LOCAL-TO-GLOBAL SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS DAVID BLANC, MARK W. JOHNSON, AND JAMES M. TURNER 0. Introduction The cohomology of diagrams arises as a natural object of study in several mat* *h- ematical contexts: in deformation theory (see [GS2 , GS1 , GGS ]), and in clas* *sifying diagrams of groups, as in [C ]. If I is the one-object category corresponding * *to a group G, a diagram X 2 CI is just an object in C equipped with a G-action, and* * its cohomology is the equivariant cohomology of [I] (cf. [P1 , x2]). On the other h* *and, for any discrete or Lie group G, let I = OG denote the orbit category of G: if X* * is a G-space, X : OG ! Top is the corresponding fixed point diagram X(G=H) := XH , and M : OG ! AbGp , is any system of coefficients, then the corresponding coh* *o- mology H(X; M) is Bredon cohomology (cf. [Ma , I,x4]). Finally, when I consi* *sts of a single arrow, and the coefficients are constant, we have the usual cohomol* *ogy of a pair. See [BG ], [DS ], [FW ], [O ], [Pa ], and [BC ] for further applicatio* *ns. 0.1. Diagrams in homotopy theory. The cohomology of diagrams also plays a major role in the Dwyer-Kan-Smith theory for the rectification of homotopy-comm* *uta- tive diagrams (cf. [DKS ] and [DF , DK ]). In fact, our interest in the subj* *ect was motivated by the related realization problem for diagrams of -algebras (graded groups with an action of the primary homotopy operations): as in the case of a single -algebra (cf. [BDG ]), the obstructions to realizing a diagram of -al* *gebras : I ! -Alg lie in appropriate cohomology groups of (see [BJT , Thm. 6.3]* *). Furthermore, given a -algebra , all distinct homotopy types realizing may be distinguished by a set of higher homotopy operations associated to a collect* *ion (Iff)ff2A of finite indexing categories Iff and homotopy-commutative diagrams Xff: Iff! hoTop , where all the spaces Xffiare wedges of spheres. Since these h* *igher operations are obstructions to the rectification of the diagrams (Xff)ff2A (a* *nd thus the associated diagrams ff:= ss*Xff: Iff! -Alg), they correspond to elemen* *ts in the cohomology of . Understanding the cohomology groups of such diagrams may therefore be helpful in algebraicizing (and organizing) the "higher -algebra" * *of a space Y , consisting of all higher homotopy operations in ss*Y . 0.2. Computing diagram cohomology. Even the cohomology of a single map may be hard to calculate (cf. [BJT , x5.16]), so some computational tools are n* *eeded. For this purpose we construct "local-to-global" spectral sequences for the coho* *mology of a diagram, which can be used to compute the cohomology of the full diagram in terms of smaller pieces. Given a small category I, a model category C (in the sense of [Q1 ]), and an * *I- diagram X 2 CI, one can define the cohomology of X with coefficients in any abelian group object Y 2 CI. For technical reasons, we shall concentrate on th* *e case where C = sA is the category of simplicial objects over some variety of univ* *ersal ___________ Date: July 9, 2007. 1 2 D. BLANC, M.W. JOHNSON, AND J.M. TURNER algebras A: since the homotopy category of simplicial groups is equivalent to t* *hat of (pointed connected) topological spaces, this actually covers all cases of inter* *est above. Some of our results are valid, however, for an arbitrary simplicial model categ* *ory C. Another reason for our interest in the "local-to-global" approach to diagram * *co- homology is that in order for the higher homotopy operation corresponding to a homotopy commutative diagram X : I ! ho Top to be defined, all lower order operations (corresponding to subdiagrams of I) must vanish coherently. Thus an * *es- sential step in a cohomological description of higher order operations is the a* *bility to piece together local data to obtain global information. 0.3. Remark. We should point out that our methods work (almost exclusively) for* * a directed indexing category I (i.e., with only identities as endomorphisms), whi* *ch is a significant restriction. However, such diagrams certainly suffice for the descr* *iption of higher homotopy operations, as above: even the linear case - when I consists * *of a single composable sequence of arrows - is of interest, since the realizabilit* *y of such a diagram is essentially equivalent to calculating higher Toda brackets. Furthe* *rmore, diagrams arising in deformation theory (indexed by the nerve of a covering) are* * of this form. Our methods, suitably modified (cf. Remark 1.7), also apply to diagr* *ams indexed by the orbit category OG of a group G. 0.4. The spectral sequences. Let C be a simplicial model category and I a directed index category, and assume given diagrams Z : I ! C, and X, Y 2 CI=* *Z, with Y an abelian group object in CI=Z. Our main results may be summarized as follows: Theorem A. There is a first quadrant spectral sequence with: Y E2s,t= Ht+s(Xj=Zj, ^OEj) =) Hs+t(X=Z; Y ) j2Jes This is constructed by taking increasing truncations of the coefficient diagr* *am Y (cf. Theorem 3.5). Here H*(X=Z, OE) denotes relative cohomology for a map of * *the coefficients (see Definition 3.1). Theorem B. There is a first quadrant spectral sequence with: E2s,t= Hs+t(js; Y ) =) Hs+t(X=Z; Y ) This spectral sequence is constructed dually to the previous one, by taking i* *ncreas- ing truncations of the source diagram X (see Theorem 3.7). Here H*(j, Y ) deno* *tes the usual cohomology of a map (or pair). Theorem C. If I is countable, then for any ordering (cs)1s=1 of the objects of * *I, there is a first quadrant spectral sequence with E2s,t= Ht+scs(X=Z; Y ) =) Hs+t(X=* *Z; Y ). This is constructed by successively omitting the objects cs from I (see The* *orem 7.7). Here H*c(X=Z, Y ) denote the local cohomology groups at an object c * *2 I (see Definition 7.4). There are versions of all three spectral sequences define for any suitable co* *ver J of I (Definition 1.1). In particular, the spectral sequences always converge if J * *is finite, hence if I itself is finite. 0.5. Other variants. Other spectral sequences for the cohomology of a diagram have appeared in the literature. One should mention the universal coefficients * *spec- tral sequence of Piacenza (see [P2 , x1]), and the local-to-global spectral seq* *uence of SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 3 Jibladze and Pirashvili (cf. [JP ]) - though the latter uses a different defi* *nition of cohomology, based on the Baues-Wirsching and Hochschild-Mitchell cohomologies of categories (cf. [BW , Mi ]). 0.6. Organization. Section 1 provides background material on diagrams, their covers, and the model category of diagrams. In Section 2 we determine when the "restriction tower" associated to a cover of the indexing category I is a tower* * of fibrations, and in Section 3 we use this to set up the first two spectral seque* *nces. The second half of the paper is devoted to the (somewhat more technical) appr* *oach based on "localizing at an object": Section 4 provides the setting, and explain* *s the method. In Section 5 we describe an auxiliary construction associated to the to* *wer of certain covers of I, and in Section 6 show that this auxiliary tower is a to* *wer of fibrations. Finally, in Section 7 we identify the fibers of the new tower, and* * obtain the third spectral sequence. 1. The category of diagrams Our object of study will be the category CI of diagrams - i.e., functors* * from a fixed small (often finite) indexing category I into a model category C. The m* *aps are natural transformations. In this section we define some concepts and intro* *duce notation related to I and CI: 1.1. Definition. Let I be any small category. By an N-indexed cover of I we mean some collection J = {J } 2N of subcategories of I, such that each arrow in I belongs to at least one J . A cover J = {J } 2N for I will be called orderable if the relation: 1 2 (Def) 9 i1 2 J 1, i2 2 J 2 9 OE : i2 ! i1 inI with i1 62 J 2 ori2 62* * J 1 . defines a partial order on N, and the partially ordered set (N, ) can be embe* *dded as a (possibly infinite) segment of (Z, ). Choosing such an embedding N Z, we * *mayS think of J as indexed by integers, and we can then filter I by setting J[n] := * * i nJi. If N is bounded below in Z we say that J is right-orderable, and if it is bound* *ed above we say it is left-orderable. 1.2. Remark. Note that the linear ordering of J (indicated by the indices) is * *not generally uniquely determined by the partial order : there may be elements of J which are not comparable under . This happens when all maps out of Jn actually land in J[k] for k < n - 1. In this case the linear ordering of Jn and J* *n-1, for example, may be switched with impunity. 