LOCALTOGLOBAL 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 oneobject category corresponding *
*to a
group G, a diagram X 2 CI is just an object in C equipped with a Gaction, 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
Gspace, 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 DwyerKanSmith theory for the rectification of homotopycomm*
*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 homotopycommutative 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 "localtoglobal" 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 "localtoglobal" 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 localtoglobal 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 BauesWirsching and HochschildMitchell 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 Nindexed 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 rightorderable, and if it is bound*
*ed above
we say it is leftorderable.
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*
*n1, 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 nonident*
*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) = deg1([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*
*n1
(discrete, by assumption) by adding all the maps from any of these objects into*
* In1.
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 OEn1 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
fflfflfflffl
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, OE1 : i ! j, Gi *
*= Z=2,
and Gj = 0. Then the indexing category I has a single new nonidentity map
f : i ! i and OEk O f = OEk ( 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 Idiagrams 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 FPsketchable 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
Gsketchable 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 Zmodules 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 In1 . ...
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 In1 ,! In and In ,! I ind*
*uce
maps aen : Mn ! Mn1 and ^aen: M ! Mn which fit into a tower:
______________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*________________________________________@
map Z(X, Y )______________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________________
NNNN _______________________________________________*
*__________________________________________________________^ae________________*
*_____________________________________________________________________________*
*___________________________
(1.22) NNN^aen+1NN __^aen____________________________________*
*_____________________________________________________________n1_____________*
*_______________________________________
NN'' aen__%%__________________________________$*
*$___________________________________________+1aeaen1
. ._._______//Mn+1 ____//_Mn ____//_Mn1 ____//_. .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=ZJn+1(XJn+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=ZJn+1(XJn+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) = """"
fflfflqi""""
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) fflfflsjO!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 Gsketchable 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
 
 
fflffloenq 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=2equivariant 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 nonnegative. 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=ZJn+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 Gsketchable 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 E2term, 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=ZJ(, Fib(^!)), (into simplicial abelian groups), *
*denoted
by H(X=Z; ^!). In particular, the ith 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 nth dimensional EilenbergMac Lane object (over Z), in which case it
is customary to reindex the relative cohomology groups so that Hn(X=Z; ^!) :=
ss0H(X=Z; ^!).
Observe, however, that our setup allows Y to consist of EilenbergMac Lane ob*
*jects
of varying dimensions, with the maps Y (f) representing cohomology operation*
*s.
In this general setting, no canonical reindexing 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
leftorderable 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! E2s2,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 E2term 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 E2term 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 rightorderable,
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)=ZJt; Y ) is just the usual*
* coho
mology of the map of diagrams jt : `tt1X ! 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 MayerVietoris 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
 
 
fflfflfflffl
0_____//Y1
for !1 : Y ! oe0Y .
12 D. BLANC, M.W. JOHNSON, AND J.M. TURNER
Thus the E2term for the spectral sequence consists of only four nontrivial*
* 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 nonidentity 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, Ik1 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 'k1 : Ik*
*1 ,! Ik
induce a finite tower of simplicial abelian groups:
'*k1
(4.6) map CIn=Z(X, Y ) ! . .!. map CIk=Z(X, Y ) ! map CIk1=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 E2term. 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 leftexact 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
righthand 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:
pn1 p1
(4.11) LIn(X, Y ) ! LIn1(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 ik1 : Ik1 ,! Ik will induce a commuting *
*square
of fibrations:
k
DCIk=Z(X, Y )______//LIk(X, Y )
ssk1 pk1
fflffl fflffl
DCIk1=Z(X, Y )_k1//_LIk1(X, Y ),
where the left vertical map ssk1 is the projection onto the appropriate fac*
*tors.
Thus we will have a homotopycommutative diagram:
Q
Fib(i*k1)_______//_i2I map C=Zi(Xi, Yi)___//HIk(X=Z, Y )
k\Ik1 c
 
  
  
