Moduli Problems for Structured Ring Spectra
P. G. Goerss and M. J. Hopkins1
1The authors were partially supported by the National Science Foundation (USA).
In this document we make good on all the assertions we made in the previous
paper "Moduli spaces of commutative ring spectra" [20] wherein we laid out a
theory a moduli spaces and problems for the existence and uniqueness of E1 
ring spectra. In that paper, we discussed the the HopkinsMiller theorem on
the LubinTate or Morava spectra En; in particular, we showed how to prove
that the moduli space of all E1 ring spectra X so that (En)*X ~= (En)*En
as commutative (En)* algebras had the homotopy type of BG, where G was
an appropriate variant of the Morava stabilizer group. This is but one point
of view on these results, and the reader should also consult [3], [38], and [41*
*],
among others.
A point worth reiterating is that the moduli problems here begin with al
gebra: we have a homology theory E* and a commutative ring A in E*E co
modules and we wish to discuss the homotopy type of the space T M (A) of all
E1 ring spectra so that E*X ~=A. We do not, a priori, assume that T M (A) is
nonempty, or even that there is a spectrum X so that E*X ~=A as comodules.
For a variety of applications we are not simply interested in this absolute
problem, but in a relative version as well. We fix an E1 ring spectrum Y and
write k = E*Y for the resulting commutative algebra in E*E comodules. Then
we may choose a morphism of commutative algebras k ! A in E*Ecomodules
and write T M (A=k) for the moduli space of Y algebras X so that E*X ~=A as
a kalgebra. The absolute case can be recovered by setting Y = S0, the zero
sphere. While we are assuming the existence of Y , we are not assuming that
T M (A=k) is nonempty or even that there exists a spectrum X with E*X ~=A.
The main results are Theorems 3.3.2, 3.3.3, and 3.3.5 which together give a
decomposition of T M (A=k) as the homotopy inverse limit of a tower of fibra
tions
. .!.T M n(A=k) ! T M n1(A=k) ! . .!.T M 1(A=k)
where
1.T M 1(A=k) is weakly equivalent to B Autk(A) where Autk(A) is the group
of automorphisms of the kalgebra A in E*Ecomodules; in particular,
T M 1(A) is is nonempty and connected;
2.for all n > 1, there is a homotopy pullback square
T M n(A=k) ______//_B Autk(A, nA)
 
 
fflffl fflffl
T M n1(A=k) _____//^Hn+2A(A=k, nA).
This last diagram needs a bit of explanation. As a graded abelian group
[ nA]k = An+k; this is a module over A in the category of E*Ecomodules.
The group Autk(A, n) is the automorphism group of the pair. If M is an
Amodule and n a nonnegative integer, there is an Andr'eQuillen cohomology
space so that
ssiHn(A=k, M) = Hni(A=k, M)
1
where H*(, ) denotes an appropriate Andr'eQuillen cohomology functor.
The group Autk(A, M) acts on Hn(A=k, M) and ^Hn(A=k, M) is the Borel con
struction of this action. Note that the fiber of T M n(A=k) ! T M n1(A=k) at
any basepoint will either be empty or will be homotopy equivalent to the space
Hn+1(A=k, nA).
What is notable about this decomposition is that the spaces B Autk(A, n)
and ^Hn+2A(A=k, nA) are determined completely by algebraic data.
By trying to lift the vertex of T M 1(A=k) up the tower, one gets an obstruc
tion theory for realizing A. The obstructions to both existence and uniqueness
lie in Andr'eQuillen cohomology groups. See Remark 3.3.7. This is surely the
same obstruction theory as in [41], although we haven't checked this.
This paper is very long  even though we consigned the applications to [20] *
*or
to an asyetnonexistent paper on elliptic cohomology and topological modular
forms. Some of this length is probably gratuitous, as we have repeated a lot of
material available elsewhere, notably [7], [10], [17], and [20]. It was tempti*
*ng
to simply point to results in all of these papers, but in the end there were too
many small details that needed reworking and, perhaps worse, the result had
all the narrative flow of a spreadsheet.
Here are some highlights of what is accomplished here. The main idea,
which goes back to Dwyer, Kan, and Stover, is to try to construct a simplicial
E1 algebra whose geometric realization will realize A. Then we use the new
simplicial direction and apply Postnikov tower techniques to get the decompo
sition of the moduli space. Making this work requires an enormous of amount
of technical detail. Specifically:
1.The resolution model category structures of [16] and [10] must be reworked
to accommodate resolving the E1 operad as well. This is necessary, in
some cases, to obtain computational control over free objects  for an ar
bitrary homology theory E*, the homology of a free E1 ring spectrum
may be hard to compute. Even more, we are not really interested in the
resolution model category itself, but a localization of it at some homol
ogy theory E*. While localization theory is highly developed [23], the
hypotheses remain fairly rigid, and this leads us into a discussion of the
pointset topology of structured ring spectra. In addition, the standard
localization theorems don't apply directly  although the techniques do.
All of this is accomplished in the first chapter.
2.The second chapter is a grabbag of essentially algebraic results. For
example, we need to have a description of comodules as diagrams in order
to prove the important Corollary 3.1.18 which allows us to identify the
module structure on nA in our Andr'eQuillen cohomology. We need a
theory of Postnikov towers for simplicial algebras in E*Ecomodules, and
for that we need a BlakersMassey excision theorem, and so on. We also
have to be a bit careful about what Andr'eQuillen cohomology actually is.
And, along the way, we discuss a spectral sequence for computing mapping
spaces.
2
3.If these results ever do get used to discuss topological modular forms, we
will need a version suitable for use when E* is pcompleted Ktheory. This
was not discussed in [20] and takes some pages to set up as well.
4.The third chapter, which is where all the theorems are, is the shortest,
and really is a recapitulation of the program set out in [7]. But, again,
there are details to be spelled out. Some of these involve the passage
to E*localization and its effect on the spiral exact sequence; another of
these is to spell out exactly what it needed for the relative case; only t*
*he
absolute case is in the literature.
Throughout this manuscript, we are working with simplicial algebras is spec
tra over a simplicial operad T . If E* is a homology theory based on a homotopy
commutative ring spectrum E so that E*E is flat over E*, then we have a theory
of E*E modules. If X is a simplicial T algebra, then E*X is an E*T algebra
in category of simplicial E*Ecomodules. One of the central difficulties we had
to confront was to find some condition on T and E*T so that we could control,
at once, the homotopical algebra of T algebras in simplicial spectra and E*T
algebras in E*Ecomodules. The condition we arrived at  that of homotopi
cally adapted operad (See Definition 1.4.16.)  is somewhat cumbersome, but it
is satisfied in all the applications we have in mind.
Many thanks to Matt Ando for carefully reading this manuscript, and many
thanks to all readers for so patiently waiting through the long gestation period
of these results.
3
Contents
1 Homotopy Theory and Spectra 5
1.1 Mapping spaces and moduli spaces . . . . . . . . . . . . . . . . . 5
1.1.1 Model category basics . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 The ground category: basics on spectra . . . . . . . . . . . . . . 13
1.3 Simplicial spectra over simplicial operads . . . . . . . . . . . . . *
*19
1.4 Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *
*. 24
1.5 Localization of the resolution model category . . . . . . . . . . . 32
2 The Algebra of Comodules 40
2.1 Comodules, algebras, and modules as diagrams . . . . . . . . . . 40
2.1.1 Comodules as productpreserving diagrams . . . . . . . . 41
2.1.2 Algebras as diagrams . . . . . . . . . . . . . . . . . . . . . 47
2.1.3 Modules as diagrams . . . . . . . . . . . . . . . . . . . . . 50
2.2 Thetaalgebras and the padic Ktheory of E1 ring spectra . . . 52
2.3 Homotopy pushouts of simplicial algebras . . . . . . . . . . . . . 59
2.4 Andr'eQuillen cohomology . . . . . . . . . . . . . . . . . . . . . . 67
2.4.1 Cohomology of algebras over operads . . . . . . . . . . . . 67
2.4.2 Cohomology of algebras in comodules . . . . . . . . . . . 72
2.4.3 The cohomology of thetaalgebras . . . . . . . . . . . . . 74
2.4.4 Computing mapping spaces  the K(1)local case . . . . . 76
2.5 Postnikov systems for simplicial algebras . . . . . . . . . . . . . . 79
3 Decompositions of Moduli Spaces 92
3.1 The spiral exact sequence . . . . . . . . . . . . . . . . . . . . . . *
*92
3.1.1 Natural homotopy groups and the exact sequence . . . . . 92
3.1.2 The module structure . . . . . . . . . . . . . . . . . . . . 98
3.2 Postnikov systems for simplicial algebras in spectra . . . . . . . .103
3.3 The decomposition of the moduli spaces . . . . . . . . . . . . . .115
4
Chapter 1
Homotopy Theory and
Spectra
1.1 Mapping spaces and moduli spaces
1.1.1 Model category basics
We will assume that the reader is familiar with basics of model categories,
cofibrantly generated model categories, and simplicial model categories. These
are adequately and thoroughly presented in many references, including [25] and
[23]. All our model categories will be, at the very least, cofibrantly generate*
*d.
This implies, in particular, that given any morphism f : X ! Y in our model
category, there are natural factorizations
j q
X _____//Z____//_X
of f where j is a cofibration and a weak equivalence and q is a fibration; ther*
*e is
also a natural factorization with j a cofibration and q and fibration and a weak
equivalence.
Less familiar, perhaps, is the notion of a cellular model category, which we
now review. The importance of this notion is that cellular model categories are
particularly amenable to localization, and this makes for a very clean theory f*
*or
us. Here are the definitions, all from [23].
1.1.1 Definition. Fix a category C with all limits and colimits. If I = {A ! B}
is some chosen set of maps in C, a presentation of a relative Icell complex
f : X ! Y consists of an ordinal number ~ = ~f and a colimit preserving
functor Y(): ~ ! C so that
1.Y0 = X;
2.for each fi there is a set of maps TfXi= {fi : A ! Yfi} with A the source
5
of a morphism in I and a pushout diagram
tTfiA__tfi_//_Yfi
 
 
fflffl fflffl
tTfiB_____//Yfi+1;
3.an isomorphism from X ! colimfi<~Yfito f : X ! Y .
The size of f : X ! Y is the cardinality of the set of cells qTfi. If X is t*
*he
initial object of C, then Y is a presented Icell complex.
We are particularly interested in the case when I generates the cofibrations.
1.1.2 Definition. A subcomplex of a presented relative Icell complex X ! Y ,
consists of a presented Icell complex X ! K so that ~K = ~Y and a natural
transformation K()! Y(): ~ ! C so that for all fi < ~, the induced map
TfKi!TfYi
is an injection and so that the induced map of pushout squares commutes. If
X is the initial object, we may write K Y .
1.1.3 Definition. Let C be a category with all colimits, W an object of C, and
I a class of morphisms in C.
1.The object W of C is small relative to I if there is a cardinal number ~
so that for every regular cardinal ~ ~ and every ~ sequence
Z0 _____//Z1____//._._._//Zff___//_. . .
of morphisms in I, the natural map
colimff<~Hom C(W, Zff) ! Hom C(W, colimff<~Zff)
is an isomorphism.
2.The object W is compact relative to I if there is a cardinal fl so that for
every presented relative Icomplex X ! Y every map from W to Y factors
through a subcomplex of size at most fl.
Recall that in any category, an effective monomorphism is a morphism which
can be written as the equalizer of a pair of parallel arrows.
1.1.4 Definition. A cellular model category C is a cofibrantly generated model
category for which there is a set of I of generating cofibrations and set J of
generating acyclic cofibrations so that
1.the domains and codomains of the elements of I are compact relative to I;
6
2.the domains of the elements of J are small relative to I; and
3.the cofibrations are effective monomorphisms.
1.1.5 Remark. Almost all of our model categories will be, in some way, based
on topological spaces  or, more exactly, compactly generated weak Hausdorff
spaces. In this case, if a morphism is a cofibration then it will be a Hurewicz
cofibration and, hence, a closed inclusion and an effective monomorphism. Fur
thermore, the domains of the generating sets I and J of cofibrations and acyclic
cofibrations will be cofibrant. Finally, if A is the domain of an object in I or
J, it will have a stronger compactness property than that required by Defini
tion 1.1.3: the functor Hom C(A, ) will commute with all filtered colimits over
diagrams of closed inclusions. Thus, many of the conditions of Definition 1.1.4
will be nearly automatic.
We next come to a slight variation on model categories. When considering
categories of simplicial algebras in spectra, we will want to stipulate that the
weak equivalences be those morphisms X ! Y so that after applying some
homology theory E*, the resulting morphism E*X ! E*Y becomes a weak
equivalence of simplicial E*modules. This won't quite be a model category
structure, for reasons which are by now familiar: pushouts along all cofibrati*
*ons
do not necessary preserve these E*equivalences  one has to assume that the
cofibration has cofibrant source. This situation arose also in [18] and [45]. T*
*he
latter source supplies an axiomatic framework (there credited to Mark Hovey,
see [26]) for coming to terms with this phenomenon. Here is the definition. We
highlight where the usual notion of a model category is weakened.
