . Furthe rmore, let A be a complete ab elian category with exact 10 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S products. W e have in each case a diagram: Fun ( F E , A ) u ′ ∗ / / lim F E # # ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Fun ( C , A ) u ′ ∗ o o lim C { { ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① A c ; ; ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① ① c c c ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● where c denotes the constant diagram functor , u ′ ∗ is pre-composition with u ′ , and the other functo rs in the diagram are the right adjoints of these , given by the limits lim F E , lim C and by u ′ ∗ = Ran u ′ . The sp ectral sequence for the derived functors of the comp o site functor lim F E ( − ) = lim C u ′ ∗ ( − ) is an Andr ´ e s pectral se quence as cons idered above. It converges to the Baues-W irsching cohomology of E with coefficients D of F un ( F E , A ) and therefor e T heorem 1.2 gives a first quadrant cohomology s pectral s e quence of the form: E p,q 2 ∼ = H p ( C , R an q u ′ ( D )) ⇒ H p + q B W ( E , D ) Identifying the terms in t he E 2 -page accor ding to Coroll ary 1.3 we therefore get: E p,q 2 ∼ = H p ( C , H q ( − /u ′ , D ◦ Q ( − ) )) ⇒ H p + q B W ( E , D ) where D ◦ Q ( − ) is the composition of functors − /u ′ Q ( − ) − → F E D − → A . In particular case s the E 2 -page may be simplified. F o r example in the case C = F B with u ′ = F u considered above, we get the following sp ectral sequence: Theorem 1.14. Let E and B be small cate gories and u : E → B a functor . Let A be a complete abelian category with exact products. Given a natural system D : F E → A on E , ther e is a first quadrant cohomolog y spe ctral sequence E p,q 2 ∼ = H p B W ( B , ( R q F u ∗ )( D )) ⇒ H p + q B W ( E , D ) which is functorial with r espect to natural transformations and where R q F u ∗ = Ran q F u is the q -th right satellite of Ran F u , the right Kan extension along F u . Using the identification of the terms in the E 2 -page as ab ove we get therefor e: Corollary 1.1 5. Let u : E → B be a functor between small categori es and A a complete abelian category with exact pr oducts. Let D : F E → A be a natural ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 11 system on E . Then ther e exists a first qu adrant cohomology spectral sequence of the form E p,q 2 ∼ = H p B W ( B , H q ( − /F u, D ◦ Q ( − ) )) ⇒ H p + q B W ( E , D ) which is functorial with res pect to natural transformations and where H q ( − /F u, D ◦ Q ( − ) ) = lim q − /F u ( D ◦ Q ( − ) ) : F B → A . For the case C = B op × B and u ′ = ˜ π ◦ F u with a natural sys tem D : F E → A on E w e get in particular the following spe ctral sequence: E p,q 2 ∼ = H p ( B op × B , R q ( ˜ π ◦ F u ) ∗ ( D )) ⇒ H p + q B W ( E , D ) which after ident ifying the various te rms involved gives a spectral sequence for Baues-W irsching cohomology in terms of Hochschild- Mitchell cohomology : Theorem 1.16. Let E and B be small cate gories and u : E → B a functor . Let A be a complete abelian category with exact products. Given a natural system D : F E → A , ther e is a first quadrant cohomolog y spectral sequence E p,q 2 ∼ = H p H M ( B , R q (( u op × u ) ◦ π ) ∗ ( D )) ⇒ H p + q B W ( E , D ) which is functori al with respe ct to n atural transformations and where R q (( u op × u ) ◦ π ) ∗ = R an q ( u op × u ) ◦ π is given as the q -th right sate llite of Ran ( u op × u ) ◦ π , the right Kan extension along ( u op × u ) ◦ π . As be fore we can identify the terms in the E 2 -page o f the spectral sequence with more concrete data and get: Corollary 1.17. Let u : E → B be a fun ctor between small cate gories. Let A be a complete abelian category with exact products. Given