On local-to-global spectral sequences for the cohomology of diagrams

LOCAL-TO-GLOBAL SPECTRAL SEQUENCES F OR THE COHOMOLOGY OF DIA GR AMS D A VID BLANC, MARK W . JOHNSON, AND JAMES M. TURN ER 0. Intr oduction The cohomology of diagr ams arises as a natural ob ject of study in sev eral math- ematical con texts: in deformation theory (see [GS2, GS1, GGS]), and in classifying diagrams of groups, as in [C]. If I is the one-ob ject category c orresp onding to a group G , a diagram X ∈ C I is just an ob j ect in C equipped with a G -action, and its cohomology is the equiv arian t cohomolo gy of [I] (cf. [P1, § 2]). On the other hand, for an y discrete or Lie group G , let I = O G denote the orbit catego r y of G : if X is a G -space, X : O G → T op is the corresp onding fixed p oin t diagram X ( G/H ) := X H , and M : O G → A bGp , is an y system of co efficien ts, then the corresp o nding coho- mology H ( X ; M ) is Bredon cohomology (cf. [Ma, I, § 4]). Finally , when I consists of a single arrow , and the co efficien ts are constan t, w e ha ve the usual cohomology of a pair. See [BG], [DS], [FW], [O], [P a ], and [BC] for further applications. 0.1 . Diagrams in homotop y theory . The cohomology of diagrams also plays a ma jor role in the Dwy er-Kan-Smith theory for the rectification of homot o p y-commu ta- tiv e diagrams (cf. [D KS] and [DF, D K]). In fact, our in terest in the sub ject was motiv a t ed b y the relat ed realization problem for diagr a ms of Π-algebras (graded groups with an action o f the primary homotopy o p erations): as in the case of a single Π-algebra (cf. [BDG]) , the o bstructions to realizing a diagr a m of Π-algebras Λ : I → Π- A l g lie in appropriate cohomology gro ups of Λ (see [BJT, Thm. 6.3 ]). F urthermore, giv en a Π-algebra Γ, all distinct homoto p y types realizing Γ ma y b e distinguished b y a set of higher homotop y op erations associat ed to a collection ( I α ) α ∈ A of finite indexing categories I α and homotop y-comm utative diagrams X α : I α → ho T op , w here all the spaces X α i are w edges of spheres. Since these higher op erations are obstructions to the rectification o f the diagrams ( X α ) α ∈ A (and th us the asso ciated diagrams Λ α := π ∗ X α : I α → Π- A l g ), they corresp ond to elemen ts in the cohomology of Γ. Understanding the cohomology gro ups of such diag r ams may therefore b e helpful in algebraicizing (and organizing) the “higher Π-a lgebra” of a space Y , consisting o f all higher homotopy op erations in π ∗ Y . 0.2 . Computing diagram cohomology . Ev en the cohomology of a single map ma y b e hard to calculate (cf. [BJT, § 5.16]), so some computational to ols are needed. F or this purp ose we construct “lo cal-to-g lobal” sp ectral sequenc es for the cohomolog y of a diag r a m, whic h can b e used to compute the cohomology of the full dia gram in terms of smaller pieces. Giv en a small category I , a mo del category C (in the sense of [Q1 ]), and an I - diagram X ∈ C I , one can define t he c ohomology of X with co efficien ts in an y Date : July 9, 2007; r evised April 3, 20 08. 1991 Mathematics Subje ct Classific ation. Prima ry: 5 5N99; secondary: 18G55,1 8G40,18 G55. Key wor ds and phr ases. Diagr ams, Andr´ e-Quillen co homology , spectral sequences, lo cal-to-glo bal. 1 2 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER ab elian group ob j ect Y ∈ C I . F or tec hnical reasons, w e shall concen tra te on the case where C = s A is the category o f simplicial ob j ects ov er some v ariet y of univ ersal algebras A : since the homotopy category of simplicial gro ups is equiv alen t to that of (p oin ted connected) t o p ological spaces, this actually cov ers a ll cases of inte rest ab o ve. Some of our results are v alid, how ev er, for an ar bitrary simplicial mo del category C . Another reason for our interes t in t he “ lo cal-to- g lobal” approa c h to diagram co- homology is that in o rder for the highe r homotop y op eration corresponding to a homotop y comm utativ e diagram X : I → ho T op to b e d e fi ne d , all lo w er order op erations (corresp onding to sub dia g rams of I ) m ust v anish c oher ently . Th us an es- sen tial step in a cohomological description of higher order op erations is the a bilit y to piece together lo cal dat a to obtain global information. 0.3. R emark. W e should p oint out tha t our metho ds w ork (almost exclusiv ely) for a dir e cte d indexing category I (i.e., with only iden tit ies as endomorphisms), whic h is a significan t restriction. Ho we v er, suc h diagrams certainly suffice for the description of higher homotopy op erations, as ab ov e: ev en the linear case – when I consists of a single comp osable sequenc e of arrows – is of in terest, since the realizabilit y of suc h a diagram is essen tially equiv alent to calculating higher T o da brac k ets. F urthermore, diagrams arising in deformation theory (indexed b y the nerv e of a co vering) are of this form. Our metho ds, suitably mo dified (cf. Remark 1.7), also apply to diagrams indexed by the orbit category O G of a gr o up G . 0.4 . The sp ect r al sequences. Let C b e a simplicial mo del catego r y and I a directed index category , and assume g iv en diagrams Z : I → C , and X , Y ∈ C I / Z , with Y a n ab elian group ob ject in C I / Z . Our main results may b e summarized as follo ws: Theorem A. T her e is a first quadr ant sp e ctr a l se quenc e with: E 2 s,t = Y j ∈ e J s H t + s ( X j / Z j , ˆ φ j ) = ⇒ H s + t ( X/ Z ; Y ) This is constructed by taking increasing truncations of t he co efficien t diagram Y (cf. Theorem 3 .5). Here H ∗ ( X/ Z , φ ) denotes relativ e cohomology for a map of the co efficien ts (see D efinition 3.1). Theorem B. T her e is a first quadr ant sp e ctr a l se quenc e with: E 2 s,t = H s + t ( η s ; Y ) = ⇒ H s + t ( X/ Z ; Y ) This spectral sequence is constructed dually to the previous one, b y taking increas- ing truncations of the sour c e dia g ram X (see Theorem 3.7). Here H ∗ ( η , Y ) denotes the usual cohomology of a map (or pair). Theorem C. If I i s c ountable, then for any or dering ( c s ) ∞ s =1 of the obje cts of I , ther e is a first quad r ant sp e ctr al se quenc e with E 2 s,t = H t + s c s ( X/ Z ; Y ) = ⇒ H s + t ( X/ Z ; Y ) . This is constructed by succes siv ely omitting the ob jects c s from I (see Theorem 7.7). Here H ∗ c ( X/ Z , Y ) denote the lo cal cohomology gro ups at an ob ject c ∈ I (see Definition 7.4). There are ve rsions of all three sp ectral seq uences defined for an y suitable co v er J of I (Definition 1.1). In particular, the sp ectral sequ ences a lwa ys conv erge if J is finite, hence if I itself is finite. SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 3 0.5 . Other v arian ts. Other sp ectral sequence s for the cohomology of a diagram ha ve app eared in the lit erature. One should mention the univ ersal co efficien ts sp ectral sequence of Piacenza (see [P2, § 1]), the the p -c hain sp ectral sequence of D a vis and L ¨ uc k (see [DL]), the equiv ar ia n t Serre sp ectral sequence of Mo erdijk and Sv ensson (see [MS], and the lo cal-to-glo bal sp ectral sequences of Jibladze and Pirashvili (see [JP]) and Robinson (see [R]) – though the la st three use a differen t definition of cohomology , based on the Baues-Wirsc hing and Ho c hsc hild-Mitc hell cohomologies of categories (cf. [BW, Mi]) . 0.6 . Orga nization. Section 1 provid es bac kgro und material o n dia grams, their co ve rs, and the mo del category o f diagra ms. In Section 2 we determine when the “restriction tow er” associated to a cov er of the in dexing category I is a tow er of fibrations, and in Section 3 we use this to set up the first t w o sp ectral sequences . The second half of the pap er is dev oted to the (somewhat more technic al) approac h based on “lo calizing at an ob ject”: Section 4 pro vides the setting, and explains the metho d. In Section 5 w e describe a n auxiliary construction asso ciated to the to we r of certain cov ers of I , a nd in Section 6 show that this auxiliary t o we r is a to wer of fibrations. Finally , in Section 7 w e identify the fib ers of the new tow er, and obtain the third sp ectral sequence. 0.7. A cknow le dgements. W e w ould lik e to thank the referee for his or her commen ts. This researc h w as supp o rted b y BSF grant 2006039 ; the third author was also sup- p orted b y NSF grant DMS-0206647 and a Calvin Researc h F ellowship (SDG). 