1.1.6 Definition. Let C be category with specified classes of weak equivalences,
fibrations, and cofibrations. Then C is a semimodel category provided the fol
lowing axioms hold:
1.The category C has all limits and colimits;
2.Weak equivalences, cofibrations, and fibrations are all closed under re
tracts; fibrations and acyclic fibrations are closed under pull back;
3.If f and g are composable morphisms and two of f, g, and gf are weak
equivalences, so is the third;
4.All cofibrations have the left lifting property with respect to acyclic fi*
*bra
tions, and all acyclic cofibrations with cofibrant source have the left
lifting property with respect to all fibrations.
5.Every morphism can be functorially factored as as a cofibration followed
by an acyclic fibration and every morphism with cofibrant source can be
functorially factored as an acyclic cofibration followed by a fibration.
Note that this should really be called a left semimodel category, as the
definition singles out cofibrations. But this is only kind of semimodel catego*
*ry
which will arise in this paper.
7
The various auxiliary notions of model category also can be similarly modi
fied. For example, we have the following.
1.1.7 Definition. A semimodel category C is cofibrantly generated if there are
sets of morphisms I and J which detect, respectively the acyclic fibrations and
the fibrations. Furthermore, the domains of the morphism in I should be small
relative to relative Icell morphisms and the domains of J should be small with
respect to relative Jcell morphisms with cofibrant source.
Here "detect" means, for example, that a morphism is an acyclic fibration if
and only if it has the right lifting property with respect to the morphisms in *
*I.
Or again, the following:
1.1.8 Definition. A semimodel category C is a simplicial semimodel category
if it simplicial in the sense of [35] xII.2, and if the following corner axiom *
*holds.
Let
map(, ) : Copx C ! sSets
denote the simplicial mapping space functor. Then if j : A ! B is a cofibration
with cofibrant source and q : X ! Y is a fibration, then
map(B, X)! map(B, Y ) xmap(A,Y )map(A, X)
is a fibration of simplicial sets which is a weak equivalence if either f is a *
*weak
equivalence or j is a weak equivalence.
This gives a working model for mapping spaces in a semimodel category;
namely, the simplicial set of maps map (X, Y ) where X is cofibrant and Y is
fibrant.
We append here a final definition, mostly because we have no other place
to put it. Let I be a small category, C any category with colimits and CI the
category of Idiagrams in C. Let Iffibe the category with same objects as I but
only identity morphisms; thus, Iffiis I made discrete. An Idiagram X : I ! C is
Ifree (or simply free) if it is the left Kan extension of some diagram X0 : If*
*fi! C.
1.1.9 Definition. Let be the ordinal number category and + the
category with same objects but only surjective morphisms. Let C be a category
and X : op! C a simplicial object. Then X is sfree if the underlying diagram
X : op+!C
is free.
The restricted diagram X : op+! C is the underlying degeneracy diagram,
and to be sfree is to say that there are objects Zk so that there are isomorph*
*isms
a
Xn = Zk
OE:n!k
where OE runs over the surjections in . Furthermore, these isomorphisms should
commute with the degeneracies. In many model categories of simplicial objects,
the cofibrant objects are retracts of sfree objects. See [35]xII.4.
8
1.1.2 Moduli spaces
We now recall some of the basic facts about DwyerKan classification spaces,
mapping spaces, and moduli spaces. In all cases, these spaces will be the nerve
(or classifying space) of some category. The subtlety in this construction will
be that often the category C to which we wish to apply the nerve functor is not
small and, therefore, we don't immediately get a simplicial set. However, there
are at least three ways to deal with this problem. The first is to notice that
the we will obtain homotopically small nerves, which determine a welldefined
homotopy type. For this, see [14]. The second is to restrict, in each case, to
a small subcategory of the category in question which is still large enough to
capture enough information to determine the correct homotopy type. In both
cases, the constructions are routine, so we employ the third solution: we ignore
the problem in order to simplify exposition.
To begin the theory, we need only consider some category C with a specified
class of weak equivalences. Later on, in order to make calculations, we will ne*
*ed
a model category or perhaps, only a semimodel category.
If C is a category with weak equivalences, the DwyerKan hammock localiza
tion LH C(X, Y ) yields a model for the space of morphisms between two objects
X and Y of C. See [13]. The following result implies that the hammock local
ization is a good model for the derived space of maps between two objects.
1.1.10 Proposition. 1.) Suppose X0 ! X and Y ! Y 0are weak equivalences
in C. Then
LH C(X, Y ) ! LH C(X0, Y 0)
is a weak equivalence.
2.) Let C be a simplicial semimodel category, and denote by
map(, ) : Copx C ! sSets
the mapping space functor. Then if X is cofibrant and Y is fibrant there is a
zigzag of weak equivalences between map (X, Y ) and LH C(X, Y ).
Proof. The first property is Proposition 3.3 of [13]. For the second statement,
we note that the argument in x7 of [15] easily adapts to the more general_semi
model category. __
For fixed X, the components ss0LH C(X, X) of LH C(X, X) form a monoid,
and we define the derived simplicial monoid of selfequivalences
(1.1.1) Aut C(X) LH C(X, X)
of X by taking those components which are invertible. We note that if X in some
semimodel category is cofibrant and fibrant, then the previous result implies
that AutC(X) is weakly equivalent to the components of map (X, X) which are
invertible.
1.1.11 Definition. Let C be a semimodel category. A category of weak equiv
alences in C is a subcategory of E of C which has the twin properties that
9
1.)if X is an object in E and Y is weakly equivalent to X, then Y 2 E;
2.)the morphisms in E are weak equivalences and if f : X ! Y is a weak
equivalence in C between objects of E, then f 2 E.
For example, E might have the same objects as C and all weak equivalences.
Let BE denote the nerve of the category E; this is the DwyerKan classifica
tion spaces, and we will refer to it as a moduli space. In fact, there is a for*
*mula
for this weak homotopy type: the following is from [14].
1.1.12 Proposition. Let E be a category of weak equivalences in some semi
model category C. Then a
BE ' BAut C(X)
[X]
where [X] runs over the weak homotopy types in E and AutC(X) is the (derived)
monoid of selfweak equivalences of X.
Proof. See x2 of [14]. The proof goes through verbatim in the more general
context. Since one of the needed references for this argument can be hard to *
* __
obtain, we will also offer an outline of the proof below in 1.1.18. *
*__
1.1.13 Example (The moduli space of an object). Fix an object X of
some semimodel category C and let E(X) be the smallest category of weak
equivalences containing X. Then E(X) has as objects all Y which are weakly
equivalent to X and as morphisms all weak equivalences Y ! Y 0. We will write
M(X) for BE(X). Then
M(X) ' B AutC(X).
1.1.14 Example (Moduli spaces for diagrams). If C is a semimodel cate
gory and I is some small indexing category, let CI be the category of Idiagrams
in C. Under many conditions, CI has a semimodel category structure with
X ! Y a weak equivalence if Xi! Yi is a weak equivalence for all i. (See [23],
among many references.) But in any case, this always yields a notion of weak
equivalence and we can talk about categories E of weak equivalences as above.
For example, let I be the category with two objects and one nonidentity arrow;
then CI is the category of arrows in C. Then we may let M(X_ Y ) denote the
classifying space of the category with objects all arrows U ! V with U weakly
equivalent to X and Y weakly equivalent to Y . This is not quite the moduli
space of arrows X ! Y ; see the next example, and Proposition 1.1.17.
1.1.15 Example (Mapping spaces as moduli spaces). Let X and Y be
two objects in a semimodel category C. We can define a space of morphisms
between X and Y as a moduli space. It is the nerve of the category E(X, Y )
whose objects are diagrams
X o'o__U _____//Vo'o__Y
10
where U ! X and Y ! V are weak equivalences. Morphisms are commutative
diagrams of the form
X oo'__ U _____//_Vo'o_Y
= ' ' =
fflffl'fflffl fflffl'fflffl
X oo___U0 _____//Vo0o__Y
in which the indicated maps are weak equivalences. Let MHom (X, Y ) denote
the moduli space of E(X, Y ). A theorem of Dwyer and Kan [13] implies that if
C is a model category, there is a natural weak equivalence
MHom (X, Y )! LH C(X, Y ).
Thus, in a simplicial model category, MHom (X, Y ) is weakly equivalent to the
derived mapping space.
1.1.16 Example (Mapping spaces in semimodel categories). Now sup
pose that C is only a semimodel category. Then the argument that the inclusion
MHom (X, Y ) ! LH C(X, Y ) is a weak equivalence will not work for all X and
Y , for at some point (see Proposition 8.2 of [13]) one must take the pushout
along an acyclic cofibration and claim it is a weak equivalence. This defect can
be remedied as follows.
First, let Cc C be the full subcategory of cofibrant objects, with the in
herited class of weak equivalences. Furthermore, if X and Y are cofibrant, let
McHom(X, Y ) be the nerve of the category of diagrams
X o'o__U _____//Vo'o__Y
where U and V are cofibrant. Then the argument cited above does show that
McHom(X, Y )! LH Cc(X, Y )
is a weak equivalence when C is a semimodel category.
Second, if X and Y are cofibrant, then functorial factorizations make it easy
to show that the inclusion
McHom(X, Y ) ! MHom (X, Y )
is a weak equivalence. Since LH Cc(X, Y ) ! LH C(X, Y ) is a weak equivalence,
by the analog of [13] 8.4, we obtain that
MHom (X, Y )! LH C(X, Y )
is a weak equivalence for X and Y cofibrant in a semimodel category C.
The relationship between the various mapping objects thus far defined is
spelled out in the following result. The proof here is a paradigm for many
similar results, and we will often refer to it in later parts of the paper.
11
1.1.17 Proposition. Suppose that X and Y are two objects in a model category
C. Then there is a homotopy fiber sequence
MHom (X, Y ) ! M(X_ Y ) ! M(X) x M(Y ).
If C is only a semimodel category, we must also assume that X and Y are
cofibrant.
Proof. This is an application of Quillen's Theorem B (see [21]), which specifies
the homotopy fiber of the morphism on nerves BF : BC ! BD induced by a
functor F : C ! D between small categories. For X 2 D, let X=F denote the
category with objects the arrows X ! F Y in D, with Y 2 C; the arrows in
X=F will be triangles induced by morphisms Y ! Y 0. If X0 ! X is a morphism
in D, we get a functor X=F ! X0=F by precomposition, and Theorem B says
that
B(X=F ) ! BC ! BD
is a fiber sequence if B(X=F ) ! B(X0=F ) is a weak equivalence of all X0 ! X.
The result now follows. The maps are the obvious ones: the morphism
MHom (X, Y ) ! M(X_ Y ) is induced by the functor that sends X U !
V Y to U ! V ; the morphism M(X_ Y ) ! M(X) x M(Y ) sends U ! V
to (U, V ). One easily checks the conditions of Theorem B, using Example_1.1.15_
or Example 1.1.16 as necessary. __
1.1.18 Example (A proof of Proposition 1.1.12). If we let MAut(X) be
the moduli space of diagrams
X o'o__U __'__//Vo'o_ X
and M(Xwe_X) the moduli space of morphisms
U __'__//V
where U and V are both weakly equivalent to X, then the kind of argument
just given provides a fiber sequence
MAut(X) _____//M(Xwe_X) __q_//_M(X) x M(X).
However, there is weak equivalence M(X) ! M(Xwe_X) sending U to 1 : U !
U, and the morphism q becomes equivalent to the diagonal. Then Proposition
1.1.12 follows once we identify MAut(X) with Aut(X). For this see [13] 6.3.
1.1.19 Example (Moduli spaces in the presence of homotopy groups).
Suppose that the semimodel category C has some specified notion of homotopy
groups ssi, i 0. Then we let M(X# Y ) denote the moduli space of arrows
f : U ! V , where
1.U is weakly equivalent to X and V is weakly equivalent to Y , and
12
2.the morphism f induces an isomorphism on ssifor all i such that ssiX and
ssiY are both nontrivial.
Note that M(X# Y ) is a (possibly empty) disjoint union of components of
M(X_ Y ), as defined in Example 1.1.14.
This kind of moduli space will be mostly used when we have a pair X and
Y where ssiY is isomorphic  but not canonically isomorphic  to ssiX whenever
ssiY is nonzero.
There are many variants on this sort of example. For example, given three
spaces, we can form M(X# Y " Z).
In a semimodel category, we will always assume we have cofibrant objects.
1.2 The ground category: basics on spectra
The whole point of this document is to produce a theory of moduli spaces of
structured ring spectra; in particular, we wish to discuss E1 ring spectra. Th*
*us
we need some category of spectra where we can work easily with operads. This
works best if the underlying category has a closed symmetric monoidal smash
product, so we will choose one of the models of spectra with this property. It
turns out that almost any of the categories of this sort built from topological
spaces (as opposed to simplicial sets) will do. For example, we could choose
the Smodules of [18] or the orthogonal spectra of [33]; however, simply to be
concrete, we will select the symmetric spectra in topological spaces, as discus*
*sed
in [33]. This category owes much to the symmetric spectra in simplicial sets, as
developed in [28], but it is not clear that the latter category satisfies Theor*
*em
1.2.3 below.
