THE REALIZATION SPACE OF A -ALGEBRA:
A MODULI PROBLEM IN ALGEBRAIC TOPOLOGY
D. BLANC, W. G. DWYER, AND P. G. GOERSS
Contents
1. Introduction 1
2. Moduli spaces 7
3. Postnikov systems for spaces 9
4. -algebras and their modules 13
5. Relative connectivity of pushouts 18
6. Postnikov systems for simplicial -algebras 22
7. Simplicial spaces and the spiral exact sequence 26
8. Postnikov systems for simplicial spaces 32
9. The main theorem 38
References 43
1. Introduction
A -algebra A is a graded group {An}n 1 with all of the primary
algebraic structure possessed by the collection of homotopy groups of
a pointed connected topological space. In particular, An is abelian
for n 2, and there are Whitehead product and composition maps
which satisfy appropriate identities (see 4.1). The basic example of a
-algebra is the homotopy -algebra ß*X of a space X.
Given an abstract -algebra A, it is tempting to ask whether it has
any topological significance. Is it possible to find a space X such that
A is isomorphic to ß*X? If such an X exists, is it unique up to weak
equivalence? These questions and others can be studied by looking
at the moduli space T M (A) of topological realizations of A, or the
realization space of A. This is defined to be the nerve or classifying
space of the category whose objects are the topological spaces X with
ß*X ' A and whose morphisms are the weak equivalences between
____________
Date: September 14, 2001.
The second and third authors were partially supported by the National Science
Foundation (USA)..
1
2 BLANC, DWYER, AND GOERSS
these spaces. Up to homotopy T M (A) can be identified (2.1) as a
disjoint union a
W~Aut h(X) ,
indexed by homotopy equivalence classes of CW-complexes X with
ß*X ' A, where W~ Auth(X) is the classifying space of the simplicial
monoid of self homotopy equivalences of X. The -algebra A can be
realized as ß*X for some X if and only if T M (A) is nonempty; the
realization is unique up to weak equivalence if and only if T M (A) is
connected.
In this paper we study T M (A). The first step is to construct partial
moduli spaces T M n(A), n 0, which fit into a tower
. .!.T M n(A) ! T M n-1(A) ! . .!.T M 1(A) ! T M 0(A)
whose homotopy limit is equivalent to T M (A). We then approach the
partial moduli spaces inductively, and show that T M n(A) is tied to
T M n-1 (A) by a simple homotopy fibre square (9.6, 9.7). The conclu-
sion is that the spaces T M n(A) are relatively accessible, and in fact
have a surprisingly cohomological flavor. In analyzing them we are
doing a type of homotopical deformation theory; the obstructions and
choices at each level lie in the Quillen cohomology groups of A, which
are the analogues for a -algebra of the Hochschild cohomology groups
of an associative ring or the Andr'e-Quillen cohomology groups of a
commutative ring.
One of the motivations for this paper is that we expect our study
of the realization space of a -algebra to serve as a blueprint for the
study of other moduli problems of a similar type. For that reason we
have tried to keep our constructions and arguments as conceptual as
possible. There are several lessons that might be learned from the pa-
per. One is the usefulness of working with moduli spaces as a whole,
rather than with their sets of components, if only because the moduli
spaces tend to fit into fibration sequences and fibre squares. This is not
a new lesson, but it comes through pretty clearly in what we do. An-
other point is the power and flexibility that can be gained by working
with simplicial resolutions of objects (in our case simplicial resolutions
of spaces) instead of with the objects themselves. Finally, on a much
more technical level, suppose that F is a functor from finite sets to
sets or spaces. The reader might be interested in how prolonging F
to the category of simplicial sets can be interpreted as taking a homo-
topy coend (5.10); this explains to the authors a family of connectivity
formulas (e.g. 5.1) which otherwise can seem mysterious.
We will now discuss our results in more detail.
MODULI PROBLEM 3
The partial moduli spaces. We first describe how the partial moduli
spaces T M n(A) arise. Any space X has a spherical resolution S(X);
this is a simplicial space whose realization is equivalent to X, and each
of whose simplicial constituents S(X)[n] is equivalent to a wedge of
spheres. In fact there is a model category structure on the category
of simplicial spaces in which the cofibrant objects are spherical; the
resolution S(X) is obtained by treating X as a constant simplicial space
and taking a cofibrant model for it. This is analogous to a standard
procedure in homological algebra. There is a model category structure
on the category of nonnegatively graded chain complexes in which the
cofibrant objects are the chain complexes of projective modules. A
projective resolution of a module M is then obtained by treating M as
a chain complex concentrated in degree 0 and taking a cofibrant model
for it.
Suppose now that A is a -algebra. Rather than directly trying
to build a space X which realizes A, we try to build the resolution
S(X). This gives some added flexibility, because inside the category
of simplicial spaces there are various types of Postnikov stages; we
concentrate on one of these types, the horizontal Postnikov stages P^*,
and attempt to construct S(X) inductively by building its Postnikov
sections ^PnS(X). It turns out that there is a simple algebraic condition
that a simplicial space Y has to satisfy in order to be of the form
^PnS(X) for some space X realizing A; if Y satisfies this property,
we say that it is a potential n-stage for A. The partial moduli space
T M n (A) is then defined to be the moduli space of all potential n-stages
for A, i.e., the nerve of the category whose objects are the simplicial
spaces which are potential n-stages for A, and whose maps are the weak
equivalences between these simplicial spaces.
Analyzing the partial moduli spaces. A module M over the -algebra
A is defined to be an abelian -algebra with a certain kind of action
by A, or equivalently as an abelian group object in the category of
-algebras over A. Associated to such a module M are cohomology
groups Hn (A; M), n 0. These cohomology groups can be described
in terms of the homotopy groups of certain simplicial sets Hn(A; M)
obtained by mapping A into Eilenberg-Mac Lane objects. The group
Hn (A; M) is given by ß0Hn(A; M), and more generally there are iso-
morphisms
ßiHn(A; M) ' Hn-i(A; M) .
By functoriality the discrete group Aut (A, M) of automorphisms of the
pair (A, M) acts on Hn(A; M), and we let H^n(A; M) denote the Borel
construction of this action. The group Aut (A, M) fixes the basepoint of
4 BLANC, DWYER, AND GOERSS
H^n(A; M) (which corresponds to the zero element of Hn (A; M)), and
this gives a natural map W~ Aut(A, M) ! H^n(A; M). The A-modules
that are interesting for our purposes are shifted copies m A of A itself.
Our main result is the following one, which is a recast version (9.7) of
Theorem 9.6. It provides an inductive approach to understanding the
partial moduli spaces T M n(A).
1.1. Theorem. Suppose that A is a -algebra. Then T M 0(A) is equiv-
alent to W~Aut (A), and for each n 1 there is a homotopy fibre square
T M n(A) --- ! W~Aut (A, nA)
? ?
? ?
y y .
T M n-1(A) --- ! H^n+2 (A; nA)
It follows immediately from the theorem that the homotopy fibre
of T M n(A) ! T M n-1(A) over any point of T M n-1(A) is equiva-
lent to the generalized Eilenberg-Mac Lane space Hn+2(A; nA) ~
Hn+1(A; nA). This space has nontrivial homotopy groups only in di-
mensions 0 through n + 1, and so the tower {T M n(A)} is a type of
modified Postnikov system for T M (A). This tower is better than the
usual Postnikov system for T M (A) in that the successive fibres de-
pend in an explicit cohomological way on A. The tower also leads to
an obstruction theory for finding a point in T M (A) ~ holim T M n (A),
i.e., an obstruction theory for finding a topological realization of A.
1.2. Theorem. Suppose that A is a -algebra, and that Y is a poten-
tial (n - 1)-stage for A. Then there is an associated element oY in
Hn+2 (A; nA), well-defined up to the action of Aut (A; nA) on this
group, such that Y lifts up to weak equivalence to a potential n-stage
for A if and only if oY = 0.
This theorem is proved by noticing that ß0H^n+2(A; nA) is the orbit
space of the action of Aut (A, nA) on Hn+2 (A; nA); by 1.1, the path
component P of T M n-1(A) corresponding to Y is the image of a com-
ponent of T M n(A) if and only if the image of P in H^n+2(A; nA) lies
in the component corresponding to the zero element of Hn+2 (A, nA).
Interpretation of the partial moduli spaces. It is natural to ask about
the conceptual nature of the partial moduli spaces T M n(A). Since a
vertex of T M n(A) is just a simplicial space with is a potential n-stage
for A, this amounts to asking what topological information relevant to
the problem of realizing A is contained in such a Y . To begin with, the
realization of Y is a connected space X<0, n+1> with ßiX<0, n+1> = Ai
for i n + 1 and vanishing homotopy in higher dimensions; this is just
the (n + 1)'st (ordinary) Postnikov stage of a potential realization of
MODULI PROBLEM 5
A. But there is more. Suppose that a and b are positive integers with
b > a and b-a n. With some simple manipulation (9.9) it is possible
to extract from Y spaces X with
(
Ai a i b
ßiX = .
0 otherwise
This X is the b'th ordinary Postnikov stage of the (a-1)'st connec-
tive cover of a potential realization of A. The various X obtained
in this way are as compatible as they can be when a and b vary; for
instance X is the (b - 1)'st Postnikov stage of X. We
interpret this to mean that giving a potential n-stage Y for A amounts
among other things to threading the constituents of A together by
k-invariants in such a way that the threads only reach a depth of n-
dimensions. These threads create genuine spaces which realize each
block of groups from A which is n dimensions or less in extent. As
the threads grow in length one dimension at time (if possible, since by
1.2 there may be obstructions) the blocks of homotopy which achieve
geometric expression also expand. In the limit, we obtain a space X
with ß*X = A.
Organization of the paper. Section 2 contains a general discussion of
moduli spaces, and x3 analyzes Postnikov theory for ordinary topolog-
ical spaces in terms of moduli. Sections 4 and 6 treat the Postnikov
theory of simplicial -algebras; this is what leads to the construction
of our algebraic invariants. There is a detour in x5 to prove a general
relative connectivity theorem that gives information about homotopy
pushouts in the category of simplicial -algebras. Sections 7 and 8
look at simplicial spaces and their Postnikov theory, and x9 contains
proofs of the main results.
1.3. Notation. We use the language of simplicial model categories
([19 ] [12 ] [15 ] [13 ]); if C is a simplicial model category and X and Y
are objects of C, then Map (X, Y ) denotes the simplicial set of maps in
C from X to Y . All of our model categories have functorial factoriza-
tions, in that a map X ! Y can be naturally factored as a cofibration
followed by an acyclic fibration, or as an acyclic cofibration followed
by a fibration. The notation Map h(X, Y ) denotes the derived mapping
complex obtained by finding a functorial cofibrant model X0 ! X for
X, a functorial fibrant model Y ! Y 0for Y , and forming Map (X0, Y 0);
the set ß0 Map h(X, Y ) of derived homotopy classes of maps is denoted
[X, Y ]. In the same way, Aut h(X) is the simplicial monoid of self ho-
motopy equivalences of some cofibrant/fibrant object weakly equivalent
6 BLANC, DWYER, AND GOERSS
to X in a functorial way. Homotopy pushouts and pullbacks are con-
structed as usual [12 , x10]; since the model categories have functorial
factorization, we can take the homotopy pushouts and pullbacks to be
functorial.
We will make use of Eilenberg-Mac Lane objects in various cate-
gories, and we will try to make notational distinctions between them.
We use W~ G for the classifying simplicial set of a group or simplicial
monoid G [17 , x21]. The notations BG(M, n), K (M, n), and B (M, n)
specify twisted Eilenberg-Mac Lane objects in, respectively, the cate-
gory of pointed spaces (3.1), simplicial -algebras (6.1), and simplicial
spaces (8.1). Here G is a group, is a -algebra, M is a module over
G or , and n denotes the dimension in which M sits`as a homotopy
object. We will also need various coproducts: is a generic coprod-
uct, t is the coproduct of sets or unpointed spaces, _ the coproduct
for pointed spaces, and * the coproduct for -algebras.