a natural system D : F E → A , ther e is a first quadrant cohomolog y spectral sequence E p,q 2 ∼ = H p H M ( B , H q ( − / ( u op × u ) ◦ π , D ◦ Q ( − ) )) ⇒ H p + q B W ( E , D ) which is functorial with res pect to natural transformations and where H q ( − / ( u op × u ) ◦ π , D ◦ Q ( − ) ) = lim q − / ( u op × u ) ◦ π ( D ◦ Q ( − ) ) : B op × B → A . In the sp e cial case that D = π ∗ M for a bimodule M : E op × E → A on E we can further identify the E 2 -term o f the s pectral sequen ce in T h e orem 1.16 to obtain a s pectral sequence for Hochschild-Mitchell cohomolog y of the form: E p,q 2 ∼ = H p H M ( B , R q ( u op × u ) ∗ ( M )) ⇒ H p + q H M ( E , M ) Similarl y , we can identify the E 2 -page in t erms of local fiber d ata as above in Corollary 1.17. This sp e ctral seque n ce can also be d erived d ir ectly fr om the cohomological Andr ´ e s pectral seque n ce of T heorem 1.2 us ing the interp retation of H ochschild-Mitchell cohomolog y as an appropriate cohomology of small categories. 12 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S Dually , using simil ar ar gument s as above we can derive homological versions of the spectral seque n ce s for Baues-W irsching homolog y of categories. Theorem 1.18. Let E and B be small cate gories and u : E → B a functor . Let A be a cocomplete abelia n category with exact copro ducts. Given a natural system D : F E → A on E , ther e is a third quadrant homology spectral sequence E 2 p,q ∼ = H B W p ( B , ( L q F u ∗ )( D )) ⇒ H B W p + q ( E , D ) which is functorial with respe ct to n atural transformat ions, where L q F u ∗ = Lan F u q is the q -th left satellit e of Lan F u , the left K an extension along F u . Pro of. Th is is simply dual to the s t atement of Theor em 1.14 and follows fr om the d u al Andr ´ e homo log y spe ctral sequence involving the highe r derived functors of colim F C n D in the d e scription of Baues-W irsching homology H B W n ( C , D ) (see [1], [16, p. 2291 ] and [18, Appendix II . 3]).  W e can furthe r identify the E 2 -page of this s pectral seque nce and g e t: Corollary 1.19. Let E and B be small catego ries and u : E → B a fu n ctor . Let A be a cocomplete abelia n category with exact copro ducts. Given a natural system D : F E → A on E , ther e is a third quadrant cohomolog y spectral sequence E 2 p,q ∼ = H B W p ( B , H q ( F u/ − , D ◦ Q ( − ) )) ⇒ H B W p + q ( E , D ) which is functorial with res pect to natural transformations and where H q ( F u/ − , D ◦ Q ( − ) ) = coli m F u/ − q ( D ◦ Q ( − ) ) : F B → A . Pro of. W e can identify the E 2 -term in the above homology s pectral sequence as follows (se e [1], [16]) Lan F u q ( D ) ∼ = colim F u/β q D ◦ Q ∗ which the n gives t he des ir ed homolog y spectral se quence.  For the special case C = B op × B and u ′ = ˜ π ◦ F u with a natural syst em D : F E → A on E we get a homolog y spectral se q u ence E 2 p,q ∼ = H p ( B op × B , L q ( ˜ π ◦ F u ) ∗ ( D )) ⇒ H B W p + q ( E , D ) which after ide n t ifying the various ter ms gives the following spectral sequence for Hochs child-Mitchell homology d u al t o the one of The orem 1.16: Theorem 1.20. Let E and B be small cate gories and u : E → B a functor . Let A be a cocomplete abelia n category with exact copro ducts. Given a natural system D : F E → A , ther e is a third quadrant homology spectral sequence E 2 p,q ∼ = H H M p ( B , L q (( u op × u ) ◦ π ∗ )( D )) ⇒ H B W p + q ( E , D ) which is functorial with respe ct to natural transformations and wher e L q (( u op × u ) ◦ π ) ∗ = Lan ( u op × u ) ◦ π q is given as the q -th left satellite of Lan ( u op × u ) ◦ π , the left Kan extension along ( u op × u ) ◦ π . ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 13 Similarl y , we can again i dent ify the terms in the E 2 -page o f the spectral sequence and obtain: Corollary 1.2 1. Let u : E → B be a functor between small categori es and A a cocomplete abelian category with exact copr oducts. Given a natural system D : F E → A , ther e is a third quadrant homology spectral sequence E 2 p,q ∼ = H H M p ( B , H q (( u op × u ) ◦ π / − , D ◦ Q ( − ) )) ⇒ H B W p + q ( E , D ) which is functorial with res pect to natural transformations and where H q (( u op × u ) ◦ π / − , D ◦ Q ( − ) ) = coli m ( u op × u ) ◦ π/ − q ( D ◦ Q ( − ) ) : B op × B → A . As above in the special case D = π ∗ M for a bimodule M : E op × E → A on E we can further identify the E 2 -term of the spectral seque nce s imila rly as in The o rem 1.16 to obtain a spe ctr al sequ e nce for H ochschild-Mitchell homology of the form: E 2 p,q ∼ = H H M p ( B , L q (( u op × u ) ∗ )( M )) ⇒ H H M p + q ( E , M ) And again we can now ide ntify the E 2 -page in te rms of local fiber data as above in Corolla ry 1.17. This sp ectral sequence could also be d erived directly fr om the homological Andr ´ e spectral sequence of Theorem 1.4 using the interpretation of Hochschild-Mi tchell homolog y as an appropriate homology of s mall categories . W ith s imilar ar guments as above, we can also derive cohomology and homology spe ctral sequences for more special coe f ficient s ystems like E - modules, local s y stems or trivial systems, which we will leave to th e interested reader . For the case of E -modules we only note that one just recovers the cohomolog y and ho mo log y Andr ´ e s pectral seque nces of Theorems 1.2 a nd 1.4. In the case of local sy stems, the spectral se quences can also be interpreted as a L eray t ype s pectral se quence for the indu ce d map B u : B E → B B be t ween t h e classifying spaces of the categories E and B . 1.3. A spectral sequence fo r the cohomolo gy of algebraic theories. Let us now cons ide r an application to the cohomology of algebraic theories . Jibladze and Pirashvili [17] constructed a general coho mology theory for algebraic theories, which gives a well-behaved cohomology theory for rings and algebras. They also indicated the importance o f having more general coe f ficient systems than just modules . One appr oach towar ds this is to interpret cohomology of algebraic theories again as an appropriate Baues-W irsching cohomolog y with natural sy stems as coeffici ents , s ee [17 , Sect. 6]. One nee ds to use these more general coeffic ient sys tems, for example, when classifying ex t ensions of algebraic theo ries. Let u s first recal l the definition of an algebraic theory : 14 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S Definition 1.22. A (finitary) algebraic theory is a category whose o bjects ar e the natur al numbers N , denote d 0 , 1 , 2 , . . . n , . . . , equipped with distinguished isomorphisms φ n : n ∼ = − → 1 n between each object n and t he product o f n copies of the object 1 . Morphisms between algebraic t heories are simply functors which are identities on objects and pr eserve finite products, i. e. functors pr ese rving the morphisms φ n for all n . The catego ry of algebraic theories will be denote d by Theories . Now let T be an algebraic theory . There is an equival ence of categories [17, Sect. 6] Ab ( Theorie s / T ) ∼ = Nat th ( T op ) where Ab ( Theories / T ) is the category of internal abelian group objects in the comma category Theories / T and Nat th ( T op ) is the category o f coproduct-pr ese r ving natural sy stems on t he opposite categ ory T op , that is, natural s ystems D such that for any diagram in T of the form y 1 x f / / y 1 × y 2 × . . . × y n p 1 2 2 ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ p n , , ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ . . . y n , where p i ar e the canonical projections, the homomorphism D ( f op ) → D ( f op ◦ p 1 op ) ⊕ · · · ⊕ D ( f op ◦ p n op ) induced by the st ructur e maps ( p op i ) ∗ of Definition 1.7 ( 1) is an isomorphism. The cohomology with coefficients M in Ab ( Theories / T ) of an algebraic theory T is defined as the Baue s -W irsching cohomo log y H ∗ ( T , M ) = H ∗ B W ( T op , ˜ M ) where ˜ M is the natural system associated to M by the above equivalence between the categ ories Ab ( Theories / T ) and Nat th ( T op ) . The coho mo log y may be calculated with the complex given in Definition 1.8 or us ing a smaller normalized complex as e x p lained in [3]. Dually , we can also de fine the homology of algebraic theo ries as H ∗ ( T , M ) = H B W ∗ ( T op , ˜ M ) . Now let u : E → B be a morphism of algebraic theo ries and M a coeffic ient system from Ab ( Theories / E ) , then it follows from Corolla ry 1.15 that there exist s a first quadrant cohomo log y spe ctral sequence E p,q 2 ∼ = H p B W ( B op , H q ( − /F u op , ˜ M ◦ Q ( − ) )) ⇒ H p + q ( E , M ) . ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 15 Dually , from Coroll ary 1.19 we get a thir d quadrant homology spectral sequence E 2 p,q ∼ = H B W p ( B op , H q ( F u op / − , ˜ M ◦ Q ( − ) )) ⇒ H p + q ( E , M ) . In general though, the E 2 -pages of thes e spe ctral seque nces are only given as general Baues-W irsching cohomology gr oups and might not always be identified with a cert ain cohomology or homology of the algebraic theory B . 2. C O N S T R U C T I N G S P E C T R A L S E Q U E N C E S F O R G R O T H E N D I E C K FI B R AT I O N S 2.1. Grothendieck fi brations and spectral sequences. In this section we will analyze t he spe ctr al se quence constructions in the particula r situation of a Grothendieck fibration o f small categories . Applying the sp ectral sequence constructions o f the preceding sections to a functor which is a Grothendieck fibration allows to identify the E 2 -pages with simpler cohomology or homology g roups keeping track of the fiber data. Given any functor u : E → B and an object b of B , recall that the fiber category E b = u − 1 ( b ) is the s ubcategory of E that fits into the pullback diagram E b / /   E u   ∗ b / / B . The objects of E b ar e those objects of E which map o nto b via the functor u and the morphisms are given by t hose which map to the ide ntity 1 b . W e now have the following notion of a fibration betwee n s mall categories due to Grothendieck [15, Expo s ´ e V I]: Definition 2.1. L e t E and B be small catego ries. A Gr othendieck fibration is a functor u : E → B such t hat the fibers E b = u − 1 ( b ) depend cont ravariantly and pseud ofunctorially on the objects b of the category B . The category E is also called a category fibered over B . There are many equivalent explicit d efinitions of a Grothendieck fibration. W e only recall her e from [1 3], that u : E → B is a Gr othend ieck fibration if for e ach object b o f B the inclusion functor from the fiber into the comma category j b : E b → b/u, e 7→ ( e, b = − → ue ) is coreflexive, i. e. has a right adjoint left inverse. W e have the following general characterization of Grothendieck fibrations: Theorem 2.2. T her e is an equivalence of 2 -categorie s Fib ( B ) ≃ ↔ PsdFun ( B op , Cat ) 16 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S between the 2 -catego ry of Grothendie ck fibrations Fib ( B ) over a category B and the 2 -catego ry of contravaria nt pseudof unctors PsdFun ( B op , Cat ) from B to the categor y C at of small categories. Pro of. F or a detailed proof we refer to [15] or [5, V ol. 2, 8.3.1]. The equivalence is ind u ced by the Gr othe ndieck construction Z : PsdFun ( B op , Cat ) → Fib ( B ) , F 7→ Z B F . And if conversely u : E → B is a Grothendieck fibration the ass ignment G : B op → Cat , b 7→ E b = u − 1 ( b ) . defines a pseud ofunctor .  