It turns out that for any of the models of spectra we might consider here,
the category of Calgebras in spectra, where C is some operad, depends only on
the weak equivalence type of C in the na"ivest possible sense, which is in sharp
distinction to the usual results about, say, spaces. (The exact result is below,
in Theorem 1.2.4.) However, the reasons for this are not very transparent,
because they are buried in the construction of the smash product. But it is
worth emphasizing this point: the smash product has the property that if X is
a cofibrant spectrum, then the evident action of the nth symmetric group on
the nfold iterated smash product of X with itself is free.
The concepts of a monoidal model category and of a module over a monoidal
category is discussed in Chapter 4.2 of [25]. Specifically, simplicial sets ar*
*e a
monoidal model category and a simplicial model category is a module category
over simplicial sets. For any category of spectra, the action of a simplicial s*
*et
K on a spectrum X should be, up to weak equivalence, given by the formula
X K = X ^ K+
whenever this makes homotopical sense. Here the functor    is geometric
realization and ()+ means adjoin a disjoint basepoint. This is the part 3.)
of the next result. Also, whatever category of spectra we have, it should be
13
amenable to localization. This happens most easily when one has a cellular
model category, an idea discussed in the previous section; see Definition 1.1.4.
Let S denote the category of symmetric spectra in topological spaces, as
developed in [33]. We fix once and for all the "positive" model category struct*
*ure
on S, as in x14 of that paper.
1.2.1 Theorem. The category S of symmetric spectra satisfies the following
conditions:
1.)The category S is a cellular simplicial model category Quillen equivalent
to the BousfieldFriedlander [11] category of simplicial spectra.
2.)The category S has a closed symmetric monoidal smash product which de
scends to the usual smash product on the homotopy category; furthermore,
with that monoidal structure, S is a monoidal model category.
3.)The smash product behaves well with respect to the simplicial structure;
specifically, if S is the unit object of the smash product, then there is a
natural monoidal isomorphism
~=
X K ! X ^ (S K).
Note that Part 1 guarantees, among other things, that the homotopy cate
gory is the usual stable category.
Proof. Symmetric spectra in spaces is not immediately a simplicial model cat
egory, but a topological model category. But any topological model category is
automatically a simplicial model category via the realization functor. The fact
that we have a cellular model category follows from Remark 1.1.5. For example,
the effective monomorphism condition follows from the fact the every Hurewicz
cofibration of topological spaces is a closed inclusion and the "Cofibration Hy*
*_
pothesis", which is 5.3 in [33]. Parts 2 and 3 can be found in [33]. *
*__
As with categories modeling the stable homotopy category one has to ex
plicitly spell out what one means by some familiar terms.
1.2.2 Notation for Spectra. The following remarks and notation will be used
throughout this paper.
1.)When referring to a spectrum, we will use the words cofibrant and cellular
interchangeably. The generating cofibrations of S are inclusions of spheres
into cells.
2.)We will write [X, Y ] for the morphisms in the homotopy category Ho (S).
As usual, this is ss0 for some derived space of maps. See point (5) below.
3.)In the category S the unit object S for the smash product ("the zero
sphere") is not cofibrant. We will write Sk, 1 < k < 1 for a cofibrant
14
model for the ksphere unless we explicitly state otherwise. In this lan
guage the suspension functor on the homotopy category is induced by
X 7! X ^ S1.
Also the suspension spectrum functor from pointed simplicial sets to spec
tra is, by axiom 3, modeled by
0 K
K 7! S0 ^ K def=S_____S0 *
Note that because the unit object S is not cofibrant, the functor S ()
is not part of a Quillen pair.
4.)Let K be a simplicial set and X 2 S. We may write X ^ K+ for the
tensor object X K. This is permissible by axiom 3 and in line with the
geometry. The exponential object in S will be written XK .
5.)We will write map (X, Y ) or map S(X, Y ) for the derived simplicial set of
maps between two objects of S. Thus, map (X, Y ) is the simplicial map
ping space between some fibrantcofibrant models ("bifibrant") models for
X and Y . This can be done functorially if necessary, as the category S is
cofibrantly generated. Alternatively, we could use some categorical con
struction, such as the moduli spaces of Example 1.1.15. Note that with
this convention
ss0map (X, Y ) = [X, Y ].
6.)We will write F (X, Y ) for the function spectrum of two objects X, Y 2 S.
The closure statement in Axiom 2 of 1.2.1 amounts to the statement that
Hom S(X, F (Y, Z)) ~=Hom S(X ^ Y, Z).
This can be derived:
map(X, RF (Y, Z)) ' map (X ^L Y, Z)
where the R and L refer to the total derived functors and map (, ) is
the derived mapping space. In particular
sskRF (Y, Z) ~=[ kY, Z].
7.)If X is cofibrant and Y is fibrant, then there is a natural weak equivalence
map (X, Y ) ' map (S0, F (X, Y ))
and the functor map (S0, ) is the total right derived functor of the sus
pension spectrum functor from pointed simplicial sets to S. Thus we could
write
map (X, Y ) ' 1 F (X, Y ).
In particular, map (X, Y ) is canonically weakly equivalent to an infinite
loop space.
15
We need a notation for iterated smash products. So, define, for n 1,
X(n)def=X^..^.Xn!.
Set X(0)= S.
This paper is particularly concerned with the existence of A1 and E1 ring
spectrum structures. Thus we must introduce the study of operads acting on
spectra.
Let O denote the category of operads in simplicial sets. Our major source
of results for this category is [38]. The category O is a cofibrantly generated
simplicial model category where C ! D is a weak equivalence or fibration if
each of the maps C(n) ! D(n) is a weak equivalence or fibration of nspaces in
the sense of equivariant homotopy theory. Thus, for each subgroup H n, the
induced map C(n)H ! D(n)H is a weak equivalence or fibration. The existence
of the model category structure follows from the fact that the forgetful functor
from operads to the category with objects X = {X(n)}n 0 with each X(n) a
nspace has a left adjoint with enough good properties that the usual lifting
lemmas apply.
If C is an operad in simplicial sets, then we have a category of Alg C of
algebras over C is spectra. These are exactly the algebras over the triple
X 7! C(X) def=_n 0C(n) n X(n).
Note that we should really write X(n) n C(n), but we don't.
The object C(*) ~=S C(0) is the initial object of AlgC . If the operad is
reduced  that is, C(0) is a point  then this is simply S itself.
If f : C ! D is a morphism of operads, then there is a restriction of struct*
*ure
functor f* : AlgD ! AlgC , and this has a left adjoint
f* def=D C () : AlgC ! AlgD
The categories AlgC are simplicial categories in the sense of Quillen and both
the restriction of structure functor and its adjoint are continuous. Indeed, if
X 2 Alg C and K is a simplicial set, and if XK is the exponential object of
K in S, then XK is naturally an object in AlgC and with this structure, it is
the exponential object in AlgC . Succinctly, we say the forgetful functor creat*
*es
exponential objects. It also creates limits and reflexive coequalizers, filter*
*ed
colimits, and geometric realization of simplicial objects.
Here is our second set of results about spectra. The numbering continues
that of Theorem 1.2.1.
1.2.3 Theorem. The category S of symmetric spectra in topological spaces has
the following additional properties.
4.)For a fixed operad C 2 O, define a morphism of X ! Y of Calgebras in
spectra to be a weak equivalence or fibration if it is so in spectra. Then*
* the
category AlgC becomes a cofibrantly generated simplicial model category.
Furthermore, Alg C has a generating set of cofibrations and a generating
set of acyclic cofibrations with cofibrant source.
16
5.)In the category Alg C, every cofibration is a Hurewicz cofibration on the
underlying spectra and, in particular, is a levelwise closed inclusion and
an effective monomorphism.
6.)Let n 1 and let K ! L be a morphism of n spaces which is a weak
equivalence on the underlying spaces. Then for all cofibrant spectra X, the
induced map on orbit spectra
K n X(n)! L n X(n)
is a weak equivalence of spectra. If K ! L is a cofibration of simplicial
sets, then this same map is a cofibration of spectra.
Proof. First, part 4.) The argument goes exactly as in x15 of [33]. The argument
there is only for the commutative algebra operad, but it goes through with no
changes for the geometric realization of an arbitrary simplicial operad.
Part 5.) follows from the Cofibration Hypothesis, [33] 5.3.
Part 6.) follows from the observation that for cofibrant X (here is where
the positive model category structure is required), the smash product X(n)is_
actually a free nspectrum. See Lemma 15.5 of [33]. __
We wonder whether this result is also true for symmetric spectra in simplici*
*al
sets. This is not immediately obvious: many of the technical arguments of [33]
use that the inclusion of a sphere into a disk is an NDRpair.
The following result emphasizes the importance part 6.) of Theorem 1.2.3.
1.2.4 Theorem. Let f : C ! D be a morphism of operads in simplicial sets.
Then the adjoint pair
f* : AlgC_____//AlgD:of*o_
is a Quillen pair. If, in addition, the morphism of operads has the the property
that C(n) ! D(n) is a weak equivalence of spaces for all n 0, this Quillen
pair is a Quillen equivalence.
Proof. The fact that we have a Quillen pair follows from the fact that the
restriction of structure functor (the right adjoint) f* : AlgD ! AlgC certainly
preserves weak equivalences and fibrations.
For the second assertion, first note that since f* creates weak equivalences,
we need only show that for all cofibrant X 2 AlgC , the unit of the adjunction
X ! f*f*X = D C X
is a weak equivalence. If X = C(X0) is actually a free algebra on a cofibrant
spectrum, then this map is exactly the map induced by f:
` (n) ` (n)
C(X0) = C(n) n X0 ! D(n) n X0 = D(X0).
n n
For this case, Axiom 6 of 1.2.3 supplies the result. We now reduce to this case.
17
Let X 2 AlgC be cofibrant. We will make use of an augmented simplicial
resolution in AlgC
Po! X
with the following properties:
i.)the induced map Po ! X from the geometric realization of Po to X is a
weak equivalence;
ii.)the simplicial Calgebra Po is sfree on a set of Calgebras {C(Zn)} where
each Zn is a cofibrant spectrum. (The notion of sfree was defined in
Definition 1.1.9.)
There are many ways to produce such a Po. For example, we could take an
appropriate subdivision of a cofibrant model for X in the resolution model cat
egory for simplicial Calgebras based on the homotopy cogroup objects C(Sn),
1 < n < 1. 1
Given Po, consider the diagram
(1.2.1) Po_____//f*f*Po
 
 
fflffl fflffl
X ______//_f*f*X
For all n, we have an isomorphism
`
Pn ~=C( Zk)
OE:[n]![k]
where OE runs over the surjections in the ordinal number category. Thus we can
conclude that Pn ! f*f*Pn is a (levelwise) weak equivalence and that both Po
and f*f*Po are Reedy cofibrant. The morphism Po ! X is a weak equivalence
by construction, and Po ! f*f*Po is a weak equivalence since geometric
realization preserves weak equivalences between Reedy cofibrant objects. Thus
we need only show that
f*f*Po! f*f*X
is a weak equivalence.
To see this, we note that since weak equivalences and geometric realizations
are created in the underlying category of spectra, it is sufficient to show f**
*Po !
f*X is a weak equivalence. However f*Po = f*Po since f* is a left adjoint.
Finally, since f* is part of a Quillen pair, it preserves weak equivalences bet*
*ween_
cofibrant objects (which is where that hypothesis is used). __
We now make precise the observation that Theorem 1.2.4 implies that the
notion of, for example, an E1 ring spectrum is independent of which E1 op
erad we choose. Actually, even more is true. Let C be an operad so that for
____________________________1
See Proposition 1.4.11. Resolution model categories are reviewed in section*
* 1.4. The
notion of Reedy cofibrant, used in the next paragraph, is discussed in the next*
* section.
18
all n, the unique map to the onepoint space C(n) ! * is a weak equivalence
(nonequivariantly). Then the obvious map C ! Comm from C to the com
mutative monoid operad satisfies the hypotheses of Theorem 1.2.4 and thus we
may conclude that AlgC is Quillen equivalent to the category of commutative
Salgebras.
1.3 Simplicial spectra over simplicial operads
Simplicial objects are often used to build resolutions  and that is our main
point here. However, given an algebra X in spectra over some operad, there
are times when we will resolve not only X, but the operad as well. The main
results of this section are that if X is a simplicial algebra over a simplicial
operad T then the geometric realization X is an algebra over the geometric
realization T  and, furthermore, that geometric realization preserves levelw*
*ise
weak equivalences between Reedy cofibrant objects, appropriately defined.
1.3.1 Remark. In what follows we are going to discuss the category sO of
simplicial operads. These are bisimplicial operads is sets, but when we say
simplicial operad, we will mean a simplicial object in O, emphasizing the second
(external) simplicial variable as the resolution variable. The first (internal)
simplicial variable will be regarded as the geometric variable.
As mentioned in the previous section, the category of operads O is a sim
plicial model category. From this one gets the Reedy model category structure
on simplicial operads sO ([37]), which are the simplicial objects in O. Weak
equivalences are levelwise and cofibrations are defined using the latching ob
jects. The Reedy model category structure has the property that geometric
realization preserves weak equivalences between cofibrant objects. It also has a
structure as a simplicial model category; for example if T is a simplicial oper*
*ad
and K is simplicial set, then
T K = {TnK}.
However, note that this module structure over simplicial sets is inherited from
O and is not the simplicial structure arising externally, as in [35], xII.2.