1. The ca tegor y of diagrams Our ob ject of study will b e the category C I of diagrams – i.e., functors from a fixed small (often finite) indexing category I into a mo del category C . The maps are natural transformations. In this section w e define some concepts and in tro duce notation related to I and C I : 1.1. D efinition. Let I b e any small catego ry . By an N -indexed c over of I w e mean some collection J = { J ν } ν ∈ N of sub categories of I , suc h that eac h arrow in I b elongs to at least one J ν . A co ver J = { J ν } ν ∈ N for I will b e called or der a ble if the relation: ν 1 ≺ ν 2 Def ⇐ ⇒ ∃ i 1 ∈ J ν 1 , i 2 ∈ J ν 2 ∃ φ : i 2 → i 1 in I with i 1 6∈ J ν 2 or i 2 6∈ J ν 1 . defines a par tial order on N , and the partially ordered set ( N , ≺ ) can b e em b edded as a (p o ssibly infinite) segmen t of ( Z , ≤ ). Choo sing suc h an em b edding N ⊆ Z , w e may think of J as indexed b y in tegers, and we can then filter I b y setting J [ n ] := S i ≤ n J i . If N is b ounded b elow in Z w e sa y t ha t J is right-or der able , and if it is b ounded ab ov e w e sa y it is left-or der able . 1.2. R emark. Note that the linear ordering of J (indicated by the indices) is not generally uniquely determined b y the partial order ≺ : there may b e elemen ts of J whic h are not comparable under ≺ . This happ ens when all maps out of J n actually land in J [ k ] for k < n − 1. In this case the linear ordering of J n and J n − 1 , for example, may b e switc hed with impunit y . 1.3 . Directed indexing c at egor ies. A dir e cte d indexing c ate gory is a s mall category I equipp ed with a map deg : Ob j( I ) → Z , suc h that for ev ery no n-iden tity 4 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER map φ : j → i in I , deg ( j ) > deg( i ). Then I is filtered b y the full sub categories I n = J [ n ] whose ob jects ha ve degree at most n . An or derable co v er J = { J n } n ∈ N for suc h an I will b e called c om p a tible (with the c hoice of deg) if there is a strictly increasing sequence of in tegers ( k n ) n ∈ N suc h that Ob j ( J n ) = deg − 1 ([ k n − 1 , k n ]). 1.4. Example. The fine c over for a directed indexing category I is defined b y letting J n b e the sub category obtained from the “difference categor ies” e J n := I n \ I n − 1 (discrete, b y a ssumption) b y adding a ll the maps from any of t hese ob jects in to I n − 1 . F or instance, if I = [ n ] is the line ar category of n comp osable maps (with degrees as lab els): n φ n − → n − 1 φ n − 1 − − − → . . . 2 φ 2 − → . . . 1 φ 1 − → 0 , then I k consists of the k arrows o n the righ t, e J k = { k } , and the fine co ver th us is J k := { φ k } . 1.5. Example. If I is the commutativ e square diagram (1.6) 4 d / / c   3 b   2 a / / 1 then e J k con tains only k , while J 2 = { a : 2 → 1 } , J 3 = { b : 3 → 1 } , and J 4 con tains b ot h c : 4 → 2 and d : 4 → 3 (since I 3 con tains b oth 2 and 3). 1.7. R emark . As noted in the intro duction, a g r o up (o r monoid) G ma y b e though t of as a category with a single ob ject. If w e start with a directed indexing catego ry I ′ , a nd for i ∈ I ′ , w e add maps g : i → i for eac h g ∈ G fo r some group G = G i (with suitable comm uta tion relations with the maps of I ′ ), we obtain a small catego r y I (no lo ng er directed) whose diagra ms describ e directed systems of group actions. Clearly , an y orderable cov er J ′ of I ′ induces an orderable cov er J of I . 1.8. Example. Let I ′ consist of t w o parallel arrows φ 1 , φ − 1 : i → j , G i = Z / 2, and G j = 0 . Then t he indexing category I has a single new non- iden tity map f : i → i and φ k ◦ f = φ − k ( k = ± 1). Compare [D]. 1.9 . Mo del categories. No w let C b e a simplicial mo del category (cf. [Q1 , II, § 1]), and let C I denote the functor category of I -diagrams in C . There are (at least) t w o relev ant simplicial mo del categor y structures o n C I : (a) F or general I a nd cofibrantly generated C , we hav e the diagr am mo del cat- egory structure, in whic h the w eak equiv alences and fibrations are defined ob ject wise, and the cofibrations are generated (under retracts, pushouts, and transfinite comp ositions) by the free maps (fr ee o n a generating cofibration a t some i ∈ I ) – cf. [H, Theorem 11.6.1]. (b) If I is a directed indexing catego ry as ab o ve, it is in particular a (one-sided) Reedy category (cf. [H, § 15.1.1]). Th us C I has a R e e dy mo del category SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 5 structure, in whic h the w eak equiv alences are defined ob jectw ise, the cofibra- tions are defined b y attac hing of a suitable latching ob ject, and the fibratio ns are defined by requiring tha t t he structure map to the matc hing ob jects are all fibrations (cf. [H, § 15.3]). 1.10. R emark. In the cases where I is a Reedy category and C is cofibrantly gen- erated, the iden tity Id : C → C is a strong Quillen functor (actually a Quillen equiv alence) betw een the t wo mo del category structures (see [H, Theorem 1 5 .6.4]), considered as a righ t adjoin t from the Reedy mo del structure to the diagra m mo del structure. As a consequence, ev ery R eedy fibration is an ob ject wise fibration ( cf. [H, Prop osition 15.3.11 ]), and con vers ely , ev ery cofibration in the diagram mo del cate- gory is a Reedy cofibration. In b o t h cases w e use the same simplicial mapping spaces map C I ( X , Y ), (sometimes denoted simply by map ( X, Y )), with (1.11) map C I ( X , Y ) n := Hom C I ( X × ∆[ n ] , Y ) . 1.12 . Diagrams ov er Z . F or a fixed ground diagram Z : I → C , the comma category C I / Z consis ts of diag rams X : I → C o v er Z – tha t is, for eac h i ∈ I w e hav e maps p i : X i → Z i , na t ur a l in I . Once again C I / Z has the tw o mo del category structures describ ed ab o v e. The simplicial mapping space map C I / Z ( X , Y ), defined as in (1.11), will usually b e denoted simply by map Z ( X , Y ). W e ma y assume that Z is Reedy fibran t, so in particular (ob ject wise) fibrant. 1.13 . Sk etchable categories. Most of our results are v alid for quite general simplicial mo del categories C . How ev er, as noted in the introduction, w e shall b e mainly interes t ed in the case where C = s A is the category of simplicial ob jects o ver some FP-sk etc ha ble c ategory A (essen tially: a category of (p ossibly g r a ded) univ ersal algebras – cf. [AR, § 1]). Note that an y suc h C is cofibran tly g enerated – in fact, a resolution mo del category (see [BJT, § 3]). Suc h an A will b e called G -sketchable if it is equipp ed with a f aithful forgetful functor to a category of graded groups (compare [BP, § 4.1]). The imp ort an t prop ert y for our purp oses is that a map f : X → Y in C is a fibration if and only if it is an epimorphism onto the basep oin t comp onen t of Y ( cf. [Q1, I I, § 3, Prop. 1]). If w e let A = G p , w e obtain the homotop y categor y of p ointed connected top olog- ical spaces (see [GJ, V, § 6]), so our assumptions cov er all the t o p ological applications men tioned in the in tro duction. In this con text we ma y need to consider diagrams o ver a fixed gro und diagram Z : follo wing [Q2, § 2] a nd [Be, § 3], for (diag r ams of simplicial ob jects in) a G -sk etchable category A , one may iden tify Z - mo dules with ab elian group ob jects o v er Z . Th us w e may be forced to w ork in C I / Z if w e w a n t to study cohomology with t wisted co efficien ts. 1.14 . Diagra m completion. An y inclusion of catego r ies J ֒ → I induces a forgetful trunc a tion functor τ = τ I J : C I → C J , and this has a right adjoin t ξ = ξ I J : C J → C I , which assigns to a diag ram Y : J → C the diagra m ξ Y : I → C with ξ Y ( i ) := lim i/J Y for eac h i ∈ I (where i/J is the ob vious subcat ego ry of the under category i/I ). Note that ξ Y ( j ) = Y j for j ∈ J . Also, if J ⊆ J ′ ⊆ I then ξ J ′ J = τ I J ′ ◦ ξ I J , ξ I J = ξ I J ′ ◦ ξ J ′ J , and τ I J = τ J ′ J ◦ τ I J ′ , so w e shall often omit the sup erscripts from these functors, with the second category understo o d f rom t he con text. The resulting monad σ J := ξ J ◦ τ J : C I → C I is called the c ompletion at J , and w e denote the augmen ta tion of the adjunction b y ω J : Y → σ J Y . 6 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER Moreo ve r, giv en a fixed Z ∈ C I , the truncation functor ˆ τ J : C I / Z → C J /τ Z also has a r ig h t adjoint ˆ ξ J : C J /τ Z → C I / Z , with the limit ˆ ξ J Y ( i ) := lim i/J Y take n o ver τ J Z (t ha t is, the diagram whose limit w e take consists of Y | i/J mapping to τ J Z , where the latter includes a lso Z i ). Th us the completion at J in C I / Z is: (1.15) ˆ σ J Y ( j ) = σ J Y ( j ) × σ J Z ( j ) Z j , where the structure map σ J q : σ J Y → σ J Z is induced b y the functoriality of limits. Once again, there will b e a n augmen tatio n ˆ ω J : Y → ˆ σ J Y . 