1.3. Directed indexing categories. A directed indexing category is a small category I equipped with a map deg: Obj(I) ! Z, such that for every non-ident* *ity map OE : j ! i in I, deg(j) > deg(i). Then I is filtered by the full subca* *tegories In = J[n] whose objects have degree at most n. An orderable cover J = {Jn}n2N for such an I will be called compatible (wi* *th the choice of deg ) if there is a strictly increasing sequence of integers (kn)* *n2N such that Obj(Jn) = deg-1([kn+1, kn]). 1.4. Example. The fine cover for a directed indexing category I is defined by l* *etting Jn be the subcategory obtained from the "difference categories" eJn:= In \ I* *n-1 (discrete, by assumption) by adding all the maps from any of these objects into* * In-1. 4 D. BLANC, M.W. JOHNSON, AND J.M. TURNER For instance, if I = [n] is the linear category of n composable maps (with d* *egrees as labels): OEn OEn-1 OE2 OE1 n -! n - 1 ---! . .2. -! . .1. -! 0 , then Ik consists of the k + 1 arrows on the right, eJk= {k}, and the fine* * cover thus is Jk := {OEk}. 1.5. Example. If I is the commutative square diagram d // 4 _____3 (1.6) c || |b| fflffl|fflffl| 2 __a_//_1 then Jek contains only k, while J2 = {a : 2 ! 1}, J3 = b : 3 ! 1, and J4 contains both c : 4 ! 2 and d : 4 ! 3 (since I3 contains both 2 and 3). 1.7. Remark. As noted in the introduction, a group (or monoid) G may be thought of as a category with a single object. If we start with a directed indexing ca* *tegory I0, and for i 2 I0, we add maps g : i ! i for each g 2 G for some group G = Gi (with suitable commutation relations with the maps of I0), we obtain a small category I (no longer directed) whose diagrams describe directed systems * *of group actions. Clearly, any orderable cover J 0 of I0 induces an orderable c* *over J of I. 1.8. Example. Let I0 consist of two parallel arrows OE1, OE-1 : i ! j, Gi * *= Z=2, and Gj = 0. Then the indexing category I has a single new non-identity map f : i ! i and OEk O f = OE-k ( k = 1). Compare [D ]. 1.9. Model categories. Now let C be a simplicial model category (cf. [Q1 , II, x1]), and let CI d* *enote the functor category of I-diagrams in C. There are (at least) two relevant simp* *licial model category structures on CI: (a)For general I and cofibrantly generated C, we have the diagram model cat- egory structure, in which the weak equivalences and fibrations are defined objectwise, and the cofibrations are generated (under retracts, pushouts,* * and transfinite compositions) by the free maps (free on a generating cofibrat* *ion at some i 2 I) - cf. [H , Theorem 11.6.1]. (b)If I is a directed indexing category as above, it is in particular a (one* *-sided) Reedy category (cf. [H , x15.1.1]). Thus CI has a Reedy model category structure, in which the weak equivalences are defined objectwise, the cof* *ibra- tions are defined by attaching of a suitable latching object, and the fib* *rations are defined by requiring that the structure map to the matching objects a* *re all fibrations (cf. [H , x15.3]). 1.10. Remark. In the cases where I is a Reedy category and C is cofibrantly gen- erated, the identity Id : C ! C is a strong Quillen functor (actually a Quil* *len equivalence) between the two model category structures (see [H , Theorem 15.6.4* *]), considered as a right adjoint from the Reedy model structure to the diagram mod* *el structure. As a consequence, every Reedy fibration is an objectwise fibration (* *cf. [H , SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 5 Proposition 15.3.11]), and conversely, every cofibration in the diagram model c* *ate- gory is a Reedy cofibration. In both cases we use the same simplicial mapping s* *paces map CI(X, Y ), (sometimes denoted simply by map (X, Y )), with (1.11) map CI(X, Y )n := Hom CI(X x [n], Y ) . 1.12. Diagrams over Z. For a fixed ground diagram Z : I ! C, the comma category CI=Z consists of diagrams X : I ! C over Z - that is, for each* * i 2 I we have maps pi : Xi ! Zi, natural in I. Once again CI=Z has the two model category structures described above. The simplicial mapping space map CI=Z(X, * *Y ), defined as in (1.11), will usually be denoted simply by map Z(X, Y ). We m* *ay assume that Z is Reedy fibrant, so in particular (objectwise) fibrant. 1.13. Sketchable categories. Most of our results are valid for quite general simplicial model categories C. However, as noted in the introduction, we shall* * be mainly interested in the case where C = sA is the category of simplicial obj* *ects over some FP-sketchable category A (essentially: a category of (possibly grade* *d) universal algebras - cf. [AR , x1]). Note that any such C is cofibrantly gen* *erated - in fact, a resolution model category (see [BJT , x3]). Such an A will be ca* *lled G-sketchable if it is equipped with a faithful forgetful functor to a category * *of graded groups (compare [BP , x4.1]). The important property for our purposes is that a* * map f : X ! Y in C is a fibration if and only if it is an epimorphism onto the ba* *sepoint component of Y (cf. [Q1 , II, x3, Prop. 1]). If we let A = Gp, we obtain the homotopy category of pointed connected topol* *og- ical spaces (see [GJ , V, x6]), so our assumptions cover all the topological ap* *plications mentioned in the introduction. In this context we may need to consider diagrams over a fixed ground diagram * *Z: following [Q2 , x2] and [Be , x3], for (diagrams of simplicial objects in) a G-* *sketchable category A, one may identify Z-modules with abelian group objects over Z. Thus we may be forced to work in CI=Z if we want to study cohomology with twisted coefficients. 1.14. Diagram completion. Any inclusion of categories J ,! I induces a forgetful truncation functor o = oIJ: CI ! CJ, and this has a right adjoint * *, = ,IJ: CJ ! CI, which assigns to a diagram Y : J ! C the diagram ,Y : I ! C with ,Y (i) := limi=JY for each i 2 I (where i=J is the obvious subcategory * *of the under category i=I). Note that ,Y (j) = Yj for j 2 J. Also, if J J0 * * I 0 I I I I J0 I J0 I then ,JJ = oJ0O ,J, ,J = ,J0O ,J , and oJ = oJ O oJ0, so we shall often * *omit the superscripts from these functors, with the second category understood from * *the context. The resulting monad oeJ := ,J O oJ : CI ! CI is called the completion at J,* * and we denote the augmentation of the adjunction by !J : Y ! oeJY . Moreover, given a fixed Z 2 CI, the truncation functor ^oJ: CI=Z ! CJ=oZ * *also has a right adjoint ^,J: CJ=oZ ! CI=Z, with the limit ^,JYi:= limi=JY take* *n over oJZ, so that the completion at J in CI=Z is: (1.15) ^oeJY (j) = oeJY (j) xoeJZ(j)Zj , where the structure map oeJq : oeJY ! oeJZ is induced by the functoriality of * *limits. Once again, there will be an augmentation ^!J: Y ! ^oeJY . 6 D. BLANC, M.W. JOHNSON, AND J.M. TURNER 1.16. Example. If I = [n] is linear (x1.4) and J = [k] is an initial (right) * *segment, then for any tower Y : [n] ! C we have: ( Yi ifi k oeJY (i) = Yk ifi k 1.17. Example. If I is the commutative square of x1.5, then oeJ3Y is the pul* *lback diagram Y2 xY1Y3 _____//Y3 | | (1.18) fflffl|| Yf(b)flffl|| Y2___Y_(a)//_Y1, while ^oeJ3Y (3) is the further pullback ^oeJ3Y (3)__//_Y2 xY1Y3 (1.19) || || fflffl| fflffl| Z4 _______//Z2 xZ1 Z3. 