Now fix a simplicial operad T = {Tn}. (At this point, T need not have any
special properties.) The free algebra functor X 7! C(X) is natural in X and
the operad C; hence, for any simplicial spectrum X we can define a bisimplicial
spectrum {Tq(Xq)}. We will denote the diagonal of this bisimplicial spectrum
by T (X). A simplicial algebra in spectra over T is a simplicial spectrum X
equipped with a multiplication map
T (X) ! X
so that the usual associativity and unit diagrams commute. In particular, if
X = {Xn}, then each Xn is a Tnalgebra. Let sAlg T be the category of
simplicial T algebras.
19
The category sAlg Tis a simplicial model category, and geometric realization
behaves well with respect to this structure. The exact result we need is below
in Theorem 1.3.4, but its complete statement requires some preliminaries.
Recall that given a morphism of operads C ! D, the restriction of structure
functor AlgD ! AlgC is continuous. This implies that if K is a simplicial set
and X 2 sAlg T, we may define X K and XK levelwise; for example,
X K = {Xn K}.
We could use this structure to define a geometric realization functor; how
ever, we prefer to proceed as follows.
If M is a module category ([25], x4.2) over simplicial sets, then the geomet*
*ric
realization functor  .  : sM ! M has a right adjoint
n
Y 7! Y = {Y }.
where n is the standard nsimplex. In particular, this applies to simplicial
operads, and we are interested in the unit of the adjunction T ! T  . If C is
any operad and Y is a Calgebra, then for all simplicial sets K, the spectrum
Y K is a CK algebra. From this it follows that Y is a simplicial C algebra.
Setting C = T  and restricting structure defines a functor
Y 7! Y : AlgT! sAlg T.
The result we want is the following.
1.3.2 Theorem. Let T be a simplicial operad and X 2 sAlg T a simplicial
T algebra. Then the geometric realization X of X as a spectrum has a natural
structure as a T  algebra and, with this structure, the functor
X 7! X
is left adjoint to Y 7! Y .
Proof. We know that for an operad C 2 O the forgetful functor from AlgC to
spectra creates geometric realization. Actually, what one proves is that if X is
a simplicial spectrum and C(X) is the simplicial Calgebra on X, then there is
a natural (in C and X) isomorphism
C(X)! C(X).
This uses a "reflexive coequalizer" argument; see Lemma II.6.6 of [18]. Now use
a diagonal argument. If T is a simplicial operad and X is a simplicial spectrum,
then, by definition,
T (X) = diag{Tp(Xq)}.
Since the functor Y 7! C(Y ) is a continuous left adjoint, taking the realizati*
*on
in the pvariable yields a simplicial object
{{To(Xq)}} ~={T (Xq)}.
20
Now take the realization in the q variable and get
T (X) ~=T (X)
using the fact about the constant case sited above. The result now follows. _*
*__
The next item to study is the homotopy invariance of the geometric realiza
tion functor in this setting. The usual result has been cited above: realization
preserves levelwise weak equivalences between Reedy cofibrant objects. The
same result holds in this case, but one must take some care when defining
"Reedy cofibrant". The difficulty is this: the definition of Reedy cofibrant in
volves the latching object, which is the colimit
LnX = colim Xm
OE:[n]![m]
where OE runs over the nonidentity surjections in the ordinal number category.
We must define this colimit if each of the Xm is an algebra over a different
operad. The observation needed is the following. Let S : I ! O be a diagram
of operads. Then an Idiagram of Salgebras is an Idiagram X : I ! S of
spectra equipped with a natural transformation of Idiagrams
S(X) ! X
satisfying the usual associativity and unit conditions. For example if I = op
one recovers simplicial Salgebras. Call the category of such AlgS .2 Then one
can form the colimit operad colimS = colimIS and there is a constant diagram
functor
AlgcolimS! AlgS
sending X to the constant Idiagram i 7! X where X gets an Si structure via
restriction of structure along
Si ! colimIS.
1.3.3 Lemma. This constant diagram functor has a left adjoint
X ! colimIX.
Despite the notation, colimIX is not the colimit of X as an I diagram of
spectra; indeed, if X = S(Y ) where Y is an Idiagram of spectra
colimIX ~=(colimIS)(colimIY ).
If T is a simplicial operad we can form the latching object
LnT = colim Tm .
OE:[n]![m]
____________________________2
This is a slight variation on the notation sAlgT. If T is a simplicial opera*
*d, this new
notation would simply have us write AlgTfor sAlgT. No confusion should arise.
21
There are natural maps LnT ! Tn of operads. If X is a simplicial T algebra
we extend this definition slightly and define
LnX = Tn LnT colim Xm
OE:[n]![m]
where, again, OE runs over the nonidentity surjections in . In short we extend
the operad structure to make LnX a Tnalgebra and the natural map LnX !
Xn a morphism of Tnalgebras.
With this construction on hand one can make the following definition. Let
T be a simplicial operad and f : X ! Y a morphism of simplicial T algebras.
Then f is a levelwise weak equivalence (or Reedy weak equivalence) if each of
the maps Xn ! Yn is a weak equivalence of Tnalgebras  or, by definition,
a weak equivalence as spectra. The morphism f is a Reedy cofibration if the
morphism of Tnalgebras
LnY tLnX Yn ! Yn
is a cofibration of Tnalgebras. The coproduct here occurs in the category of
Tnalgebras. (Fibrations are then determined; they have a description in terms
of matching objects. See [23], x15.1.) The main result is then:
1.3.4 Theorem. With these definitions, and the levelwise simplicial structure
defined above, the category sAlg T becomes a simplicial cellular model category.
Furthermore,
1.the geometric realization functor    : sAlg T ! AlgTsends levelwise
weak equivalences between Reedy cofibrant objects to weak equivalences;
and
2.any Reedy cofibration in sAlg T is a Hurewicz cofibration in spectra at ea*
*ch
simplicial level; in particular, it is an effective monomorphism.
Proof. The standard argument for the existence of a Reedy model category
structure (see [23] x15.6, for example) easily adapts to this situation; one ne*
*ed
only take care with latching objects, and we have described these in some detail
above. The same reference also supplies arguments to show that the model
category structure is cellular. See [23] x15.7. That it is a simplicial model
category is an easy exercise.
To prove point 1.), note that the right adjoint to geometric realization Y 7!
Y preserves fibrations and weak equivalences when considered as a functor to
sS, hence it has the same properties when considered as a functor to sAlg T.
Thus geometric realization is part of a Quillen pair. For point 2.), one checks
that a Reedy cofibration X ! Y in sAlg T yields a (Quillen) cofibration of
Tnalgebras Xn ! Yn for all n. This can be done by adapting the argument of_
Proposition 15.3.11 of [23]. Now apply Theorem 1.2.3. __
Now let us next spell out the kind of simplicial operads we want might want.
One example is, obviously, the constant simplicial operad T on the commutative
22
monoid operad or, perhaps, an E1 operad in O. Then sAlg T will simply be
simplicial commutative algebras (or E1 algebras) in spectra. However, there
are times when this might be too simplistic.
If E* is the homology theory of a homotopy commutative ring spectrum and
C is an operad in O, one might like to compute E*C(X). This might be quite
difficult, unless E*X is projective as an E* module and ss0C(q) is a free qset
for all q. Thus we'd like to resolve a general operad C using operads of this s*
*ort.
If T is a simplicial operad and E is a commutative ring spectrum in the
homotopy category of spectra, then E*T is a simplicial operad in the category of
E*modules. The category of simplicial operads in E*modules has a simplicial
model category structure in the sense of xII.4 of [35], precisely because there
is a free operad functor. Cofibrant objects are retracts of diagrams which are
"free" in the sense of [35]; meaning the underlying degeneracy diagram is a free
diagram of free operads. Free operads are discussed in detail in the appendix
to [38].
Given an operad C 2 O, we'd like to consider simplicial operads T of the
following sort:
1.3.5 Theorem. Let C 2 O be an operad. Then there exists an augmented
simplicial operad
T ! C
so that
1.T is Reedy cofibrant as a simplicial operad;
2.For each n 0 and each q 0, ss0Tn(q) is a free qset;
3.The map of operads T  ! C induced by the augmentation is a weak
equivalence;
4.If E*C(q) is projective as an E* module for all q, then E*T is cofibrant as
a simplicial operad in E* modules and E*T ! E*C is a weak equivalence
of operads in that category.
This theorem is not hard to prove, once one has the explicit construction
of the free operad; for example, see the appendix to [38]. Indeed, here is a
construction: first take a cofibrant model C0 for C. Then, if FO is the free
operad functor on graded spaces, one may take T to be the standard cotriple
resolution of C0. What this theorem does not supply is some sort of uniqueness
result for T ; nonetheless, what we have here is sufficient for our purposes.
Note that if C is the commutative monoid operad, then we can simply take
T to be a cofibrant model for C in the category of simplicial operads and run
it out in the simplicial (i.e., external in the sense of Remark 1.3.1) directio*
*n.
Then T is, of course, an example of an E1 operad; furthermore, E*T will be a
simplicial E1 operad in E*modules in the sense of Definition 2.3.8.
23
1.4 Resolutions
Building on the results of the last section, we'd like to assert the following.
Fix a homology theory E*. Let X be a simplicial algebra over a simplicial
operad T . Then, perhaps under hypotheses on T , we would like to assert there
is a simplicial T algebra Y and a morphism of T algebras Y ! X so that
a.) Y  ! X is a weak equivalence and b.) E*Y is cofibrant as an E*T
algebra. The device for this construction is an appropriate Stover resolution
([46],[16],[17]) and, particularly, the concise and elegant paper of Bousfield *
*[10].3
We explain some of the details in this section.
We begin by specifying the building blocks of our resolutions. We fix a
spectrum E which is a commutative ring object in the homotopy category of
spectra. Let D(.) denote the SpanierWhitehead duality functor.
1.4.1 Definition. A homotopy commutative and associative ring spectrum E
satisfies Adams's condition if E can be written, up to weak equivalence, as a
homotopy colimit of a filtered diagram of finite cellular spectra Effwith the
properties that
1.E*DEffis projective as an E*module; and
2.for every module spectrum M over E the K"unneth map
[DEff, M] ! Hom E*(E*DEff, M*)
is an isomorphism.
This is the condition Adams (following Atiyah) wrote down in [1] to guar
antee that the (co)homology theory over E has K"unneth spectral sequences. If
M is a module spectrum over E, then so is every suspension or desuspension of
M; therefore, one could replace the source and target of the map in part 2.) of
this definition by the corresponding graded objects.
Many spectra of interest satisfy this condition; for example, if E is the
spectrum for a Landweber exact homology theory, it holds. (This is implicit in
[1], and made explicit in [39].) In fact, the result for Landweber exact theori*
*es
follows easily from the example of MU, which, in turn, was Atiyah's original
example. See [2]. Some spectra do not satisfy this condition, however  the
easiest example is HZ.
We want to use the spectra DEffas detecting objects for a homotopy theory,
but first we enlarge the scope a bit.
1.4.2 Definition. Define P(E) = P to be a set of finite cellular spectra so that
1.the spectrum S0 2 P and E*X is projective as an E*module for all X 2 P;
2.for each ff there is finite cellular spectrum weakly equivalent to DEffin *
*P;
____________________________3
Bousfield's paper is written cosimplicially, but the arguments are so catego*
*rical and so
clean that they easily produce the simplicial objects we require.
24
3.P is closed under suspension and desuspension;
4.P is closed under finite coproducts (i.e, wedges); and
5.for all X 2 P and all Emodule spectra M the K"unneth map
[X, M] ! Hom E*(E*X, M*)
is an isomorphism.
The E2 or resolution model category which we now describe uses the set P
to build cofibrations in simplicial spectra and, hence, some sort of projective
resolutions.
Because the category of spectra has all limits and colimits, the category of
simplicial spectra is a simplicial category in the sense of Quillen using exter*
*nal
constructions as in xII.4 of [35]. However, the Reedy model category structure *
*on
simplicial spectra is not a simplicial model category using the external simpli*
*cial
structure; for example, if i : X ! Y is a Reedy cofibration and j : K ! L is a
cofibration of simplicial sets, then
i j : X L tX K Y K ! Y L
is a Reedy cofibration, it is a levelwise weak equivalence if i is, but it is *
*not
necessarily a levelwise weak equivalence if j is.
The following ideas are straight out of Bousfield's paper [10].
1.4.3 Definition. Let Ho (S) denote the stable homotopy category.
1.)A morphism p : X ! Y in Ho (S) is Pepi if p* : [P, X] ! [P, Y ] is onto
for each P 2 P.
2.)An object A 2 Ho (S) is Pprojective if
p* : [A, X]! [A, Y ]
is onto for all Pepi maps.
3.)A morphism A ! B of spectra is called Pprojective cofibration if it has
the left lifting property for all Pepi fibrations in S.
The classes of Pepi maps and of Pprojective objects determine each other;
furthermore, every object in P is Pprojective. Note however, that the class
of Pprojectives is closed under arbitrary wedges. The class of Pprojective
cofibrations will be characterized below; see Lemma 1.4.7.
1.4.4 Lemma. 1.) The category Ho (S) has enough Pprojectives; that is, for
every object X 2 Ho (S) there is a Pepi Y ! X with Y a Pprojective.