1.16. Example. If I = [ n ] is linear ( § 1.4) and J = [ k ] is an initial (righ t) segmen t, then for an y to wer Y : [ n ] → C w e ha ve: σ J Y ( i ) = ( Y i if i ≤ k Y k if i ≥ k 1.17. Example. If I is the comm utative square o f § 1.5, then σ I 3 Y is the pullbac k diagram (1.18) Y 2 × Y 1 Y 3 / /   Y 3 Y ( b )   Y 2 Y ( a ) / / Y 1 , while ˆ σ I 3 Y (3) is the further pullbac k (1.19) ˆ σ I 3 Y (3) / /   Y 2 × Y 1 Y 3   Z 4 / / Z 2 × Z 1 Z 3 . 1.20. Example. If I = ∆ ′ ⊆ ∆ op is the indexing category fo r restricted simplicial ob jects Y (without degeneracies), and J is its truncation to dimensions < n , then σ J Y ( n ) = M n Y is the classical matching ob ject of [BK, X, § 4.5] 1.21 . Maps of diagrams. Giv en a fixed Reedy fibrant ground diagram Z : I → C , consider the simplicial mapping space map Z ( X , Y ) as in § 1.12 for X , Y ∈ C I / Z , where X is cofibran t and Y is fibrant. In the cases of interes t to us, Y will b e an ab elian group ob ject in C I / Z , so the homotop y groups of map Z ( X , Y ) a re the cohomology gro ups of X with co efficien ts in Y (see [BJT, § 5] for further details). In order to build our restriction to wer, w e need an appropriate orderable cov er J o f I ( § 1.1), yielding a filtration I ⊇ . . . ⊇ I n ⊇ I n − 1 ⊇ . . . . Let M n := map C I n /τ n Z ( τ n X , τ n Z ) for each n ∈ N , where τ n X is the restriction of a diag r a m X ∈ C I to I n . The inclusions I n − 1 ֒ → I n and I n ֒ → I induce maps ρ n : M n → M n − 1 and ˆ ρ n : M → M n whic h fit into a tow er: (1.22) map Z ( X , Y ) ˆ ρ n +1 ' ' N N N N N N N N N N N ˆ ρ n % % ˆ ρ n − 1 $ $ . . . / / M n +1 ρ n +1 / / M n ρ n / / M n − 1 ρ n − 1 / / . . . M 0 SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 7 with (1.23) map Z ( X , Y ) ∼ = lim n M n . 2. A to wer of fibra tions T o determine when (1.22) is a to we r of fibrat io ns (so that (1 .2 3) is a ho mo t o p y limit), we need the f ollo wing: 2.1. Definition. Let I be an indexing category , C a mo del categor y , a nd Z ∈ C I . Giv en an orderable co v er J = { J ν } ν ∈ N of I with asso ciated filtratio n ( I n ) = ( J [ n ]) n ∈ Z , let τ k : C I → C I k and τ m k : C I m → C I k denote the truncation functors, with adjo in ts indexed accordingly . A diagram Y ∈ C I / Z is called J - fibr ant if for eac h n ∈ Z , the augmen t a tion ˆ ω n +1 : τ n +1 Y → ˆ σ n +1 n Y = ˆ σ I n +1 I n Y is a fibration in C I n +1 /σ n +1 n Z = C I n +1 /σ I n +1 I n Z . 2.2. R emark. Because w e assumed the degree is strictly decreasing, I n +1 and I are the same so far a s the augmen tation map ˆ ω n +1 is concerned. Thus if w e assume for simplicit y tha t I = I n +1 , then ˆ ω n +1 ma y b e iden tified with its adjoint map Y → ˆ σ n Y in C I n +1 /σ n +1 n Z = C I /σ n Z . 2.3. Prop osition. Assume J = { J ν } ν ∈ N is an or der able c over of I , X ∈ C I / Z is c ofibr ant, and Y ∈ C I / Z is a J -fibr ant ab elian gr oup obje ct. Then F n +1 → M n +1 ρ n +1 − − − → M n is a fibr a tion se quenc e of simplicial ab elian gr o ups for e ach n ∈ Z , and the fib er F n +1 is map C J n +1 / Z | J n +1 ( X | J n +1 , Fib( ω n +1 )) . Her e Fib( ω n +1 ) denotes the fib er (in C I n +1 /σ n +1 n Z ) of the augmen tation ω n +1 : τ n +1 Y → σ n +1 n Y = σ I n +1 I n Y . Pr o of. Assume for simplicity that I = I n +1 (= J [ n + 1]), with τ n = τ I n : C I → C I n and σ n (= σ J [ n ] ) the completion at I n (= J [ n ]) (as in Remark 2.2). Then there is a natural adjunction isomorphism: map C I n /τ n Z ( τ n X , τ n Y ) = map C I /σ n Z ( X , ˆ σ n Y ) , under whic h ρ n is iden tified with the map induced in map σ n Z ( X , − ) b y ˆ ω n +1 : Y → ˆ σ n Y . This ˆ ω n +1 is a fibration in C I /σ n Z b y Definition 2.1, and th us induces a fibration of mapping spaces, with fib er map σ n Z ( X , Fib( ˆ ω n +1 )). Th us, it suffices to iden tify the fib er instead as map C J n +1 / Z | J n +1 ( X | J n +1 , Fib( ω n +1 )). Ho we v er, since ˆ ω n +1 ( i ) : Y i → ˆ σ n Y ( i ) is the iden tit y for i ∈ I n , the diagram Fib( ˆ ω n +1 ) : I → C is trivial (o ver Z ) when restricted to I n , and since J w as orderable, an y map f : X = τ n +1 X → Fib( ˆ ω n +1 ) is determined uniquely b y its restriction to J n +1 – in fact, t o the discrete sub category e J n +1 := J n +1 \ I n . The fact that Y is an ab elian group ob ject in C I / Z implies, b y definition, that for eac h i ∈ I there is a commuting triangle: (2.4) Z i =   s i / / Y i q i   ~ ~ ~ ~ ~ ~ ~ Z i , 8 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER natural in I . Th us Fib( ˆ ω n +1 )( j ) for j ∈ J n +1 is by definition the pullbac k of : (2.5) Z j   σ n s j ◦ ω Z ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Id + + W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W Y j ˆ ω ( ω, q j ) / / ˆ σ n Y j = σ n Y ( j ) × σ n Z ( j ) Z j , and w e readily chec k that t his is the same as Fib( ω n +1 )( j ), whic h is the pullbac k of: (2.6) σ n Z ( j ) σ n s ( j )   Y j ω Y / / σ n Y ( j ) .  2.7 . Direc t ed indexing diagrams. W e shall no w see ho w Prop osition 2.3 a pplies when J is an orderable co v er of a directed indexing category I (see § 1.3) . Recall that in the Reedy mo del category structure (cf. § 1.9) on C I , a map f : X → Y is a fibration if and only if (2.8) X j ( f ,p ) − − → Y j × σ n Y ( j ) σ n X ( j ) is a fibration in C for ev ery j ∈ Ob j I with deg ( j ) = n + 1, where σ n = σ I n is the completion at I n . In C I / Z we m ust replace σ n b y ˆ σ n ( § 1.14), of course. 2.9. Lemma. If I is a dir e cte d indexi n g c ate gory, any R e e dy fibr ant Y ∈ C I / Z is J -fibr ant for the fine c over of I ( § 1. 4). Pr o of. Once again w e assume I = I n +1 ( § 2.2), so we mu st sho w that ˆ ω n +1 : Y → ˆ σ n Y is a fibration in C I /σ n Z . Since ˆ ω n +1 is the iden tity for j ∈ I n , consider j ∈ e J n +1 := I n +1 \ I n . Since Y is Reedy fibrant in C I / Z , q : Y → Z is a Reedy fibration in C I , and since J is fine, this means that Y j ( ω n +1 ,q j ) − − − − − → σ n Y ( j ) × σ n Z ( j ) Z j = ˆ σ n Y ( j ) = ˆ σ n Y ( j ) × ˆ σ n Y ( j ) ˆ σ n Y ( j ) is a fibrat io n in C – whic h sho ws that ( 2 .8) indeed holds for eac h j ∈ I .  2.10. Prop osition. L et C = s A for so m e G -sketcha b l e c ate go ry A ( § 1.13), an d let J = { J ν } ν ∈ N b e an or der able c ove r of a dir e cte d indexing c ate gory I , with Z ∈ C I R e e dy fi br a n t. Then a ny ab elian g r oup obje ct Y ∈ C I / Z is we akly e quiva l e nt to a fibr ant (obje ctwise) ab elian gr oup o bje ct w hich is J -fibr ant. Pr o of. Because I is dir ected, we ma y construct the desired J - fibra n t replacemen t ¯ Y – a n ab elian gro up ob ject in C I / Z – by induction on the degree of j ∈ I . Moreo ve r, w e assumed that Z is Reedy fibran t, so in particular ob jectw ise fibran t (see Remark 1.10). Note that an y ab elian group ob ject p : V → Z in C I / Z is (ob ject wise) fibrant, since p has a section b y (2.4) and § 1.13; hence p has the righ t lifting pro p ert y with resp ect to any acyclic cofibration. W e assume by induction on deg( j ) = n + 1 tha t b oth ¯ ω n +1 ( j ) : ¯ Y j → ˆ σ n ¯ Y ( j ) and ¯ q j : ¯ Y j → Z j are fibrations in C . Since for eac h j , σ n Y ( j ) is defined as a limit, and an ab elian g roup ob ject structure on an y V is a map V × Z V → V (ov er SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 9 Z ), b y functorialit y (and comm utativity ) of limits w e see that σ n q : σ n ¯ Y → σ n Z is an ab elian group ob ject, to o – so σ n q is an ob jectw ise fibration in C I . But ˆ σ n ¯ Y j π Z / /   Z j   σ n ¯ Y ( j ) σ n q / / σ n Z ( j ) is a pullback square, b y definition, so π Z is a fibration in C by base change. In the induction step, for each j of degree n + 1, w e factor: ¯ ˆ ω j : ¯ Y j → ˆ σ n ¯ Y ( j ) = σ n ¯ Y ( j ) × σ n Z ( j ) Z j as ¯ Y j ֒ → ¯ Y ′ j ¯ ω ′ j − → ˆ σ ¯ Y ( j ) (an acyclic cofibra tion follo w ed b y a fibration), and replace ¯ Y j b y ¯ Y ′ j . Both ¯ ω ′ j and ¯ q j := π Z ◦ ¯ ω ′ j : ¯ Y j → Z j are then fibrations in C , as required.  2.11. R emark. This actually works for some o r derable co v ers of indexing categories whic h are not directed. F or example, if w e use the fine cov er J for an indexing category I constructed a s in § 1.7, w e can still change any Y in t o a J -fibrant one by induction on the degree in I ′ , since w e ha v e not introduced any new ob jects 2.12. Example. In Example 1.8, fo r an y Y ∈ C I , σ Y is giv en by: σ Y ( j ) = Y i × Y i − → − → Y i = σ Y ( i ) , with horizon tal maps Y ( φ ± 1 ) the t wo pro jections, and f : σ Y ( j ) → σ Y ( j ) the switc h map. T o mak e this J -fibran t for the ob vious (fine) co v er, w e just ha v e to c ho o se ¯ Y so that ˆ ω : ¯ Y j → σ ¯ Y ( j ) is a Z / 2-equiv ariant fibration. 