1.4. Simplicial objects. A simplicial object X in a category C is a func-
tor from op to C, where is the simplicial category [17 ]. Equivalently,
X is a collection X[n], n 0 of objects of C, together with face maps
di : X[n] ! X[n - 1] and degeneracy maps si : X[n] ! X[n + 1] which
satisfy the standard simplicial identities. Note that we write X[n] to
distinguish the simplicial grading of X from a possible internal grad-
ing associated to the individual objects of C. We identify C with the
category of constant simplicial objects in C, i.e., simplicial objects in
which the face and degeneracy maps are identities.
1.5. Simplicial disks and spheres. Our basic reference for simplicial sets
and their model category structure is [13 ]. It is convenient to have
fixed models for simplicial disks and spheres. The standard simplicial
model for the n-sphere is cSn = n=@ n (the letter "c" stands for
combinatorial). It is natural to take as a model for the n-disk the
combinatorial simplex n itself, so that the sphere cSn is obtained from
the disk by collapsing out the boundary. This convention is slightly
awkward, because the boundary @ n is not combinatorially isomorphic
to cSn-1 (although these two complexes are weakly equivalent). To
avoid this awkwardness we let 0nbe the contractible subcomplex of
n obtained by taking the union of all faces of the top-dimensional
simplex except the 0'th face, and we take as our simplicial model for
the n-disk the quotient cDn = n= 0n. The inclusion of the 0'th face
in n induces a map n-1 ! cDn which is constant on @ n-1 and
passes to an inclusion cSn-1 ! cDn. This gives a cofibration sequence
of pointed simplicial sets
cSn-1 ! cDn ! cSn .
MODULI PROBLEM 7
2. Moduli spaces
Here we define moduli spaces, and recall some of the properties of
moduli spaces which arise from model categories. For our purposes, a
moduli space is always the nerve [3, XI.2] of some category. The reader
may be worried by the fact that the categories we consider in this
connection are usually large, in the sense that the collection of objects
forms a proper class instead of a set. The nerve of such a category is not
strictly speaking a simplicial set. There are two ways to deal with this.
One is to notice that the nerves we make use of are homotopically small
[5] and so determine well-defined ordinary homotopy types. Another
is to restrict in each case to a small subcategory of the category in
question, a subcategory which is still large enough to have a nerve of
the correct homotopy type; e.g., in the case of a model category C,
restrict to some small model subcategory of C containing some desired
set of objects. The issues here are routine, and we will suppress them
in order to avoid cluttering the exposition.
2.1. Moduli spaces for objects. A category with weak equivalences
is a pair (C, W) consisting of a category C together with a subcategory
W which contains all of the isomorphisms of C. The morphisms of W
are called weak equivalences. The basic examples are model categories,
which come equipped with such subcategories of weak equivalences as
part of the model category structure. Two objects X and Y of C
are said to be weakly equivalent if they are related by the equivalence
relation generated by the existence of a weak equivalence f : X ! Y .
If X is an object of a category with weak equivalences, the moduli
space M(X) is defined to be the nerve of the subcategory of C con-
sisting of all objects weakly equivalent to X together with the weak
equivalences between them. By definition M(X) is connected. The
main general theorem about it is the following.
2.2. Theorem. [7, 2.3] Suppose that C is a simplicial model category
and that X is an object of C. Then there is a natural weak equivalence
M(X) ~ W~Aut h(X).
If {Xff} is a set of objects in a category with weak equivalences,
then M{Xff} denotes the nerve of the category consisting of all ob-
jects weakly equivalent to one of the Xff's, together with the weak
equivalences between these objects.
2.3. Moduli spaces for diagrams. Suppose that C is a category with
weak equivalences and that D is some small category. The functor
category CD is in a natural way a category with weak equivalences,
where a natural transformation between functors is a weak equivalence
8 BLANC, DWYER, AND GOERSS
if for each object in D it gives a weak equivalence in C. For instance,
if D is a category with two objects and one nonidentity map between
them, we obtain the category of arrows in C. Given a map f : X ! Y
f
in C, we let M(X -! Y ) = M(f) denote the moduli space of f inside
the category of arrows. More generally, M(X _ Y ) denotes the moduli
space of all arrows X0 ! Y 0, where X0 is weakly equivalent to X and
Y 0is weakly equivalent to Y . If C is`a model category, X is cofibrant,
and Y is fibrant, then M(X _ Y ) is M(f), where f ranges over
weak equivalence classes of maps X ! Y . The indexing set here is not
quite homotopy classes of maps (see 2.10).
If C is a category with some specified notion of homotopy groups or
homotopy objects ßi, i 0, then for convenience we let M(X # Y )
denote the moduli space of arrows f : X0 ! Y 0, where X0 is weakly
equivalent to X, Y 0is weakly equivalent to Y , and f induces isomor-
phisms on ßi for all i with the property that ßiX and ßiY are both
nontrivial. Note that M(X # Y ) is a (possibly empty) union of com-
ponents of M(X _ Y ).
We use similar notation for moduli spaces of pairs of arrows. For
instance M(X _ Y " Z) denotes the moduli space of all diagrams
U ! V W in which U, V and W are weakly equivalent to X, Y and
Z respectively, and the map W ! V has the appropriate isomorphism
property on homotopy.
2.4. Function spaces as moduli. We also need to express derived
function complexes as moduli spaces. If X and Y are two objects of a
model category C, let MHom (X, Y ) denote the nerve of the category
whose objects are diagrams X U ! V Y in which the maps U !
X and Y ! V are weak equivalences. The morphisms are commutative
diagrams
X - - ~- U --- ! V - -~- Y
? ? ? ?
(2.5) =?y ~ ?y ?y~ ?y=
X - - ~- U0 --- ! V 0--~- Y
in which the indicated maps are identities or weak equivalences.
2.6. Theorem. [6, 4.7] [5, 1.1] Suppose that C is a simplicial model
category and that X and Y are objects of C. Then MHom (X, Y ) is in
a natural way weakly equivalent to the simplicial set Map h(X, Y ).
2.7. Remark. One can consider a similar category whose objects are
the smaller diagrams X- ~ U ! Y ; this is the full subcategory of the
above given by diagrams in which the map Y ! V is required to be the
identity. We denote the nerve of this category MfHom(X, Y ). If Y is a
MODULI PROBLEM 9
fibrant object of C, then the inclusion MfHom(X, Y ) ! MHom (X, Y ) is
a weak equivalence. This follows from the arguments of [6, 7.2].
2.8. Relationships between moduli spaces. Suppose that X and Y
are two objects of a model category C. There is a map MHom (X, Y ) !
M(X _ Y ) given by the functor which sends a diagram X U !
V Y to the diagram U ! V . The composite of this with the
obvious projection M(X _ Y ) ! M(X) x M(Y ) is again given by a
functor, and this is connected to the constant functor with value (X, Y )
by a chain of two natural transformations. This induces a map from
MHom (X, Y ) to the homotopy fibre of the projection.
2.9. Theorem. Suppose that X and Y are two objects of a model cat-
egory C. The sequence
p
MHom (X, Y ) ! M(X _ Y ) -! M(X) x M(Y )
is a homotopy fibre sequence, in the sense that the natural map from
MHom (X, Y ) to the homotopy fibre of p is a weak equivalence.
Proof. This follows from Quillen's Theorem B [18 ], given the observa-
tion, immediate from 2.6, that weak equivalences X ! X0 and Y 0! Y
induce a weak equivalence MHom (X, Y ) ! MHom (X0, Y 0).
2.10. Remark. Theorem 2.9 indicates that in the model category case
the set which indexes the components of M(X _ Y ) is the set of
homotopy classes of maps from X to Y , modulo the action on the one
hand of the self homotopy equivalences of X and on the other of the
self homotopy equivalences of Y .
2.11. Remark. The proof of 2.9 gives many other similar results. For
instance, given three objects X, Y , Z in an appropriate model category,
there is a natural homotopy fibre sequence
MHom (X, Y ) ! M(X _ Y " Z) ! M(X) x M(Y " Z) .
3. Postnikov systems for spaces
In this section we sketch an approach to the Postnikov theory of
pointed topological spaces which is based on the use of moduli spaces.
Our object is to establish some notation and provide a context for what
we do later on. We assume that the spaces are pointed and usually (for
convenience) that they are connected. The category of pointed topo-
logical spaces has its usual model category structure [19 , II.3] [12 , x8]
in which weak equivalences are weak homotopy equivalences, fibrations
are Serre fibrations, and cofibrations are retracts of relative cell com-
plexes.
10 BLANC, DWYER, AND GOERSS
Postnikov systems. Attaching an (n + 2)-cell to a space X by a map
f : Sn+1 ! X has no effect on the homotopy of X in dimensions n,
and clearly kills off the class represented by f in ßn+1X. Now attach
cells of dimension (n + 2) and greater to X by all possible attaching
maps to obtain an inclusion X X1, repeat the process to obtain
X1 X2, repeat again, etc., and let PnX = [kXk. There is a map
X ! PnX which induces isomorphisms on ßi for i n, and ßiPnX ' 0
for i > n. The construction of PnX is functorial in X and preserves
weak equivalences, and so it induces a map M(PnX) ! M(Pn-1X).
3.1. Eilenberg-Mac Lane objects. If G is a group, we say that a space X
is of type B G if ß1X is isomorphic to G and the higher homotopy of X
vanishes. Suppose that M is a G-module. We say that a map X ! Y
is of type B G(M, n), n 2, if X is of type B G, ß1Y ' G, ßnY ' M
(as a G-module), all other homotopy groups of Y vanish, and the map
X ! Y gives an isomorphism on ß1. Sometimes we say for short that
the target space Y is of type B G(M, n).
The difference construction. Suppose that f : Y ! X is a map of
spaces. Consider the pushout C of the diagram X0 Y 0! (P1X)0
obtained by using some functorial construction to replace Y by a cofi-
brant space and the two maps Y ! X and Y ! P1X by cofibrations.
There is a commutative diagram
Y - -~- Y 0 --- ! (P1X)0
? ? ?
(3.2) f?y ?y ?y n(f)
X - -~- X0 --- ! Pn+1C
We denote the vertical map on the right by n(f); its source is sn(f)
and its target is tn(f).
The following is easy to prove by calculating that, in the above situ-
ation, if X ! Y is surjective on ß1 then the universal cover of C is the
homotopy cofibre of the map X~ ! ~Y, where ~Y is the universal cover
of Y and X~ is the pullback of the cover ~Y to X.
3.3. Proposition. Suppose that f : Y ! X is a map of spaces whose
homotopy fibre F is (n - 1)-connected, n 1. Let M = ßnF and if
n = 1 assume that M is abelian. Then M is naturally a G-module for
G = ß1F , and n(f) is a map of type B G(M, n + 1). If ßiF vanishes
except for i = n, then the right-hand square in 3.2 is a homotopy fibre
square.
Existence and uniqueness of Eilenberg-Mac Lane objects. It is easy to
construct spaces of type B G by hand (take a wedge of circles indexed
by a set of generators for G, attach a 2-cell for each relation between
MODULI PROBLEM 11
the generators, and apply the functor P1) or by taking the geometric re-
alization of W~G. A simple argument gives that these spaces are unique
up to weak equivalence. We let B G denote a generic cofibrant space of
this type. It follows from obstruction theory or covering space theory
that Aut h(B G) is homotopically discrete and that its group of compo-
nents is Aut (G). Another way to express this is to say that the moduli
space of all spaces of type B G is weakly equivalent to W~Aut (G). The
next proposition extends this to higher Eilenberg-Mac Lane objects.
If G is a group and M is a G-module, we write Aut (G, M) for the
group of pairs (ff, fi), where ff is an automorphism of G and fi is an
ff-linear automorphism of M. This is the same as the group of auto-
morphisms of the split short exact sequence
0 -! M - ! G o M --! G -! 0 .
3.4. Proposition. Let G be a group, M a G-module, and n 2 and
integer. Then the moduli space of all maps of type B G(M, n) is weakly
equivalent to W~ Aut(G, M).
3.5. Remark. In particular the moduli space is nonempty and con-
nected, and so spaces or maps of type B G(M, n) exist and are unique
up to weak equivalence. We denote a generic space of this type by
BG(M, n).