2.) Let X be a Pprojective object. Then E*X is a projective E*module,
and the K"unneth map
[X, M]! Hom E*(E*X, M*)
is an isomorphism for all Emodule spectra M.
25
Proof. For part 1.) we can simply take
Y = Pq2Pf:qP!XP
where f ranges over all maps P ! X in Ho (S). Then, for part 2.), we note
that the evaluation map Y ! X has a homotopy section if X is Pprojective. __
Then the result follows from the properties of the elements of P. __
We can now specify the Presolution model category structure. Recall that
a morphism f : A ! B of simplicial abelian groups is a weak equivalence if
f* : ss*A ! ss*B is an isomorphism. Also f : A ! B is a fibration if the induced
map of normalized chain complexes Nf : NA ! NB is surjective in positive
degrees. The same definitions apply to simplicial Rmodules or even graded
simplicial Rmodules over a graded ring R. A morphism is a cofibration if it is
injective with levelwise projective cokernel.
1.4.5 Definition. Let f : X ! Y be a morphism of simplicial spectra. Then
1.)the map f is a Pequivalence if the induced morphism
f* : [P, X]! [P, Y ]
is a weak equivalence of simplicial abelian groups for all P 2 P;
2.)the map f is a Pfibration if it is a Reedy fibration and f* : [P, X]! [P*
*, Y ]
is a fibration of simplicial abelian groups for all P 2 P;
3.)the map f is a Pcofibration if the induced maps
Xn tLnX LnY ! Yn, n 0,
are Pprojective cofibrations.
Then, of course, the theorem is as follows.
1.4.6 Theorem. With these definitions of Pequivalence, Pfibration, and P
fibration, the category sS becomes a simplicial model category.
The proof is given in [10]. We call this the Presolution model category
structure. It is cofibrantly generated; furthermore there are sets of generating
cofibrations and generating acyclic cofibrations with cofibrant source. An obje*
*ct
is Pfibrant if and only if it is Reedy fibrant. We will see below, in Theorem
1.4.9  using the case where T is the identity operad  that this model category
structure on sS is, in fact, cellular.
The next result gives a characterization of Pcofibrations.
Call a morphism X ! Y of spectra Pfree if it can be written as a compo
sition
q
X ___i_//X q F_____//Y
where i is the inclusion of the summand, F is cofibrant and Pprojective, and q
is an acyclic cofibration. The following is also in [10]. Another characterizat*
*ion
of cofibrations can be obtained from the Lemma 1.4.10, which displays a set of
generating cofibrations.
26
1.4.7 Lemma. A morphism X ! Y of spectra is a Pprojective cofibration if
and only if it is a retract of Pfree map.
1.4.8 Remark. At this point we can explain one of the reasons for using the
models P to define the resolution model category. Suppose X ! Y is a weak
equivalence between cofibrant objects in the Presolution model category. Then
for each of the spectra DEffwe have an isomorphism
~= q
f* : ssp[ qDEff, X] ! ssp[ DEff, Y ].
However, if E*() is our chosen homology theory
sspEqX~=colimffssp(Eff)qX
~=colimffssp[ qDEff, X].
In particular, if X ! Y is a Pequivalence of simplicial spectra, then
E*X ! E*Y
is a weak equivalence of simplicial E*modules. Also note that if X ! Y is a
Pcofibration, then E*X ! E*Y is a cofibration of simplicial E* modules. This
follows from Lemma 1.4.7.
For a Reedy cofibrant simplicial spectrum X or, more generally a proper4
simplicial object X, there is a spectral sequence
(1.4.1) sspEqX =) Ep+qX.
This is, of course, the standard homology spectral sequence of a simplicial spe*
*c
trum. If X ! Y is an Pequivalence of Reedy cofibrant simplicial spectra, then
we get isomorphic E* homology spectral sequences.
The Presolution model category structure can be promoted to a model
category for simplicial algebras over a simplicial operad. Fix a simplicial ope*
*rad
T and let sAlg T be the category of algebras over T . This category has an
external simplicial structure; indeed, if K is a simplicial set and X 2 sAlg T,
one has
(1.4.2) (X K)n = qKTnXn.
n
The superscript Tn is indicates that the coproduct is taken in the category of
Tn algebras. The simplicial set of maps is defined again by
[n] 7! Hom sAlgT(X n, Y ).
We say that a morphism X ! Y of simplicial T algebras is a Pfibration or P
equivalence if the underlying morphism of simplicial spectra is. Then we have
the Presolution model category structure on sAlg T. We will discuss cofibratio*
*ns
below when we have more hypthoses.
____________________________4
An object is proper if the inclusions of the latching objects LnX ! Xn are H*
*urewicz
cofibrations.
27
1.4.9 Theorem. With these definitions, the category sAlg T becomes a simpli
cial cellular model category.
Proof. The existence of the simplicial model category structure is the standard
lifting argument. (See [21] xII.2 for the case of simplicial model categories, *
*or [23]
x11.3. for a more general statement.) Since sAlg T is a simplicial category, in
the sense of Quillen, the categorysAlg T has a functorial path object. Since the
forgetful functor to sS creates filtered colimits in sAlg T, we need only suppl*
*y a
Pfibrant replacement functor for sAlg T. However, every Reedy fibrant object
in sAlg Twill be Pfibrant, and the sAlg Tin its Reedy model category structure
is cofibrantly generated, so we can choose a Reedy fibrant replacement functor.
This will do the job. Note that this model category is cofibrantly generated,
again by the standard lifting arguments.
To get that the model category is cellular, first note that since every Reedy
weak equivalence is Pequivalence and every Reedy acyclic fibration is a P
acyclic fibration, every Pcofibration will be Reedy cofibration, and hence a
spacewise closed inclusion, by Theorem 1.3.4. Since sS, in its Presolution
model category structure has a set of generating cofibration A ! B with
cofibrant source, so does sAlg T; indeed, the generators will be of the form_
T (A) ! T (B). To complete the argument, we apply Remark 1.1.5. __
We now give a set of generating cofibrations for sAlg T. This will be impor
tant when discussing the size of cell complexes in localization arguments. Reca*
*ll
that we have fixed our set P(E) = P of projectives: see 1.4.2.
1.4.10 Lemma. Fix a set of J of generating acyclic cofibrations for S. The
Pmodel category structure on sAlg T has, as a set I of generating cofibrations,
the morphisms
T (Aj n qAj @ n Bj @ n) ! T (Bj n)
where Aj ! Bj is a morphism in J and the morphisms
T (P @ n) ! T (P n)
where P 2 P.
Proof. A morphism X ! Y is an acyclic fibration if and only if it is a Reedy
fibration and (by virtue of the spiral exact sequence, Theorem 3.1.4) the induc*
*ed
morphism of underived mapping spaces
sAlg T(T (P ), X)! sAlg T(T (P ), Y )
is an acyclic fibration of simplicial sets. The result follows by an_adjointne*
*ss
argument. __
1.4.11 Proposition. For each X 2 sAlg T there is a natural Pequivalence
PT(X) ! X
so that
28
1.)PT(X) is cofibrant in the Presolution model category structure on sAlg T;
2.)the underlying degeneracy diagram of PT(X) is of the form T (Z) where Z
is free as a degeneracy diagram and each Zn is a wedge of elements of P.
Proof. The object PT(X) is produced by taking an appropriate subdivision (for
example the big subdivision of [9] xXII.3, Example 3.4) of a cofibrant_model for
X. __
The following result has content because it is not at all obvious that a P
cofibrant algebra in sAlg T is Reedy cofibrant when regarded as a spectrum.
1.4.12 Corollary. Suppose that T is a simplicial operad. Let X be a P
cofibrant simplicial T algebra in sAlg T. Then for any homology theory E*,
there is strongly convergent first quadrant spectral sequence
sspEqX =) Ep+qX.
Proof. We may assume that X is of the form stipulated by Proposition 1.4.11.
Then we claim that X is, in fact, Reedy cofibrant when regarded as a simplicial
spectrum. This is routine, if tedious, and we leave the details to the reader.
There are two key observations. First, if T is a Reedy cofibrant operad, then
for each n, the bisimplicial set T (n) is Reedy cofibrant. This is because all
bisimplicial sets are Reedy cofibrant. Second, if C is any operad and Z1 and Z2
are spectra, then there is a decomposition formula
C(Z1 q Z2) ~=q C(n + m) m x n Z(m)1^ Z(n)2.
___
To make constructive use of the Presolution model category structure on
sAlg T, we impose a further condition.
1.4.13 Definition. An operad C is adapted to E* if there is a triple CE on
E*modules so that
1.if X is a Calgebra in spectra, then E*X is naturally a CE algebra in
E*modules;
2.if Z is a cofibrant spectrum such that E*Z is projective as an E*module,
then the natural map of CE algebras
CE (E*Z)! E*C(Z)
is an isomorphism.
There is a simplicial version, also: a simplicial operad T is adapted to E* *
*if
there is a triple TE on simplicial E*modules so that
3.if X is a simplicial T algebra in spectra, then E*X is naturally a TE 
algebra in simplicial E*modules;
29
4.if Z is a Reedy cofibrant spectrum such that E*Z is a cofibrant simplicial
E*module, then the natural map of TE algebras
TE (E*Z)! E*T (Z)
is an isomorphism.
Here are some basic examples. There are more below in Remark 1.4.17.
1.4.14 Example. 1.) If C is an operad adapted to E, then the C, regarded as
a constant simplicial operad, is adapted as a simplicial operad to E.
2.) By the results of section 2.2 below, any E1 operad is adapted to p
complete Ktheory.
3.) If C is any operad so that ss0C(k) is a free k set for all k, then C
is adapted to any Adamstype homology theory. This means, specifically, that
any A1 operad is adapted to E. More generally, if T is a simplicial operad so
that for all k and n, the set ss0Tn(k) is a free kset, then T is adapted as a
simplicial operad to E.
In the following result, we make a cardinality statement about relative cell
complexes. The generating set I of cofibrations is that of Lemma 1.4.10.
1.4.15 Lemma. Suppose T is a simplicial operad adapted to E and suppose
f : X ! Y is a cofibration with cofibrant source in sAlg T with its Presolution
model category structure. The f is a retract of a morphism g : X ! Z with the
following property:
(*)The underlying morphism of degeneracy diagrams for E*g is isomorphic
to a morphism of the form
E*X __i__//E*X q TE (M)
where M is sfree on a projective E*module.
Furthermore, g has a presentation as a relative Icell complex with fl cells, t*
*hen
M has a set of generators as an E*module of cardinality fl.
Proof. All acyclic cofibrations in spectra have a strong deformation retraction.
This follows from Theorem 14.1 (see also Theorem 6.5) of [33]. This implies
that if we have pushout diagram in simplicial T algebras of the form
T (Aj n qAj @ n Bj @ n) ____//_X
 
 
fflffl fflffl
T (Bj n)_______________//Y
then, at every simplicial level k, we have that Xk ! Yk is a homotopy equiv
alence. In particular E*Xk ~=E*Yk. On the other hand, if we have pushout
30
diagram of the form
T (P @ n)_____//X
 
 
fflffl fflffl
T (P n)_____//_Y
then, at every simplicial level k, we have that Yk ~=Xk q Tk(qIkP ) for some
finite indexing set Ik and this decomposition respects the degeneracies.
If f : X ! Y is any cofibration with cofibrant source, then f is a retract of
a cofibration g : X ! Z built by the small object argument from the generating
cofibrations. This, in turn, is a retract of a cofibration g0: X0 ! Z0 so that *
*X0
is built by the small object argument from the initial object S and Z0 is built
from X0 by the small object argument. The conclusion (*) for S ! X0 and
hence for g0 by observations of the previous paragraph. Then (*) holds for_g
because it is a retract of g0. __
To go further we have to assume that our the category of TE algebras has
good homotopical behavior. This is encoded in the following definition.
1.4.16 Definition. Let E* be a homology theory so that E*E is flat over E*
and let T be a simplicial operad adapted to E. Then we will say that T is
homotopically adapted to E if:
1.the triple TE on simplicial E*modules lifts to a triple on E*Ecomodules;
2.the category of simplicial TE algebras in E*modules supports the structu*
*re
of a simplicial model category where a morphism is a weak equivalence or
fibration if and only if it is so as as a simplicial E*module;
3.the category of simplicial TE algebras in the category of E*Ecomodules
supports the structure of a simplicial model category such that the forget
ful functor to TE algebras in E*modules creates weak equivalences and
preserves fibrations.
1.4.17 Remark. This definition is rather complicated; however, our three main
examples will all produce homotopically adapted operads. But let us first say
that what is needed in the next section is only part (2) of this definition. The
rest becomes crucial later.
1.If E* is any Adamstype homology theory with E*E flat over E*, then
the associative monoid operad is homotopically adapted to E*. Then TE
will be the simplicial associative algebra triple. The necessary model cat
egory structure on simplicial associative E*algebras is the one supplied
by Quillen in [35]xII.4 and the model category structure on simplicial as
sociative algebras in E*Ecomodules appeared in [19]. (See the beginning
of section 2.5 for a more thorough review of the comodule case.)