Let u : E → B be a Grothendieck fibration and D : F E → A be a natural sys t em on t he category E , where A is a complet e abelian catego ry with exact products. W e g e t a local system H q B W ( G ( − ) , D | F E ( − ) ) : B → A from the asso ciated pseudo functor G : B op → Cat by assigning t o every o bject b of the category B the q -th Baues-W irsching cohomology of the category G ( b ) H q B W ( G ( − ) , D | F E ( − ) ) : B → A , b 7→ H q B W ( G ( b ) , D | F E b ) with coefficients in the natural sys t em D | F E b : F E b → A . For each object b of the base category B we have a cartesian diagram E b =   j b / / b/u Q b   E b   i b / /   E u   ∗ b / / B . Let R b denote t he right adjoint functor of the inclusion functor j b and let D b denote the natural sy stem on b/u g iven by t he following compo sition, D b : F ( b/u ) F R b − − − → F E b F i b − − → F E D − → A . Now we get a firs t qu adrant cohomolog y s pectral se quence from the cohomology spe ctral s equence for the Grothendieck construction of Pirashvili -Redo ndo [23] and the equivalence betwee n Grothendieck fibrations and pseud ofunctors: E p,q 2 ∼ = H p ( B , H q B W ( − /u, D ( − ) )) ⇒ H p + q B W ( E , D ) This spe ctr al s e quence is functorial with respect to 1 -morphisms, i.e. natural transformations between Grothendieck fibrations. T o summariz e, we have const ructed the following cohomology spectral sequence for general Grothendieck fibrations of small catego ries: ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 17 Theorem 2.3. L et E and B be small categories an d let u : E → B be a Gro thendieck fibr ation. G iven a natural system D : F E → A on E , wher e A is a complete abelian categ ory with exact products, ther e is a first quadrant spe ctral sequence E p,q 2 ∼ = H p ( B , H q B W ( − /u, D ( − ) )) ⇒ H p + q B W ( E , D ) which is functorial with res pect to 1 -morphi sms of Gro thendieck fibrations. In order to identify the E 2 -term o f this sp ectral seq u ence with local data of the fibe r category we need th e following definition, which corr espo nds to the p roperty of h-locality introduced in [23]. Definition 2.4. A natural sys tem D : F E → A is ca lled local if the adjoint functor R b of the inclusion functor j b : E b → b/u induces an isomorphism in Baues -W irsching cohomolog y H q B W ( b/u, D b ) ∼ = H q B W ( E b , D ◦ F i b ) for e very q and e very object b o f the base categ ory B , i. e. we h ave a natural isomorphism of local coefficient sy s tems H q B W ( − /u, D ( − ) ) ∼ = H q B W ( E ( − ) , D ◦ F i ( − ) ) . Identifying the E 2 -page of t he above s p ectral sequence , w e now have the following cohomolog y spectral se quence: Theorem 2.5. Let E and B be small categories and u : E → B be a Gr othendieck fibration. Given a local n atural syste m D : F E → A on E , wher e A is a complete abelian category with exact pr oducts, ther e is a first quadrant spectral sequence E p,q 2 ∼ = H p ( B , H q B W ( E ( − ) , D ◦ F i ( − ) )) ⇒ H p + q B W ( E , D ) with the local coefficient system H q B W ( E ( − ) , D ◦ F i ( − ) ) : B → A , b 7→ H q B W ( E b , D ◦ F i b ) . Furthermor e, th e spectral sequence is functorial with resp ect to 1 -morphisms of Gro thendieck fibrations. Pro of. Th is follows immediately from the construction of the coho mo log y spectral s e quence in Theorem 2.3 and t he definition o f a local natural syste m D : F E → A on the total categ ory E .  