2.13 . The dual construction. The approach describ ed ab ov e is clearly b est suited t o directed indexing categories I where the degree function is non-negativ e. In the in v erse case, the dual appro ac h ma y b e preferable: Giv en a small indexing category I and a sub category J , the truncation functor τ = τ I J : C I → C J also ha s a left adjoin t ζ = ζ I J : C J → C I , whic h a ssigns to a diagram X : J → C the diagram ζ X : I → C with ζ X ( i ) := colim J /i X for eac h i ∈ I . W e denote the resulting comonad on C I b y θ J = ζ J ◦ τ J . Note that if X ∈ C I / Z , then θ J X comes equipped with a map to θ J Z ∈ C I / Z , so w e do not need the analog ue of (1.15). W e then sa y that a diagram X ∈ C I / Z is J - c ofi b r ant fo r an orderable cov er J if for eac h n ∈ Z , the coa ugmen tation η n +1 : θ n +1 n X = θ I n +1 I n X → τ n +1 X is a cofibration in C I n +1 /τ n +1 Z . W e then ha v e: 2.14. P rop osition. Assume J = { J ν } ν ∈ N is an or der able c over of I , X ∈ C I / Z is J -c ofibr ant, and Y ∈ C I / Z is a fibr ant ab eli a n gr oup obje ct. Th en F n +1 → map C I n +1 /τ n +1 Z ( τ n +1 X , τ n +1 Y ) ρ n +1 − − − → map C I n /τ n Z ( τ n X , τ n Y ) is a fibr a tion se quenc e of simplicial ab elian gr o ups for e ach n ∈ Z , and the fib er F n +1 is map C J n +1 / Z | J n +1 (Cof ( η n +1 ) , Y | J n +1 ) . 10 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER Her e Cof ( η n +1 ) denotes the c ofib er (over τ n +1 Z ) of the c o augmen tation η n +1 : θ n +1 n X → τ n +1 X . Pr o of. Dual to that of Prop osition 2.3  Note that if I is a directed ind exing category , w e need no s p ecial a ssumptions on X , Y ∈ C I / Z (or C ) in order for the dual of Prop osition 2.10 to hold, since all colimits a re ov er Z to b egin with. Thus , w e can again build J -cofibrant replacemen ts b y induction on degree to yield the following: 2.15. P rop osition. L et C = s A for some G -sketcha b l e c ate gory A , and l e t J = { J ν } ν ∈ N b e an or de r able c over of a d i r e cte d indexin g c ate gory I . Then any X ∈ C I / Z is we akly e quivalen t to a c ofibr ant obje ct (with r esp e ct to the mo del structur e of § 1.9(a)), which is J -c ofib r ant. 3. The tw o trunca tion s pectral sequence s As noted ab o ve, f o r a suitable mo del category C and an y indexing category I , giv en Z ∈ C I and X , Y ∈ C I / Z with X cofibran t and Y a fibrant a b elian group ob ject, the homotopy groups of map Z ( X , Y ) are the cohomology groups H ∗ ( X/ Z , Y ) (suitably indexed). Th us if J is some orderable co v er of I suc h that Y is J -fibrant, the homotop y sp ectral sequenc e for the to we r o f fibratio ns (cf. [GJ, VI I, § 6]) o f (fibran t) simplicial sets (1 .22) yields a sp ectral sequence with E 2 k ,n = π k + n Fib( ρ n ) = ⇒ π k + n map Z ( X , Y ). T o iden t if y the E 2 -term, we need the follo wing: 3.1. Definition. Consider an orderable cov er J = { I ′ , J } of a diagram I (where w e hav e in mind I = I n +1 , I ′ = I n , and J = J n +1 ). If Y is an ab elian group ob ject in C I / Z whic h is J -fibran t, then w e hav e a fibration sequence Fib( ˆ ω ) → Y ˆ ω − → ˆ σ Y , of ab elian group ob jects ov er Z , where ˆ σ is the completion at I ′ . W e define the r elative c ohom o lo g y of the pair ( I , J ) to b e t he total left de- riv ed functor of Hom C J / Z | J ( − , Fib( ˆ ω )) , (in to simplicial ab elian groups), denoted b y H ( X/ Z ; ˆ ω ). In particular, the i - th r e l a tive c oho m olo gy gr o up for ( I , J ) is H i ( X/ Z ; ˆ ω ) := π i H ( X/ Z ; ˆ ω ). 3.2. R ema rk. Note that in most applications the ab elian group ob j ect Y ∈ C I / Z will b e a n n -th dimensional Eilenberg-Mac Lane ob ject (o v er Z ), in whic h case it is customary to re- index t he relative cohomology groups so that H n ( X/ Z ; ˆ ω ) := π 0 H ( X/ Z ; ˆ ω ). Observ e, ho we v er, that our setup allo ws Y to consist o f Eilenberg-Mac La ne ob j ects of v arying dimensions, with the maps Y ( f ) represen ting cohomology op erations. In this general setting, no canonical re-indexing exists. 3.3. F act. Give n I , J, I ′ and Y , Z as ab ove, for any (c ofibr ant) X ∈ C I / Z ther e is a long exact se quenc e in c ohomo lo gy (3.4) → H i (( X/ Z ) | J ; ˆ ω ) → H i ( X/ Z ; Y ) → H i (( X/ Z ) | I ′ ; Y | I ′ ) → H i +1 (( X/ Z ) | J ; ˆ ω ) → 3.5. Theorem. F or any simplicial mo del c ate gory C , dir e cte d indexing c a te gory I , and diagr ams Z : I → C , X ∈ C I / Z , ab eli a n gr oup obje ct Y ∈ C I / Z , an d left-or der a ble c over J of I ther e is a first quadr ant sp e ctr al s e q uenc e w i th: E 2 s,t = H t + s (( X/ Z ) | J t ; ˆ ω ) = ⇒ H s + t ( X/ Z ; Y ) SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 11 and d 2 : E 2 s,t → E 2 s − 2 ,t +1 . Pr o of. Replace Z b y a weakly equiv alen t Reedy fibrant diag ram in C I , then X b y a w eakly equiv alen t cofibran t ob ject in C I / Z , and then using Prop osition 2.10 to replace Y b y a w eakly equiv alent J -fibran t ab elian group ob ject in C I / Z . Prop o- sition 2.3 then implies tha t (1.22) is a tow er of fibrations, and the ass o ciated homotop y sp ectral sequence has the sp ecified relative cohomology groups as t he ho- motop y groups o f the fib ers ( which are the E 2 -term of the sp ectral sequence, in our indexing).  The sp ectral sequence need not con v erge, in g eneral, without some cohomological connectivit y assumptions o n the sub diag r ams ( unless the cov er J is finite, of course). 3.6. R emark. If J is the fine co ver, the E 2 -term simplifies to: E 2 s,t = Y j ∈ e J t H t + s ( X j / Z j , ˆ φ j ) , where ˆ φ j : Y j → lim φ : j → i Y i is the structure map. Using the approac h of § 2.13 , w e also obta in a dual sp ectral sequenc e: 3.7. Theorem. F or C , I , Z , X , and Y as in The or em 3. 5, and J right-or der able, ther e is a first quadr ant sp e ctr al se quenc e with: E 2 s,t = H s + t ( η t ; Y ) = ⇒ H s + t ( X/ Z ; Y ) . 3.8. R emark . Note that H ∗ ( η t ; Y ) := H ∗ (Cof ( η t ) / Z | J t ; Y ) is just the usual coho- mology of the map of diagrams η t : θ t t − 1 X → τ s X ( see § 2.13). This fits into the usual long exact sequence of a pair, dual to that of (3.4). When X is cofibra nt, Z and Y are constant, and colim I X = ho colim I X – for example, when I is a partially ordered set, s o colim I X = S i ∈ I X i – then H ∗ ( X/ Z ; Y ) = H ∗ (colim I X/ Z ; Y ), and the dual sp ectral seqeunc e is simply the usual Ma y er-Vietoris sp ectral sequence for the cov er X of colim I X (cf. [Se , § 5], and compare [BK, XI I, 4.5], [V, § 10], and [Sl]) . 3.9. Example. Let I b e the comm uting square a s in Example 1.5: Giv en a diagram of ab elian gro up ob jects Y : I → C , the succ essiv e fib ers Fib( ω n +1 ) (see Prop o sition 2.3) are: Ker( Y ( c ) ) ∩ Ker( Y ( d )) / /   0   0 / / 0 for ω 4 : Y = τ 4 Y → σ 3 Y ; Ker( Y ( b ))   0 / / 0 for ω 3 : τ 3 Y → σ 2 Y ; Ker( Y ( a )) / / 0 for ω 2 : τ 2 Y → σ 1 Y ; and the single o b ject Y 1 for ω 1 : τ 1 Y → σ 0 Y . 12 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER Th us the E 2 -term fo r the sp ectral sequence consists of only four non-trivial lines: (3.10) E 2 s,t ∼ =              H s +4 ( X 4 ; Ker( Y ( c )) ∩ Ker( Y ( d ))) if t = 4 ; H s +3 ( X 3 ; Ker( Y ( b ))) if t = 3 ; H s +2 ( X 2 ; Ker( Y ( a ))) if t = 2 ; H s +1 ( X 1 ; Y 1 ) if t = 1 ; 0 otherwise. If w e had used the fine co ver, by Remark 3.6 we w ould instead ha ve: E 2 s,t ∼ =          H s +3 ( X 4 ; Ker( Y ( c )) ∩ Ker( Y ( d ))) if t = 3; H s +2 ( X 3 ; Ker( Y ( a ))) ⊕ H s +2 ( X 2 ; Ker( Y ( b ))) if t = 2; H s +1 ( X 1 ; Y 1 ) if t = 1; 0 otherwise. 3.11. R emark. The square can b e thoug ht of as a single morphism in the category of arro ws, so that w e could analyze it as in [BJT, § 4], where H ∗ ( X ; Y ) is shown to fit into a long exact sequence with ordinary cohomology groups in the remaining t w o slots. See § 7.11 b elow. 4. An appro ach through local cohomology The tow ers of Section 2 w ere constructed b y co v ering a given indexing category I b y truncated subcatego ries, obtained b y omitting success iv e initial (or terminal) ob jects. W e now presen t an alternativ e approac h, using sub categories obtained by omitting internal ob jects of I . As we shall see, the resulting tow ers differ in nature from those considered ab o ve. 4.1. Definition. An indexing category I will b e called str ongly dir e cte d if: i. It is dir e cte d in the sense of having no maps f : i → i but the iden tity . ii. It has a nonempt y we akly initial sub category (necessarily discrete) consisting of all ob jects with no incoming maps, as w ell as a nonem pt y we akly final sub category consisting of all ob jects with no outgoing maps. iii. It is lo c al ly fi n ite (that is, all Hom-sets a r e finite). iv. I (tha t is, its underlying undirected g r a ph) is c onne cte d . 