1.20. Example. If I = 0 op is the indexing category for restricted simplic* *ial objects Y (without degeneracies), and J is its truncation to dimensions < n, t* *hen oeJY (n) = MnY is the classical matching object of [BK , X, x4.5] 1.21. Maps of diagrams. Given a fixed Reedy fibrant ground diagram Z : I ! C, consider the simplicial mapping space map Z(X, Y ) as in x1.12 for X, Y 2 C* *I=Z, where X is cofibrant and Y is fibrant. In the cases of interest to us, Y will be an abelian group object in CI=Z, * *so the homotopy groups of map Z(X, Y ) are the cohomology groups of X with coefficient* *s in Y (see [BJT , x5] for further details). In order to build our restriction tower* *, we need an appropriate orderable cover J of I (x1.1), yielding a filtration I . . . In In-1 . ... Let Mn := map CIn=onZ(onX, onZ) for each n 2 N, where onX is the restriction of a diagram X 2 CI to In. The inclusions In-1 ,! In and In ,! I ind* *uce maps aen : Mn ! Mn-1 and ^aen: M ! Mn which fit into a tower: ______________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________________________________@ map Z(X, Y )______________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________________ NNNN _______________________________________________* *__________________________________________________________^ae________________* *_____________________________________________________________________________* *___________________________ (1.22) NNN^aen+1NN __^aen____________________________________* *_____________________________________________________________n-1_____________* *_______________________________________ NN'' aen__%%__________________________________$* *$___________________________________________+1aeaen-1 . ._._______//Mn+1 ____//_Mn ____//_Mn-1 ____//_. .M.0 * * n with (1.23) map Z(X, Y ) ~= limMn . n 2. A tower of fibrations To determine when (1.22) is a tower of fibrations (so that (1.23) is a homo* *topy limit), we need the following: SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 7 2.1. Definition. Let I be an indexing category, C a model category, and Z 2 CI. Given an orderable cover J = {J } 2N of I with associated filtration (In) = (J[n])n2Z, let ok : CI ! CIk and omk : CIm ! CIk denote the truncation fun* *ctors, with adjoints indexed accordingly. A diagram Y 2 CI=Z is called J -fibrant i* *f for each n 2 Z, the augmentation ^!n+1: on+1Y ! ^oen+1nY = ^oeIn+1InY is a fib* *ration in CIn+1=oen+1nZ = CIn+1=oeIn+1InZ. 2.2. Remark. Because we assumed the degree is strictly decreasing, In+1 and I* * are the same so far as the augmentation map ^!n+1 is concerned. Thus if we assume for simplicity that I = In+1, then ^!n+1 may be identified with its adjoint* * map Y ! ^oenY in CIn+1=oen+1nZ = CI=oenZ. 2.3. Proposition. Assume J = {J } 2N is an orderable cover of I, X 2 CI=Z is cofibrant, and Y 2 CI=Z is a J -fibrant abelian group object. Then aen+1 Fn+1 ! Mn+1 ---! Mn is a fibration sequence of simplicial abelian groups for each n 2 Z, and the * *fiber Fn+1 is map CJn+1=Z|Jn+1(X|Jn+1, Fib(!n+1)). Here Fib(!n+1) denotes the fib* *er (in CIn+1=oen+1nZ) of the augmentation !n+1 : on+1Y ! oen+1nY = oeIn+1InY . Proof.Assume for simplicity that I = In+1(= J[n + 1]), with on = oIn : CI ! * *CIn and oen(= oeJ[n]) the completion at In(= J[n]) (as in Remark 2.2). Then the* *re is a natural adjunction isomorphism: map CIn=onZ(onX, onY ) = map CI=oenZ(X, ^oenY ) , under which aen is identified with the map induced in map oenZ(X, -) by * *^!n+1: Y ! ^oenY . This !^n+1 is a fibration in CI=oenZ by Definition 2.1, and thus* * induces a fibration of mapping spaces, with fiber map oenZ(X, Fib(^!n+1)). Thus, it suffices to identify the fiber instead as map CJn+1=Z|Jn+1(X|Jn+1, F* *ib(!n+1)). However, since ^!n+1(i) : Yi ! ^oenY (i) is the identity for i 2 In, the * *diagram Fib(^!n+1) : I ! C is trivial (over Z) when restricted to In, and since J * *was orderable, any map f : X = on+1X ! Fib(^!n+1) is determined uniquely by its restriction to Jn+1 - in fact, to the discrete subcategory eJn+1:= Jn+1 \ * *In. The fact that Y is an abelian group object in CI=Z implies, by definition,* * that for each i 2 I there is a commuting triangle: si Zi_____//Yi " (2.4) = ||"""" fflffl|qi"""" Zi , natural in I. Thus Fib(^!n+1)(j) for j 2 Jn+1 is by definition the pullback* * of: Zj QQWWWW QQWWWWWWW | QQQQQQ WWWWIdWWW (2.5) fflffl||sjO!ZQ((QQQWWWWWWWWW Y _^!_//_ WWWW++W j (!,qj)^oenYj = oenY (j) xoenZ(j) Zj , 8 D. BLANC, M.W. JOHNSON, AND J.M. TURNER and we readily check that this is the same as Fib(!n+1)(j), which is the pull* *back of: oenZ(j) (2.6) |oens(j)| ! fflffl| Yj__Y__//oenY (j) . 2.7. Directed indexing diagrams. We shall now see how Proposition 2.3 applies when J is an orderable cover of a directed indexing category I (see x1.3). Recall that in the Reedy model category structure (cf. x1.9) on CI, a map * *f : X ! Y is a fibration if and only if (f,p) (2.8) Xj - -! Yj xoenY (j)oenX(j) is a fibration in C for every j 2 Obj I with deg(j) = n + 1, where oen = * *oeIn is the completion at In. In CI=Z we must replace oen by ^oen(x1.14), of cours* *e. 2.9. Lemma. If I is a directed indexing category, any Reedy fibrant Y 2 CI=Z * * is J -fibrant for the fine cover of I (x1.4). Proof.Once again we assume I = In+1 (x2.2), so we must show that ^!n+1: Y ! ^oenY is a fibration in CI=oenZ. Since ^!n+1is the identity for j 2 In, * *consider j 2 eJn+1:= In+1 \ In. Since Y is Reedy fibrant in CI=Z, q : Y ! Z is a R* *eedy fibration in CI, and since J is fine, this means that (!n+1,q) Yj -----! Zj xoenZ(j)oenY (j) = ^oenY (j) = ^oenY (j) x^oenY (j)^oe* *nY (j) is a fibration in C - which shows that (2.8) indeed holds for each j 2 I. 2.10. Proposition. Let C = sA for some G-sketchable category A (x1.13), and l* *et J = {J } 2N be an orderable cover of a directed indexing category I, with Z * *2 CI Reedy fibrant. Then any abelian group object Y 2 CI=Z is weakly equivalent t* *o a fibrant (objectwise) abelian group object which is J -fibrant. Proof.Because I is directed, we may construct the desired J -fibrant replacemen* *t ~Y - an abelian group object in CI=Z - by induction on the degree of j 2 I. Moreover, we assumed that Z is Reedy fibrant, so in particular objectwise fibra* *nt (see Remark 1.10). Note that any abelian group object p : V ! Z in CI=Z is (objectwise) fibrant, since p has a section by (2.4) and x1.13; hence p has th* *e right lifting property with respect to any acyclic cofibration. We assume by induction on deg(j) = n + 1 that both ~!n+1(j) : ~Yj! ^oen~Y* *(j) and ~qj: ~Yj! Zj are fibrations in C. Since for each j, oenY (j) is defi* *ned as a limit, and an abelian group object structure on any V is a map V xZ V ! V (o* *ver Z), by functoriality (and commutativity) of limits we see that oenq : oenY~ ! * *oenZ is an abelian group object, too - so oenq is an objectwise fibration in CI. * *But ssZ ^oen~Yj_____//_Zj | | | | fflffl|oenq fflffl| oenY~(j)___//_oenZ(j) SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 9 is a pullback square, by definition, so ssZ is a fibration in C by base chan* *ge. In the induction step, for each j of degree n + 1, we factor: ~^!j : ~Yj! ^oen~Y(j) = oenY~(j) xoenZ(j)Zj as 0 ~Yj,! ~Yj0~!j-!^oe~Y(j) (an acyclic cofibration followed by a fibration), and replace ~Yj by ~Yj0. B* *oth ~!0j and ~qj:= ssZ O ~!0j: ~Yj! Zj are then fibrations in C, as required. 2.11. Remark. This actually works for some orderable covers of indexing categor* *ies which are not directed. For example, if we use the fine cover J for an indexi* *ng category I constructed as in x1.7, we can still change any Y into a J -fibrant * *one by induction on the degree in I0, since we have not introduced any new objects 2.12. Example. In Example 1.8, for any Y 2 CI, oeY is given by: oeY (j) = Yix Yi -!-!Yi = oeY (i) , with horizontal maps Y (OE 1) the two projections, and f : oeY (j) ! oeY (j* *) the switch map. To make this J -fibrant for the obvious (fine) cover, we just have* * to choose ~Y so that ^!: ~Yj! oeY~(j) is a Z=2-equivariant fibration. 2.13. The dual construction. The approach described above is clearly best suited to directed indexing cate* *gories I where the degree function is non-negative. In the inverse case, the dual appr* *oach may be preferable: Given a small indexing category I and a subcategory J, the truncation functor o = oIJ: CI ! CJ also has a left adjoint i = iIJ: CJ ! CI, which assigns to* * a diagram X : J ! C the diagram iX : I ! C with iX(i) := colimJ=iX for each i 2 I. We denote the resulting comonad on CI by `J = iJ O oJ. Note t* *hat if X 2 CI=Z, then `JX comes equipped with a map to `JZ 2 CI=Z, so we do not need the analogue of (1.15). We then say that a diagram X 2 CI=Z is J -cofibrant for an orderable cover* * J if for each n 2 Z, the coaugmentation jn+1 : `n+1nX = `In+1InX ! on+1X is a cofibration in CIn+1=on+1Z. We then have: 2.14. Proposition. Assume J = {J } 2N is an orderable cover of I, X 2 CI=Z is J -cofibrant, and Y 2 CI=Z is a fibrant abelian group object. Then aen+1 Fn+1 ! map CIn+1=on+1Z(on+1X, on+1Y ) ---! map CIn=onZ(onX, onY ) is a fibration sequence of simplicial abelian groups for each n 2 Z, and the * *fiber Fn+1 is map CJn+1=Z|Jn+1(Cof (jn+1), Y |Jn+1). Here Cof(jn+1) denotes the cofiber (over on+1Z) of the coaugmentation j* *n+1 : `n+1nX ! on+1X. Proof.Dual to that of Proposition 2.3 Note that if I is a directed indexing category, we need no special assumptions on X, Y 2 CI=Z (or C) in order for the dual of Proposition 2.10 to hold, sin* *ce all colimits are over Z to begin with. Thus, we can again build J -cofibrant replac* *ements by induction on degree to yield the following: 10 D. BLANC, M.W. JOHNSON, AND J.M. TURNER 2.15. Proposition. Let C = sA for some G-sketchable category A, and let J = {J } 2N be an orderable cover of a directed indexing category I. Then any X* * 2 CI=Z is weakly equivalent to a cofibrant object (with respect to the model st* *ructure of x1.9(a)), which is J -cofibrant. 