Sketch of proof. Let Mn, n 2, denote the moduli space of all maps
X ! Y of type B G(M, n). There is a map Mn ! Mn+1 induced by
the functor which sends X ! Y to n(X ! P1X). There is also a map
Mn+1 ! Mn induced by the functor which sends X ! Y to the homo-
topy pullback of X ! Y X. Both composite functors are connected
to the respective identity functors by chains of natural transformations,
and so these maps of moduli spaces are weak equivalences. Similar
constructions give a weak equivalence M2 ~ M(B (G o M)--! B G),
where this last denotes the moduli space of maps U ! V with a section
V ! U, such that U and V have no higher homotopy groups, and such
that on the level of ß1 the map U ! V with its section gives a diagram
of groups isomorphic to G o M--! G. Now compute directly that this
last moduli space is weakly equivalent to W~Aut (G, M).
3.6. Cohomology of spaces. Consider a space B G(M, n), n 2. Then
P1 BG(M, n) ~ B G, and so we write the map from this space to its
first Postnikov stage as B G(M, n) ! BG. Given another space X over
BG (i.e. with a map X ! BG), we define HnG(X; M) by
HnG(X; M) ' [X, BG(M, n)]BG
where the symbol on the right denotes derived (1.4) homotopy classes
of maps from X to B G(M, n) in the model category of spaces over
12 BLANC, DWYER, AND GOERSS
BG [12 , 3.11]. Let HnG(X; M) denote Map hBG(X, BG(M, n)), so that
HnG(X; M) is ß0 of this space. The homotopy fibre squares
BG(M, n - 1) --- ! BG
? ?
? ?
y y
B G --- ! BG(M, n)
give natural weak equivalences HnG(X; M) ~ Hn-1G(X; M), so that
there are isomorphisms
(
Hn-iG(X; M) 0 i n - 2
ßiHnG(X; M) ' .
0 i > n
We use this formula to define HiG(X; M) for i = 0, 1; because we are
working with pointed maps these turn out to be what would normally
be called reduced twisted cohomology groups.
Classification of Postnikov stages. Suppose that X is a space with X ~
Pn-1X, n 2, and that M is a module over G = ß1X. If Y is a space,
we write Y ~ X + (M, n) if PnY ~ Y , Pn-1Y ~ X, and ßnY ' M
as a module over G, where this module isomorphism is realized with
respect to some isomorphism ß1Y ' G. We write M(X + (M, n)) for
the moduli space of all spaces Y of this type.
3.7. Proposition. Suppose that X is a space with X ~ Pn-1X, n 2
and that M is a module over G = ß1X. Then there is a natural weak
equivalence of moduli spaces
M(X + (M, n)) ~ M(X # BG(M, n + 1) " BG) .
3.8. Remark. The arrows # on the right indicate maps which induce
isomorphisms on appropriate homotopy groups (2.3); in this case it is
just isomorphisms on ß1.
Proof. There is a functor in one direction which given a space Y ~
X + (M, n) constructs the diagram (Pn-1Y )0! tn(f) sn(f) from
3.2, where f is the map Y ! Pn-1Y . There is a functor in the other
direction which given U ! V W of type X # BG(M, n + 1) " BG
constructs the space Y ~ X + (M, n) which is the homotopy pullback
of U ! V W . Both composites are connected to the corresponding
identity functors by chains of natural transformations, and so they
induce weak equivalences on the moduli spaces.
3.9. Interpretation. Let X, G and M be as above. According to 3.7,
3.4, and 2.11, there is a fibration sequence
(3.10) Map h1(X, BG(M, n + 1)) ! M(X + (M, n)) ! M(X) x ~W .
MODULI PROBLEM 13
where = Aut (G, M) and the object on the left is the union of the
components of Map h(X, BG(M, n)) giving maps which induce isomor-
phisms on ß1. It is easy to identify this subcomplex as tffHGX(n +
1; Mff), where ff runs through the isomorphisms ß1X ! G and Mff
is the module over ß1X determined by M and ff. Each space Y ~
X + (M, n) determines an element of
G n+1
ß0 tffHX (n + 1; Mff) ' tffHG (X; Mff)
modulo the action of ß0 Auth(X)xAut (G, M) on this set; this is the k-
invariant kn(Y ), in its genuinely invariant form. Correspondingly, each
k-invariant gives rise to a space Y . Note that 3.7 not only classifies
spaces of type X + (M, n), but also determines their self-equivalences.
The reader might want to compare fibration 3.10 with the corre-
sponding fibration
Map h0(X, Bfl(M, n + 1))u ! Mu(X + (M, n)) ! Mu(X)
from [9]. Here fl = Aut (M), Map h0(-, -)u denotes an appropriate set of
components of the space of unpointed maps, and Mu is the unpointed
moduli space. The appearance of the extra factor in the base of the our
fibration 3.10 is explained by the fact that for us the target of the k-
invariant map is BG(M, n+1), G = ß1X, while in [9] it is Bfl(M, n+1),
fl = Aut (M); the extra factor allows for potential automorphisms of
M which are not induced by elements of G.
4. -algebras and their modules
Here we explore -algebras, simplicial -algebras, and modules over
them. This is in preparation for a discussion in x6 of their cohomology.
4.1. -algebras. Let be the full sub-category of the homotopy cat-
egory of pointed spaces closed under isomorphism and containing the
wedges of spheres
Sn1 _ . ._.Snk
with ni 1. A -algebra is a product-preserving functor
: op -! S ,
or equivalently a contravariant functor ! S which takes wedges to
products. This condition and the Hilton-Milnor Theorem imply that
is determined by the sets n = (Sn), n 1 and the following
additional data:
(1) a group structure on n which is abelian for n > 1;
(2) composition maps (Sn, Sk) x k = ßn(Sk) x k ! n;
(3) Whitehead product maps [ , ] : n x k ! n+k-1 ;
14 BLANC, DWYER, AND GOERSS
(4) a 1-module structure on each abelian group n, n > 1.
There are relations among these structures; for example, (4) is redun-
dant, since for x 2 1 and a 2 n,
ax = [a, x] + a
where + is the group operation on n. The relations are classical,
but are complicated to write down [4]. We omit them, as the exact
formulas are unnecessary for our purposes. But recall that composition
is not additive: if {!} is a basis for the free Lie algebra over Z on two
generators, then for x, y 2 k, k > 1, and ff 2 ßkSn, we have
X
(4.2) (x + y) O ff = x O ff + y O ff + !(x, y) O H!(ff)
!
where the sum is over elements ! of length greater than 1, we write
!(x, y) for the corresponding iterated Whitehead product, and H! is
the associated higher Hopf invariant [22 , xXI.8.5]. We may at times
take to be the graded group { n} together with this additional struc-
ture; however, we will often stipulate -algebras by displaying the func-
tor
U 7! (U)
from op to the category of sets. In particular, we will often write U
for an object in the category . -algebras form a category, in which
the morphisms are natural transformations of functors.
4.3. Example. If X is a pointed space, there is a -algebra ß*X given
by the functor which sends U 2 to the set [U, X] of homotopy classes
of pointed maps from U to X. Note that ß*(X)n = ßnX, and that this
functor does not include ß0X. The -algebra ß*X captures the homo-
topy groups of X and all of the primary operations tying these groups
together. The construction ß*(-) gives a functor from the homotopy
category of pointed spaces to the category of -algebras.
The category of -algebras is a category of universal algebras and
has all limits and colimits. We write 0 for the trivial object, which
can be described as ß*X for X a one-point space. This object is both
initial and terminal, and the category of -algebras is pointed in the
sense that the unique map from the initial object to the terminal object
is an isomorphism.
4.4. Simplicial -algebras. As usual, a simplicial -algebra A is a
simplicial object (1.4) in the category of -algebras. The -algebra
A[n] is the portion of A in simplicial degree n, and A[n]i is the group
(abelian if i > 1) which is the i'th constituent of the -algebra A[n].
We write Ai for the associated simplicial group which in simplicial
dimension n contains the group A[n]i. Each simplicial group Ai has
MODULI PROBLEM 15
homotopy groups ß*Ai, which can be computed from the associated
normalized (Moore) complex N(Ai) [17 , 17.3, 22.1]. We let ß*A denote
the collection of all of these homotopy groups.
4.5. Model category structure. By Quillen [19 , xII.4], there is a standard
simplicial model category structure on the category of simplicial -
algebras. In this structure, a map f : A ! B is a weak equivalence if
and only if it is a weak equivalence of graded simplicial groups, i.e., if
and only if ß*A ! ß*B is an isomorphism. Every object is fibrant, and
a map A ! B is a fibration if for each i the induced map N(Ai) !
N(Bi) is surjective in degrees 1 and above. A map is a cofibration if
and only if it is a retract of a map which is "free" in the sense of [19 ,
xII.4]. To define these free maps, note that the forgetful functor from
-algebras to graded sets has a left adjoint F with
F (V*) ~=ß*(_n _x2Vn Sn) ' *n *x2Vn ß*Sn.
Then a morphism A ! B of simplicial -algebras is free if for each
n 0 there is a graded set Vn B[n], closed under the degeneracy
maps in B, such that
B[n] ~=A[n] * F (Vn).
Suppose that A is a simplicial -algebra and K is a simplicial set. The
simplicial structure on the category of simplicial -algebras is given by
letting K A be the simplicial object with (K A)[n] = *s2K[n]A[n].
4.6. Cells. Suppose in the above situation that K is a pointed sim-
plicial set. In this case we write K ~A = (K ~A)=(* ~A), where the
quotient is taken in the category of simplicial -algebras. The pairs
(cDi+1~ ß*Sj, cSi~ ß*Sj), i 0, j 1, are called cells, and a simplicial
-algebra is cellular if it can be constructed from a trivial simplicial
-algebra by attaching cells, perhaps transfinitely often. Any cellular
simplicial -algebra is cofibrant, any simplicial -algebra has a func-
torial cellular approximation, and any cofibrant simplicial -algebra is
a retract of a cellular one.
Cells are attached to A by elements in ß*A, in that [cSn ~ß*Sj, A]
is isomorphic to ßnAj. Note that in fact for each n 0, ßnA is a
-algebra, given as a functor (4.1) by the formula
(4.7) (ßnA)(U) = [cSn ~ß*U, A], U 2 .
4.8. Abelian -algebras; modules. A -algebra M is abelian if the
map M x M ! M given in each gradation by group multiplication
is a map of -algebras. This is equivalent to saying that M admits
the structure of an abelian group object in the category of -algebras,
or more concretely to saying that all of the Whitehead products in
16 BLANC, DWYER, AND GOERSS
M vanish [1]. The full-subcategory of -algebras consisting of abelian
-algebras is an abelian category.
As in any category of universal algebras, the notion of module is a
relativization of this concept.
4.9. Definition. Given a -algebra , a -module is an abelian group
object in the category of -algebras over .
More explicitly, a -module amounts to a split short exact sequence
of -algebras
(4.10) 0 -! M - ! EM --! -! 0
in which M is an abelian -algebra. A morphism of -modules is a map
of split sequences which is the identity on . We will sometimes identify
a -module with M and leave the short exact sequence understood; in
particular, we usually write M ! N for a morphism of -modules.
Since the graded constituents of a -algebra are already groups, it is
easy to see that an abelian group object in the category of -algebras
over is the same as a group object in this category.
Modules via actions. A -module M gives rise to a type of action of
on M. To see this, observe that the splitting of EM ! determines,
for each U 2 , an isomorphism of sets
EM (U) ~= (U) x M(U).
This means that for each map f : V ! U in , the morphism EM (f) :
EM (U) ! EM (V ) is determined by an action map
(4.11) OEf : (U) x M(U) ! M(V )
subject to the conditions
(1) OEf(0, x) = M(f)(x), and
(2) OEgOf(a, x) = OEg( (f)a, OEf(a, x)).
It is even possible to go in the other direction. Given maps 4.11 subject
to the indicated conditions, we can form a -algebra o M which lies
in a split sequence
0 -! M - ! o M --! -! 0
and so define a -module structure on M. If M began life as a -
module, there is an isomorphism of -algebras EM ~= o M, making
the evident diagram of split sequences commute.
4.12. Modules via split cofibration sequences. A split cofibration se-
quence in a pointed model category C is a diagram
A --! B -! C
MODULI PROBLEM 17
in C such that the objects involved are cofibrant, A ! B ! C is a
cofibration sequence, and the left-hand maps exhibit A as a retract
of B. Suppose that there are functors OE, _ : ! C which take on
cofibrant values and preserve coproducts up to weak equivalence. Then
there are -algebras MX and X associated to any object X of C and
given by the formulas
X (U) = [OEU, X] MX (U) = [_U, X] .