4.2. Definition. W e refer to ( C , I , Z , X , Y ) as admissib l e if: (a) C is a simplicial mo del category; (b) I is strongly directed; (c) Z ∈ C I is Reedy fibrant (hence ob ject wise fibrant); (d) X , Y ∈ C I / Z with X cofibrant and Y a fibran t ab elian group ob ject. 4.3. Definition. F or any categories C and I and diagrams Z ∈ C I and X , Y ∈ C I / Z , the pro duct of simplicial sets D C I / Z ( X , Y ) := Y i ∈ I map C / Z i ( X i , Y i ) . will b e called the sp ac e of discr ete tr a nsformations from X to Y o v er Z . W e shall generally abbreviate this to D Z ( X , Y ). Note that these are maps o f functors only for the discrete indexing category I δ , with no non-iden tity maps. SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 13 4.4 . The primary to wer. In the spirit of Section 1, for an y finite indexing category I we construct a finite sequence of f ull sub categories (4.5) I 1 ⊂ I 2 ⊂ . . . I n = I of I , starting with I 1 , whose ob jects are the we akly initial and final sets. As b efore, this can b e done in sev eral w a ys (ultimately yielding v arian t spectral sequence s). In a n y case, we can refine (4.5) so that for each k , I k − 1 is o btained from I k b y omitting a single in ternal ob ject i k (where internal means that it is neither weakly initial nor w eakly final). If ( C , I , Z , X, Y ) is admissible, the inclusions of categories ι k − 1 : I k − 1 ֒ → I k induce a finite tow er of simplicial a b elian gr o ups: (4.6) map C I n / Z ( X , Y ) → . . . → map C I k / Z ( X , Y ) ι ∗ k − 1 − − → map C I k − 1 / Z ( X , Y ) → . . . , analogous to (1.22). 4.7 . The auxiliary fibration. Unfo rtunately , (4.6) is not, in g eneral, a to we r of fibrations, so w e cannot use it directly to obtain a useable sp ectral sequence for the cohomology of a diag r am. T o do so, w e must replace it (up to homotopy ) by a tow er of fibrations, with map Z ( X , Y ) as its homotop y inv erse limit. The resulting spectral sequence (abutting to the ho mo t o p y gro ups o f map Z ( X , Y )), will hav e the homotopy groups of the homotopy fib ers of the maps ι ∗ k as its E 2 -term. In fact, instead of constructing t he replacemen t directly , we mak e use of the following observ atio n: F or an y indexing category I and diagrams X, Y : I → C , the set Nat C I ( X , Y ) of diagram maps (natural transformations) from X to Y fits into an equalizer diag r a m: (4.8) Nat C I ( X , Y ) ֒ → Y i ∈ I Hom C ( X i , Y i ) − → − → Y i,j ∈ I Y η ∈ Hom I ( i,j ) Hom C ( X i , Y j ) . Here the tw o parallel arrows map to eac h factor indexed b y η : i → j in I b y the appropriate pro j ection, follow ed by either Y ( η ) ∗ : Hom C ( X i , Y i ) → Hom C ( X i , Y j ), or X ( η ) ∗ : Hom C ( X j , Y j ) → Hom C ( X i , Y j ), resp ectiv ely . In the case where Y is an ab elian group ob ject in C I (or C I / Z ), this describ es Nat C I ( X , Y ) as the k ernel of the difference ξ of the t wo para llel arro ws. By con- sidering mapping spaces ra ther than Hom-sets, w e obtain a left-exact sequence of simplicial ab elian gro ups: (4.9) 0 → map ( X , Y ) → D ( X , Y ) ξ − → Y i,j ∈ I Y η : i → j map ( X i , Y j ) , and similarly for map Z ( X , Y ). Ho we v er, (4.9) is not generally a fibration sequence, except when the underlying graph of I is a tree (the pro of of [BJT, Prop. 4.23], where I consists of a single map, generalizes to this case). Nev ertheless, for strongly directed indexing categories I (Definition 4 .1), w e can define a subspace L I ( X , Y ) ( see Definition 5 .5) inside the righ t- hand space of (4.9), suc h that ξ factors through a fibration Ψ (see Lemma 5.9 b elo w), and: (4.10) 0 → map Z ( X , Y ) → D Z ( X , Y ) Ψ − → L I ( X , Y ) is thus a fibration sequence. 14 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER F or suc h an I we obtain an auxiliary t ow er: (4.11) L I n ( X , Y ) p n − 1 − − − → L I n − 1 ( X , Y ) → . . . → L I 2 ( X , Y ) p 1 − → L I 1 ( X , Y ) (see § 5.10). W e shall sho w that the maps p k are fibrations (see Prop osition 6 .2 ), with a fib er whic h we iden tify as F k := H I k c ( X/ Z , Y ) (cf. Definition 7.4). 4.12 . The a uxiliary fib ers. Since all of these constructions will b e natural, for eac h k the inclusion of catego ries i k − 1 : I k − 1 ֒ → I k will induce a comm uting square of fibrations: D C I k / Z ( X , Y ) Ψ k / / π k − 1   L I k ( X , Y ) p k − 1   D C I k − 1 / Z ( X , Y ) Ψ k − 1 / / L I k − 1 ( X , Y ) , where the left v ertical map π k − 1 is the pro jection on to the appropriate factors. Th us w e will ha ve a homotopy-comm utativ e diag ram: (4.13) Fib( i ∗ k − 1 ) / /   Q i ∈ I k \ I k − 1 map C / Z i ( X i , Y i ) / /   H I k c ( X/ Z , Y )   map C I k / Z ( X , Y ) / / i ∗ k − 1   D C I k / Z ( X , Y ) Ψ k / / π k − 1   L I k ( X , Y ) p k − 1   map C I k − 1 / Z ( X , Y ) / / D C I k − 1 / Z ( X , Y ) Ψ k − 1 / / L I k − 1 ( X , Y ) in whic h a ll ro ws and columns are fibration sequenc es up to homotopy . Since the homotop y groups of Π i map C / Z i ( X i , Y i ) are a direct pro duct o f cohomol- ogy groups of the individual spaces in the diag ram X , the top row o f (4.13) allows us to iden tify the successiv e homotop y fib ers of maps of the primary tow er (4.6 ) in terms of those of the auxiliary tow er (4.11). T aking k = n , w e see also that map Z ( X , Y ) is in fact the homotopy limit of the primary to wer. 4.14 . A mo dified primary t o wer. Using standard metho ds, we can c hange (4.6) in to a tow er with the same homotopy limit, but simpler successiv e fibers: F or 1 ≤ k ≤ n w e define q k : D Z ( X , Y ) → L I k ( X , Y ) to b e the comp osite fibration: D Z ( X , Y ) Ψ I − → L I ( X , Y ) p k ◦ ... ◦ p n − 1 − − − − − − → L I k ( X , Y ) , and denote the fib er of q k b y E I C I k / Z ( X , Y ). The induced ma ps r k : E I k Z ( X , Y ) → E I k − 1 Z ( X , Y ) then fit into a tow er: (4.15) E I n Z ( X , Y ) r n − 1 − − → . . . r 2 − → E I 2 Z ( X , Y ) r 1 − → E I 1 Z ( X , Y ) . As in § 4.12, w e see tha t the ho mo t o p y fib er of r k is the lo op space of the fib er F k := H I k c ( X/ Z , Y ) of p k , while the homotop y limit of (4.15) is E I Z ( X , Y ) = map Z ( X , Y ). Therefore, if w e tak e the homotopy sp ectral sequence fo r the to w er (4.15), rather than that for (4.6), w e get the same abutment, and a closely related E 2 -term. 4.16. Definition. F or ( C , I , Z , X , Y ) as ab ov e and J a sub catego r y of I , w e denote b y E J C I / Z ( X , Y ) = E J Z ( X , Y ) the sub-simplicial set o f D Z ( X , Y ) consisting of SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 15 transformations whic h are natura l when restricted to J -diagrams. In other w ords, these are elemen ts σ of D Z ( X , Y ) whic h make (4.17) X i σ i   X ( f ) / / X j σ j   Y i Y ( f ) / / Y j comm ute, for a ny morphism f : i → j in J . F or example, E I 1 Z ( X , Y ), consists of those transformations whic h are natural only with resp ect to morphisms of maximal length. On the other hand, E I Z ( X , Y ) is simply map Z ( X , Y ). Note that an y inclusion of sub categories J ′ → J of I induces an injection of simplicial sets r J J ′ : E J Z ( X , Y ) → E J ′ Z ( X , Y ), since an y transformation natur a l o v er J m ust b e natura l ov er the sub category J ′ . 4.18. Lemma . F or ( I k ) n k =1 as in (4.5) , we c an iden tify E I k Z ( X , Y ) of § 4.14 with E I k C I / Z ( X , Y ) , and r k : E I k Z ( X , Y ) → E I k − 1 Z ( X , Y ) with r I k I k − 1 . Pr o of. F ollows from Definition 4.16.  5. The A uxiliar y To we r Supp ose ( C , I , Z , X , Y ) is admissible. In order to construct the auxiliary tow er (4.11), w e need a num b er of definitions: 5.1. Definition. Assuming ( C , I , Z , X , Y ) is admissible: a) F or an y comp o sable sequence f • of k non-identit y morphisms in I (i.e., a k -simplex of the reduced nerv e of I , N ( I ), where identities are excluded) its diagonal mapping space is M ( f • ) := map Z t ( f k ) ( X s ( f 1 ) , Y t ( f k ) ) , In particular, for f : a → b in I w e ha v e M ( f ) := ma p Z b ( X a , Y b ). b) F or each k ≥ 1, let Diag k Z ( X , Y ) := Y f • ∈ N ( I ) k M ( f • ). In particular, w e denote Diag 1 Z ( X , Y ) = Q f ∈ I M ( f ) b y Diag Z ( X , Y ). c) Any map in to the pro duct D iag k Z ( X , Y ) is define d b y specifying it s pro jection on to eac h factor M ( f • ), indexed b y f • ∈ N ( I ) k . In particular, w e ha ve t w o maps of in terest Diag k − 1 Z ( X , Y ) → D iag k Z ( X , Y ): (i) X ∗ , for whic h the f • -comp onen t is the comp osite Diag k − 1 Z ( X , Y ) pro j − − → M ( f 2 , . . . , f k ) X ( f 1 ) ∗ − − − − → M ( f • ) . (ii) Y ∗ , for whic h the f • -comp onen t is the comp osite Diag k − 1 Z ( X , Y ) pro j − − → M ( f 1 , . . . , f k − 1 ) Y ( f k ) ∗ − − − − → M ( f • ) d) By iterating the maps Φ 1 := Y ∗ + X ∗ : Diag k − 1 Z ( X , Y ) → Diag k Z ( X , Y ) for v ario us k > 1 w e obtain maps: Φ j : Diag k Z ( X , Y ) → Diag k + j Z ( X , Y ) 16 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER for eac h j ≥ 1. Setting Φ 0 := Id : Diag 1 Z ( X , Y ) → D iag 1 Z ( X , Y ), we ma y com bine these to define: Φ : Diag Z ( X , Y ) → n Y k =1 Diag k Z ( X , Y ) . F or an y f • ∈ N ( I ) k , w e write Φ f • for Φ comp osed with the pro jection on to M ( f • ). e) F or an y f • = ( f 1 , . . . , , f k ) ∈ N ( I ) k , let c ( f • ) := f k ◦ f k − 1 ◦ . . . ◦ f 1 denote the comp osition in I . W e then hav e a map κ f • : Q n k =1 Diag k Z ( X , Y ) → M ( c ( f • )), whic h is just the pro jection onto M ( f • ) = − → M ( c ( f • )). 