fflffl fflffl fflffl
(4.13) map CIk=Z(X, Y )_________//_DCIk=Z(X, Y_)___k____//LI
k(X, Y )
i*k1 ssk1 pk1
fflffl fflffl fflffl
map CIk1=Z(X, Y )_______//_DCIk1=Z(X, Y_)_k1__//LIk1(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...Opn1
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 ) ! EIk1Z(X, Y ) then fit into a tower:
rn1 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
E2term.
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 subsimplicial set of DZ(X, Y ) consist*
*ing of
SPECTRAL SEQUENCES FOR THE COHOMOLOGY OF DIAGRAMS 15
transformations which are natural when restricted to Jdiagrams. 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 ) ! EIk1Z(X, Y ) with rIkIk1.
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 nonidentity morphisms in I (i.e.,*
* a
ksimplex 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 k1Z(X, Y )! DiagkZ(X*
*, Y ):
(i) X*, for which the focomponent is the composite
proj X(f1)*
Diagk1Z(X, Y )! M(f2, . .,.fk) ! M(fo) .
(ii) Y*, for which the focomponent is the composite
proj Y (fk)*
Diagk1Z(X, Y )! M(f1, . .,.fk1) ! M(fo)
d) By iterating the maps 1 := Y* + X* : Diagk1Z(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:= fkOfk1O. .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*)ji(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(fi1O. .O.f1) = Y (glO. .O.gi+1)'giX(gi1O. .*
*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 Lconstruction 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 KJdiagrams from , : VI,J! VJ,
obtained by projecting the larger products DiagkZ(X, Y ) onto DiagkZJ(XJ, 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 Jnatural 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, fk1, . .,.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(fi1O . .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 welldefined (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
overcategory Fi=e (which is nonempty 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 \ Jk1 consists of a single object of I for each k 2 N. Let C = *
*sA
for some Gsketchable 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 ith 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'eQuillen 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! E2s2,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 E2term, 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 E2term 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 nonidentity 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==
fflfflOEOE=
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 E2term 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
fff2iffiffif111
p ffifflfflffifflffl11
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 =ZI0, then (7.12) yields a fibration sequence:
,
map (X, Y ) ! map (Xn, Yn) x map (oX, oY ) ! map (Xn, Yn1)
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, Yn1) _____________//_map(Xn, Yn1)________________//_*
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 EilenbergMoore spectral sequence*
*", Bull. AMS
78 (1972), No. 5, pp. 784786.
[BW] H.J. Baues & G. Wirsching, "The cohomology of small categories", , J. Pur*
*e Appl. Alg. 38
(1985), pp. 187211.
[Be] J.M. Beck, "Triples, algebras and cohomology", Repr. Theory Appl. Cats. 2*
* (2003), pp. 159.
[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. 6171.
[BC] D.J. Benson & J.F. Carlson, "Diagrammatic methods for modular representat*
*ions and co
homology", Comm. Algebra 15 (1987), 53121.
[Bl] D. Blanc, "Higher homotopy operations and the realizability of homotopy g*
*roups", Proc.
London Math. Soc. 70 (1995), pp. 214240.
[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. 857892.
[BJT] D. Blanc, M.W. Johnson, & J.M. Turner, "On Realizing Diagrams of algebr*
*as", Algebraic
& Geometric Topology 6 (2006), pp. 763807
[BP] D. Blanc & G. Peschke, "The fiber of functors between categories of algeb*
*ras", J. Pure &
Appl. Alg. 207 (2006), pp. 687715.
[Bo] A.K. Bousfield, "On the homology spectral sequence of a cosimplicial spac*
*e", Amer. J.
Math. 109 (1987), No. 2, pp. 361394.
[BK] A.K. Bousfield & D.M. Kan, Homotopy Limits, Completions, and Localization*
*s, Springer
Lec. Notes Math. 304, BerlinNew 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. 643680.
[D] T. Datuashvili, "Cohomologically trivial internal categories in categorie*
*s of groups with
operations", Appl. Categ. Structures 3 (1995), pp. 221237.
26 D. BLANC, M.W. JOHNSON, AND J.M. TURNER
[DF] E. DrorFarjoun, "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, BerlinNew York, 1987, pp. 93134.
[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, "HochschildMitchell cohomology of simplicial cate*
*gories and
the cohomology of simplicial diagrams of simplicial sets", Nederl. Akad. *
*Wetensch. Indag.
Math. 50 (1988), pp. 111120.
[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. 524.
[FW] H.R. Fischer & F.L. Williams, "BorelLePotier diagramscalculus of their *
*cohomology bun
dles", Tohoku Math. J. (2) 36 (1984), pp. 233251.
[GGS] M. Gerstenhaber, A. Giaquinto & S.D. Schack, "Diagrams of Lie algebras", *
*J. Pure &
Appl. Alg. 196 (2005), pp. 169184.
[GS1] M. Gerstenhaber & S.D. Schack, "On the deformation of algebra morphisms a*
*nd diagrams",
Trans. Amer. Math. Soc. 279 (1983), pp. 150.
[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. 11264.
[GJ] P.G. Goerss & J.F. Jardine, Simplicial Homotopy Theory, Prog. in Math. 17*
*4, Birkh"auser,
BostonStuttgart, 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. 253296.
[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*
*. 1161.
[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, BerlinNew York, 1969, pp. 480489.
[Pa] P. Pave~si'c, "Diagram cohomologies using categorical fibrations", J. Pur*
*e Appl. Algebra 112
(1996), pp. 7390.
[P1] R.J. Piacenza, "Cohomology of diagrams and equivariant singular theory", *
*Pac. J. Math.
91 (1980), pp. 435443.
[P2] R.J. Piacenza, "Diagrams of simplicial sets, complexes and bundles", Tamk*
*ang J. Math.
15 (1984), pp. 8394.
[Q1] D.G. Quillen, Homotopical Algebra, Springer Lec. Notes Math. 20, BerlinN*
*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. 6587.
[Se] G.B. Segal, "Categories and cohomology theories", Topology 13 (1974), pp.*
* 293312.
[Sl] J. S_lomi'nska, "Some spectral sequences in Bredon cohomology", Cahiers T*
*op. G'eom. Diff.
Cat. 33 (1992), pp. 99133.
[V] R.M. Vogt, "Homotopy limits and colimits", Math. Z. 134 (1973), pp. 1152.
Department of Mathematics, University of Haifa, 31905 Haifa, Israel
Email address: blanc@math.haifa.ac.il
Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA
Email address: mwj3@psu.edu
Department of Mathematics, Calvin College, Grand Rapids, MI 49546, USA
Email address: jturner@calvin.edu