Dually , we can also derive a homology version of the spectral seque n ce for a Grothendieck fibration by invoking the obvious d ual notions and constructions involved: Theorem 2.6. Let E and B be small categories and u : E → B be a Gr othendieck fibration. Given a colocal n atural system D : F E → A on E , w her e A is a co complete abelian cate gory with exa ct copr oducts, there is a third qu adrant spectra l s equence E 2 p,q ∼ = H p ( B , H B W q ( E ( − ) , D ◦ F i ( − ) )) ⇒ H B W p + q ( E , D ) 18 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S with the local coefficient system H B W q ( E ( − ) , D ◦ F i ( − ) ) : B → A , b 7→ H B W q ( E b , D ◦ F i b ) . Furthermor e, th e spectral sequence is functorial with resp ect to 1 -morphisms of Gro thendieck fibrations. As in the preceding se ctions we can sp ecializ e these cohomology and homology spe ctr al se quences for coef ficient sys tems of bimodules , modules, l ocal sy s tems or trivial sys t ems. The associated spe ctral sequences will then for e xample in the case of bimodules r elate the Hochschild-Mitchell cohomology or homology groups in a Grothendieck fibration. W e leave it to the interested reader to w ork out the de tails. 2.2. App lications and examples. In this final section we will g ive interpretations and applications of particular spectral s e quences, which were studied before in the literature. Formulated here in the language of Grothendieck fibrations, the cohomology s p ectral sequ ence of Theo rem 2.3 can also be se e n as an equivalent version of t he Pirashvili-Redondo spectral se quence, which converges t o the Baues-W irsching coh o mology of the Grothendieck construction of a p seudofuncto r (see [23], [24]). And this spectral se quence again gene ralizes several o t hers. Let us study so me important e xamples. Example 2.7. Let C be a small category with an action of a gr oup G . The group G can be considered as a small category w ith a s ingle object ⋆ and the group G as morphism s et. A G -action on C is then s imply given as a functor F : G op → Cat , F ( ⋆ ) = C . W e can vi ew F as a pseud ofunctor and the Grothendieck construction giv es a small category R G F . The above equivalence between ps e udofunctors and Gr othe n d ieck fibrations therefore gives a Grothendieck fibration o f small categories u : Z G F → G. Given in addition a natural s ystem D : R G F → Ab , we get t herefor e a first quadrant cohomology spectral s equences of the form: E p,q 2 ∼ = H p ( G, H q B W ( − /u, D ( − ) )) ⇒ H p + q B W ( Z G F , D ) . Finally , we can ident ify the fiber data in this partic ular s ituation, which allows to rewrite the E 2 -page and get: E p,q 2 ∼ = H p ( G, H q B W ( C , D )) ⇒ H p + q B W ( Z G F , D ) . This can be interpreted a s a catego r ical version of the classical Cartan-Leray spectral sequence, which was also derived as a special case in [23, R e mark 5.4] and for local coefficient syste ms in [11, P roposition 5.4.2]. ANDR ´ E SPECTRAL SEQUENCES FOR B AUES-WIRSCHING COHOMOLOGY 19 Let us also stud y the particular situation of a k -linear category with a G -action, which is of import ance in r epresentation theory . L et k be a field and C be a k -linear categ ory , i. e . a small category whe re morphism s ets ar e k -vector s paces and composition is k -bilinear . Assume now that a group G is acting freely on C . As before we can take the Grothendieck construction which in this situation gives the quo tient or orbit category R G F = C /G . So we ge t as a special case a Cartan-Leray spectral s equence, which calc ulates the Baues-W irsching coho mo log y of the quotient category: E p,q 2 ∼ = H p ( G, H q B W ( C , D )) ⇒ H p + q B W ( C /G, D ) . For bimodule coefficients M of Fun ( C /G, Ab ) , this spectral sequence is the Cartan-Leray spectral se quence of Ciblis-Redondo [10, Theorem 