5.2. R emark. If ( g , f ) ∈ N ( I ) 2 , is a comp osable pair in I , then by definition of Φ w e ha v e Φ ( g, f ) = Y ( f ) ◦ Φ g + Φ f ◦ X ( g ) . More generally , if h • = ( g • , f • ) ∈ N ( I ) k + j is the concaten tation of g • ∈ N ( I ) k and f • ∈ N ( I ) j , then: (5.3) Φ ( g • ,f • ) = Y ( c ( f • )) ∗ Φ g • + X ( c ( g • )) ∗ Φ f • . Note also t ha t ( Y ∗ + X ∗ ) ◦ ( Y ∗ + X ∗ ) = Y ∗ Y ∗ + Y ∗ X ∗ + X ∗ X ∗ : Diag k Z ( X , Y ) → Diag k +2 Z ( X , Y ) and so inductive ly: (5.4) Φ j = ( Y ∗ + X ∗ ) j = Σ j i =0 ( Y ∗ ) j − i ( X ∗ ) i : D iag k Z ( X , Y ) → D iag k + j Z ( X , Y ) . 5.5. Definition. Let K I denote t he indexing category with • o b jects: 0 , 1 , a nd Arr( I ) := N ( I ) 1 , • mo r phisms: one arrow φ : 0 → 1 , and an arrow k f • : 1 → c ( f • ) ∈ Arr( I ) for eac h f • ∈ N ( I ). If ( C , I , Z , X, Y ) is admissible, define a diag ram of simplicial ab elian groups V I : K I → s A by setting V I ( 0 ) = Diag Z ( X , Y ), V I ( 1 ) = Q n k =1 Diag k Z ( X , Y ), and V I ( f ) = M ( f ), with V I ( φ ) = Φ and V I ( k f • ) = κ f • . Then set L I ( X , Y ) := lim K I V I . This limit can b e described more concretely as follow s: write Indec ( I ) for the collection of indecomp o sable maps in I , and let L I ( X , Y ) denote the subspace of Q f ∈ Indec ( I ) M ( f ) consisting of tuples ϕ • satisfying (5.6) k X i =0 Y ( f k ◦ · · ·◦ f i +1 ) ϕ f i X ( f i − 1 ◦ · · ·◦ f 1 ) = l X i =0 Y ( g l ◦ · · ·◦ g i +1 ) ϕ g i X ( g i − 1 ◦ · · ·◦ g 1 ) whenev er c ( f • ) = c ( g • ). 5.7. Lemma. The simpli c ial ab elia n gr oup L I ( X , Y ) is isomorphic to L I ( X , Y ) . Pr o of. The limit condition for ϕ ∈ L I ( X , Y ) implies that the v alue of ϕ f for any decomp osable f is uniquely determined b y the v a lues o f ϕ f i for f i indecomp o sable, b y the recursiv e f orm ula (5.3).  5.8. R emark. As a consequence of the previous lemma, for (full) sub categories J ⊂ I w e ha v e natural inclusion maps i J : L J ( X , Y ) → Q f ∈ Indec ( J ) M ( f ). SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 17 W e now in ves tigate the prop erties of L I ( X , Y ) a nd its asso ciated fibrations. First, note that there are tw o maps X ∗ , Y ∗ : D Z ( X , Y ) → D iag Z ( X , Y ), which pro ject to precomposition and p ostcomp osition respective ly on appropriate factors and we sho w: 5.9. Lemma. The differ enc e map ξ := Y ∗ − X ∗ : D Z ( X , Y ) → D iag Z ( X , Y ) f a ctors thr ough a map Ψ : D Z ( X , Y ) → L I ( X , Y ) w i th kernel map Z ( X , Y ) . Pr o of. Note that the sum (5.4), a pplied to an elemen t in the image of the difference map Y ∗ − X ∗ : D Z ( X , Y ) → D iag Z ( X , Y ) , is telescopic, so w e are left with: ( Y ∗ ) k − ( X ∗ ) k . Since X a nd Y a re in C I , for an y f • ∈ N ( I ) k the comp osite: D Z ( X , Y ) → D iag Z ( X , Y ) → n Y k =1 Diag k Z ( X , Y ) κ f • − − → M ( c ( f • )) sends an y σ to Y ( f ) σ s ( f ) − σ t ( f ) X ( f ). As a consequence , w e get an iden tical v alue for an y g • ∈ N ( I ) j with c ( f • ) = c ( g • ). Th us, the univ ersal prop ert y of the limit implies the difference map factors t hr o ugh the limit L I ( X , Y ). T o identify the ke rnel of Ψ , w e instead consider the difference map: Y ∗ − X ∗ : D Z ( X , Y ) → D iag Z ( X , Y ) . Clearly Ψ( σ ) = 0 if and only if Y ( f ) σ s ( f ) − σ t ( f ) X ( f ) = 0 , fo r ev ery morphism f in I – that is, precisely when σ is a natural transformation of C I . Since b oth X and Y are diagrams o v er Z , and each σ f is a map o ve r Z f , σ is in that case actually a natural tra nsformation ov er Z .  5.10. Not ation. In or der to describe the b ehavior of the L -construction with respect to the inclusion of a sub category ι : J → I , note that w e can define t w o differen t diagrams of simplicial a b elian gr oups indexed by K J (Definition 5 .5): One is V J , whose limit is L J ( X , Y ). The second, whic h w e denote by V I ,J , has V I ,J ( 0 ) = Diag Z ( X , Y ), V I ,J ( 1 ) = Q n k =1 Diag k Z ( X , Y ), as for V I , (and V I ,J ( f ) = M ( f ) for f ∈ Arr( J )). If w e set L I ,J ( X , Y ) := lim K J V I ,J , w e see that there is a canonical map τ : L I ( X , Y ) → L I ,J ( X , Y ) (since few er constrain ts ar e imp osed in defining the second limit as a subset o f Q f ∈ Indec ( I ) M ( f )). On the other hand, we ha ve a morphism of K J -diagrams f rom ξ : V I ,J → V J , obtained b y pro jecting the la rger pro ducts D ia g k Z ( X , Y ) onto Diag k Z | J ( X | J , Y | J ) for eac h k ≥ 1 . This induces a map on the limits ξ ∗ : L I ,J ( X , Y ) → L J ( X , Y ), and w e define the r estriction map ( p =) p I J : L I ( X , Y ) → L J ( X , Y ) to b e p I J := ξ ∗ ◦ τ . Finally , note that there is an ob vious restriction map r : D C I / Z ( X , Y ) → D C J / Z ( X , Y ), whic h is simply the pro jection on to the factors indexed by Arr( J ). F rom the definitions it is clear that the diagra m: (5.11) D C I / Z ( X , Y ) Ψ I / / r   L I ( X , Y ) p I J   D C J / Z ( X , Y ) Ψ J / / L J ( X , Y ) . comm utes. 18 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER The k ernel of p I J ◦ Ψ I will be the same as the k ernel of Ψ J ◦ r I J , b y the comm utativity of (5.11). Ho w ev er, b y L emma 5.9, t he k ernel of Ψ J is the space of J -natural transformations. Thus the k ernel of the comp osite p I J ◦ Ψ I will b e the space D C J / Z ( X , Y ). 5.12. Lemma. Given J ⊆ I an d f ∈ Indec ( J ) with f = c ( f • ) for f • = ( f k , f k − 1 , . . . , f 1 ) ∈ N ( I ) k with f i ∈ Indec ( I ) ( i = 1 , . . . k ), the fol lowing diagr am c ommutes: (5.13) L I ( X , Y ) p I J / / i I   L J ( X , Y ) i J   Y f ∈ Indec ( I ) M ( f ) pro j   Y f ∈ Indec ( J ) M ( f ) pro j   M ( f 1 ) × . . . × M ( f k ) Φ k f • / / M ( f ) wher e the maps i I and i J ar e the inclusions of § 5.8. Pr o of. Supp ose ϕ • is an elemen t of L I ( X , Y ), while f = c ( f • ) is a maximal decomp osition (so eac h f i is indecomp osable). Then ϕ f lies in Diag 1 Z ( X , Y ), so Φ ϕ f = ϕ f lands in M ( f ). How ev er, ( ϕ f k , . . . , ϕ f 1 ) ∈ M ( f k ) × · · · × M ( f 1 ) maps to Σ k i =0 Y ( f k ◦ · · · ◦ f i +1 ) ϕ f i X ( f i − 1 ◦ · · · ◦ f 1 ) also in M ( c ( f • )) = M ( f ). Th us, ϕ • ∈ L I ( X , Y ) = L I ( X , Y ) (see Lemma 5 .7) implies the v alue of ϕ f for a ny decomp osable f is uniquely determined b y the v a lues o f ϕ f i for f i indecomp o sable, using formula (5.6).  Note that if f is also indecomp osable in I , the b ottom map of (5.13) is Id : M ( f ) → M ( f ) . The c hoice of decomp osition of f in I is also ir r elev an t, b y Definition 5.5. 6. Fibra tions in the A uxiliar y To wer As noted in § 4.7, t he auxiliary tow er (4.11) w as constructed with t w o goals in mind: to replace (4.6) b y a to w er of fibrations (with the same homotop y limit), and to iden t if y the homoto p y fib ers o f the success iv e maps in (4.6). In this section w e sho w that the map Ψ of L emma 5.9 is indeed a fibrat io n, a nd that the a uxiliary to wer is a to wer of fibrat io ns. First, w e need the follo wing: 6.1. Definition. An y strongly directed indexing category I has t w o filtra tions, defined inductiv ely: a) The filtration {F i } n i =0 on I is defined b y decomp osition length fro m the left, so F 0 consists o f w eakly initial ob jects in I and F n +1 consists of indecomp o sable maps with sources in F n , (including their t a rgets). b) The filtration {G i } n i =0 is similarly defined b y decomp osition length from the righ t, so G 0 consists of the we akly terminal ob jects in I and G n +1 consists of indecomp osable maps with targets in G n , (including their sources). 