3. The two truncation spectral sequences As noted above, for a suitable model category C and any indexing category I, * *given Z 2 CI and X, Y 2 CI=Z with X cofibrant and Y a fibrant abelian group object, the homotopy groups of map Z (X, Y ) are the cohomology groups H*(X=Z, Y ) (suitably indexed). Thus if J is some orderable cover of I such that Y is J -fi* *brant, the homotopy spectral sequence for the tower of fibrations (cf. [GJ , VII, x6]) of * *(fibrant) simplicial sets (1.22) yields a spectral sequence with E2k,n= ssk+n Fib(aen* *) =) ssk+n map Z(X, Y ). To identify the E2-term, we need the following: 3.1. Definition. Consider an orderable cover J = {I0, J} of a diagram I (whe* *re we have in mind I = In+1, I0 = In, and J = Jn+1). If Y is an abelian gro* *up object in CI=Z which is J -fibrant, then we have a fibration sequence Fib(^!) ! Y -^!!^oeY , of abelian group objects over Z, where ^oeis the completion at I0. We define the relative cohomology of the pair (I, J) to be the total left * *de- rived functor of Hom CJ=Z|J(-, Fib(^!)), (into simplicial abelian groups), * *denoted by H(X=Z; ^!). In particular, the i-th relative cohomology group for (I, J)* * is Hi(X=Z; ^!) := ssiH(X=Z; ^!). 3.2. Remark. Note that in most applications the abelian group object Y 2 CI=Z will be an n-th dimensional Eilenberg-Mac Lane object (over Z), in which case it is customary to re-index the relative cohomology groups so that Hn(X=Z; ^!) := ss0H(X=Z; ^!). Observe, however, that our setup allows Y to consist of Eilenberg-Mac Lane ob* *jects of varying dimensions, with the maps Y (f) representing cohomology operation* *s. In this general setting, no canonical re-indexing exists. 3.3. Fact. Given I, J, I0 and Y, Z as above, for any (cofibrant) X 2 CI=Z * * there is a long exact sequence in cohomology (3.4) ! Hi((X=Z)|J; ^!) ! Hi(X=Z; Y ) ! Hi((X=Z)|I0; Y |I0) ! Hi+1((X=Z)|J; ^!) ! 3.5. Theorem. For any simplicial model category C, directed indexing category I, and diagrams Z : I ! C, X 2 CI=Z, abelian group object Y 2 CI=Z, and left-orderable cover J of I there is a first quadrant spectral sequence with: E2s,t= Ht+s((X=Z)|Jt; ^!) =) Hs+t(X=Z; Y ) and d2 : E2s,t! E2s-2,t+1. Proof.Replace Z by a weakly equivalent Reedy fibrant diagram in CI, then X by a weakly equivalent cofibrant object in CI=Z, and then using Proposition 2.10* * to replace Y by a weakly equivalent J -fibrant abelian group object in CI=Z. Pr* *opo- sition 2.3 then implies that (1.22) is a tower of fibrations, and the associ* *ated homotopy spectral sequence has the specified relative cohomology groups as the * *ho- motopy groups of the fibers (which are the E2-term of the spectral sequence, i* *n our indexing). SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 11 The spectral sequence need not converge, in general, without some cohomologic* *al connectivity assumptions on the subdiagrams (unless the cover J is finite, of c* *ourse). 3.6. Remark. If J is the fine cover, the E2-term simplifies to: Y E2s,t= Ht+s(Xj=Zj, ^OEj) , j2Jet where ^OEj: Yj ! lim Yi is the structure map. OE:j!i Using the approach of x2.13, we also obtain a dual spectral sequence: 3.7. Theorem. For C, I, Z, X, and Y as in Theorem 3.5, and J right-orderable, there is a first quadrant spectral sequence with: E2s,t= Hs+t(jt; Y ) =) Hs+t(X=Z; Y ) . 3.8. Remark. Note that H*(jt; Y ) := H*(Cof (jt)=Z|Jt; Y ) is just the usual* * coho- mology of the map of diagrams jt : `tt-1X ! osX (see x2.13). This fits into* * the usual long exact sequence of a pair, dual to that of (3.4). When X is cofibrant, Z and Y are constant, and colimIX = hocolimSIX - for example, when I is a partially ordered set, so colimIX = i2IXi - then H*(X=Z; Y ) = H*(colimIX=Z; Y ), and the dual spectral seqeunce is simply the usual Mayer-Vietoris spectral sequence for the cover X of colimIX (cf. [Se, * *x5], and compare [BK , XII, 4.5], [V , x10], and [Sl]). 3.9. Example. Let I be the commuting square as in Example 1.5: Given a diagram of abelian group objects Y : I ! C, the successive fibers Fib(!n+1) (see Proposition 2.3) are: Ker(Y (c)) \ Ker(Y (d))___//_0 | | | | fflffl| fflffl| 0 ______________//_0 for !4 : Y ! oe3Y ; Ker(Y (b))_=__//_Ker(Y (b)) | | | | fflffl| fflffl| 0 ____________//_0 for !3 : o3Y ! oe2Y ; Ker(Y (a))____//_0 | =|| || fflffl| |fflffl Ker(Y (a))____//_0 for !2 : o2Y ! oe1Y ; and 0_____//0 | | | | fflffl|fflffl| 0_____//Y1 for !1 : Y ! oe0Y . 12 D. BLANC, M.W. JOHNSON, AND J.M. TURNER Thus the E2-term for the spectral sequence consists of only four non-trivial* * lines: 8 s+4 >>H (X4; Ker (Y (c)) \ Ker(Y (d)))ift = 4; >> >> >>Hs+1(X ; Y ) ift = 1; >: 1 1 0 otherwise. If we had used the fine cover, by Remark 3.6 we would instead have: 8 >>Hs+3(X4; Ker (Y (c)) \ Ker(Y (d))) ift = 3; >< Hs+2(X3; Ker (Y (a))) Hs+2(X2; Ker (Y (b)))ift = 2; E2s,t~= >>Hs+1(X1; Y1) ift = 1; >: 0 otherwise. 3.11. Remark. The square can be thought of as a single morphism in the category* * of arrows, so that we could analyze it as in [BJT , x4], where H*(X; Y ) is sho* *wn to fit into a long exact sequence with ordinary cohomology groups in the remaining* * two slots. See x7.11 below. 4.An approach through local cohomology The towers of Section 2 were constructed by covering a given indexing category I by truncated subcategories, obtained by omitting successive initial (or termi* *nal) objects. We now present an alternative approach, using subcategories obtained * *by omitting internal objects of I. As we shall see, the resulting towers differ in* * nature from those considered above. 4.1. Definition. An indexing category I will be called strongly directed if: i.It is directed in the sense of having no maps f : i ! i but the identit* *y. ii.It has a nonempty weakly initial subcategory (necessarily discrete) consi* *sting of all objects with no incoming maps, as well as a nonempty weakly final subcategory consisting of all objects with no outgoing maps. iii.It is locally finite (that is, all Hom -sets are finite). iv.I (that is, its underlying undirected graph) is connected. 4.2. Definition. We refer to (C, I, Z, X, Y ) as admissible if: (a) C is a simplicial model category; (b) I is strongly directed; (c) Z 2 CI is Reedy fibrant (hence objectwise fibrant); (d) X, Y 2 CI=Z with X cofibrant and Y a fibrant abelian group object. 4.3. Definition. For any categories C and I and diagrams Z 2 CI and X, Y 2 CI=* *Z, the product of simplicial sets Y DCI=Z(X, Y ) := map C=Zi(Xi, Yi) . i2I will be called the space of discrete transformations from X to Y over Z. We shall generally abbreviate this to DZ(X, Y ). Note that these are maps * *of functors only for the discrete indexing category Iffi, with no non-identity m* *aps. SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 13 4.4. The primary tower. In the spirit of Section 1, for any finite indexing category I we construct a* * finite sequence of full subcategories (4.5) I1 I2 . .I.n= I of I, starting with I1, whose objects are the weakly initial and final sets. As before, this can be done in several ways (ultimately yielding variant spec* *tral sequences). In any case, we can refine (4.5) so that for each k, Ik-1 is obt* *ained from Ik by omitting a single internal object ik (where internal means that * *it is neither weakly initial nor weakly final). If (C, I, Z, X, Y ) is admissible, the inclusions of categories 'k-1 : Ik* *-1 ,! Ik induce a finite tower of simplicial abelian groups: '*k-1 (4.6) map CIn=Z(X, Y ) ! . .!. map CIk=Z(X, Y ) --! map CIk-1=Z(X, Y ) ! . * *.,. analogous to (1.22). 4.7. The auxiliary fibration. Unfortunately, (4.6) is not, in general, a towe* *r of fibrations, so we cannot use it directly to obtain a useable spectral sequence * *for the cohomology of a diagram. To do so, we must replace it (up to homotopy) by a tow* *er of fibrations, with map Z (X, Y ) as its homotopy inverse limit. The resulting* * spectral sequence (abutting to the homotopy groups of map Z (X, Y )), will have the homo* *topy groups of the homotopy fibers of the maps qj as its E2-term. In fact, instea* *d of constructing the replacement directly, we make use of the following observation: For any indexing category I and diagrams X, Y : I ! C, the set Nat CI(X, Y )* * of diagram maps (natural transformations) from X to Y fits into an equalizer diagr* *am: Y Y Y (4.8) Nat CI(X, Y ) ,! Hom C(Xi, Yi) -!