In order to show that MX is in a natural way a module over X it
is enough to prove that MX is abelian for each X, and to construct
objects _+ U which fit into split cofibration sequences
OEU --! _+ U - ! _U
which are natural in U. For the split sequences encoding the module
structure (4.10) can be constructed by mapping this split cofibration
sequence into objects X of C. Note that in order to show that MX is
an abelian -algebra for each X, it is enough by Yoneda's lemma to
show that _U is a cogroup object in the homotopy category of C in a
way which is natural in U.
4.13. Examples. A -algebra is not a module over itself, unless is
abelian. However, we may define new -algebras n by the functor
on op
U 7! (Sn ^ U).
This mimics topology: nß*X ~=ß* nX. For n 1, n is a -
module. To see this, define a -algebra n+ by
U 7! (Sn+^ U)
where the (-)+ denotes adding a disjoint basepoint. Then there is a
split sequence
0 -! n -! n+ --! -! 0
which gives a canonical -module structure on n . These module
structures are central to what follows in this paper; they arise from the
fact that in the homotopy category of pointed spaces, Sn+for n 1 is a
cogroup object in the category of spaces under S0. Note that if X is a
space then n+ß*X is naturally isomorphic to the homotopy -algebra
of the space of all (not necessarily pointed) maps Sn ! X.
If we have a morphism M ! N of -modules, the ordinary kernel
K is a -module; the necessary total space EK for the split sequence
is the pull-back of EM ! EN - s . If M is a -module, so is + M;
18 BLANC, DWYER, AND GOERSS
the total space of the split sequence is defined by the pull-back square
E +M --- ! + EM
? ?
? ?
y y .
---s! +
Consequently, if M is a -module, M is a module: it is the kernel
of n+M ! M. It is easy to check that the -module structure on n
described above is the same as that obtained inductively by starting
with the given -module structure on and making the identification
n ' ( n-1 ).
4.14. Homotopy group modules. For n 1, cSn is a cogroup object in
the homotopy category of pointed simplicial sets, and there is a split
cofibration sequence
(4.15) cS0 --! cSn+- ! cSn
of pointed simplicial sets, where (-)+ denotes adding a disjoint base-
point. Tensoring this with ß*U for U 2 (4.7) gives the structure
necessary (4.12) to show that for any simplicial -algebra A, ßnA is
abelian for n 1 and is naturally a module over ß0A.
5. Relative connectivity of pushouts
In this section we give a partial calculation of the homotopy type
of the homotopy pushout of a diagram of simplicial -algebras (5.1).
This is along the lines of [21 , 1.10, 3.6], but we work in more generality
and remove some simple connectivity hypotheses.
To express the result we will introduce a slightly unorthodox notion
of connectivity. If f : A ! B is a map of simplicial sets, the cellu-
lar connectivity of f, denoted ~(f) (or ~(B, A) if f is understood), is
the greatest integer n such that f can be obtained up to weak equiv-
alence by taking A (or a fibrant representative) and attaching cells of
dimension n and above. If f is a weak equivalence, then ~(f) = 1.
More precisely, ~(f) = n if and only if all of the homotopy fibres of f
are (n - 2)-connected, and at least one of the homotopy fibres is not
(n - 1)-connected. The numbers here are potentially confusing. One
rough way to remember them is to keep in mind that if A and B are
1-connected and A is a subcomplex of B, then ~(B, A) is the lowest
dimension in which B=A has nontrivial homology (or homotopy).
If f : A ! B is a map of simplicial -algebras or of graded sim-
plicial sets, we let ~(B, A) denote the minimum value of the numbers
~(Bn, An), n 1. In the statement of the following proposition the
MODULI PROBLEM 19
symbol [h denotes homotopy pushout in the category of graded sim-
plicial sets, while *h is homotopy pushout in the category of simplicial
-algebras.
5.1. Proposition. Suppose that B A ! C is a two-source of sim-
plicial -algebras. Then
~(B *hAC , B [hAC) ~(B, A) + ~(C, A) .
We will deduce 5.1 from some very general observations. A finite
graded set is a graded set which is finite in every gradation and empty
in all but a finite number of gradations. Consider a functor F from the
category of finite graded sets to the category of graded simplicial sets.
There is a standard way to prolong F to a functor on the category of
all graded sets by setting
(5.2) F (T ) = colimS T F (S) ,
where the colimit is taken over the category of finite graded subsets of
T . The functor F can be further prolonged to a functor on the category
of graded simplicial sets by setting
(5.3) F (X) = diag(n 7! F (X[n]) .
Here diag is the diagonal or realization functor from the category of
bisimplicial sets to the category of simplicial sets [13 , IV.1]. The argu-
ment of diag in the above formula is a graded bisimplicial set, but the
diagonal is to be taken gradation by gradation. In each of the following
statements the functor F involved is prolonged like this to a functor
on the category of graded simplicial sets.
5.4. Proposition. Any functor F from finite graded sets to graded
simplicial sets respects cellular connectivity, in the sense that for any
map X ! Y of graded simplicial sets there is an inequality
~(F (Y ), F (X)) ~(Y, X) .
5.5. Proposition. Any functor F from finite graded sets to graded
simplicial sets preserves homotopy pushouts in the stable range, in the
sense that for any two-source Y X ! Z of graded simplicial sets
there is an inequality
h h
~ F (Y [X Z), F (Y ) [F(X) F (Z) ~(Y, X) + ~(Z, X) .
We also need the following lemma, which can be proved by the same
sort of gluing argument used in the proof of [13 , IV.1.7].
5.6. Lemma. Suppose that X ! Y is a map of bisimplicial sets, and
that n is an integer such that ~(Y [i], X[i]) n for all i 0. Then
~(diag(Y ), diag(X)) n.
20 BLANC, DWYER, AND GOERSS
Proof of 5.1. This is similar to the second part of the proof of [21 , 3.6].
First, some background. Let F denote the free functor from graded
sets to -algebras, prolonged degreewise to be a functor from graded
simplicial sets to simplicial -algebras. For any simplicial -algebra
D there is a bar resolution B(D) [21 , 3.2]; this is a bisimplicial -
algebra, i.e. a simplicial object in the category simplicial -algebras,
with B(D)[n] = F n+1(D). Let D~ = diag (B(D)). By [21 , 3.2], D~
is a cofibrant simplicial -algebra; more generally, if D ! D0 is a
map of simplicial -algebras which is an injection of underlying graded
simplicial sets, then the maps B(D)[n] ! B(D0)[n] and the diagonal
map ~D! ~D0are both cofibrations. There is a natural weak equivalence
~D! D.
Now for the proof. By adjusting the objects up to weak equivalence,
we can assume that the maps A ! B and A ! C are cofibrations of
simplicial -algebras and hence injections on underlying graded simpli-
cial sets. The simplicial -algebra ~Ais cofibrant and the induced maps
~A! ~Band ~A! ~Care cofibrations; hence there are weak equivalences
B *hAC ~ ~B*A~C~ = diag B(B) *B(A)B(C)
(5.7) h h
B [A C ~ ~B[A~C~ = diag B(B) [B(A)B(C) .
Let U = B(A), V = B(B), W = B(C). By 5.4 and induction on n,
there are inequalities
~(V [n], U[n])= ~(F n+1(B), F n+1(A)) ~(B, A)
~(W [n], U[n])= ~(F n+1(C), F n+1(A)) ~(C, A)
and hence by 5.5 inequalities
~ ((V *U W )[n], (V [U W )[n]) ~(B, A) + ~(C, A) .
Note in this connection that because of the fact that F (as a left ad-
joint) preserves colimits, there is a natural isomorphism (V *U W )[n] '
F ((V [U W )[n - 1]). The result follows from 5.6 and 5.7.
For the sake of clarity we will prove 5.5 and 5.4 in the ungraded
case (i.e. with the word "graded" deleted from the statements); the
modifications necessary to pass to the graded case are notational.
Suppose that D be a small category and that F and G are respec-
tively covariant and contravariant functors from D to simplicial sets.
We denote the coend [16 , IX.6] of the bifunctor GxF by GxD F . This
is the coequalizer of a more or less evident pair of maps
a a
G(d0) x F (d) ' G(d) x F (d)
d!d0 d
MODULI PROBLEM 21
where the coproduct in the range is indexed by the objects in D and
the coproduct in the domain by the arrows. This coequalizer diagram
is the low degree part of the bisimplicial set B(F, D, G) (cf. [14 , x3])
with
a
(5.8) B(F, D, G)[k] = G(dk) x F (d0) ,
d0!...!dk
where the coproduct is indexed by the k-simplices of the nerve of D.
We will denote the diagonal of this bisimplicial set by G xhDF and call
it the homotopy coend of the bifunctor G x F . There is an obvious
map
(5.9) G xhDF ! G xD F.
Let F be the category of finite sets. Suppose that F is a functor from
finite sets to simplicial sets, prolonged as in 5.2 and 5.3 to a functor of
simplicial sets. As remarked in [21 , 1.1], this prolonged functor can be
expressed by the formula
F (X) = X* xF F
where X is a simplicial set and X* is the contravariant functor on F
which sends S to XS . The observation we begin with is that this coend
is actually equivalent to the corresponding homotopy coend.
5.10. Proposition. Suppose that F is a functor from finite sets to
simplicial sets. Then for any simplicial set X the natural map
X* xhFF ! X* xF F = F (X)
is a weak equivalence.
Proof. We consider the map 5.9 for an arbitrary contravariant functor
G from F to sets or simplicial sets. It is easy to see that the map
is a weak equivalence if G is representable, that is, if G has the form
Hom (-, T ) for some object T of F; in this case both domain and range
are equivalent to F (T ) [14 , 3.1(5)]. Since filtered colimits preserve
weak equivalences [3, XII.3.6] and all of the constructions in question
commute with filtered colimits, the map 5.9 is clearly an equivalence if
G is a filtered colimit of representable functors. It now follows from a
diagonal argument that 5.9 is a weak equivalence if each of the functors
G(-)[n] is a filtered colimit of representable functor; to obtain this use
[13 , IV.1.7] and the fact that 5.9 is the diagonal of a map of bisimplicial
sets which in degree n contains the map G(-)[n] xhFF ! G(-)[n] xF F .
But observe that any set is the filtered colimit of its finite subsets, so
that the functor on F sending S to X[n]S = Hom S(S, X[n]) is indeed
a filtered colimit of representable functors.
22 BLANC, DWYER, AND GOERSS
The following is an exercise in elementary homotopy theory.
5.11. Lemma. Suppose that Y X ! Z is a two-source of simpli-
cial sets in which the maps are injective (so that the homotopy pushout
agrees with the ordinary pushout). Then for any n 0 there are in-
equalities
~(Y n, Xn ) ~(Y, X)
~((Y [X Z)n, Y n[Xn Zn ) ~(Y, X) + ~(Z, X)
Proof of 5.4 (ungraded case). By 5.10, ~(F (Y ), F (X)) is the same as
the cellular connectivity of the map X* xhFF ! Y *xhFF . This map
can be realized as the diagonal of a map of bisimplicial sets (5.8) which
in degree k is constructed as a disjoint union of maps of the form
XS x F (T ) ! Y Sx F (T ). It follows from the first inequality of 5.11
that ~(Y Sx F (T ), XS x F (T )) ~(Y, X). Since taking disjoint unions
does not lower cellular connectivity, the desired result follows from
5.6.
Proof of 5.5 (ungraded case). We can assume that X ! Y and X ! Z
are injections, so that the pushoutiof the two-source is thejsame as the
homotopy pushout. By 5.10, ~ F (Y [hXZ), F (Y ) [hF(X)F (Z) is the
same as the cellular connectivity of the map
(Y *xhFF ) [X*xhFF (Z* xhFF ) ! (Y [X Z)* xhFF .
By definition (5.8) and inspection, this map is realized as the diagonal
of a map of bisimplicial sets which in degree k is constructed as a
disjoint union of maps of the form
(Y S[XS ZS ) x F (T ) ! (Y [X Z)S x F (T ) .
It follows from the second inequality of 5.11 that the cellular connec-
tivity of this last map is at least ~(Y, X) + ~(Z, X), and as in the proof
above the desired result is now a consequence of 5.6.