6.2. Prop osition. I f ( C , I , Z , X , Y ) is a d missible, the induc e d di ffer enc e map: Ψ : D Z ( X , Y ) → L I ( X , Y ) SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 19 of L emma 5.9 is a fibr ation of s i m plicial ab elian gr oups. Pr o of. By [Q1, I I, § 3, Prop. 1], it s uffices to sho w that Ψ surjects on to the zero comp onen t of L I ( X , Y ). Thus, g iven 0 ∼ ϕ • ∈ L I ( X , Y ), we m ust pro duce an elemen t σ • ∈ D Z ( X , Y ) with Ψ( σ • ) = ϕ • ; i.e., for ev ery f : a → b in I we wan t: (6.3) σ b ◦ X ( f ) = Y ( f ) ◦ σ a − ϕ f . Note that since Y is an ab elian gr o up ob ject in C I / Z , the zero map X → Y is the unique map in C I / Z that factors through the section s : Z → Y (whic h exists b y (2.4) and § 1.13). W e pro ceed b y induction on the filtration {F i } n i =0 of I of Definition 6.1. T o b egin, fo r eac h c ∈ F 0 , w e may c ho o se σ c : X c → Y c to b e 0. Assume b y induction that w e hav e constructed maps σ c : X c → Y c for eac h c ∈ F i , satisfying (6.3) fo r ev ery f in F i , and with each σ c ∼ 0. Note that for an y f : b → c , in F i +1 the map: (6.4) ν ( f ) := Y ( f ) ◦ σ b − ϕ f : X b → Y c is well-defin ed (since necessarily b ∈ F i ). This is our candidate for σ c ◦ X ( f ), and ν ( f ) ∼ − Y ( f ) ◦ σ b ∼ 0 b y the assumption on ϕ together with the induction h yp othesis (considering naturality of the section Z → Y ). Moreo ve r, giv en an y g : a → b (necessarily in F i ), w e ha ve ϕ g = Y ( g ) ◦ σ a + σ b ◦ X ( f ) by (6.3), so from ϕ • ∈ L J ( X , Y ) it fo llows that: ν ( f ◦ g ) = Y ( f ◦ g ) ◦ σ a − ϕ f ◦ g = Y ( f ◦ g ) ◦ σ a − [ Y ( f ) ◦ ϕ g + ϕ f ◦ X ( g )] = Y ( f ◦ g ) ◦ σ a − [ Y ( f ) ◦ ( Y ( g ) ◦ σ a − σ b ◦ X ( g )) + ϕ f ◦ X ( g ) ] = ν ( f ) ◦ X ( g ) . (6.5) No w giv en c ∈ F i +1 \ F i , set: ˆ X c := colim b ∈ I /c X b . Since X ∈ C I is cofibrant, it is Reedy cofibran t ( § 1.10), whic h implies that the canonical map ε c : ˆ X c → X c is a cofibration. Moreov er, (6.5) implies that the maps ν ( f ) defined ab ov e induce a map ˆ ν c : ˆ X c → Y c . Since all the maps in question are n ullhomo t opic by construction, the diagram: ˆ X c ˆ ν c A A A A A A A ε c / / X c 0   Y c comm utes up t o homotop y . Hence b y [BJT, Cor. 4.20] t here is a map σ : X c → Y c in C / Z c making the diagram (6.6) ˆ X c ˆ ν c A A A A A A A ε c / / X c σ   Y c 20 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER comm ute, and w e choose this to b e σ c . By construction σ c ◦ X ( f ) = ν ( f ) for ev ery f : b → c , so (6.3 ) is satisfied. This completes t he induction.  6.7. Prop osition. If ( C , I , Z , X , Y ) is admissible, let J b e a sub c ate gory of I o b - taine d by omitting a terminal o bje ct c . Then the r e striction m a p p I J : L I ( X , Y ) → L J ( X , Y ) is a fibr ation. Pr o of. As in the previous pro of, w e mus t inductiv ely define a lift σ • ∈ L I ( X , Y ) for a nullhomotopic ϕ • ∈ L J ( X , Y ). Under these conditions, p I J is simply a forgetful functor, so this means σ g = ϕ g for g a morphism of J and w e m ust define σ ℓ : X d → Y c whenev er ℓ : d → c is a morphism in I , in a manner compatible with the definition of ϕ • . Note that ϕ • determines the comp osite Y ( f ) ◦ Φ I g • =: ψ ( g • , f ). F ollowing the approach of the previous pro of, w e will define ν ( g • , f ) for an y e g • − → d f − → c in I , where f is indecomp osable, so as to satisfy three pro p erties: First, w e require that our c ho ices b e c oher ent : (6.8) ν ( g • ◦ h • , f ) = ν ( g • , f ) ◦ X ( c ( h • )) , whic h will allo w us to build a homotop y commutativ e triangle using a colimit con- struction. Second, w e need our choice s t o b e c ons i s tent : (6.9) ν ( g • , f ) = ν ( g ′ • , f ′ ) + ψ ( g • , f ) − ψ ( g ′ • , f ′ ) whenev er f ◦ g • = f ′ ◦ g ′ • in I , whic h is needed so that we ev en tually obtain an elemen t σ • ∈ L I ( X , Y ). In fact, our construction will also w ork when g • = ∅ , whic h will yield σ ( f ) = ν ( ∅ , f ). Finally , w e require that eac h ν ( g • , f ) ∼ 0. W e no w pro ceed to choo se ν ( g • , f ) for e g • − → d f − → c with e ∈ F i (Definition 6.1) by induction on i ≥ 0: F or each ℓ : e → c in I with e ∈ F 0 , choose some decomp osition e g • − → d f − → c (with ℓ = c ( g • , f ) and f indecomp osable), and an arbitrary n ullhomotopic 0 = ν ( g • , f ) : X e → Y c . F or any o ther decomp osition ℓ = c ( g ′ • , f ′ ), the map ν ( g ′ • , f ′ ) is then determined by (6.9). Assume that ν has b een defined for eve ry e ∈ F i so that (6.8) and (6.9) hold (wherev er applicable). F o r eac h e ∈ F i +1 \ F i and map ℓ : e → c , consider the o ver-category F i /e ( whic h is non-empt y b y definition of F i +1 ) and set ˆ X e := colim a ∈F i /e X a . Because the diagram X is cofibran t, hence Reedy cofibran t ( § 1.10) in C I , the canonical map ε e : ˆ X e ֒ → X e is a cofibrat io n. Again c ho ose some decomp o sition e g • − → d f − → c of ℓ . The maps ν ( g • ◦ h • , f ) : X a → Y c , for eac h comp osable sequence h • : a → e in F i /e induce a (necessarily n ullhomotopic) map ˆ µ e : ˆ X e → Y c b y (6.8). Since: ˆ X e ε e / / ˆ ν ( g • ,f ) A A A A A A A A X e 0   Y c SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 21 then comm utes up to homotop y , w e apply [BJT, Cor. 4.20 ] to find ˆ X e ε e / / ˆ ν ( g • ,f ) A A A A A A A A X e ν ( g • ,f )   Y c making the diagram comm ute. F or a n y other decomp osition e g ′ • − → d ′ f ′ − → c of ℓ , use ( 6 .9) to define ν ( g ′ • , f ′ ). This completes the induction step. W e hav e th us defined ν ( g • , f ) : X e → Y c satisfying (6.8) and (6.9 ) for ev ery e g • − → d f − → c in I /c . In pa rticular, w e can c ho ose σ ( f ) = ν ( ∅ , f ) : X d → Y c for eac h indecomp osable f : d → c in I and see that σ • ∈ L I ( X , Y ) (b y Lemma 5.7) is the desired lift.  6.10. Corollary . I f ( C , I , Z , X , Y ) is admissible, let J b e a ful l sub c ate gory of I obtaine d by omitting an obje ct c such that al l maps out of c ar e inde c omp osable. Then p I J : L I ( X , Y ) → L J ( X , Y ) is a fibr ation. Pr o of. As in the pro of o f Prop osition 6.7 w e can construct σ ( f ) f or eac h f : d → c . in I , suc h that w e ha ve ˆ ν : colim d ∈ I /c X d → Y c , as w ell as ˆ ǫ c : colim d ∈ I /c X d → X c . F or any g : c → b , in I ( indecomp osable b y ass umption), we also hav e a ma p ˆ ϕ : colim d ∈ I /c X d → X b induced by ϕ • . Note that by (5.3) w e mus t hav e: σ ( g ) ◦ X ( ˆ ǫ c ) = Φ I ( g, f ) − Y ( g ) ◦ σ ( f ) = ˆ ϕ − Y ( g ) ◦ ˆ ν , and since X ( f ) is a cofibration, w e ma y c ho ose the extension σ ( g ) as in (6.6).  6.11. Definition. If I is a stro ng ly directed indexing category , let J = { J k } k ∈ N b e a fine o rderable cov er ( § 1.4) o f I sub ordinate to the filtra t ion G (D efinition 6.1), suc h that J k \ J k − 1 consists of a single ob j ect of I for eac h k ∈ N . Let C = s A for some G -sk etchable category A ( § 1 .1 3), with Z ∈ C I fibran t. A fibrant a b elian group ob ject Y ∈ C I / Z is called str ongly fibr ant if it is J -fibrant with resp ect to the mo del category structure of § 1.9(a). 6.12. R emark. Note that this definition is indep enden t of the c hoice of the refinemen t J o f G . F orthermore, by Prop osition 2.10 , any ab elian group ob ject Y ∈ C I / Z is w eakly equiv alen t to one whic h is strongly fibrant. 6.13. Prop osition. Supp ose ( C , I , Z , X , Y ) is admissib le, and that Y is str ongly fibr ant. Assume that J is obtaine d fr om I by omitting an obje ct c such that al l maps into c ar e inde c omp osable. Then the r estriction map p I J : L I ( X , Y ) → L J ( X , Y ) is a fibr ation. Pr o of. Dual to the pro ofs of Prop osition 6.7 and Corollary 6.10. The strong fibrancy is needed since in the mo del category we use for diagrams ordinary fibrancy is merely ob ject wise, while strong fibrancy is dual to R eedy cofibrancy for our purp oses.  6.14. Prop osition. If ( C , I , Z , X , Y ) is admissible, Y is str ongly fibr ant, and J is obtaine d fr om I by omitting a ny obje ct c , then the r estriction ma p p I J : L I ( X , Y ) → L J ( X , Y ) is a fibr ation. 22 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER Pr o of. Consider any comp osable sequenc e: (6.15) d h • − → c g − → b f • − → a in I . As ab ov e, 0 ∼ ϕ • ∈ L J ( X , Y ) will determine t he map (6.16) ψ ( h • , g , f • ) := Y ( c (( g , f • ))) ◦ Φ I h • + Φ I f • ◦ X ( c (( h • , g ))) and w e use ν ( h • , g , f • ) : X d → Y a , t o denote the candidate for Y ( c ( f • )) ◦ σ ( g ) ◦ X ( c ( h • )) whic h we will construct. As b efore we require c oh e r en c e : (6.17) ν ( h • ◦ ℓ • , g , k • ◦ f • ) = Y ( c ( k • )) ◦ ν ( h • , g , f • ) ◦ X ( c ( ℓ • )) for any e ℓ • − → d h • − → c g − → b f • − → a k • − → z in I ; and c ons i s tenc y : (6.18) ν ( h ′ • , g ′ , f ′ • ) = ψ ( h • , g , f • ) + ν ( h • , g , f • ) − ψ ( h ′ • , g ′ , f ′ • ) whenev er c ( h ′ • , g ′ , f ′ • ) = c ( h • , g , f • ). W e c ho ose the maps ν satisfying (6.17) and (6.18) by t w o success ive inductions: • The first is b y induction on i , the filtra tion degree of d in {F i } m i =0 (b y comp osition length from the left): this is done as in the pro of of Prop osition 6.7, un til finally w e ha ve ν ( h, g , f • ) for ev ery d h − → c g − → b f • − → a , whe re h is indecomp osable a nd a is terminal in I (by coherence this extends bac k to an y d h • − → c ) . • The second is b y induction on j , the filtration degree of a in {G j } n j =0 (b y comp osition length from the right), as in the pro of o f Prop osition 6 .1 3 (whic h is wh y we need Y to b e strongly fibran t). A t the end of the t wo induction pro cesses w e ha v e c hosen ν ( h, g ) : X d → Y b for h and g indecomp osable. W e can then c ho ose σ ( h ) = ν ( h ) : X d → Y c as in the last step o f the pro of of Prop o sition 6.7, and finally c ho ose σ ( g ) = ν ( g ) : X c → Y b as in the pro of of Coro llary 6.10. This completes the construction of a lift σ • ∈ L I ( X , Y ) for ϕ • as required.  