-! Hom C(Xi, Yj) . i2I i,j2Ij2HomI(i,j) Here the two parallel arrows map to each factor indexed by j : i ! j in I by* * the appropriate projection, followed by either Y (j)* : Hom C(Xi, Yi) ! Hom C(Xi, Y* *j), or X(j)* : Hom C(Xj, Yj) ! Hom C(Xi, Yj), respectively. In the case where Y is an abelian group object in CI (or CI=Z), this desc* *ribes NatCI(X, Y ) as the kernel of the difference , of the two parallel arrows. B* *y con- sidering mapping spaces rather than Hom -sets, we obtain a left-exact sequence* * of simplicial abelian groups: , Y Y (4.9) 0 ! map (X, Y ) ! D(X, Y ) -! map (Xi, Yj) , i,j2Ij:i!j and similarly for map Z(X, Y ). However, (4.9) is not generally a fibration sequence, except when the underl* *ying graph of I is a tree (the proof of [BJT , Prop. 4.23], where I consists of a si* *ngle map, generalizes to this case). Nevertheless, for strongly directed indexing catego* *ries I (Definition 4.1), we can define a subspace LI(X, Y ) (see Definition 5.5) ins* *ide the right-hand space of (4.9), such that , factors through a fibration (see Lem* *ma 5.9 below), and: (4.10) 0 ! map Z(X, Y ) ! DZ(X, Y ) -! LI(X, Y ) is thus a fibration sequence. 14 D. BLANC, M.W. JOHNSON, AND J.M. TURNER For such an I we obtain an auxiliary tower: pn-1 p1 (4.11) LIn(X, Y ) ---! LIn-1(X, Y )! . .!. LI2(X, Y ) -! LI1(X, Y ) (see x5.10). We shall show that the maps pk are fibrations (see Proposition 6* *.2), with a fiber which we identify as Fk := HIkc(X=Z, Y ) (cf. Definition 7.4). 4.12. The auxiliary fibers. Since all of these constructions will be natural,* * for each k the inclusion of categories ik-1 : Ik-1 ,! Ik will induce a commuting * *square of fibrations: k DCIk=Z(X, Y )______//LIk(X, Y ) ssk-1|| pk-1|| fflffl| fflffl| DCIk-1=Z(X, Y )_k-1//_LIk-1(X, Y ), where the left vertical map ssk-1 is the projection onto the appropriate fac* *tors. Thus we will have a homotopy-commutative diagram: Q Fib(i*k-1)_______//_i2I map C=Zi(Xi, Yi)___//HIk(X=Z, Y ) k\Ik-1 c | | | | | | | | fflffl| fflffl| fflffl| (4.13) map CIk=Z(X, Y )_________//_DCIk=Z(X, Y_)___k____//LI k(X, Y ) i*k-1|| ssk-1|| |pk-1| fflffl| fflffl| fflffl| map CIk-1=Z(X, Y )_______//_DCIk-1=Z(X, Y_)_k-1__//LIk-1(X, Y ) in which all rows and columns are fibration sequences up to homotopy. Since the homotopy groups of imap C=Zi(Xi, Yi) are a direct product of coho* *mol- ogy groups of the individual spaces in the diagram X, the top row of (4.13) al* *lows us to identify the successive homotopy fibers of maps of the primary tower (4.* *6) in terms of those of the auxiliary tower (4.11). Taking k = n, we see also * *that map Z(X, Y ) is in fact the homotopy limit of the primary tower. 4.14. A modified primary tower. Using standard methods, we can change (4.6) into a tower with the same homotopy limit, but simpler successive fibers: For 1 k n we define qk : DZ(X, Y ) ! LIk(X, Y ) to be the composite fibration: I pkO...Opn-1 DZ(X, Y ) -! LI(X, Y ) ------! LIk(X, Y ), and denote the fiber of qk by EICIk=Z(X, Y ). The induced maps rk : EIkZ(X, Y ) ! EIk-1Z(X, Y ) then fit into a tower: rn-1 r2 I2 r1 I1 (4.15) EInZ(X, Y ) --! . . .-! EZ (X, Y ) -! EZ (X, Y ) . As in x4.12, we see that the homotopy fiber of rk is the loop space of the f* *iber Fk := HIkc(X=Z, Y ) of pk, while the homotopy limit of (4.15) is EIZ(X, * *Y ) = map Z(X, Y ). Therefore, if we take the homotopy spectral sequence for the to* *wer (4.15), rather than that for (4.6), we get the same abutment, and a closely * *related E2-term. 4.16. Definition. For (C, I, Z, X, Y ) as above and J a subcategory of I, we * *denote by EJCI=Z(X, Y ) = EJZ(X, Y ) the sub-simplicial set of DZ(X, Y ) consist* *ing of SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 15 transformations which are natural when restricted to J-diagrams. In other word* *s, these are elements oe of DZ(X, Y ) which make X(f) Xi ____//_Xj (4.17) oei|| |oej| fflffl| fflffl| Yi_Y_(f)//_Yj commute, for any morphism f : i ! j in J. For example, EI1Z(X, Y ), consists of those transformations which are natur* *al only with respect to morphisms of maximal length. On the other hand, EIZ(X, Y ) * *is simply map Z(X, Y ). Note that any inclusion of subcategories0 J0 ! J of I induces an injection * *of simplicial sets rJJ0: EJZ(X, Y ) ! EJZ(X, Y ), since any transformation natur* *al over J must be natural over the subcategory J0. 4.18. Lemma. For (Ik)nk=1 as in (4.5), we can identify EIkZ(X, Y ) of x4.1* *4 with EIkCI=Z(X, Y ), and rk : EIkZ(X, Y ) ! EIk-1Z(X, Y ) with rIkIk-1. Proof.Follows from Definition 4.16. 5.The Auxiliary Tower Suppose (C, I, Z, X, Y ) is admissible. In order to construct the auxiliar* *y tower (4.11), we need a number of definitions: 5.1. Definition. Assuming (C, I, Z, X, Y ) is admissible: a) For any composable sequence fo of k non-identity morphisms in I (i.e.,* * a k-simplex of the reduced nerve of I, N (I), where identities are exclud* *ed) its diagonal mapping space is M(fo) := map Zt(fk)(Xs(f1), Yt(fk)) , In particular, for f : a ! b in I we haveY M(f) := map Zb(Xa, Yb). b) For each k 1, let DiagkZ(X, Y ) := M(fo). In particular, we Q fo2 N(I)k denote Diag1Z(X, Y )= f2I M(f) by DiagZ (X, Y ). c) Any map into the product Diag kZ(X, Y )is defined by specifying its proje* *ction onto each factor M(fo), indexed by fo 2 N (I)k. In particular, we have two maps of interest Diag k-1Z(X, Y )! DiagkZ(X* *, Y ): (i) X*, for which the fo-component is the composite proj X(f1)* Diagk-1Z(X, Y )--! M(f2, . .,.fk) ----! M(fo) . (ii) Y*, for which the fo-component is the composite proj Y (fk)* Diagk-1Z(X, Y )--! M(f1, . .,.fk-1) ----! M(fo) d) By iterating the maps 1 := Y* + X* : Diagk-1Z(X, Y )! DiagkZ(X, Y ) for various k > 1 we obtain maps: j : DiagkZ(X, Y )! Diagk+jZ(X, Y ) 16 D. BLANC, M.W. JOHNSON, AND J.M. TURNER for each j 1. Setting 0 := Id : Diag1Z(X, Y )! Diag1Z(X, Y ), we m* *ay combine these to define: Yn : DiagZ (X, Y )! DiagkZ(X, Y ). k=1 For any fo 2 N (I)k, we write fo for composed with the projection onto M(fo). e) For any fo = (f1, . .,., fk) 2 N (I)k, let c(fo)Q:= fkOfk-1O. .O.f1 deno* *te the composition in I. We then have a map ~fo: nk=1DiagkZ(X, Y )! M(c(fo)), which is just the projection onto M(fo) =-!M(c(fo)). 5.2. Remark. If (g, f) 2 N (I)2, is a composable pair in I, then by definitio* *n of we have (g,f)= Y (f) O g + f O X(g) . More generally, if ho = (go, fo) 2 N (I)k+j is the concatentation of go 2 N* * (I)k and fo 2 N (I)j, then: (5.3) (go,fo)= Y (c(fo))* go + X(c(go))* fo . Note also that (Y* + X*) O (Y* + X*) = Y*Y* + Y*X* + X*X* : DiagkZ(X, Y )! Diagk+2Z(X, Y ) and so inductively: (5.4) j = (Y* + X*)j = ji=0(Y*)j-i(X*)i: DiagkZ(X, Y )! Diagk+jZ(X, Y ). 5.5. Definition. Let KI denote the indexing category with o objects: 0, 1, and Arr(I) := N (I)1, o morphisms: one arrow OE : 0 ! 1, and an arrow kfo : 1 ! c(fo) 2 Arr(I) for each fo 2 N (I). If (C, I, Z, X, Y ) is admissible, define a diagram of simplicialQabelian * *groups VI : KI ! sA by setting VI(0) = Diag Z(X, Y ), VI(1) = nk=1Diag kZ(X, Y,) and VI(f) = M(f), with VI(OE) = and VI(kfo) = ~fo. Then set LI(X, Y ):= limKI VI. This limit can be described more concretely as follows: write Indec(I) for* * the collectionQof indecomposable maps in I, and let LI(X, Y ) denote the subspace* * of f2Indec(I)M(f) consisting of tuples 'o satisfying X k X l (5.6) Y (fkO. .O.fi+1)'fiX(fi-1O. .O.f1) = Y (glO. .O.gi+1)'giX(gi-1O. .* *O.g1) i=0 i=0 whenever c(fo) = c(go). 5.7. Lemma. The simplicial abelian group LI(X, Y ) is isomorphic to LI(X, Y * *). Proof.The limit condition for ' 2 LI(X, Y ) implies that the value of 'f fo* *r any decomposable f is uniquely determined by the values of 'fi for fi indecomposab* *le, by the recursive formula (5.3). 