6.Postnikov systems for simplicial -algebras
In this section we study Postnikov systems for simplicial -algebras
in a way which is largely parallel to the study of Postnikov systems
for topological spaces in x3. In the course of this we develop a notion
of cohomology for simplicial -algebras. This differs from the notion
of cohomology for -algebras considered by the second author and
Kan in [8] in that more general coefficients are allowed. In [8] the
coefficients are "strongly abelian" -algebras in which both Whitehead
products and compositions are trivial; here we accept arbitrary abelian
MODULI PROBLEM 23
-algebras, in which the Whitehead products vanish but compositions
may be nontrivial.
Postnikov systems. Suppose that X is a simplicial -algebra. Attach-
ing an (n + 2)-cell cDn+2 ~ß*Sk to X via a map f : cSn+1 ~ß*Sk ! X
has no effect on ßiX for i n, and clearly kills of the class repre-
sented by f (4.7) in (ßn+1X)k. Now attach cells of dimension (n + 2)
and greater to X by all possible attaching maps to obtain an inclusion
X X1, repeat the process to obtain X1 X2, repeat again, etc., and
let PnX = [jXj. There is a map X ! PnX which induces isomor-
phisms on ßi for i n, and ßiPnX ' 0 for i > n. The construction
of PnX is functorial in X, and there is a natural map PnX ! Pn-1X
which respects the inclusions of X in these two simplicial -algebras.
6.1. Eilenberg-Mac Lane objects. If is a -algebra, we say that a sim-
plicial -algebra X is of type K if ß0X ' and the higher homotopy
of X is trivial. Suppose that M is a -module. We say that a map
X ! Y is of type B (M, n) n 1, if X is of type K , ß0Y ' ,
ßnY ' M (as a -module), all other homotopy of Y is trivial, and the
map X ! Y gives an isomorphism on ß0. Sometimes we will say for
short that the target Y is of type K (M, n).
The difference construction. Suppose that f : Y ! X is a map of
simplicial -algebras. Consider the pushout C of the diagram X0
Y 0! (P0X)0obtained by using some functorial construction to replace
Y by a cofibrant object and the two maps Y ! X and Y ! P0X by
cofibrations. There is a commutative diagram
Y - -~- Y 0 --- ! (P0X)0
? ? ?
(6.2) f?y ?y ?y n(f)
X - -~- X0 --- ! Pn+1C
in which the vertical map on the right is n(f). The source (P0X)0 of
n(f) is sn(f), and the target Pn+1C is tn(f).
6.3. Proposition. Suppose that f : Y ! X is a map of simplicial
-algebras which is an isomorphism on ß0 and whose homotopy fibre
F is (n - 1)-connected, n 1. Let M = ßnF . Then M is naturally a
-module for = ß0X and n(f) is a map of type K (M, n + 1). If
ßiF vanishes except for i = n, then the right-hand square in 6.2 is a
homotopy fibre square.
We need a modified form of 3.3. A map f : A ! B of connected
simplicial sets is simple if its homotopy fibre is connected and ß1A acts
trivially on the homotopy groups of the homotopy fibre.
24 BLANC, DWYER, AND GOERSS
6.4. Proposition. Let f : A ! B be a simple map of connected simpli-
cial sets with homotopy fibre F . Assume that ßiF is trivial for i < n,
n 1, and let M = ßnF . Let be the mapping cone of f, and Pn+1
its (n + 1)'st Postnikov stage in the category of simplicial sets. Then
Pn+1 is a simplicial set of type K(M, n + 1). If the homotopy of F
vanishes except in dimension n, then the sequence A ! B ! Pn+1 is
a homotopy fibre sequence.
Proof of 6.3. This follows from 5.1 and 6.4. Clearly ~((P0X)0, Y 0) 2
and ~(X0, Y 0) 2. Let = (P0X)0[Y 0X0; this is a homotopy pushout
in the category of graded simplicial sets. By 5.1, ~(C, ) 2 (here C
is from 6.2). It is easy to see that up to weak equivalence applying
Pn+1 to a simplicial -algebra commutes with taking the underlying
graded simplicial set. But ß0 ' ß0C ' , and it follows from 6.4 that
ßiPn+1C vanishes except for the fact that it is isomorphic to if i = 0
and to M if i = n + 1. Thus M is naturally a -module (4.14) and
Pn+1C is of type K (M, n + 1). (This last deduction involves applying
6.4 componentwise to a map Y 0! X0 of graded disconnected simplicial
sets which is an isomorphism on ß0; note that P0X is homotopically
discrete, so that is essentially obtained by taking componentwise
mapping cones of Y 0! X0. The map Y 0! X0 is componentwise
simple because Y 0and X0, as simplicial -algebras, are actually graded
simplicial groups.) The final statement again follows from 6.4, since
taking the homotopy pullback of a two-sink of simplicial -algebras
commutes up to weak equivalence with passing to underlying graded
simplicial sets.
Existence and uniqueness of Eilenberg-Mac Lane objects. The -algebra
, considered as a constant simplicial object, is of type K . Moreover,
if X is any simplicial -algebra of type K then the natural map from
X to its -algebra of components gives a weak equivalence X ~ . It
is easy to deduce from this that the moduli space of all simplicial -
algebra's of type K is connected and weakly equivalent to W~Aut ( ).
We will denote a generic simplicial -algebra of this type by K .
6.5. Proposition. Let be a -algebra and M a -module. Then for
each n 1 the moduli space of all maps of type K (M, n) is weakly
equivalent to W~ Aut( , M).
6.6. Remark. In particular, the moduli space is nonempty and con-
nected, so objects or maps of type K (M, n) are unique up to weak
equivalence. We will denote a generic simplicial -algebra of this type
by K (M, n).
Proof. Let Mn be the moduli space of maps of type K (M, n). As in
the proof of 3.4, the difference construction 6.3 gives weak equivalences
MODULI PROBLEM 25
Mn ! Mn+1, n 1. Let M0 be the moduli space M(K oM --! K ),
i.e, the moduli space of maps U ! V of simplicial -algebras with a
section V ! U such that U and V have trivial higher homotopy and on
ß0 the map with its section gives a diagram of -algebras isomorphic
to o M--! . It is easy to see that M0 is weakly equivalent to
~WAut ( , M). The functor which assigns to a map U ! V of type
K (M, n) the homotopy pullback of U ! V U gives a map M1 !
M0, but in contrast to the situation in the proof of 3.4, the difference
construction does not give an inverse. Instead we proceed as follows.
Given U--! V of type K oM --! K , write 0 = ß0V , 0o M0 = ß0U
and form the map 0! 0oW~ M0 of type K (M, 1). This construction
is functorial and gives a map M0 ! M1. The composite M0 !
M1 ! M0 is clearly an equivalence because the underlying functor
is connected to the identity by natural transformations. The same is
true of the other composite; the key observation is this. Suppose that
U ! V is a map of type K (M, 1), which we can assume to be a
fibration, and let U*Vbe the simplicial object which in simplicial degree
n contains the n-fold fibre power of U over V . The diagonal of this
bisimplicial -algebra maps to V by a weak equivalence, but it also
maps to the simplicial -algebra obtained by applying ß0 degreewise;
this simplicial -algebra is exactly ß0V o ~Wß1V .
6.7. Cohomology of -algebras. We follow 3.6. Consider an Eilenberg-
Mac Lane object K (M, n), n 1. Then P0K (M, n) ~ K , and
so we write the map from this object to its zeroth Postnikov stage as
K (M, n) ! K . Given another simplicial -algebra X over K , we
define Hn (X; M) by
Hn (X; M) = [X, K (M, n)]K
where the symbol on the right denotes derived homotopy classes of
maps in the category of simplicial -algebras over K . Let Hn(X; M)
denote Map hK(X, K (M, n)), so that the set of components of this
space is Hn (X; M). As in 3.6, there are isomorphisms
(
Hn-i(X; M) 0 i n - 1
ßiHn(X; M) ' .
0 i > n
We use this formula to define H0(X; M).
Classification of Postnikov stages. Suppose that X is a simplicial -
algebra with X ~ Pn-1X, n 1 and that M is a module over ß0X.
If Y is a simplicial -algebra, we write Y ~ X + (M, n) if PnY ~ Y ,
Pn-1Y ~ X, and ßnY is isomorphic to M as a module over ß0Y , where
the isomorphism is realized by some isomorphism ß0X ! ß0Y . We
26 BLANC, DWYER, AND GOERSS
write M(X + (M, n)) for the moduli space of all simplicial -algebras
of type X + (M, n).
The following result is proved in the same way as 3.7, with 6.3 re-
placing 3.2 in the argument.
6.8. Theorem. Suppose that X is a simplicial -algebra with X ~
Pn-1X, n 1. Let = ß0X, and let M be a module over . Then
there is an natural weak equivalence
M(X + (M, n)) ~ M(X # K (M, n + 1) " K ) .
6.9. Remark. The arrows # on the right indicate maps which induce
isomorphisms on appropriate homotopy groups (2.3); in this case it is
just isomorphisms on ß0. The remarks at the beginning of 3.9 can be
repeated almost verbatim here.
7.Simplicial spaces and the spiral exact sequence
In [10 ] and [11 ], Kan and Stover and the second author of this pa-
per developed a model category structure on the category of simplicial
pointed topological spaces which is adapted to making spherical res-
olutions of ordinary spaces that mirror resolutions of their homotopy
-algebras. In this section we spell out what we need from these pa-
pers and extend the theory in some ways (7.13). All of our topological
spaces have basepoints; we sometimes take this for granted and refer
to "spaces" instead of to öp inted spaces".
7.1. The Reedy model structure. To begin with, the category of
simplicial spaces acquires a Reedy model category structure [20 ] [10 ,
2.4] [15 , 5.2.5] from the usual model category structure (x3) on the
category of pointed spaces. A map X ! Y of simplicial spaces is a
Reedy weak equivalence if X[n] ! Y [n] is a weak equivalence for all
n 0, a Reedy fibration if X[0] ! Y [0] is a fibration and, for all n 1,
the map
X[n] ! Y [n] xMnY MnX
is a fibration. Here MnX is the nth matching space:
MnX = lim X[m]
ffi:[m]![n]
where OE runs over injections in the ordinal number category with
m < n. Cofibrations are defined symmetrically: X ! Y is a Reedy
cofibration if X[0] ! Y [0] is a cofibration and for n 1,
`
X[n] LnY ! Y [n]
LnX
MODULI PROBLEM 27
is a cofibration. Here LnX is the latching space
LnX = colim X[m]
_:[n]![m]
where _ runs over the surjections in the ordinal number category with
m < n. This Reedy model structure has the desirable property that
the geometric realization functor X 7! |X| preserves weak equivalences
between cofibrant objects [11 , x4].
7.2. The E2 model structure. The E2 model category structure is
built from the Reedy model category structure. If X is a simplicial
pointed space, we let ß*X denote the simplicial -algebra obtained by
applying the functor ß* degreewise to X.
7.3. Definition. Define a morphism f : X ! Y of simplicial pointed
spaces to be
(1) an E2-equivalence if ß*(f) is a weak equivalence of simplicial
-algebras (4.4);
(2) an E2-fibration, if f is a Reedy fibration and ß*(f) is a fibration
of simplicial -algebras (4.4); and
(3) an E2-cofibration if f is a retract of an S1-free map; here f
is S1-free if there is a CW complex Zn Y [n] which has the
homotopy type of a wedge of spheres Sk, k 1, and
`
(X[n] LnY ) _ Zn ! Y [n]
LnX
is an acyclic cofibration.
The category of simplicial spaces has a standard simplicial structure
in the sense of Quillen [19 , xII.2]; if K is a simplicial set and X is
simplicial space, then K X is the simplicial space with
(K X)[n] = _x2K[n]X[n] .
The Reedy model category structure on simplicial spaces does not ex-
tend to a simplicial model category structure with respect to this sim-
plicial structure: if X ! Y is a Reedy cofibration and K ! L is a
cofibration of simplicial sets, then
X L _X K Y K ! Y L
is a Reedy cofibration which is Reedy acyclic if X ! Y is a Reedy weak
equivalence, but pretty evidently need not be a Reedy weak equivalence
if K ! L is a weak equivalence of simplicial sets. The main result of
[10 ] is:
7.4. Proposition. With notions of E2-equivalence, E2-fibration, and
E2-cofibration just given, and with the simplicial structure described
28 BLANC, DWYER, AND GOERSS
above, the category of simplicial spaces becomes a cofibrantly generated
simplicial model category.