31
2.Again let E* be any Adamstype homology theory with E*E flat over
E*. Let C be an E1 operad in the category of simplicial sets; thus C(k)
is contractible and has a free kaction. Then let T be the resulting
simplicial operad obtained by running C out in the external5 simplicial
direction. Then T  ~=C and E*T is an E1 operad in E*modules. (See
Definition 2.3.8) and TE is the free simplicial E1 algebra triple. Again
the necessary model category structure on E1 algebras is the one supplied
by Quillen in [35]xII.4 and the model category structure on E1 algebras
in E*Ecomodules appeared in [19].
3.Let K* be pcompleted Ktheory, and T the commutative monoid operad,
so that T algebras are simplical commutative Salgebras. Then TE is the
free thetaalgebra functor. The details of this example, including the fact
T is homotopically adapted to K* appear in section 2.3.
The following result is an immediate consequence of Lemma 1.4.15 given
Quillen's characterization ([35]xII.4) of cofibrations as retracts of "free" ma*
*ps.
1.4.18 Corollary. Suppose the simplicial operad T is homotopically adapted to
E. Then the functor
E* : sAlg T! sAlg TE
sends weak equivalences to weak equivalences and cofibrations with cofibrant
source to cofibrations.
1.4.19 Example. Suppose we fix an operad C 2 O and a simplicial resolution
T ! C of C as in Theorem 1.3.5. If X is an Calgebra, then X can be regarded
as a constant object in sAlg T and, hence, we have the resolution PT(X) ! X
of Proposition 1.4.11. Then PT(X) is Pcofibrant in sAlg T. Since Remark
1.4.8 implies that the augmentation ss*E*PT(X) ! E*X is an isomorphism,
the previous result and Example 1.4.14.3 imply that E*PT(X) is a cofibrant
replacement for E*X in simplicial E*T algebras. (Here we are using the model
category structure on simplicial E*algebras of [35]xII.4.) Furthermore we can
use the E* homology spectral sequence of Corollary 1.4.12 to conclude
ss*E*PT(X) ~=E*X.
1.5 Localization of the resolution model cate
gory
In the previous section, we developed the resolution model category of spectra,
or simplicial T algebras, based on some set of projectives P. In particular, we
were interested in the set P = P(E) arising from an Adamstype homology
theory, as in Definition 1.4.2. This resolution model category has the type of
cofibrant objects we'd like, but  as the reader may have surmised  we are not
____________________________5
See Remark 1.3.1 for the meaning of "external".
32
primarily interested in the Pequivalence classes of objects in simplicial spec*
*tra
or simplicial T algebras, but in certain types of E*equivalences. There does *
*not
appear to be a model category with these cofibrations and weak equivalences;
therefore, we settle for a semimodel category, as in the next result. It is a
localization of the one supplied in Theorem 1.4.9.
The material of this section developed out of some conversations with Phil
Hirschhorn.
The rest of this section will be devoted to proving the following result. The
notion of semimodel category was discussed in Section 1.1, and the definition *
*of
what it means for an operad to be homotopically adapted to a homology theory
is in the Definitions 1.4.13 and 1.4.16.
1.5.1 Theorem. Suppose that T is a simplicial operad homotopically adapted
to the homology theory E. Then the category sAlg T supports the structure of a
cofibrantly generated simplicial semimodel category so that
1.)a morphism f : X ! Y is an E*equivalence if
ss*E*(f) : ss*E*X ! ss*E*Y
is an isomorphism;
2.)a morphism is an E*cofibration if it is a Pcofibration; and
3.)a morphism is an E*fibration if it has the right lifting property with re
spect to all morphisms which are at once an E*equivalence and an E*
cofibration.
Since, by Remark 1.4.8, every Pequivalence in sAlg T is an E*equivalence,
this semimodel category structure can be produced using the localization tech
nology of Bousfield, et al., with variations which have previously been confron*
*ted
in [18], xVIII.1. There are many minute details, and we vary somewhat from
the canonical path  as mapped out in [23]  but the route is familiar.
To begin, let E* be our chosen Adamstype homology theory, and let Ch E*
denote the category of nonnegatively graded chain complexes over E*. Then
we have a functor
hE def=NE*() : sS ! ChE*
given sending a simplicial spectrum X to the normalized complex NE*(X).
Note that we have the H*hE (X) = ss*E*X. The following is obvious, and
included only to ground the argument.
1.5.2 Lemma. The functor hE : sS ! Ch E* has the following properties:
i.)If X ! Y is a Pequivalence, then hE (X) ! hE (Y ) is a homology iso
morphism, and if * is the initial object then hE (*) = 0.
ii.)If i 7! Xi is a filtered diagram of Reedy cofibrant objects, then
colimhE (Xi) ! hE (colimXi)
is an isomorphism.
33
iii.)If A ! B is a Pcofibration, then hE (A) ! hE (B) is an injection. If
A ! X is any other map, then the resulting diagram
hE (A)_______//_hE (X)
 
 
fflffl fflffl
hE (B)_____//hE (B tA X)
is a pushout square.
As a remark on this result, we note that items ii.) and iii.) together imply
that if {Xff} is any set of Pcofibrant objects in sS, then the evident map
a
ffhE (Xff)! hE ( Xff)
is an isomorphism. Note also that the hypothesis on the initial object is redun
dant; the empty diagram is filtered, so ii.) implies hE (*) = 0.
The functor ss*E*() has some of the usual properties of a homology functor.
For example, if A ! B is a Pcofibration with cofibrant source, we can define
ss*E*(B, A) = H*(hE (B)=hE (A))
and we have a long exact sequence of a pair, by Definition 1.5.2.iii. The same
item also yields a MayerVietoris sequence.
We now begin to set up the localization argument. In order for this to work,
we need to know that intersections of subcomplexes exist. Here are the details.
Suppose we are given some category C and a set of maps I in C. Then, in
Definitions 1.1.1 and 1.1.2 we wrote down the definition of Icell complexes and
subcomplexes. Given two such subcomplexes K, L X, we would like to define
K \ L with the property that
TfK\Li= TfKi\ TfLi.
(This is called the combinatorial intersection in [18] xIII.2 and simply the in*
*ter
section in [23]). The difficulty is to show that (K \ L)(): ~ ! C exists. Using
transfinite induction, we can assume (K \L)fiexists and to define (K \L)fi+1we
need to be able to complete the following diagram for every element of TfKi\TfL*
*i:
(1.5.1) A TTTT5H
55H TTT
55H H TTTTT
55 H$$ TTTTT
55(K \ L) __TTT**T//_L
55 fi fi
55  
55 
aeae5fflffl fflffl
Kfi________//Xfi.
We will say that intersections of subcomplexes exist if for some set I of gener*
*ating
cofibrations of C we can solve this problem and produce K \ L. The reason we
went to all the trouble to specify that our various categories of simplicial sp*
*ectra
were cellular model categories was so that we could apply the following result.
34
1.5.3 Lemma. Let C be a cellular model category. Then intersections of sub
complexes exist.
Proof. See [23] x14.2. The proof is straightforward: the effective monomorphism
condition and a diagram chase shows that the square of diagram 1.5.1 is_a pull
back diagram. __
To prove Theorem 1.5.1, we use a standard technique for constructing cofi
brantly generated model categories: Theorem 2.1.19 of [25] (but see also the
identical Theorem 13.4.1 of [23] which credits this result to Dan Kan). The
exact statement will be incorporated in the proof below, but one begins by
specifying a class of weak equivalences and sets of maps I and J which will
generate the cofibrations and acyclic cofibrations respectively. Then one has
to show these maps satisfy certain properties. In this case the class of weak
equivalences will be the ss*E*() isomorphisms and, since sAlg T (in the P
resolution model category structure) is already cofibrantly generated, I will be
a generating set for the cofibrations. The issue is to supply J, and for this we
use an analog of the BousfieldSmith argument (cf. [23] x4.5). This comes down
to a cardinality argument, so we begin by spending a paragraph or so to specify
some cardinals.
We choose, as our generating set I def=IT of cofibrations of sAlg T, in the
Presolution model category structure, the morphisms of Lemma 1.4.10. These
are all of the form
T (A)! T (B)
where A ! B are generating cofibrations for sS in its Presolution model cat
egory structure. By the properties of a cofibrantly generated model category
(see Definition 2.1.3 of [25]), there is a cardinal number ~ so that the domain
of every morphism of IT is ~small relative to the class of cofibrations. This *
*is
the first cardinal we need.
We first record the following result. This is where the effective monomor
phism condition on cofibrations in cellular model categories arises.
1.5.4 Lemma. 1.) Every ITcell of a relative IT complex in sAlg T is contained
in a relative subIcell complex of size at most ~.
2.) Every ITcomplex of sAlg T is the filtered colimit of its subcomplexes of
size at most ~.
Proof. The first statement is Lemma 13.5.8 of [23]. The second statement_fol
lows from the first. __
The second cardinal we need is supplied by the following result.
We will assume for the rest of the section that we are working with a sim
plicial operad T homotopically adapted to E.
1.5.5 Lemma. There is a cardinal j so that if X is ITcell complex of size fl
in sAlg T, then ss*E*(X) has at most jfl elements.
35
Proof. By Lemma 1.4.15 the underlying degeneracy diagram of X has the prop
erty that
E*X ~=TE (M)
where M is sfree on a graded projective E*module with generating set of
cardinality fl. Furthermore, the triple TE has the property that
~=
TE (E*Z) ____//_E*T (Z)
whenever Z is Reedy cofibrant and E*Z is levelwise projective. We use these
two formulas to bound the cardinality of E*X.
Since M is levelwise projective, it is a retract of a degeneracy diagram F
of free E*modules with a generating set of the same cardinality fl. Thus we
may assume M is actually sfree on a graded free E*modules. By fixing a
set of generators we obtain an isomorphism of degeneracy diagrams M ~=E*Z
where Z is itself sfree on a graded spectrum which is a wedge of spheres in ea*
*ch
degree. Furthermore the cardinality of that set of spheres is fl. Thus we need
only bound the cardinality of E*T (Z).
If U is a graded simplicial set, we denote the card(U) to be the cardinality
of the union of all the sets that make up U. Since we are only trying to find a
bound, we will assume all cardinals are infinite.
For any operad C in simplicial sets and any spectrum W with E*W free as
an E*module there is a first quadrant spectral sequence
H*( k, E*(C(k)) (E*W ) k) =) E*(C(k) k W (k)).
From this it follows that
card[E*(C(k) k W (k))] card(E*(C(k)) . card(E*W ).
Thus, for our simplicial operad T and our chosen simplicial spectrum Z, we
have
card(E*T (Z)) card(E*T ) . card(E*Z).
But card(E*Z) card(E*) . fl. Thus we may take j card(E*(T )) . card(E*)._
__
Now let be any infinite cardinal greater than j~. Note that depends
only on IT, E*(), and T . Here is our variant of Bousfield's key lemma. See
Lemma X.3.5 of [21].
1.5.6 Lemma. If X ! Y is an inclusion of ITcell complexes in sAlg T such
that ss*E*(Y, X) = 0, then there exists a subcomplex D Y satisfying the
following conditions:
1.)D is of size less than ;
2.)D is not in X; and,
3.)h*(D, D \ X) = 0.
36
Proof. This argument is by now classic, and we won't repeat it. Bousfield's
original argument goes through verbatim, using the existence of intersections_of
subcomplexes. See [23]. __
This immediately allows one to prove the following result:
1.5.7 Lemma. Suppose q is a morphism in sAlg T with the right lifting property
with respect to any inclusion j : A ! B of ITcell complexes with B of size at
most and ss*E*(j) an isomorphism. Then q has the right lifting property with
respect to any inclusion of ITcell complexes which is a ss*E*() isomorphism.
Proof. This is a Zorn's lemma argument, and also classic. See [23], Lemma_2.4.8
or Lemma X.2.14 of [21]. __
Now let JT be a set of representatives for the isomorphism classes of inclu
sions A ! B of ITcell complexes with B of size at most and which induce an
isomorphism on ss*E*(). Recall that a JTcofibration in sAlg T is a morphism
in the class of maps containing JT and closed under retract, coproduct, cobase
change, and sequential colimits.
1.5.8 Lemma. Suppose that A ! B is a Pcofibration with Pcofibrant source
in sAlg T and a ss*E*()isomorphism. Then A ! B is a JTcofibration.
Proof. Recall (from [25]) that a JTinjective is any morphism with the right
lifting property with respect to all the elements of JT. Suppose, for a moment,
that we can show that A ! B has the left lifting property with respect to all
JTinjectives. Then, using the small object argument, we can factor A ! B as
j p
A _____//E____//_B
where j is a JTcofibration and p is a JTinjective. A standard argument now
shows A ! B is a retract of j, which is all that is required.
We now must show that A ! B has the left lifting property with respect to
all JTinjectives.
We start by choosing a cellular approximation A"! B"to A ! B. Thus,
A"! "Bis an inclusion of ITcell complexes and there is a commutative square
"A_____//A
 
 
fflfflfflffl
"B_____//B
with the horizontal maps weak equivalences. Note that A"! "Bis a ss*E*()
isomorphism. Now consider a lifting problem
"A_____//A____//X??