6.19. Corollary . Supp ose ( C , I , Z , X , Y ) is admissible, Y is str on g l y fibr a nt, and J is any ful l sub c ate gory of I with the same we akly initial and final obje cts. Then the r e s triction map p : L I ( X , Y ) → L J ( X , Y ) is a fibr ation Pr o of. By induction on the n umber of o b jects in I \ J , using Prop osition 6.14.  7. Identifying the Fibe rs As we ha ve just seen, if I is a go o d indexing category , under o ur standard as- sumptions on Z , X , and Y the auxiliary t ow er (4.11) is a to we r of fibrations of simplicial ab elian groups. It remains to iden tify the fib ers of the restriction maps p : L I ( X , Y ) → L J ( X , Y ), for a sub category J of I ; this will allow us to determine those of the primary to w er (4 .6) (or, more directly , those of the mo dified tow er (4.15)). W e consider only the case when I \ J consists o f a single in ternal ob ject c . 7.1. Lemma. If ( C , I , Z , X, Y ) is a d m issible and Y is str ongly fibr ant, then ϕ • ∈ Ker( p ) ⊆ L I ( X , Y ) if and only if SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 23 a) ϕ f = 0 for e ac h morphism f of I which do es not b e gin or end in c . b) fo r any d g − → c f − → b in I w i th f an d g inde c omp osabl e : (7.2) Y ( f ) ◦ ϕ g + ϕ f ◦ X ( g ) = 0 , Pr o of. This follows from Lemma 5.12.  7.3. R emark. The lemma implies that ( ϕ f , − ϕ g ) defines a map fr o m X ( g ) to Y ( f ). Note also that if ϕ f is an a r ro w o ve r Z t ( f ) , the same is true o f its negativ e; the remainder of the dia gram for a map ov er Z ( f ) already comm utes b ecause X and Y are diagrams ov er Z . Th us ( ϕ f , − ϕ g ) is a map of arrow s o v er Z ( f ). 7.4. Definition. If ( C , I , Z , X, Y ) is admissible, w e define the lo c al c ohomolo gy of X ∈ C I / Z at an o b ject c ∈ I , denoted b y H c ( X/ Z , Y ), to b e the total deriv ed functors into simplicial ab elian groups of map φ c ( ψ c , ρ c ) a pplied to X , where ψ c : ho colim d ∈ I /c X d → X c , ρ c : Y c → ho lim b ∈ c/I Y b , and φ c : Z c → holim b ∈ c/I Z b , are the structure maps. The i -th lo c al c ohomo l o gy gr oup of X ∈ C I / Z at c is defined to b e H i c ( X/ Z , Y ) := π i H c ( X/ Z , Y ). 7.5. R e m ark. In man y cases, the lo cal cohomology at c can b e iden tified explicitly as the Andr´ e-Quillen cohomology o f an appropriate (small) dia g ram. 7.6. Prop osition. I f ( C , I , Z , X , Y ) is admissible, Y is str ongly fibr ant, and J = I \ { c } , then K er ( p ) is w e akl y e quivalent (as a simplic i al ab elian gr oup) to H c ( X/ Z , Y ) . Pr o of. T o obt a in the total deriv ed functors, in this case, w e mus t replace X b y a w eakly equiv alent cofibrant, hence Reedy cofibrant ob ject, whic h implies that ho colim d ∈ I /c X d is simply the colimit, and ψ c is a cofibration. By Remark 6.12, we can replace Y b y a w eakly equiv alen t strongly fibran t ab elian group ob ject in C I / Z , whic h implies that holim b ∈ c/I Y b is the limit, and ρ c is a fibration. With these c ho ices, H I c ( X/ Z , Y ) is simply the mapping space map φ c ( ψ c , ρ c ), whic h is isomorphic t o K er ( p ) in Lemma 7.1 (using the sign of Remark 7 .3).  7.7. Theorem. If ( C , I , Z , X , Y ) is admiss i ble, for a ny or d ering ( c i ) ∞ i =1 of the obje cts of I , ther e is a natur al first quadr ant sp e ctr al se q uenc e with: E 2 s,t = H s +1 c t ( X/ Z ; Y ) = ⇒ H s + t +1 ( X/ Z ; Y ) , with d 2 : E 2 s,t → E 2 s − 2 ,t +1 . Pr o of. W e ma y replace Y b y a w eakly equiv alent strongly fibran t ab elian group ob ject, b y Remark 6.1 2. By Coro llary 6 .19, (4 .15) is t hen a tow er of fibrations, so it has an asso ciated homotopy sp ectral sequence. T o iden tify the E 2 -term, note t ha t the homotop y groups of t he homotop y fib ers o f the to wer are the lo cal cohomology gro ups in Prop o sition 7.6, suitably indexed (see Remark 3.2).  7.8. R emark. Note that p I J : L I ( X , Y ) → L J ( X , Y ) is a fibration for any full sub category J ⊆ I with the same w eakly initial and final ob jects (Corollary 6.19), and w e can similarly describ e the fib er of p I J as a sort of lo cal cohomology H I J ( X/ Z , Y ), and th us iden tify the E 2 -term of the sp ectral sequence o bt a ined from a fairly arbitrary co v er of I . W e shall not attempt to define H I J ( X/ Z , Y ) in general. O bserv e, ho we v er, that if J is discrete (i.e., there are no non-iden t it y maps b etw een its ob j ects c 1 , . . . , c n ), 24 D. BLANC, M.W. JOHNS ON, AND J.M. TURNER then (7.9) H I J ( X/ Z , Y ) ∼ = n Y i =1 H c i ( X/ Z , Y ) . 7.10. Example. F or the comm uting square of Example 3.9, w e no w get a cov er for I consisting of I 3 = I , I 2 = I \ { 3 } – i.e., a comm uting triangle: 4 c   b ◦ d   = = = = = = = 2 a / / d I 1 = { 4 a ◦ c − − → 1 } , and I 0 = { 4 } . Giv en a diagram o f ab elian group ob jects Y : I → C , t he lo cal cohomology groups whic h form the E 2 -term of the sp ectral sequence of Theorem 7.7 are: E 2 s,t ∼ =          H s +3 ( X ( d ); Y ( b )) if t = 2; H s +2 ( X ( c ); Y ( a )) if t = 1; H s +1 ( X 4 ; Y 1 ) if t = 0 ; 0 otherwise. Once more we could unite the first and second ro ws by omitting I 2 from our co ve r, as in Example 3.9, b y (7.9). 7.11 . A comparison. In the simplest case, when I = [ 1 ] (a single map): X 2 f 2   p 2                   X φ / / X 1 f 1   p 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Y 2 q 2 ~ ~ } } } } } } } } Y φ / / Y 1 q 1 ! ! C C C C C C C C Z 2 Z φ / / Z 1 , w e ha v e the “defining fibration sequence”: (7.12) map ( X , Y ) → map ( X 2 , Y 2 ) × map ( X 1 , Y 1 ) ξ − → ma p ( X 2 , Y 1 ) of [BJT, Prop. 4.20 ] (where a ll mapping spaces are tak en in the appropriate comma categories). Pro jecting the total space of (7.12) on to t he second factor yields the following in terlo c king diagram of horizontal and ve rtical fibration sequenc es: (7.13) map ( X 2 , Fib( Y φ )) i ∗   / / map ( X, Y ) / /   map ( X 1 , Y 1 ) Id   map ( X 2 , Y 2 ) φ ∗   / / map ( X 2 , Y 2 ) × map ( X 1 , Y 1 ) π / / ξ   map ( X 1 , Y 1 )   map ( X 2 , Y 1 ) Id / / map ( X 2 , Y 1 ) / / ∗ SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 25 W e see that the sp ectral sequence of Theorem 3.5 reduces to the long exact sequence in homoto p y for the t o p horizon t al fibratio n sequence in (7.1 3) , while the lo ng exact sequence of F act 3.3 is obtained fro m the left v ertical fibration sequenc e in (7.13). 7.14. R emark. This actually w orks for a ny linear order I = [ n ] ( § 1.4): Giv en X , Y ∈ C I / Z , if we set I ′ := [ n − 1 ] (so J := { n φ n − → n − 1 } ) and let τ = τ I I ′ : C I / Z → C I ′ / Z | I ′ , then (7.12) yields a fibration sequence: map ( X, Y ) → map ( X n , Y n ) × map ( τ X , τ Y ) ξ − → ma p ( X n , Y n − 1 ) whic h again induces a in terlo cking diagram o f fibrations: map ( X n , Fib( Y φ n )) i ∗   / / map ( X , Y ) / /   map ( τ X , τ Y ) Id   map ( X n , Y n ) ( φ n ) ∗   / / map ( X n , Y n ) × map ( τ X , τ Y ) π / / ξ   map ( τ X , τ Y )   map ( X n , Y n − 1 ) Id / / map ( X n , Y n − 1 ) / / ∗ as in (7.13). Note that the long exact sequence s in homotopy (i.e., cohomology) of the cen tral v ertical fibrations (for v arious v alues of n ) provide an alternative inductiv e approac h to calculating the cohomology of X , whic h can again b e formalized in a sp ectral sequence (though in this case the fib ers are the unknow n quantit y). 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V og t, “Homotopy limits and colimits”, Math. Z. 134 (1 973), pp. 11 -52. SPECTRAL SEQUENCES FOR THE COHO MOLOGY OF DIAGRAMS 27 Dep ar tment of Ma thema tics, University of Haif a, 31905 Haif a, Israel E-mail addr ess : bla nc@mat h.haif a.ac.il Dep ar tment of Ma thema tics, Penn St a te Al toona, Al toona, P A 1 6601, USA E-mail addr ess : mwj 3@psu. edu Dep ar tment of Ma thema tics, Cal vin College, Grand Rapids, MI 49546, USA E-mail addr ess : jtu rner@c alvin. edu