From now on, when we refer to cofibrations, fibrations, and weak
equivalences between simplicial spaces, we will unless otherwise speci-
fied be referring to the E2-model structure.
7.5. Remark. Note that an object is E2-fibrant if and only if it is Reedy
fibrant. If X is E2-cofibrant, it is also Reedy cofibrant, although not
vice versa (cf 7.8).
7.6. The functor ß* preserves homotopy pushouts. If f : X ! Y is
an E2-cofibration, then ß*(f) is a cofibration of simplicial -algebras.
Suppose that X Y ! Z is a two-source of simplicial pointed spaces
in which the objects are E2-cofibrant and the maps are E2-cofibrations,
and let C be the pushout of the square. Then ß*C is the pushout of
ß*X ß*Y ! ß*Z (in each simplicial degree, the pushout process
just involves wedging on spheres). It follows that the functor ß* from
simplicial spaces to simplicial -algebras preserves homotopy pushouts.
7.7. The functor ß* often preserves homotopy fibres. Let f : X ! Y
be an E2-fibration with fibre F . If ß*(f) is surjective, then clearly the
fibre of ß*(f) is exactly ß*F . By 4.4 and the definition of E2-fibration,
ß*(f) is surjective if and only if the map ß0ß*X ! ß0ß*Y is surjective.
It follows that for such maps f, the functor ß* preserves (homotopy)
fibres.
7.8. Cells. If X is a simplicial space and K is a simplicial set with
basepoint *, we define K ~X to be the quotient (K X)=(* X).
The bigraded spheres Si,jare defined by Si,j= cSi~ Sj, and the corre-
sponding disks by Di,j= cDi~ Sj. Say that a simplicial space is cellular
if it is constructed from the trivial simplicial space by attaching cells
(Di+1,j, Si,j), i 0, j 1. Then any cellular simplicial space is E2-
cofibrant, any simplicial space has a functorial cellular approximation,
and any cofibrant simplicial space is a retract of a cellular one.
7.9. Homotopy groups and the spiral exact sequence. If X is a
Reedy cofibrant simplicial space, there is a first quadrant (homology)
spectral sequence converging to ß*|X| with E2i,j= ßißjX [2] [11 , 8.3].
This explains the term "E2 model category structure": a map X ! Y
of simplicial spaces is an E2 weak equivalence if and only if it induces
an isomorphism on these E2-pages. We will write ^ffliX = ßiß*X for
the i'th column of this E2-term. By 4.5 and 4.14, ^ffliX is a -algebra
which for i 1 is naturally a module over ^ffl0X. By definition, a map
X ! Y is an E2 weak equivalence if and only if it induces isomorphisms
^ffl*X ' ^ffl*Y .
MODULI PROBLEM 29
The notion of cellular simplicial space (7.8) suggests another notion
of homotopy; if X is a simplicial space we define ßi,jX, i 0, j 1 by
ßi,jX = ßiMap h(Sj, X) ' [Si,j, X]
where the symbol on the right denotes derived homotopy classes of
maps in the E2 model category. These are the bigraded homotopy
groups of X. Let ^ßiX = ßi,*X. The objects ^ßiX (i 0) have formal
properties very similar to those of ^ffliX.
7.10. Proposition. Suppose that X is a simplicial space. Then ^ßiX
is a -algebra, which for i 1 is a module over ^ß0X. A map X !
Y of simplicial spaces is a weak equivalence if and only if it induces
isomorphisms ^ßiX ! ^ßiY , i 0.
Proof. It is easy to see that ^ßiX is exhibited as a -algebra by the
functor which sends U 2 to ßiMap h(U, X) = [cSi~ U, X]. The
module structure arises (4.12) from the fact that for i 1, cSi+ is a
cogroup object in the homotopy category of simplicial sets under cS0
with cSi+= cS0 ' cSi. The last statement is from [11 , 5.3].
The objects ^ffliX and ^ßiX are related by a long exact sequence, called
the spiral exact sequence.
7.11. Proposition. [11 , 7.2, 8.1] Suppose that X is a simplicial space.
Then there is a natural isomorphism ^ß0X ' ^ffl0X of -algebras, as well
as a long exact sequence of -algebras
. .!.^ffln+1X ! ^ßn-1X ! ^ßnX ! ^fflnX ! . .!.^ß1X ! ^ffl1X ! 0 .
7.12. Structure of the spiral exact sequence. All of the con-
stituents of the spiral exact sequence are naturally modules over ^ß0X:
^ßnX by 7.10, ^ßn-1X by 7.10 and 4.13, and ^fflnX by 4.14 and the iso-
morphism ^ffl0X ' ^ß0X given by 7.11. In the rest of this section we will
prove the following proposition.
7.13. Proposition. With respect to the module structures described
above, the spiral exact sequence 7.11 is an exact sequence of ß^0X-
modules.
This will be proved in stages.
7.14. Proposition. The homomorphisms ^ßiX ! ^ffliX from 7.11 are
maps of modules over ^ß0X.
Proof. By definition [11 ] these homomorphisms are obtained from the
isomorphisms ß*(cSi~ U) ' cSi~ ß*U, U 2 ; these give maps
(^ßiX)(U) = [cSi~ U, X] ! [cSi~ ß*U, ß*X] = (^ffliX)(U) .
For i = 0 we obtain the isomorphism ^ß0X ' ^ffl0X. Let Q be the
split cofibration sequence from 4.15. Then the corresponding maps
30 BLANC, DWYER, AND GOERSS
[Q ~U, X] ! [Q ~ß*U, ß*X] provide morphisms of split sequences (4.10)
which show that ^ßiX ! ^ffliX is a map of ^ß0X-modules.
To go any further, we need more information about how to repre-
sent the constituents of the spiral exact sequence in the E2 homotopy
category. This information is in [11 , 7.4], but we have to examine it in
some detail because we need a relative version.
If X is a space, the pointed cylinder IX is the pushout of the diagram
* *xI ! X xI, where I = [0, 1]; the cone CX is then (IX)=(X x1).
There is a natural inclusion X ! CX given by x 7! (x, 0), and the
quotient CX=X is the suspension X.
If X is a simplicial space, we write D^nX = cDn ~X and ^ nX =
cSn ~X. It is easy to see [10 , 4.1] that D^nX is always E2-contractible,
in the sense that it is E2 weakly equivalent to a trivial simplicial space
with one point in each simplicial degree.
The representing objects. Suppose that U 2 , and that n 2 is an
integer. We wish to construct a simplicial space ~ n-2 U by considering
the following diagram
^ n-2U --- ! ^ n-2CU --- ! ^n-2 U
? ? ?
= ?y ~?y ~ ?y .
^ n-2U --- ! ^Dn-1CU --- ! ~n-2 U
The top row is a sequence of simplicial spaces which in each simplicial
degree gives a cofibration sequence of spaces, and ~ n-2 U is defined so
that the same is true of the bottom row. (These are not E2-cofibration
sequences; for instance, the left hand horizontal maps do not induce
injections on ß*. In spite of the notation, ~ n-2 U is a functor of U, not
of U.) It is clear that the vertical arrows are Reedy equivalences, and
therefore E2-equivalences; in effect, ~ n-2 U is obtained from ^ n-2 U
by wedging on some number of copies of CU in each simplicial degree.
The following is clear from the definitions (4.13).
7.15. Proposition. If X is a simplicial space, the -algebra ^ßn-2X
is represented by the functor
U 7! [ ~n-2 U, X] ' [ ^n-2 U, X] .
Notice that there is a natural map
fi : ^ n-1U = ^Dn-1U= ^n-2U ! ^Dn-1CU= ^n-2U = ~ n-2 U .
MODULI PROBLEM 31
Now we construct a simplicial space _nU by considering the following
diagram
^ n-1U --ff-! D^nU --- ! ^ nU
? ? ?
(7.16) fi?y ?y ?y
~ n-2 U ---fl! _nU --- ! ^ nU
The object _nU is defined so that the left hand square is a pushout
square. Since the map ff is an E2-cofibration and both of the objects
on the left are E2-cofibrant, the rows of this diagram are E2-cofibre
sequences.
7.17. Proposition. [11 , 7.5] For any simplicial space X and integer
n 2, the -algebra ^fflnX is given by the functor
U 7! [_nU, X] .
7.18. Remark. The functor ^fflnX is representable by U 7! _nU for n 2,
and by U 7! ^ 0U for n = 0. It does not appear that ^ffl1X is repre-
sentable in a similar way.
Now we can prove 7.13. The terminal homomorphism ^ß1X ! ^ffl1X
is a ^ß0X-module map by 7.14; this proposition also handles the other
maps ^ßnX ! ^fflnX. Suppose n 2. According to [11 ], the homomor-
phism ^fflnX ! ^ßn-2X is induced (via 7.15) by the map fl in 7.16,
and the homomorphism ^ßn-2X ! ^ßn-1X is similarly induced by fi.
Now let F be one of the functors of U which appears in 7.16, or the
functor given by U 7! ^ n-2 U. Let C(F ) be the pointed simplicial
space F (S0); true, S0 is not an object of , but the construction of
F (S0) still makes sense. For each one of these functors F it is clear
that there are isomorphisms
F (U) ' C(F ) ^ U
where the object on the right is obtained by taking the simplicial space
C(F ) and smashing it in each degree with U. To each F there is
naturally associated a split diagram
S0 --! C(F )+ - ! C(F )
where C(F )+ is obtained by adding a disjoint basepoint in each degree
to C(F ). Smashing these diagrams with U 2 and mapping into X
produces the maps of split sequences (4.10) required to show that the
homomorphisms in question are maps of modules over ^ß0X (cf. 4.12).
32 BLANC, DWYER, AND GOERSS
8.Postnikov systems for simplicial spaces
In this section we set up a theory of Postnikov systems for simplicial
spaces, which is parallel to the Postnikov theories in x3 and x6. The
new ingredient is 8.15, which essentially gives a functorial relationship
between geometric k-invariants for simplicial spaces and algebraic k-
invariants for the associated simplicial -algebras.
Postnikov systems. Suppose that X is a simplicial space. Attaching a
cell (see 7.8) (Dn+2,k, Sn+1,k) of horizontal dimension (n + 2) to X via a
map f : Sn+1,k ! X has no effect on ^ßiX for i n, and clearly kills off
the class represented by f in ßn+1,kX. Now attach cells of horizontal
dimension (n + 2) and greater to X by all possible attaching maps
and perform a functorial fibrant replacement to obtain an inclusion
X X1, repeat the process to obtain X1 X2, repeat again, etc.,
and let P^nX = [jXj. (We use the notation P^nX to distinguish this
construction from PnX, which is the result of applying the topological
Postnikov construction Pn in each dimension to the simplicial space X.
The üf nctorial fibrant replacement" involves taking an object Z and
finding a functorial acyclic cofibration Z ! Z0 such that Z0 is fibrant;
it is necessary here because in the E2 model category not every object
is fibrant.) There is a map X ! ^PnX which induces isomorphisms on
^ßifor i n, and ^ßi^PnX is trivial for i > n. The construction of P^nX
is functorial in X, and there is a natural map P^nX ! P^n-1X which
respects the inclusions of X in these two simplicial spaces.
8.1. Eilenberg-Mac Lane objects. If is a -algebra, we say that a
simplicial space X is of type B if ^ß0X ' and ^ßiX is trivial for
i > 0. Suppose that M is a -module. We say that a map X ! Y is
of type B (M, n) n 1, if X is of type B , ^ß0Y ' , ^ßnY ' M (as
a -module), all other homotopy of Y is trivial, and the map X ! Y
gives an isomorphism on ^ß0. Sometimes we will say for short that the
target Y is of type B (M, n).
8.2. Remark. Recall that taking homotopy groups gives a functor ß*
from simplicial spaces to simplicial -algebras. Let f : X ! Y be a
map of type B (M, n). It turns out that ß*(f) is not in general a map
of type K (M, n). In fact, according to the spiral exact sequence, there
are isomorphisms
8 8
>< i = 0 >: >:
0 otherwise 0 otherwise
MODULI PROBLEM 33
The difference construction. Suppose that f : Y ! X is a map of
simplicial spaces. Consider the pushout C of the diagram X0 Y 0!