  
  q
fflffl fflffl fflffl
"B_____//B____//Y
37
where q is a Jinjective. By the previous lemma, we can produce a map "B! X
solving the outer lifting problem, hence a map "BtA"A ! X solving the lifting
problem under A. Since A is cofibrant, the induced map B"tA"A! B is a
homotopy equivalence; hence we have a weak equivalence between cofibrant
objects in the category of objects under A and over Y . Also, q : X ! Y
is a fibrant object in the same category, since any JTinjective is a fibration.
The original lifting problem is then solved by the following standard fact about
model categories: if C ! C0 is a weak equivalence between cofibrant objects
and C ! E is a morphism to a fibrant object, then there is a morphism C0!_E_
so that the composite C ! C0! E is homotopic to the original map. __
1.5.9 Remark. The model category sAlg T is hardly ever left proper. If it
were, we could immediately conclude that the map
"BtA"A! B
was a weak equivalence for any A and, thus, drop the hypothesis that A be
cofibrant. Then we would obtain a model category, rather than a semimodel
category in Theorem 1.5.1. This will happen, for example, in the case of the
identity operad; that is, when sAlg T = sS.
Our final technical lemma is a closure property for ss*E*()equivalences.
1.5.10 Lemma. Every JTcofibration with cofibrant source is an ITcofibration
and a ss*E*()equivalence.
Proof. Since every morphism in JT is an ITcofibration, every JTcofibration
is an IT cofibration. So we must prove that every JTcofibration is a ss*E()
equivalence. It is sufficient to show that
1.an arbitrary coproduct of elements of JT is a ss*E*()equivalence; and
2.if X oo___A __j__//B is a twosource of T algebras with A and X cofi
brant and j a cofibration and a ss*E*()equivalence, then X ! X qA B
is a ss*E*() equivalence.
Then, since ss*E*() commutes with filtered colimits, the result will follow.
For (1), let A ! B be a morphism in JT. Since this is a cofibration with
cofibrant source, Lemma 1.4.18 implies that E*A ! E*B is a cofibration of
TE algebras with cofibrant source. It is also, by assumption, a weak equivalen*
*ce
of TE algebras. Lemma 1.4.15 and the definition of what it means for an operad
to be adapted (Definition 1.4.13) next imply that if {Ai ! Bi} is a set of
morphisms JT, then
E*(qiA)! E*(qiB)
is isomorphic to
qiE*(A)! qiE*(B)
where the coproduct now is in TE algebras. Since the acyclic cofibrations are
closed under coproduct, we have that qiA ! qiB is a ss*E*()equivalence.
38
For (2), we note that the pushout X qA B is homotopy equivalent to the
homotopy pushout, which can be computed as the geometric realization of the
bar construction B(X, A, B). Here B(X, A, B) is the simplicial T algebra which,
at level n is the coproduct
B(X, A, B)n = X q Aq..q.An!qB.
The geometric realization is created in spectra and, hence, is isomorphic to the
diagonal of the bisimplicial spectrum B(X, A, B). We conclude that there is a
spectral sequence
sspssqE*B(X, A, B) =) ssp+qE*(X qA B).
Here we filter first by the external simplicial degree coming from the bar con
struction.
To finish, we assert that an argument very similar to that give for (1) impl*
*ies
that the natural map
ss*E*B(X, A, A)n !ss*B(X, A, B)
is an isomorphism. Then the spectral sequence just constructed shows ss*E*X_!
ss*E*(X qA B) is an isomorphism. __
1.5.11 Proof of Theorem 1.5.1. We specify the weak equivalences in sAlg T
to be the ss*E*isomorphisms. As above, we let IT be a generating set for the
cofibrations and we let JT a set of representatives for the isomorphism classes
of inclusions A ! B of ITcell with B of size at most fl and which induce an
isomorphism on ss*E*(). We now must show
o both IT and JT permit the small object argument;
o every JTcofibration with cofibrant source is both an ITcofibration and
an ss*E*()equivalence;
o every morphism with the right lifting property with respect to IT has the
right lifting property with respect to JT and is a ss*E*()equivalence;
o every map with cofibrant source which is both an ITcofibration and a
ss*E*()equivalence is a JTcofibration.
The first statement follows from the assumption that sAlg T is cofibrantly
generated, the second holds by Lemma 1.5.10, the third holds because ss*E*()
takes Pweak equivalences to isomorphisms, and the fourth point is Lemma
1.5.8.
39
Chapter 2
The Algebra of Comodules
2.1 Comodules, algebras, and modules as dia
grams
In Section 1.4 we introduced the notion of a homology theory E* which satis
fied a condition developed by Atiyah and Adams. (See Definition 1.4.1.) This
condition was the basis for the development of our simplicial resolutions. Now,
if E*E happens to be flat over E*, then the pair (E*, E*E) forms a Hopf alge
broid and, for any spectrum X, the module E*X is a comodule over this Hopf
algebroid. The purpose of section is to connect these two notions.
Specifically, we prove a variant of Giraud's Theorem (cf. [4] x6.8) to show
that the category of comodules over a Hopf algebroid of Adams type is equiva
lent to a category of diagrams. In particular, we will embed comodules into a
category of contravariant functors (i.e., presheaves) on some indexing category,
and show that comodules are exactly those presheaves which satisfy a conti
nuity (or sheaf) condition. We then use this to characterize various algebraic
structures in comodules in terms of such structures on presheaves.
This is section is somewhat long, mostly because of a large number of routine
 but not completely trivial  lemmas. It is included so we can discuss the kin*
*d of
algebra and module structure supported by the spiral exact sequence in section
3.1. For this application, the key result is Theorem 2.1.13 and its analog for
algebras and modules. See Corollary 2.1.21.
In this section and throughout this paper, (A, ) will be a graded Hopf
algebroid and the category C will denote left comodules. But note that the
conjugation in a Hopf algebroid induces an equivalence of categories between
left and right comodules. As a bit of notation, if N is a comodule, then kN,
k 2 Z, is the evident shifted comodule and
Hom C (M, N) = {C ( k, N)}
will denote the graded Amodule of comodule homomorphisms from M to N.
Similarly, if we need it, we will write Hom A(M, N) for the graded Amodule of
40
Amodule homomorphisms.
2.1.1 Comodules as productpreserving diagrams
2.1.1 Definition. A Hopf algebroid (A, ) is of Adamstype if
1.)The left unit jL : A ! makes a flat Amodule;
2.)There is filtered system of sub comodules i which are finitely gen
erated and projective over A and so that
colim i!
is an isomorphism.
2.1.2 Definition. A generating system J of comodules is a diagram of
comodule maps over
Cj !
so that the objects Cj are finitely generated and projective over A and the ind*
*uced
map colimJ Cj ! is an isomorphism of comodules.
2.1.3 Example. Thus if (A, ) is of Adams type, then it has a generating
system. Furthermore any diagram of comodules Cj ! over so that each
of the Cj is finitely generated and projective over A and which contains the
diagram of inclusions i ! will be a generating system. For example, we
could take as a generating system the diagram category which consists of one
representative for each isomorphism class of comodule morphisms C ! with
C finitely generated and projective over A. Morphisms would be commutative
triangles. This generating system is maximal, in an obvious sense, and closed
under the following tensor product operation. If C1 ! and C2 ! are in the
system, then the composition
C1 A C2! A m!
where m is the Hopf algebroid multiplication.
If N is a comodule which is finitely generated over A, let
DN = Hom A(N, A)
be the dual comodule. The comodule structure is that associated to the right
comodule structure of [36] Lemma A.1.16.
2.1.4 Remark. Let J = {Cj ! } be a generating system. Then, because the
comodules Cj are finitely generated and projective as Amodules, the natural
map Cj ! D(DCj) is an isomorphism of comodules. From this is follows that
for all comodules M there are natural isomorphisms
(2.1.1) colimJHom (DCj, M) ~=colimCHom (A, Cj A M) ~=M.
41
The following Lemma explains the term "generating" system.
2.1.5 Lemma. Let Cj ! be a generating system of modules. Then the
comodules kDCj are projective as Amodules and generate the category of
comodules.
Proof. In [19], x3 we showed that the comodules kD i generate. The same
argument works here. See also [27] for a cleanedup version of this proof._The_
essential fact is Equation 2.1.1. __
2.1.6 Remark. In his paper on model category structures on categories of
chain complexes in comodules [27], Mark Hovey has given a much more elegant
discussion of the role of generating systems of comodules than we have given
here. This ad hoc discussion predates his, however, and we're too tired to rewr*
*ite
at this point. It will do for now.
Let Cj ! be any generating system of comodules and let P be the full
subcategory of C which contains the objects kDCj and which is closed under
finite direct sums. Now consider the category Pre(P) of contravariant functors
F : Pop! Mod A.
Among all such functors, we single out the fullsubcategory Sh(P) of functors
which satisfy the following sheaf condition: if Q ! P is a surjection, then
(2.1.2) F (P )____//F (Q)___//_//_F (Q xP Q)
is an equalizer diagram. We will call the objects of Sh(P) sheaves.1 The inclu
sion functor Sh (P) ! Pre(P) has a left adjoint L; thus LF is the associated
sheaf. We give a concrete description of LF in the proof of Lemma 2.1.8 below.
We are mainly concerned not so much with sheaves and presheaves as the
following full subcategories.
2.1.7 Definition. Let Pre+ (P) denote the contravariant functors
F : P ! Sets
which preserves finite products in the following sense: if P ~=P1 P2, then the
natural map
F (P ) ! F (P1) x F (P2)
is an isomorphism. Morphisms in Pre+ (P) are morphisms of diagrams; hence
Pre+ (P) is a fullsubcategory of the category of Pre (P). Let Sh+ (P) be the
be the full subcategory of Pre+ (P) of objects satisfying the sheaf condition of
Equation 2.1.2; this, in turn, is a fullsubcategory of Sh(P).
____________________________1
This nomenclature can be justified by introducing a suitable topology; howev*
*er, we fore
bear.
42
Note that there is a Yoneda embedding
y* : C  !Pre+ (P)
sending a comodule M to the functor
P 7! C (P, M).
This is, in fact, a sheaf. If this is not completely obvious see the next lemma.
2.1.8 Lemma. 1.) Every object in Pre+ (P) the graded set
F ( *P ) = {F ( kP )}
has a natural structure as an Amodule.
2.) If F 2 Pre+ (P), and LF is the associated sheaf of Amodules, then
LF 2 Sh+ (P).
3.) If M 2 C is comodules, then y*M 2 Pre+ (P) is sheaf.
Proof. The first statement follows from the fact that, since F 2 Pre+ (P) pre
serves products, F ( *P ) is a right module over the graded ring End (P ) =
Hom C(P, P ), hence, an Amodule. Furthermore, the actions on Hom C (P, Q)
of End (P ) and End (Q) on the left and right, respectively, give Hom C (P, Q)
the identical structures as an Amodule; hence, any morphism P ! Q gives a
morphism F (Q) ! F (P ) of Amodules.
For the second statement, let F be a presheaf. Define a new presheaf L0F
by
(L0F )(P ) = colimQiPF (Q)
where the colimit is over all epimorphisms in P and the colimit is in Amodules.
If P 0! P is a morphism in P, then (L0F )(P ) ! (L0F )(P 0) is defined by using
the maps P 0xP Q ! P 0. If P = P1 P2 and Q ! P is an epimorphism, then
Q ~=(P1 xP Q) (P2 xP Q).
This equation and the fact that finite sums and products in Amodules are
isomorphic, imply that if F 2 Pre+ (P), then so is L0F . As usual, LF =
L0(L0F ).
For part 3, we use that colimits and finite limits in C are created in A
modules. Thus every epimorphism of comodules is, in fact, an effective epimor
phism. In formulas, this means that if Q ! P is an epimorphism of comodules,
then
Q xP Q _____////_Q_//P
is a coequalizer diagram. ___
The next result discusses limits and colimits in Pre+ (P).
43
Recall the a reflexive coequalizer in any category C is a coequalizer diagram
__d0_//
X1 _____//X0____//_X
d1
which can be equipped with a "degeneracy" s0 : X0 ! X1 so that d0s0 = d1s0 =
1.
2.1.9 Lemma. 1.) The categories Pre+ (P) and Sh+ (P) are complete and
cocomplete.
2.) Reflexive coequalizers in Pre+ (P) are created in Pre(P).
3.) The objects y*P , with P 2 P, generate Pre+ (P) and Sh+ (P).
4.) The inclusions functors Pre+ (P) ! Pre (P) and Sh+ (P) ! Sh (P)
have left adjoints. In fact, Pre+ (P) is a category of algebras over a triple on
Pre (P).
Proof. Limits and colimits in Pre (P) are constructed objectwise or "point
wise". Since reflexive coequalizers in sets commute with products, point 2.)
follows. For point 1.) note that limits and colimits in Pre+ (P) can be formed
levelwise in Amodules; then limits in Sh+ (P) can are created in Pre+ (P)
and colimits using sheafification and Lemma 2.1.8.2. For point 3, note that if
F 2 Pre+ (P), then the evident map
M M
y*P ! F
P x2F(P)
is an epimorphism in Pre+ (P). If F is a sheaf, we can sheafify the source
of this morphism. Finally point 4 follows from point 3 and the special adjoint
functor theorem; the fact that we have a category of algebras follows from_Beck*
*'s
Theorem, Theorem 10 of [4] x3.3. __
2.1.10 Lemma. The functor y* : C ! Pre+ (P) has a left adjoint y* and this
left adjoint restricts to a left adjoint to the induced functor y* : C ! Sh+ (*
*P).