5.8. Remark. As a consequence of the previous lemma,Qfor (full) subcategories * *J I we have natural inclusion maps iJ : LJ (X, Y )! f2Indec(J)M(f). SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 17 We now investigate the properties of LI(X, Y ) and its associated fibratio* *ns. First, note that there are two maps X*, Y* : DZ(X, Y ) ! Diag Z(X, Y ), which project to precomposition and postcomposition respectively on appropriate facto* *rs and we show: 5.9. Lemma. The difference map , := Y*- X* : DZ(X, Y ) ! DiagZ (X, Y ) factors through a map : DZ(X, Y ) ! LI(X, Y ) with kernel map Z(X, Y ). Proof.Note that the sum (5.4), applied to an element in the image of the diff* *erence map Y* - X* : DZ(X, Y ) ! DiagZ (X, Y ), is telescopic, so we are left with: (Y*)k - (X*)k. Since X and Y are in CI, * * for any fo 2 N (I)k the composite: Yn ~ fo DZ(X, Y ) ! DiagZ (X, Y )! DiagkZ(X, Y )--! M(c(fo)) k=1 sends any oe to Y (f)oes(f)- oet(f)X(f). As a consequence, we get an identica* *l value for any go 2 N (I)j with c(fo) = c(go). Thus, the universal property of the* * limit implies the difference map factors through the limit LI(X, Y ). To identify the kernel of , we instead consider the difference map: Y* - X* : DZ(X, Y ) ! DiagZ (X, Y ). Clearly (oe) = 0 if and only if Y (f)oes(f)- oet(f)X(f) = 0, for every mo* *rphism f in I - that is, precisely when oe is a natural transformation of CI. Sinc* *e both X and Y are diagrams over Z, and each oef is a map over Zf, oe is in that* * case actually a natural transformation over Z. 5.10. Notation. In order to describe the behavior of the L-construction with re* *spect to the inclusion of a subcategory ' : J ! I, note that we can define two diff* *erent diagrams of simplicial abelian groups indexed by KJ (Definition 5.5): One is VJ, whose limit is LJ (X, Y ).Q The second, which we denote by VI* *,J, has VI,J(0) = Diag Z(X, Y ), VI,J(1) = nk=1 DiagkZ(X, Y ), as for VI, (* *and VI,J(f) = M(f) for f 2 Arr(J)). If we set LI,J(X, Y ):= limKJ VI,J, we see* * that there is a canonical map o : LI(X, Y )! LI,J(X, Y ) (sinceQfewer constraints a* *re imposed in defining the second limit as a subset of f2Indec(I)M(f)). On the other hand, we have a morphism of KJ-diagrams from , : VI,J! VJ, obtained by projecting the larger products DiagkZ(X, Y ) onto DiagkZ|J(X|J, Y* * |J) for each k 1. This induces a map on the limits ,* : LI,J(X, Y )! LJ (X, Y ),* * and we define the restriction map (p =)pIJ: LI(X, Y )! LJ (X, Y ) to be pIJ:= ,* * *O o. Finally, note that there is an obvious restriction map r : DCI=Z(X, Y ) ! DCJ* *=Z(X, Y ), which is simply the projection onto the factors indexed by Arr(J). From the definitions it is clear that the diagram: I DCI=Z(X, Y )_____//_LI(X, Y ) (5.11) r|| |pIJ| fflffl| |fflffl DCJ=Z(X, Y ) ____//_LJ (X, Y ). J commutes. 18 D. BLANC, M.W. JOHNSON, AND J.M. TURNER The kernel of pIJO I will be the same as the kernel of J O rIJ, by the commutativity of (5.11). However, by Lemma 5.9, the kernel of J is the sp* *ace of J-natural transformations. Thus the kernel of the composite pIJO I will b* *e the space DCJ=Z(X, Y ). 5.12. Lemma. Given J I and f 2 Indec(J) with f = c(fo) for fo = (fk, fk-1, . .,.f1) 2 N (I)k with fi2 Indec(I) (i = 1, . .k.), the followin* *g diagram commutes: pIJ LI(X, Y )____________//LJ (X, Y ) |iI| |iJ| Y fflffl| Y |fflffl (5.13) M(f) M(f) f2Indec(I) f2Indec(J) |proj |proj | | fflffl| kfo |fflffl M(f1) x . .x.M(fk) ________//_M(f) where the maps iI and iJ are the inclusions of x5.8. Proof.Suppose 'o is an element of LI(X, Y ), while f = c(fo) is a maxim* *al decomposition (so each fi is indecomposable). Then 'f lies in Diag1Z(X, Y * *), so 'f = 'f lands in M(f). However, ('fk, . .,.'f1) 2 M(fk) x . .x.M(f1) maps to ki=0Y (fk O . .O.fi+1)'fiX(fi-1O . .O.f1) also in M(c(fo)) = M(f). Th* *us, 'o 2 LI(X, Y ) = LI(X, Y ) (see Lemma 5.7) implies the value of 'f for any decomposable f is uniquely determined by the values of 'fi for fi indecomposab* *le, using formula (5.6). Note that if f is also indecomposable in I, the bottom map of (5.13) is Id* * : M(f) ! M(f). The choice of decomposition of f in I is also irrelevant, by Defi* *nition 5.5. 6. Fibrations in the Auxiliary Tower As noted in x4.7, the auxiliary tower (4.11) was constructed with two goals* * in mind: to replace (4.6) by a tower of fibrations (with the same homotopy limit* *), and to identify the homotopy fibers of the successive maps in (4.6). In this * *section we show that the map of Lemma 5.9 is indeed a fibration, and that the auxilia* *ry tower is a tower of fibrations. First, we need the following: 6.1. Definition. Any strongly directed indexing category I has two filtrations,* * defined inductively: a) The filtration {Fi}ni=0 on I is defined by decomposition length from the left, so F0 consists of weakly initial objects in I and Fn+1 consis* *ts of indecomposable maps with sources in Fn, (including their targets). b) The filtration {Gi}ni=0 is similarly defined by decomposition length fro* *m the right, so G0 consists of the weakly terminal objects in I and Gn+1 co* *nsists of indecomposable maps with targets in Gn, (including their sources). 6.2. Proposition. If (C, I, Z, X, Y ) is admissible, the induced difference * *map: : DZ(X, Y ) ! LI(X, Y ) SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 19 of Lemma 5.9 is a fibration of simplicial abelian groups. Proof.By [Q1 , II, x3, Prop. 1], it suffices to show that surjects onto the z* *ero component of LI(X, Y ). Thus, given 0 ~ 'o 2 LI(X, Y ), we must produce an element oeo 2 DZ(X, Y ) with (oeo) = 'o; i.e., for every f : a ! b in I * *we want: (6.3) oebO X(f) = Y (f) O oea - 'f . Note that since Y is an abelian group object in CI=Z, the zero map X ! Y * * is the unique map in CI=Z that factors through the section s : Z ! Y (which exi* *sts by (2.4) and x1.13). We proceed by induction on the filtration {Fi}ni=0 of I of Definition 6.1. * * To begin, for each c 2 F0, we may choose oec : Xc ! Yc to be 0. Assume by induction that we have constructed maps oec : Xc ! Yc for each c 2 Fi, satisfying (6.3) for every f in Fi, and with each oec ~ 0. Note t* *hat for any f : b ! c, in Fi+1 the map: (6.4) (f) := Y (f) O oeb- 'f : Xb ! Yc is well-defined (since necessarily b 2 Fi). This is our candidate for oec O* * X(f), and (f)~ -Y (f) O oeb ~ 0 by the assumption on ' together with the inducti* *on hypothesis (considering naturality of the section Z ! Y ). Moreover, given any g : a ! b (necessarily in Fi), we have 'g = Y (g) O* * oea + oebO X(f) by (6.3), so from 'o 2 LJ (X, Y ) it follows that: (f O g)= Y (f O g) O oea - 'fOg = Y (f O g) O oea - [Y (f) O 'g + 'f O X(g)] (6.5) = Y (f O g) O oea - [Y (f) O (Y (g) O oea - oebO X(g)) + 'f O * *X(g)] = (f)O X(g). Now given c 2 Fi+1\ Fi, set: ^Xc:= colimXb . b 2 I=c Since X 2 CI is cofibrant, it is Reedy cofibrant (x1.10), which implies th* *at the canonical map "c : X^c! Xc is a cofibration. Moreover, (6.5) implies that t* *he maps (f) defined above induce a map ^ c: X^c ! Yc. Since all the maps in question are nullhomotopic by construction, the diagram: X^c _"c_//_Xc AA AA |0 ^ cA__AAfflffl|| Yc commutes up to homotopy. Hence by [BJT , Cor. 4.20] there is a map oe : Xc ! Yc in C=Zc making the diagram X^c _"c_//_Xc AA (6.6) AAA oe|| ^ cA__Afflffl| Yc 20 D. BLANC, M.W. JOHNSON, AND J.M. TURNER commute, and we choose this to be oec. By construction oec O X(f) = (f) f* *or every f : b ! c, so (6.3) is satisfied. This completes the induction. 6.7. Proposition. If (C, I, Z, X, Y ) is admissible, let J be a subcategory * *of I ob- tained by omitting a terminal object c. Then the restriction map pIJ: LI(X, Y * *)! LJ (X, Y ) is a fibration. Proof.As in the previous proof, we must inductively define a lift oeo 2 LI(X, * *Y ) for a nullhomotopic 'o 2 LJ (X, Y ). Under these conditions, pIJ is simply* * a forgetful functor, so this means oeg = 'g for g a morphism of J and we must define oe` : Xd ! Yc whenever ` : d ! c is a morphism in I, in a manner compatible with the definition of 'o. Note that 'o determines the composi* *te Y (f) O Igo=: _ (go, f). Following the approach of the previous proof, we will define (go, f) for a* *ny go f e -! d -! c in I, where f is indecomposable, so as to satisfy three properties: First, we require that our choices be coherent: (6.8) (go O ho,)f= (go, f)O X(c(ho)) , which will allow us to build a homotopy commutative triangle using a colimit co* *n- struction. Second, we need our choices to be consistent: (6.9) (go, f)= (g0o,)f0+ _ (go, f)- _ (g0o, f0)wheneverf O go = f0O g0oinI* * , which is needed so that we eventually obtain an element oeo 2 LI(X, Y ). In f* *act, our construction will also work when go = ;, which will yield oe (f)= (;, * *f). Finally, we require that each (go, f)~ 0. go f We now proceed to choose (go, f) for e -! d -! c with e 2 Fi (Definiti* *on 6.1) by induction on i 0: go f For each ` : e ! c in I with e 2 F0, choose some decomposition e -! d -* *! c (with ` = c(go, f) and f indecomposable), and an arbitrary nullhomotopic 0* * = (go, f): Xe ! Yc. For any other decomposition ` = c(g0o, f0), the map (* *g0o,)f0 is then determined by (6.9). Assume that has been defined for every e 2 Fi so that (6.8) and (6.9) h* *old (wherever applicable). For each e 2 Fi+1\ Fi and map ` : e ! c, consider t* *he over-category Fi=e (which is non-empty by definition of Fi+1) and set X^e* *:= colima2Fi=eXa. Because the diagram X is cofibrant, hence Reedy cofibrant (x1.1* *0) in CI, the canonical map "e : ^Xe,! Xe is a