(P0X)0obtained by using some functorial construction to replace Y by
an E2-cofibrant space and the two maps Y ! X and Y ! P0X by
E2-cofibrations. There is a commutative diagram
Y - -~- Y 0 --- ! (P^0X)0
? ? ?
(8.3) f?y ?y ?y n(f)
X - -~- X0 --- ! ^Pn+1C
in which the vertical map on the right is denoted n(f). The source
(P^0X)0 of n(f) is sn(f), and the target P^n+1C is tn(f).
8.4. Proposition. Suppose that f : Y ! X is a map of simplicial -
algebras which is an isomorphism on ^ß0and whose homotopy fibre F
has ^ßiF trivial for i < n (n 1). Let M = ^ßnF . Then M is naturally
a -module for = ^ß0X and n(f) is a map of type B (M, n + 1). If
^ßiF vanishes except for i = n, then the right-hand square in 8.3 is a
homotopy fibre square.
Proof. This is very much along the lines of the proof of 6.3. Let
Fi ~ ß*F be the homotopy fibre of ß*Y 0 ! ß*X0. By the spiral
exact sequence, ßiFi = ^ffliF is trivial for i < n and isomorphic to M
for i = n. Diagram 8.3 gives a homotopy pushout diagram
ß*Y 0 --- ! ß*(P^0X)0
? ?
? ?
y y
ß*X0 --- ! ß*C
Let Fi0be the homotopy fibre of the right-hand map. The techniques in
the proof of 6.3, which involve using 5.1 to relate a homotopy pushout of
simplicial -algebras to the corresponding homotopy pushout of simpli-
cial sets, show that the map ßiFi ! ßiFi0is an isomorphism for i n.
Let F 0be the homotopy fibre of (P0X)0 ! C, so that Fi0= ß*F 0.
Again, the spiral exact sequence gives that ^ßiF 0is trivial for i < n
and isomorphic to M for i = n. A homotopy exact sequence argument
shows that n(f) is of type B (M, n + 1) for an appropriate action of
on M. It is straightforward to check the homotopy pullback condi-
tion.
8.5. Mapping into Eilenberg-Mac Lane objects. We wish to study spaces
of maps from simplicial spaces into Eilenberg-Mac Lane objects. Con-
sider an Eilenberg-Mac Lane map f : B ! B (M, n) with n > 1; we
can assume that the target is fibrant. It follows from 6.3 that if n > 1
34 BLANC, DWYER, AND GOERSS
then n-1(ß*f) is a map of type K (M, n) (note that the difference
construction here is taken in the category of simplicial -algebras).
Assigning to a diagram X- ~ U ! B (M, n) of simplicial spaces the
associated diagram ß*X- ~ ß*U ! tn(ß*f) ~ K (M, n) gives a nat-
ural map (cf. 2.7)
(8.6) n(X) : MfHom(X, B (M, n)) ! MfHom(ß*X, K (M, n)) .
8.7. Proposition. The map n(X) is a weak equivalence of simplicial
sets for all simplicial spaces X and all n 2.
8.8. Remark. By a slightly more elaborate construction, it is possible
to produce an equivalence for n = 1.
Proof of 8.7. It is enough to check the cases in which X is a sphere
Si,j. The reason for this is that the domain of n(X) is equivalent
to Map h(X, B (M, n)) and the range to Map h(ß*X, K (M, n)) (2.7,
2.5); since the functor ß* takes E2-homotopy pushouts to homotopy
pushouts of simplicial -algebras, it follows that the domain and range
of n(X) take homotopy pushouts (in X) to homotopy pullbacks. So
if n(X) is a weak equivalence for spheres, it is a weak equivalence
for any simplicial space Y which can be constructed from spheres by
a finite number of homotopy pushouts. To pass to arbitrary X, note
that any simplicial space X is up to weak equivalence a filtered colimit
of such Y , and that both the domain and range of n(X) take filtered
colimits in X to homotopy limits of simplicial sets.
So we restrict attention to the bigraded spheres. Each Si,jis a
cogroup object in the E2-homotopy category of simplicial spaces, while
ß*Si,jis a cogroup object in the category of simplicial -algebras. It is
easy to check that n(X) commutes up to homotopy with the induced
multiplications on the spaces involved. This means that in order to
prove that n(Si,j) is a weak equivalence it is enough to show that it
induces an isomorphism on ordinary homotopy groups, including ß0; it
is not necessary to check all possible basepoints.
By inspection, ß0 n(Sn,j) is an isomorphism; both domain and range
are isomorphic to Mj. This implies that n(Sn,j) is a weak equiva-
lence, since the higher homotopy groups of the domain (isomorphic to
ßn+k,jB (M, n)) and of the range (isomorphic to (ßn+k K (M, n))j) are
trivial. Since Si,jis the E2-suspension of Si-1,j, it follows as above that
n(Si,j) ~ n(Si-1,j). By induction and the fact that the domain and
range of n(Si,j) are connected for i > 0, i 6= n, it is easy to conclude
that n(Si,j) is a weak equivalence for i > 0, and that ßk n(S0,j) is an
isomorphism for k > 0. But ß0 n(S0,j) is a map j ! j, and it is
easy to see by inspection that this is the identity.
MODULI PROBLEM 35
8.9. Existence of Eilenberg-Mac Lane objects. The easiest way to do this
seems to be with generators and relations. To construct a simplicial
space of type B , start with the wedge W = _j 1 _x2 j S0,j; it is clear
that ^ß0W is the free -algebra on the underlying graded set of . Now
attach a one-cell for each relation in some presentation of , and apply
the functor P^0to obtain an object W 0of type B . Since ^ß0W 0' ,
there is a map ff : ß*W 0! K which is an isomorphism on ß0. To
construct a map of type B (M, n), n 1, start with W 0and add on the
wedge _i 1_x2Mi Sn,ito obtain Z, so that ß*Z is the coproduct of ß*W 0
with *i 1 *x2Mi cSn ~ß*Si. . There is a retraction Z ! W 0obtained
by mapping the wedge factors trivially; let F be the homotopy fibre.
Consider the diagram
ß*F --- ! ß*Z --- ! ß*W 0
? ? ?
fl?y fi?y ff?y
K0(M, n) --- ! K (M, n) --- ! K
in which both rows are fibre sequences; here fi is obtained by mapping
a factor cSn ~ß*Si of ß*Z indexed by x 2 Mi so as to represent the
element x 2 ßnK0(M, n) ' M. This gives an epimorphism
^ßnF -'!^fflnF ! M .
We now attach (n+1)-cells to Z to kill off the kernel of this epimorphism
and apply the functor P^n to obtain Z0. It is routine to check that
W 0! Z0 is of type B (M, n).
8.10. Uniqueness of Eilenberg-Mac Lane objects. Recall from above that
if f is of type B (M, n) then n-1(ß*f) is of type K (M, n).
8.11. Proposition. Let be a -algebra, M a -module, and n 1 an
integer. Let Mn denote the moduli space of all maps of type B (M, n).
Then the functor n-1(ß*) induces a weak equivalence
Mn ! M(K # K (M, n)) .
8.12. Remark. By 6.5, the moduli space on the right is equivalent to
~WAut ( , M). In particular, the moduli space is connected.
Proof. We first handle the case M = 0; it is easy to see that this
amounts to showing that the functor P0ß* induces a weak equivalence
from the moduli space of all objects of type B to M(K ). In view
of 2.2, it is enough to show that B is unique up to weak equiv-
alence, that Aut h(B ) is homotopically discrete, and that the map
ß0 Auth(B ) ! Aut ( ) obtained by recording the effect of a self-map
on ß0 is an isomorphism.
36 BLANC, DWYER, AND GOERSS
Suppose that X is a fibrant object of type B and let W be as in
8.9. By the construction of W it is possible to obtain a map W ! X
which is an isomorphism on ß0; this will induce equivalences W 0=
^P0W ! P^0X X. This shows that there is only one such X up to
weak equivalence. The same kind of argument shows that ß0 Auth(X)
maps surjectively to Aut ( ). Pick such an X which is fibrant and
cofibrant, and in particular constructed by cell attachment. Attaching
a cell (Di+1,j, Si,j) to an object Y to get Y 0gives a homotopy fibre
sequence
Map h(Y 0, X) ! Map h(Y, X) ! Map h(Si,j, X)
in which the base space is contractible for i > 0 and homotopically
discrete for i = 0 (its homotopy groups are ßi+*,jX). Moreover, the
map from [S0,j, X] to the set of -algebra maps ^ß0S0,j! ^ß0X is an
isomorphism. A formal inductive argument now shows that for any Y ,
the space Map h(Y, X) is homotopically discrete and the map [Y, X] !
Hom (^ß0Y, ^ß0X) is injective. The case Y = X of this is what we are
looking for.
Now we consider the case of a general M. For any simplicial model
category C, there is an induced simplicial model category structure on
the category of arrows in C, in which a morphism
A --u-! B
? ?
ff?y fi?y
C --v-! D
from u to v is a weak equivalence (resp. fibration) if ff and fi are
weak equivalences (resp. fibrations) in`C, and a cofibration if ff is a
cofibration in C and the natural map C A B ! D is a cofibration in
C. We use this when C is the E2 model category structure on simplicial
spaces in order to have an explicit way (2.2) to identify the moduli space
of a map. Let f be a map of type B (M, n). What we have to prove is
that f is unique up to weak equivalence, that Aut h(f) is homotopically
discrete, and that the natural map ff : ß0 Auth(f) ! Aut ( , M) is an
isomorphism. Uniqueness of f and surjectivity of ff are proved as above
using the explicit models from 8.9. Write f : B ! X. We can assume
that f is obtained by starting with the identity map B ! B and
attaching cells to the target of dimension n and higher. An inductive
argument, exactly the same as above, shows that if g : B ! Y is a
map obtained in this way, then Map h(g, f) is homotopically discrete,
and the natural map [g, f] ! Hom ( , )xHom (^ßnY, ^ßnX) is injective.
Applying this in the case Y = X finishes the proof.
MODULI PROBLEM 37
For convenience, we will denote Eilenberg-Mac Lane objects by B
and B (M, n).
Classification of Postnikov stages. Suppose that X is a simplicial space
with X ~ ^Pn-1X and that M is a module over ^ß0X. If Y is a simplicial
space, we write Y ~ X + (M, n) if P^nY ~ Y , P^n-1Y ~ X, and ^ßnY
is isomorphic to M as a module over ^ß0X, where the isomorphism
is realized with respect to some isomorphism ^ß0Y ' ^ß0X. We write
M(X + (M, n)) for the moduli space of all simplicial spaces of type
X + (M, n). The following result is proved in the same way as 6.8.
8.13. Theorem. Suppose that X is a simplicial space with X ~ ^Pn-1X,
n 1. Let = ^ß0X, and let M be a module over . Then there is an
natural weak equivalence
M(X + (M, n)) ~ M(X # B (M, n + 1) " B ) .
8.14. Remark. The arrows # on the right indicate maps which induce
isomorphisms on ^ßifor appropriate i (2.3); in this case it is just iso-
morphisms on ^ß0. Again, the remarks at the beginning of 3.9 could be
repeated here with some slight modifications.
The fundamental homotopy fibre square. The following theorem is at
the basis of our classification result.
8.15. Theorem. Suppose that X is a simplicial space, is a -algebra,
and M is a -module. Then for any n 2 there is a natural homotopy
fibre square
M(X _ B (M, n) " B ) -- - ! M(ß*X _ K (M, n) " K )
? ?
? ?
y y
M(X) -- - ! M(ß*X)
8.16. Remark. The moduli spaces on the left here involve simplicial
spaces, and the ones on the right simplicial -algebras. The vertical
arrows are induced by the obvious functors which take a diagram and
select the first component; the lower horizontal arrow is induced by the
functor ß*. The upper horizontal arrow is induced (as in 8.5) by the
f
functor which takes a diagram U ! V- W to the diagram
ß*U ! tn-1(ß*f) sn-1(ß*f) .
Proof of 8.15. Consider the commutative square
M(X _ B (M, n) " B ) --- ! M(ß*X _ K (M, n) " K )
? ?