3.11] (see al so [23, Remark 5.3]), which calculates the Hochschild-Mitchell cohomology of the quot ient category C /G . W e can also derive dual homology vers ions of t h e above cohomolog y spectral sequences , but will leave these st raightforward constructions to the interested reader . Let us finally mention the case of cohomology and homology with loca l or t rivial coefficient systems, which correspond to singular coho mology and ho mology of the associated classifying s paces with local or constant coeffic ients. Example 2.8. Let u : E → B be a Grothendieck fibration and L : π 1 E → Ab a local sy stem on E . Then we get the following first quadrant cohomology spectral seq u ence from Theorem 2.5: E p,q 2 ∼ = H p ( B B , H q ( B E ( − ) , L ( − ) )) ⇒ H p + q ( B E , L ) with the local sy stem H q ( B E ( − ) , L ( − ) ) : π 1 B → Ab as coefficients. Dually , we have a thir d quadrant h o mology spe ctral seque n ce fr om Theorem 2.6: E 2 p,q ∼ = H p ( B B , H q ( B E ( − ) , L ( − ) )) ⇒ H p + q ( B E , L ) with the local sy stem H q ( B E ( − ) , L ( − ) ) : π 1 B → Ab as coefficients. Specializing even further , using trivial systems o f coefficients in a constant abelian group A , i. e. functors A : 1 → Ab , we get immediately the following first quadrant cohomology spectral se q u ence: E p,q 2 ∼ = H p ( B B , H q ( B E ( − ) , A )) ⇒ H p + q ( B E , A ) And dually again, we have a third quadrant homology spectr al sequence: E 2 p,q ∼ = H p ( B B , H q ( B E ( − ) , A )) ⇒ H p + q ( B E , A ) In general, starting with a Grothendieck fibration u : E → B o f small categor ies and applying the cla ssifying space functor B ( − ) does no t give a fibration or quasi-fibration of topological sp aces. But del Hoyo 20 IMMA G ´ AL VEZ-CARRILLO, FRANK NEUMANN, AND ANDREW TONK S [11, Theorem 5.3 .1] s howed that if u : E → B is moreover a Quill en fibration of small ca teg ories, which means essentially that al l the classifying spaces of th e fibers E b ar e weakly homot opy equivalent, t h e n applying a fiber ed version B f ( − ) of the classifying s pace functor does indeed give a quasi-fibration of topo logical spaces and t herefor e the particular spectral sequences above ar e Serre type spectral sequ ences. Acknowledgements. T h e first author was partially su pported by the grants MTM2010- 15831, MTM2010-206 92, and SGR-1092-20 09, and the third author by MTM2010-1 5831 and SGR-1 19-2009 . The second autho r likes to thank the Centre de Recerca Matem ` atica (CRM) in Bellaterra, Spain for inviting him du ring t h e researc h programme H omotopy Theory and Higher Categories (HOCA T). R E F E R E N C E S [1] M. Andr ´ e, Limites et fibr ´ es, C. R. A cad. Sci. Paris, S ´ er . 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Categorical Al gebra (La Jolla, Calif. , 196 5) , Springer , New Y o rk, 1965, 331–335 . [30] C. W eibel, Homotopy ends an d Thomason model cat egor ies, Selecta Mathematica , New ser . 7 (2001), 533– 564. D E PA R TA M E N T D E M AT E M ` A T I C A A P L I C A D A I I I , U N I V E R S I T AT P O L I T ` E C N I C A D E C A TA L U N YA , E S C O L A D ’ E N G I N Y E R I A D E T E R R A S S A , C A R R E R C O L O M 1 , 0 8 2 2 2 T E R R A S S A ( B A R C E L O N A ) , S P A I N E-mail address : m.imma culada.galvez@ upc.edu D E PA R T M E N T O F M AT H E M AT I C S , U N I V E R S I T Y O F L E I C E S T E R , U N I V E R S I T Y R O A D , L E I C E S T E R L E 1 7 R H , E N G L A N D , U N I T E D K I N G D O M E-mail address : fn8@mc s.le.ac.uk F A C U LT Y O F C O M P U T I N G , L O N D O N M E T R O P O L I TA N U N I V E R S I T Y , 1 6 6 – 2 2 0 H O L - L O WA Y R O A D , L O N D O N N 7 8 D B , U N I T E D K I N G D O M E-mail address : a.tonk s@londonmet.ac .uk