cofibration. go f Again choose some decomposition e -! d -! c of `. The maps (go O ho,)* *f: Xa ! Yc, for each composable sequence ho : a ! e in Fi=e induce a (necessa* *rily nullhomotopic) map ^~e: ^Xe! Yc by (6.8). Since: X^e _"e_//_Xe AA AAA |0 ^ (go,f)__AAAfflffl|| Yc SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 21 then commutes up to homotopy, we apply [BJT , Cor. 4.20] to find X^e _"e_//_Xe AA AAA | (go,f) ^ (go,f)__AAAfflffl|| Yc making the diagram commute. g0o 0 f0 0* * 0 For any other decomposition e -! d -! c of `, use (6.9) to define (go* *, f). This completes the induction step. We have thus defined (go, f): Xe ! Yc satisfying (6.8) and (6.9) for ev* *ery go f e -! d -! c in I=c. In particular, we can choose oe (f)= (;, f): Xd ! Yc * * for each indecomposable f : d ! c in I and see that oeo 2 LI(X, Y ) (by Lemma 5* *.7) is the desired lift. 6.10. Corollary. If (C, I, Z, X, Y ) is admissible, let J be a full subcateg* *ory of I obtained by omitting an object c such that all maps out of c are indecomposable. Then pIJ: LI(X, Y )! LJ (X, Y ) is a fibration. Proof.As in the proof of Proposition 6.7 we can construct oe (f) for each f : * *d ! c. in I, such that we have ^: colimd2I=cXd ! Yc, as well as ^fflc: colimd2I=cXd* * ! Xc. For any g : c ! b, in I (indecomposable by assumption), we also have a map ^': colimd2I=cXd ! Xb induced by 'o. Note that by (5.3) we must have: oe (g)O X(f^flc) = I(g,f)- Y (g) O oe (f) = ^'- Y (g) O ^ , and since X(f) is a cofibration, we may choose the extension oe (g) as in * *(6.6). 6.11. Definition. If I is a strongly directed indexing category, let J = {Jk}k* *2N be a fine orderable cover (x1.4) of I subordinate to the filtration G (Definition * *6.1), such that Jk \ Jk-1 consists of a single object of I for each k 2 N. Let C = * *sA for some G-sketchable category A (x1.13), with Z 2 CI fibrant. A fibrant abe* *lian group object Y 2 CI=Z is called strongly fibrant if it is J -fibrant with re* *spect to the model category structure of x1.9(a). 6.12. Remark. Note that this definition is independent of the choice of the ref* *inement J of G. Forthermore, by Proposition 2.10, any abelian group object Y 2 CI=Z * *is weakly equivalent to one which is strongly fibrant. 6.13. Proposition. Suppose (C, I, Z, X, Y ) is admissible, and that Y is st* *rongly fibrant. Assume that J is obtained from I by omitting an object c such that all* * maps into c are indecomposable. Then the restriction map pIJ: LI(X, Y )! LJ (X, Y )* * is a fibration. Proof.Dual to the proofs of Proposition 6.7 and Corollary 6.10. The strong fibr* *ancy is needed since in the model category we use for diagrams ordinary fibrancy is * *merely objectwise, while strong fibrancy is dual to Reedy cofibrancy for our purposes. 6.14. Proposition. If (C, I, Z, X, Y ) is admissible, Y is strongly fibrant* *, and J is obtained from I by omitting any object c, then the restriction map pIJ: LI(X, * *Y )! LJ (X, Y ) is a fibration. 22 D. BLANC, M.W. JOHNSON, AND J.M. TURNER Proof.Consider any composable sequence: g fo (6.15) d ho-!c -! b -! a in I. As above, 0 ~ 'o 2 LJ (X, Y ) will determine the map (6.16) _ (ho, g, fo):= Y (c((g, fo))) O Iho+ IfoO X(c((ho, g))) and we use (ho, g, fo): Xd ! Ya, to denote the candidate for Y (c(fo)) O o* *e (g)O X(c(ho)) which we will construct. As before we require coherence: (6.17) (ho O `o, g, ko)O fo= Y (c(ko)) O (ho, g, fo)O X(c(`o)) for any g fo ko e `o-!d ho-!c -! b -! a -! z in I; and consistency: (6.18) (h0o, g0,)f0o= _ (ho, g, fo)+ (ho, g, fo)- _ (h0o, g0, f0o) whenever c(h0o, g0,)f0o= c(ho, g, fo). We choose the maps satisfying (6.17) and (6.18) by two successive inducti* *ons: o The first is by induction on i, the filtration degree of d in {Fi}mi=0 * *(by composition length from the left): this is done as in the proof of Propos* *ition g fo 6.7, until finally we have (h, g, fo)for every d -h!c -! b -! a, whe* *re h is indecomposable and a is terminal in I (by coherence this extends back * *to any d ho-!c). o The second is by induction on j, the filtration degree of a in {Gj}nj=0 * *(by composition length from the right), as in the proof of Proposition 6.13 (* *which is why we need Y to be strongly fibrant). At the end of the two induction processes we have chosen (h, g): Xd ! Yb f* *or h and g indecomposable. We can then choose oe (h)= (h): Xd ! Yc as in the last step of the proof of Proposition 6.7, and finally choose oe (g)= (g): Xc ! Y* *b as in the proof of Corollary 6.10. This completes the construction of a lift oeo 2 L* *I(X, Y ) for 'o as required. 6.19. Corollary. Suppose (C, I, Z, X, Y ) is admissible, Y is strongly fibran* *t, and J is any full subcategory of I with the same weakly initial and final objects. Th* *en the restriction map p : LI(X, Y )! LJ (X, Y ) is a fibration Proof.By induction on the number of objects in I \ J, using Proposition 6.14. 7. Identifying the Fibers As we have just seen, if I is a good indexing category, under our standard as- sumptions on Z, X, and Y the auxiliary tower (4.11) is a tower of fibrations* * of simplicial abelian groups. It remains to identify the fibers of the restrictio* *n maps p : LI(X, Y )! LJ (X, Y ), for a subcategory J of I; this will allow us to det* *ermine those of the primary tower (4.6) (or, more directly, those of the modified to* *wer (4.15)). We consider only the case when I \ J consists of a single internal * *object c. 7.1. Lemma. If (C, I, Z, X, Y ) is admissible and Y is strongly fibrant, th* *en 'o 2 Ker(p) LI(X, Y ) if and only if SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 23 a) 'f = 0 for each morphism f of I which does not begin or end in c. g f b) for any d -! c -! b in I with f and g indecomposable: (7.2) Y (f) O 'g + 'f O X(g) = 0 , Proof.This follows from Lemma 5.12. 7.3. Remark. The lemma implies that ('f, -'g) defines a map from X(g) to Y (f). Note also that if 'f is an arrow over Zt(f), the same is true of it* *s negative; the remainder of the diagram for a map over Z(f) already commutes because X and Y are diagrams over Z. Thus ('f, -'g) is a map of arrows over Z(f). 7.4. Definition. If (C, I, Z, X, Y ) is admissible, we define the local coho* *mology of X 2 CI=Z at an object c 2 I, denoted by Hc(X=Z, Y ), to be the total derived functors into simplicial abelian groups of map OEc(_c, aec) applied to* * X, where _c : hocolimd2I=cXd ! Xc, aec : Yc ! holimb2c=IYb, and OEc : Zc ! holimb2c=* *IZb, are the structure maps. The i-th local cohomology group of X 2 CI=Z at c is de* *fined to be Hic(X=Z, Y ) := ssiHc(X=Z, Y ). 7.5. Remark. In many cases, the local cohomology at c can be identified explici* *tly as the Andr'e-Quillen cohomology of an appropriate (small) diagram. 7.6. Proposition. If (C, I, Z, X, Y ) is admissible, Y is strongly fibrant,* * and J = I \ {c}, then Ker(p) is weakly equivalent (as a simplicial abelian group* *) to Hc(X=Z, Y ). Proof.To obtain the total derived functors, in this case, we must replace X by a weakly equivalent cofibrant, hence Reedy cofibrant object, which implies th* *at hocolimd2I=cXd is simply the colimit, and _c is a cofibration. By Remark 6.12, we can replace Y by a weakly equivalent strongly fibrant abelian group ob* *ject in CI=Z, which implies that holim b2c=IYb is the limit, and aec is a fibratio* *n. With these choices, HIc(X=Z, Y ) is simply the mapping space map OEc(_c, aec), * *which is isomorphic to Ker(p) in Lemma 7.1 (using the sign of Remark 7.3). 7.7. Theorem. If (C, I, Z, X, Y ) is admissible, for any ordering (ci)1i=1 * *of the objects of I, there is a natural first quadrant spectral sequence with: E2s,t= Hs+1ct(X=Z; Y ) =) Hs+t+1(X=Z; Y ) , with d2 : E2s,t! E2s-2,t+1. Proof.We may replace Y by a weakly equivalent strongly fibrant abelian group ob* *ject, by Remark 6.12. By Corollary 6.19, (4.15) is then a tower of fibrations, so i* *t has an associated homotopy spectral sequence. To identify the E2-term, note that t* *he homotopy groups of the homotopy fibers of the tower are the local cohomology gr* *oups in Proposition 7.6, suitably indexed (see Remark 3.2). 