? ?
y y
M(X) x M(B (M, n) " B ) --- ! M(ß*X) x M(K (M, n) " K )
38 BLANC, DWYER, AND GOERSS
in which the second factor of the lower horizontal arrow is induced
by the difference construction (8.5). The lower spaces are connected,
and by 2.11, 2.7, and 8.7 the induced map on vertical fibres is a weak
equivalence. Note in this connection that with the help of functorial
factorization it is easy to replace the upper left hand moduli space
by an equivalent moduli space of diagrams U ! V W in which
the simplicial space V equivalent to B (M, n) is fibrant. The proof is
finished by observing that the map
M(B (M, n) " B ) ! M(K (M, n) " K )
is a weak equivalence (8.11).
9. The main theorem
Recall that if A is a -algebra, the moduli space T M (A) of realiza-
tions of A is defined by
a
T M (A) = M(X) ,
where X ranges over weak equivalence classes of (pointed) topological
spaces with ß*X ' A. In this section we give the main structure
theorems for this moduli space.
9.1. Definition. Suppose that X is a simplicial space. We say that X is
a potential n-stage for the -algebra A if the following three conditions
are satisfied:
o ^ß0(X) is isomorphic to A as a -algebra,
o ^ßi(X) ' 0 for i > n, and
o ^ffli(X) ' 0 for 1 < i n + 1.
The partial moduli space or partial realization space T M n(A) is defined
to be the moduli space of all simplicial spaces which are potential n-
stages for A.
9.2. Remark. It follows from the spiral exact sequence that a potential
n-stage X for A has ^ßiX ' iA for 0 i n, ^ßiX = 0 for i > n,
^ffliX ' 0 for i 6= 0, n + 2, ^ffl0X ' A, and ^ffln+2X ' n+1A.
The above definition makes sense for n = 1 (the simplicial space
X involved would have ^ß0X ' A and ^ffliX ' 0 for i > 0). Our first
theorem says that the potential 1-stages for A are essentially the same
as realizations of A.
9.3. Theorem. The geometric realization functor induces a weak equiv-
alence T M 1(A) ! T M (A).
MODULI PROBLEM 39
Proof. Let F be the functor which assigns to a potential 1-stage Y for
A the geometric realization |Y c|, where Y cis some functorial cofibrant
approximation to Y ; by inspection of the homotopy spectral sequence
of a realization (7.9) [11 , 8.3], F (Y ) is a topological realization of A.
Let G be the functor which assigns to such a topological realization
X the constant simplicial space given by X; it is easy to see directly
that G(X) is a potential 1-stage for A. The two composites GF
and F G are connected to the respective identity functors by chains of
natural transformations which are weak equivalences, and so induce
weak equivalences of the moduli spaces.
It is easy to see from 7.11 that if X is a potential n-stage for A and
m < n, then the horizontal Postnikov section P^mX is a potential m-
stage for A, In particular the functor ^Pn-1induces a map T M n(A) !
T M n-1 (A). Our next theorem gives an expression for T M 1 (A) in
terms of these maps. Let holim R denote the derived homotopy limit
functor for diagrams of simplicial sets; this is the functor obtained by
replacing the diagram in some functorial way by a diagram of fibrant
simplicial sets, and applying the ordinary homotopy limit functor of
[3].
9.4. Theorem. There is a natural weak equivalence of simplicial sets
T M 1 (A) ~ holimRnT M n(A) .
Proof. This follows from [7]; the main result there is stated for simplicial
sets, but the arguments apply to any cofibrantly generated simplicial
model category with arbitrary small limits and colimits. The main
result of [7] is applied in exactly the same as in [7, 4.6].
This reduces the study of T M 1(A) to the study of the individual
spaces T M n(A), together with the maps between them. We begin
with T M 0(A). The following is clear from 6.5, since T M 0(A) is the
moduli space of all simplicial spaces of type BA .
9.5. Theorem. The space T M 0(A) is naturally weakly equivalent to
BAut (A).
In this statement, Aut (A) denotes the discrete group of -algebra
automorphisms of A; in particular, the theorem states that T M 0(A)
is an Eilenberg-Mac Lane space of type K(ß, 1) for ß = Aut (A).
The next theorem analyzes the difference between T M n(A) and
T M n-1 (A).
40 BLANC, DWYER, AND GOERSS
9.6. Theorem. Suppose that n 1. Then there is a natural homotopy
fibre square
T M n(A) --- ! M(A # KA ( nA, n + 2))
? ?
P^n-1?y ?y
T M n-1(A) --- ! M(A # KA ( nA, n + 2) Ä )
The vertical map on the right is induced by the functor which takes
a map U ! V and repeats it to obtain U ! V U. The other two
maps in the square are constructed below.
9.7. Interpretation. According to 2.11 and 6.5, the space Z = M(A #
KA ( nA, n+2) Ä ) fibres over ~WAut (A)xW~ Aut (A, nA) with fibre
a
(9.8) Hn+1A(A; nA) ,
f
where the coproduct is taken over the set of all isomorphisms A !
ß0KA ( nA, n + 2). It is clear that Aut (A) acts simply transitively on
this set, and it follows that Z fibres over W~ Aut (A, nA) with fibre
Hn+1A(A; nA). In this way each potential (n - 1)-stage Y for A, i.e.,
each vertex of T M n-1(A), determines an element oY in Hn+2A(A; nA)
modulo the action of Aut (A, nA). This element (which can be iden-
tified with the k-invariant (6.8) of the simplicial -algebra ß*Y ) is the
obstruction to lifting Y to a potential n-stage. Let T M n(A)Y denote
the moduli space of all potential n-stages X for A with P^n-1X ~ Y .
If oY is nontrivial, then T M n(A)Y is empty, otherwise (given that
Hn+1A(A; nA) ~ HnA(A; nA)), there is a fibration sequence
HnA(A; nA) ! T M n(A)Y ! M(Y ) .
On the level of ß0 this can be interpreted as saying that weak equiv-
alence classes of lifts of Y to a potential n-stage for A correspond to
trivializations of oY ; of course the sequence also indicates how the space
of such trivializations contributes to the spaces of self-equivalences of
these lifts.
9.9. Potential n-stages. Suppose that Y is a potential n-stage for A; we
can assume that Y is cofibrant as a simplicial space. According to 9.2,
the homotopy spectral sequence for ß*|Y | (7.9) has only two nontrivial
columns at the E2-page: ^ffl0Y ' A in column E20,*and ^ffln+2Y ' n+1A
in column E2n+2,*. It follows from the description of the spectral se-
quence in [11 , 8.3] that the differential dn+2 maps column n+2 as much
as possible isomorphically to column 0. Consequently, ßi|Y | is trivial
for i n + 2, and ßi|Y | ' Ai for i n + 1. But more is true. Let P mY
be the simplicial space obtained by applying the (m - 1)-connective
MODULI PROBLEM 41
cover functor degreewise to Y . The spectral sequence of P mY can be
computed by a naturality argument, and it follows that ßi|P mY | is
trivial for i n + m + 1 or for i < m, and that ßi|P mY | ' Ai for
the remaining values of i. In particular, the algebraic constituents of A
are knitted together by Y in a way which is much more comprehensive
than is reflected by the single ordinary Postnikov stage |Y |.
The rest of this section is taken up with the proof of 9.6.
The first step is to analyze the difference between potential n-stages
for A and potential (n-1)-stages. Suppose that X is a potential n-stage
for A. According to 9.1 and the spiral exact sequence, ^ßnX ' nA. Let
Y = ^Pn-1X. Then Y is a potential (n - 1)-stage for A, and according
to 8.3, after adjusting X and Y up to weak equivalence there is a
homotopy pullback square
X --- ! BA
? ?
? ?
(9.10) uy vy
f n
Y --- ! BA ( A, n + 1)
in which the maps f and v give isomorphisms on ^ß0. We now determine
how to reverse this construction.
9.11. Proposition. Suppose that Y is a potential (n - 1)-stage for
A (n 1) and that X lies in a homotopy fibre square of the form
9.10. Then X is a potential n-stage for A if and only if the map g :
ß*Y ! KA ( nA, n + 1) corresponding (8.6) to f is a weak equivalence
of simplicial -algebras.
Proof. The main thing to prove in showing that X is a potential n-
stage for A is that ^ffliX vanishes for i = n, n + 1; the other conditions
are simple to check. The homotopy fibre F of v is of type B0( nA, n).
Consequently, ^ffliF vanishes unless i is n or n + 2, and the long exact
^ffl*-homotopy sequence of u (7.7) degenerates around dimension n into
the exact sequence
0 ! ^ffln+1X ! ^ffln+1Y ! ^fflnF ! ^fflnX ! 0 .
Thus X is a potential n-stage if and only if the connecting homomor-
phism ^ffln+1Y ! ^fflnF ' nA is an isomorphism. A naturality argument
identifies this connecting homomorphism with the map ßn+1ß*Y !
nA induced by g. Since ß0(g) is an isomorphism by assumption, and
both domain and range of g have trivial homotopy except in dimensions
0 and n + 1, the result follows.
42 BLANC, DWYER, AND GOERSS
Suppose that Y is a potential (n - 1)-stage for A. We write X ~
Y ( nA, n) if X is a potential n-stage for Y and P^n-1X ~ Y . The
space M(Y ( nA, n)) is the moduli space of all such X.
9.12. Proposition. Suppose that Y is a potential (n - 1)-stage for A
(n 1). Then there is a natural homotopy fibre square
M(Y ( nA, n)) --- ! M(ß*Y # KA ( nA, n + 1) " KA )
? ?
^Pn-1?y ?y .
M(Y ) --i*-! M(ß*Y )
9.13. Remark. As usual, # signifies maps which induce isomorphisms
on appropriate homotopy groups; in the case ß*Y # KA ( nA, n + 1)
these isomorphisms are such that the map is an equivalence. The right
vertical arrow in the square is induced by the functor which takes a
diagram U ! V W of simplicial -algebras and selects the first
component. As would be revealed by unraveling the proof, the up-
per horizontal arrow is induced by two applications of the difference
construction, one in the category of simplicial spaces (8.4) to obtain
Y ! BA ( nA, n + 1), and the second in the category of simplicial
-algebras (8.5) to obtain ß*Y ! KA ( nA, n + 1).
Proof of 9.12. We let M = nA and m = n + 1. There is a square
M(Y (M, n)) --- ! M(Y -! BA (M, m) " BA )
? ?
P^n-1?y ?y
M(Y ) --=-! M(Y )
whose upper arrow is a weak equivalence obtained by using 9.11 to se-
lect appropriate components of the weak equivalence from 8.13. Here
-! denotes maps which correspond via 8.5 to weak equivalences ß
*Y !
KA (M, m). Passing to appropriate components with 8.15 gives a ho-
motopy fibre square
M(Y -! BA (M, m) " BA ) --- ! M(ß*Y # KA (M, m) " KA )
? ?
? ?
y y .
M(Y ) --i*-! M(ß*Y )
Combining these squares finishes the proof.
MODULI PROBLEM 43
Proof of 9.6. For any -algebra , -module M, and m 1 there is a
commutative diagram
(9.14)
M(K (M, m) " K ) -- ~-! M(K (M, m + 1) " K )
? ?
? ?
y y
M(K + (M, m)) -- ~-! M(K # K (M, m + 1) " K )
in which the horizontal arrows are equivalences obtained with the dif-
ference construction; see the proof of 6.5 for the upper arrow and 6.8
for the lower one. Clearly, this is a homotopy fibre square. Suppose
that Y is a potential (n - 1)-stage for A. Let = A, M = nA, and
m = n + 1. Then M(ß*Y ) is one component of M(K + (M, m)).
Moreover, the map M(K (M, n)) ! M(ß*Y # KA ( nA, n + 1)) ob-
tained by sending a map U V to U -=!U V is a weak equivalence
(a homotopy inverse is given by the functor sending U ! V W to
V W ). Combining this observation with 9.12 and 9.14 then gives a
homotopy fibre square
` n
M(Y ?( A, n)) --- ! M(K (M, m + 1)?" K )
? ?
y y
`
M(Y ) --- ! M(K # K (M, m + 1) " K )
which is the one we are looking for, since the left vertical arrow is
T M n (A) ! T M n-1(A).
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Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel
Dept. of Mathematics, University of Notre Dame, Notre Dame, IN
46556 USA
Dept. of Mathematics, Northwestern University, 2033 Sheridan
Road Evanston, IL 60208 USA
E-mail address: blanc@math.haifa.ac.il
E-mail address: dwyer.1@nd.edu
E-mail address: pgoerss@math.nwu.edu