7.8. Remark. Note that pIJ: LI(X, Y ) ! LJ (X, Y ) is a fibration for any full subcategory J I with the same weakly initial and final objects (Corollary 6.19), and we can similarly describe the fiber of pIJ as a sort of local coho* *mology HIJ(X=Z, Y ), and thus identify the E2-term of the spectral sequence obtained* * from a fairly arbitrary cover of I. We shall not attempt to define HIJ(X=Z, Y ) in general. Observe, however, * *that if J is discrete (i.e., there are no non-identity maps between its objects c1,* * . .,.cn), 24 D. BLANC, M.W. JOHNSON, AND J.M. TURNER then nY (7.9) HIJ(X=Z, Y ) ~= Hci(X=Z, Y ). i=1 7.10. Example. For the commuting square of Example 3.9, we now get a cover for I consisting of I3 = I, I2 = I \ {3} - i.e., a commuting triangle: 4 = == c|| =bOd== fflffl|OEOE= 2 __a_//_d I1 = {4 aOc--!1}, and I0 = {4}. Given a diagram of abelian group objects Y : I ! C, the local cohomology grou* *ps which form the E2-term of the spectral sequence of Theorem 7.7 are: 8 >>Hs+3(X(d); Y (b)) ift = 2; >< Hs+2(X(c); Y (a)) ift = 1; E2s,t~= >>Hs+1(X4; Y1) ift = 0; >: 0 otherwise. Once more we could unite the first and second rows by omitting I2 from our cover, as in Example 3.9, by (7.9). 7.11. A comparison. In the simplest case, when I = [1] (a single map): XOE X2 ____//_X1 ffi 111 ff|f2|iffiffif1||11 p ffifflffl|ffifflffl|11 2ffiffiY____//Y 1p11 ffiffi" 2Y OE 1CC111 ffiffi"" CCC11 ffiffiq2""" q1 CC11 ffiffi"""" ZOE !,,!C Z2 _____________________//Z1 , we have the "defining fibration sequence": , (7.12) map (X, Y ) ! map (X2, Y2) x map (X1, Y1) -! map (X2, Y1) of [BJT , Prop. 4.20] (where all mapping spaces are taken in the appropriate co* *mma categories). Projecting the total space of (7.12) onto the second factor yields the foll* *owing interlocking diagram of horizontal and vertical fibration sequences: map (X2, Fib(Y OE))___________//map(X, Y )____________//map(X1, Y1) i*|| || |Id| fflffl| fflffl| ss fflffl| (7.13) map (X2, Y2)_______//_map(X2, Y2) x map (X1, Y1)___//map(X1, Y1) OE*|| |,| | fflffl| Id fflffl| fflffl|| map (X2, Y1)______________//_map(X2, Y1)________________//* SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 25 We see that the spectral sequence of Theorem 3.5 reduces to the long exact se* *quence in homotopy for the top horizontal fibration sequence in (7.13) , while the lon* *g exact sequence of Fact 3.3 is obtained from the left vertical fibration sequence in * *(7.13). 7.14. Remark. This actually works for any linear order I = [n] (x1.4): OEn Given X, Y 2 CI=Z, if we set I0 := [n - 1] (so J := {n -! n - 1}) and* * let 0 o = oII0: CI=Z ! CI =Z|I0, then (7.12) yields a fibration sequence: , map (X, Y ) ! map (Xn, Yn) x map (oX, oY ) -! map (Xn, Yn-1) which again induces a interlocking diagram of fibrations: map (Xn, Fib(Y OEn))____________//map(X, Y )_____________//map(oX, oY ) i*|| || Id|| fflffl| fflffl| ss fflffl| map (Xn, Yn) _______//_map(Xn, Yn) x map (oX, oY )____//map(oX, oY ) | (OEn)*|| ,|| | fflffl| Id fflffl| fflffl|| map (Xn, Yn-1) _____________//_map(Xn, Yn-1)________________//_* as in (7.13). Note that the long exact sequences in homotopy (i.e., cohomology* *) of the central vertical fibrations (for various values of n) provide an alternative in* *ductive approach to calculating the cohomology of X, which can again be formalized in a spectral sequence (though in this case the fibers are the unknown quantity). References [AR] J.V Ad'amek & J Rosick'y, Locally presentable and accessible categories, * *Cambridge U. Press, Cambridge, UK, 1994. [A] D.W. Anderson, "A generalization of the Eilenberg-Moore spectral sequence* *", Bull. AMS 78 (1972), No. 5, pp. 784-786. [BW] H.J. Baues & G. Wirsching, "The cohomology of small categories", , J. Pur* *e Appl. Alg. 38 (1985), pp. 187-211. [Be] J.M. Beck, "Triples, algebras and cohomology", Repr. Theory Appl. Cats. 2* * (2003), pp. 1-59. [BG] B. Bendiffalah & D. Guin, "Cohomologie de diagrammes d'alg`ebres triangul* *aires", In Collo- quium on Homology and Representation Theory (Vaquer'ias, 1998), Bol. Acad* *. Nac. Cienc. (C'ordoba 65 (2000), pp. 61-71. [BC] D.J. Benson & J.F. Carlson, "Diagrammatic methods for modular representat* *ions and co- homology", Comm. Algebra 15 (1987), 53-121. [Bl] D. Blanc, "Higher homotopy operations and the realizability of homotopy g* *roups", Proc. London Math. Soc. 70 (1995), pp. 214-240. [BDG] D. Blanc, W.G. Dwyer & P.G. Goerss, "The realization space of a -algebra* *: a moduli problem in algebraic topology", Topology 43 (2004), pp. 857-892. [BJT] D. Blanc, M.W. Johnson, & J.M. Turner, "On Realizing Diagrams of -algebr* *as", Algebraic & Geometric Topology 6 (2006), pp. 763-807 [BP] D. Blanc & G. Peschke, "The fiber of functors between categories of algeb* *ras", J. Pure & Appl. Alg. 207 (2006), pp. 687-715. [Bo] A.K. Bousfield, "On the homology spectral sequence of a cosimplicial spac* *e", Amer. J. Math. 109 (1987), No. 2, pp. 361-394. [BK] A.K. Bousfield & D.M. Kan, Homotopy Limits, Completions, and Localization* *s, Springer Lec. Notes Math. 304, Berlin-New York, 1972. [C] A.M. Cegarra, "Cohomology of diagrams of groups. The classification of (c* *o)fibred categor- ical groups", Int. Math. J. 3 (2003), pp. 643-680. [D] T. Datuashvili, "Cohomologically trivial internal categories in categorie* *s of groups with operations", Appl. Categ. Structures 3 (1995), pp. 221-237. 26 D. BLANC, M.W. JOHNSON, AND J.M. TURNER [DF] E. Dror-Farjoun, "Homotopy and homology of diagrams of spaces", in H.R. M* *iller & D.C. Ravenel, eds., Algebraic Topology (Seattle, Wash., 1985), Springer L* *ec. Notes Math. 1286, Berlin-New York, 1987, pp. 93-134. [DS] G. Dula & R. Schultz, "Diagram cohomology and isovariant homotopy theory"* *, Mem. Amer. Math. Soc. 110, Providence, RI, 1994. [DK] W.G. Dwyer & D.M. Kan, "Hochschild-Mitchell cohomology of simplicial cate* *gories and the cohomology of simplicial diagrams of simplicial sets", Nederl. Akad. * *Wetensch. Indag. Math. 50 (1988), pp. 111-120. [DKS] W.G. Dwyer, D.M. Kan, & J.H. Smith, "Homotopy commutative diagrams and th* *eir real- izations", J. Pure & Appl. Alg. 57 (1989), pp. 5-24. [FW] H.R. Fischer & F.L. Williams, "Borel-LePotier diagrams-calculus of their * *cohomology bun- dles", Tohoku Math. J. (2) 36 (1984), pp. 233-251. [GGS] M. Gerstenhaber, A. Giaquinto & S.D. Schack, "Diagrams of Lie algebras", * *J. Pure & Appl. Alg. 196 (2005), pp. 169-184. [GS1] M. Gerstenhaber & S.D. Schack, "On the deformation of algebra morphisms a* *nd diagrams", Trans. Amer. Math. Soc. 279 (1983), pp. 1-50. [GS2] M. Gerstenhaber & S.D. Schack, "Algebraic cohomology and deformation theo* *ry", in M. Gerstenhaber & M. Hazewinkel, eds., Deformation theory of algebras and* * structures and applications (Il Ciocco, 1986) NATO ASI, Series C 247, Kluwer, Dordre* *cht, 1997, pp. 11-264. [GJ] P.G. Goerss & J.F. Jardine, Simplicial Homotopy Theory, Prog. in Math. 17* *4, Birkh"auser, Boston-Stuttgart, 1999. [H] P.S. Hirschhorn, Model Categories and their Localizations, AMS, Providenc* *e, RI, 2002. [I] S. Illman, Equivariant singular homology and cohomology, I, Number 156 in* * Mem. AMS 156, Am. Math. Soc., Providence, RI, 1975. [JP] M.A. Jibladze & T.I. Pirashvili, "Cohomology of algebraic theories", J. A* *lg. 137 (1991), No. 2, pp. 253-296. [Ma] J.P. May, Equivariant homotopy and cohomology theory, Reg. Conf. Ser. Mat* *h. 91, Am. Math. Soc., Providence, RI, 1996. With contributions by M. Cole, G. Comez* *a"na, S. Costeno- ble, A.D. Elmendorf, J.P.C. Greenlees, L.G. Lewis, Jr., R.J. Piacenza, G.* *V. Triantafillou, and S. Waner. [Mi] B. Mitchell, "Rings with several objects", Advances in Math. 8 (1972), pp* *. 1-161. [O] P. Olum, "Homology of squares and factoring of diagrams", in Category The* *ory, Homol- ogy Theory and their Applications, III (Battelle Institute Conference, Se* *attle, WA, 1968) Springer, Berlin-New York, 1969, pp. 480-489. [Pa] P. Pave~si'c, "Diagram cohomologies using categorical fibrations", J. Pur* *e Appl. Algebra 112 (1996), pp. 73-90. [P1] R.J. Piacenza, "Cohomology of diagrams and equivariant singular theory", * *Pac. J. Math. 91 (1980), pp. 435-443. [P2] R.J. Piacenza, "Diagrams of simplicial sets, complexes and bundles", Tamk* *ang J. Math. 15 (1984), pp. 83-94. [Q1] D.G. Quillen, Homotopical Algebra, Springer Lec. Notes Math. 20, Berlin-N* *ew York, 1963. [Q2] D.G. Quillen, "On the (co-)homology of commutative rings", Applications o* *f Categorical Algebra, Proc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, pp. 65-87. [Se] G.B. Segal, "Categories and cohomology theories", Topology 13 (1974), pp.* * 293-312. [Sl] J. S_lomi'nska, "Some spectral sequences in Bredon cohomology", Cahiers T* *op. G'eom. Diff. Cat. 33 (1992), pp. 99-133. [V] R.M. Vogt, "Homotopy limits and colimits", Math. Z. 134 (1973), pp. 11-52. Department of Mathematics, University of Haifa, 31905 Haifa, Israel E-mail address: blanc@math.haifa.ac.il Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA E-mail address: mwj3@psu.edu Department of Mathematics, Calvin College, Grand Rapids, MI 49546, USA E-mail address: jturner@calvin.edu