Proof. This is formal. If M an Amodule and P is a comodule, define a new
comodule M A P as the evident Amodule with coproduct
M A P M_P!M A A P tP! A M A P.
There is an adjoint isomorphism
Hom A(M, Hom C (P, N)) ~=Hom C (M A P, N).
This immediately implies that y* is the coend
Z P
y*(F ) = F (P ) A P
for F either a sheaf or a presheaf. ___
44
2.1.11 Lemma. The Yoneda embedding functor
y* : C ! Sh+ (P)
is exact.
Proof. It suffices to show that y* preserves monomorphisms and epimorphisms.
It clearly preserves monomorphisms. So let q : M ! N be an epimorphism of
comodules. The induced map of sheave y*M ! y*N is an epimorphism if for
all
f 2 (y*N)(P ) = C (P, N)
there is an epimorphism j : Q ! P in P and an element
g 2 (y*M)(Q) = C (Q, M)
so that
y*(q)(g) = qg = fj = j*(f) 2 C (Q, N) = (y*N)(Q).
Form the pullback P xN M and note that the induced map P xN M ! P is a
surjection. Since the comodules DCj generate the category of comodules, there
is an epimorphism Q0! P xN M for some, possibly infinite, sum of comodules
of the form DCj. However, since P is finitely generated, there is a finite sub*
*sum
Q Q0so that the composite
Q! P xN M ! P
remains surjective. The resulting map
Q! P xN M ! M
is the morphism g required. ___
2.1.12 Proposition. Let Cj ! be a generating system of comodules and let
y* : C ! Sh+ (P)
be the associated Yoneda embedding. Then y* is an equivalence of categories.
Proof. Since all of the objects of P are finitely generated, the functor y* com
mutes with sums. The previous lemma shows that y* is exact. Next we show
that y* is full and faithful; that is, we need to show that
(2.1.3) C (Y, X) ! Sh+ (y*Y, y*X)
is an isomorphism. Regard the source and target as functors of Y . If Y is
an object in P, this map is an isomorphism by the Yoneda Lemma. Since y*
preserves sums, it is an isomorphism if Y is sum of objects of P. More generall*
*y,
we can write Y as part of an exact sequence
Y1! Y0! Y ! 0
45
where Y0 and Y1 are sums of the generators DCj, which are in P. Since y* is
exact, Equation 2.1.3 is an isomorphism for Y as well.
To finish the argument, we need to know that for every sheaf F in Sh+ (P)
there is an object M 2 C and an isomorphism of sheaves y*M ~= F . For
this category Lemma 2.1.9 implies that every sheaf is a colimit of representable
sheaves. Since y* preserves products, this implies there is a short exact seque*
*nce
of sheaves
y*Y1 f!y*Y2 ! F ! 0
where Y1 is a sum of objects in P. Since y* is full and faithful, there is a
morphism g : Y1 ! Y2 so that y*(g) = f. Let M be the cokernel of g. Then_the_
exactness of y* implies y*M ~=F . __
We can use Theorem 2.1.12 to give a formula for the left adjoint to the
Yoneda embedding y* : C ! Pre+ (P).
2.1.13 Theorem. Let J be our fixed generating system for comodules, re
garded as a category of comodules over . If F 2 Pre+ (P) then there is a
natural isomorphism of Amodules
(2.1.4) y*(F ) ~=colimCF (DCj).
Proof. We simply define a functor from Pre+ (P) to Amodules by the formula
2.1.4. This functor is exact, since the category C is filtered. By Remark 2.1.4,
there is a natural isomorphism
(y*M) ~=colimJHom (A, Cj A M) ~=M.
Now, since y* preserves sums, we can write any object F 2 Pre+ (P) in a
reflexive coequalizer diagram in Pre+ (P)
(2.1.5) y*M1 _____////_y*M0__//F
where Mi is a sum of objects in P. Since y* is full and faithful, and reflexive
coequalizers in C are created in sets (or Amodules), this implies there is a
reflexive coequalizer diagram
M1 _____////_M0__// (F ).
Since reflexive coequalizers in C are created in Amodules (or even sets) this
supplies (F ) with a natural structure as a comodule; furthermore, if F =
y*M, then this structure is isomorphic to the original structure on M.
Now, the fact the is a functor yields a natural map
Pre+ (P)(F, y*M)! C ( (F ), M).
If F = y*N this is an isomorphism. Then an exactness argument using the
reflexive coequalizer diagram 2.1.5 yields that this map is an isomorphism for
all F . Thus, the uniqueness of adjoints supplies a natural isomorphism_ (F ) ~=
y*F . __
46
2.1.14 Remark. The associated sheaf functor L : Pre+ (P) ! Sh+ (P) has a
formula in terms of comodules. Indeed, if F is a presheaf
L(F ) = y*y*F = y*colimF ( *DCi)
using Proposition 2.1.12 and Lemma 2.1.10.
2.1.2 Algebras as diagrams
We would now like to expand the notions of the previous subsection in order to
encompass algebras and modules over algebras in comodules. This is the point
of this theory, for we are working to put a module structure into the spiral ex*
*act
sequence.
Let be a triple in C and Alg or simply Alg as the category of algebras
over in comodules. We will assume that preserves surjections. Let Cj !
be a generating system of of comodules and let (P) be the full subcategory
of Alg which contains the objects ( kDCj) and which is closed under finite
coproducts and finite limits.
2.1.15 Definition. 1.) Let Pre+ ( (P)) denote the contravariant functors
F : (P)op! Sets
which preserve finite products; that is, which send finite coproducts to finite
products.
2.) Let Sh+ ( (P)) be the fullsubcategory of Pre+ ( (P)) containing the
functors which for which
F (P )____//F (Q)___//_//_F (Q xP Q)
is an equalizer for all surjections Q ! P in (P)).
Note that there is a Yoneda embedding
y* : Alg ! Sh+( (P))
sending B to the representable functor Alg (, B). Note also that the functor
P ! (P) sending P ! (P ) defines a restriction functor
r* : Pre+ ( (P))! Pre+ (P).
2.1.16 Lemma. Reflexive coequalizers and filtered colimits in Pre+ ( (P)) ex
ist and are created in Pre+ (P).
Proof. Reflexive coequalizers in Pre+ (P) are constructed pointwise in sets.
See Lemma 2.1.9. Thus, if we have parallel arrows X1 ____//_//_X0which can be
47
given a degeneracy, we can form the equalizer X in Pre+ (P). Then we have,
for each f : (Q) ! (P ) in (P) a diagram
X1( (Q)) _____////_X0( (Q))ffl//_X(O(Q))
  O
  O
fflffl fflfflffl fflffl
X1( (P ))_____////_X0( (P_))//_X( (P )).
The solid vertical arrows are induced by f and the fact that X1 and X0 are in
Pre+ ( (P)). The dotted vertical arrow exists because the horizontal rows are
equalizer diagrams of sets. We have a functor X on (P)op because each of the
maps ffl is a surjection. Finally, X preserves products because it is the equal*
*izer
in Pre+ (P).
The same argument works for filtered colimits, which are also constructed_
pointwise in sets. __
2.1.17 Lemma. The category Pre+ ( (P)) has all coproducts. Furthermore, if
{Aff= (Pff)} is a set of free objects of (P), then
ty*Aff~=y*(tAff).
Proof. We first show that the Yoneda embedding preserves coproducts. This
goes in several steps. First note that y* (P1) t y* (P2) ~=y*( (P1) t (P2)).
This is a consequence of the following isomorphism, where F 2 Pre+ ( (P)):
Pre+ ( (P))(y*( (P1) t (P2)), F~)=F ( (P1) t (P2))
~=F (P1) x F (P2)
Next, note that y* commutes with filtered colimits, since each of the objects in
(P) is small. It follows immediately that y* commutes with all coproducts.
To complete the existence of coproducts in Pre+ ( (P)) we use that any
object Fff2 Pre+ ( (P) fits into a reflexive coequalizer diagram
ty* (Qj,ff)____////_ty* (Pi,ff)//_F.
Taking the coproduct of such diagrams and applying Lemma 2.1.16 finishes_the
argument. __
2.1.18 Lemma. 1.) The categories Pre+ ( (P)) and Sh+ ( (P)) are complete
and cocomplete.
2.) The restriction functor r* : Pre+ ( (P)) ! Pre+ (P) has a left adjoint
r* with the property that if P 2 P is a generating comodule, then there is a
natural isomorphism
r*y*P ~=y* (P ).
3.) The category of Pre+ ( (P)) is the category of algebras for some triple
on Pre+ (P).
4.) The Yoneda embedding y* : Alg ! Pre+ ( (P)) has a left adjoint y*.
48
Proof. Part 1 follows from the previous two lemmas and the fact that limits are
created in Pre+ (P).
For Part 2, note that if F 2 Pre+ ( (P)), then
Pre+( (P))(y*P, r*F ) ~=r*F (P ) = F ( (P )).
We can take r*y*P = y* (P ) as the definition. To define r*G for general
G 2 Pre+ (P), write
G = colimy*P ! G y*P
and set
r*G = colimy*P ! G y* (P ).
Part 3 now follows from Lemma 2.1.16 and Beck's Theorem. See [4], x3.3.
The triple has underlying functor r*r*.
For Part 4, the adjoint y* can be written down as a coend; compare_Lemma_
2.1.10. __
The first part of this last lemma implies that the category Pre+ ( (P)) has
an initial object. In fact, one can take that object to be y* 0, where 0 is the
is the initial object in Alg .
We now turn to the relationship between the category of sheaves and the
category of algebras in comodules.
2.1.19 Lemma. The Yoneda embedding
y* : Alg ! Sh+( (P))
preserves reflexive coequalizers and coproducts.
Proof. It is a consequence of Lemma 2.1.16 that the reflexive coequalizers in
Sh+ ( (P)) are created in Sh+ (P). Now apply Lemma 2.1.11 to get the first
half of the statement. For the part about coproducts, use that every object
A 2 Alg fits into a reflexive coequalizer diagram
X1 _____////_X0_//_A
where Xi is a coproduct of objects of the form (P ) 2 (P). This is because
those objects generate the category Alg . Now apply Lemma 2.1.17 and the __
fact the y* preserves reflexive coequalizers. __
2.1.20 Proposition. The Yoneda embedding functor
y* : Alg ! Sh+( (P))
is an equivalence of categories.
Proof. The argument is essentially the same as that for Theorem 2.1.12, al
though the two arguments there using exact sequences must replaced by argu __
ments using reflexive coequalizers. The details are routine. __
49
As in Lemma 2.1.13, this result can be used to give a formula for the adjoint
to the Yoneda embedding y* : Alg ! Pre+ ( (P)).
2.1.21 Corollary. Let J be our fixed generating system for comodules, re
garded as a category of comodules over . Then, if F 2 Pre+ ( (P)) then there
is a natural isomorphism of Amodules
(2.1.6) y*(F ) ~=colimJF ( * (DCj)).
Proof. The argument is the same as for Lemma 2.1.13; one defines an auxiliary
functor by the formula of 2.1.21 and uses a succession of reflexive coequaliz*
*er_
arguments to show that is must be the adjoint. __
2.1.3 Modules as diagrams
In this section we talk about modules over algebras and how they can be de
scribed in terms of the presheaves.
We fix an object F in Pre+ ( (P)). Then an abelian object over F is a
morphism in G ! F in Pre+ ( (P)) so that the functor
Hom Pre+( (P))=F(, G) : Pre+ ( (P))op! Sets
has a chosen lift to abelian groups. As usual, this means that there are specif*
*ied
maps
~ : G xF G! G and e : F ! G
over F satisfying the evident associative, commutative, and unital diagrams.
Let Abpre+ ( (P))=F be the evident category of abelian objects over F .
2.1.22 Example. 1.) Let 2 Alg ; then the notion of an abelian object
q : E ! over can be defined the same way. However, if M is the kernel
of q, then E ~=M as Amodules, and we may as well write M o for the
algebra E. We will call M an module. Note that y*(M o ) ! y* is an
abelian group object over y* .
In the same way, abelian objects over a fixed object F 2 Pre+ ( P)) can be
identified with modules of the following sort.
2.1.23 Definition. Let F 2 Pre+ ( (P)). Then we specify an F module M by
the following data:
1.)an object M 2 Pre+ (P); and
2.)for each f : (Q) ! (P ) a map of sets
OEf : M(P ) x F ( (P ))! M(Q)
subject to the conditions that
a.)if f = (f0), then OEf(x, a) = M(f0)x;
50
b.)for any composable pair of arrows in (P),
OEgf(x, a) = OEf(OEg(x, a), F (g)a);
c.)for all a 2 F ( (P )), the function OEf(, a) is a homomorphism of abelian
groups.
The F modules form a category Mod F(P) in the obvious way.
2.1.24 Remark. If M is an F module, we form a new object M o F of
Pre+ ( (P)) by setting
(M o F )( (P )) = M(P ) x F ( (P ))
and for any morphism f : (Q) ! (P ), we set
(M o F )(f)(x, a) = (OEf(x, a), F (f)a).
The fact that M preserves coproducts and conditions a.) and b.) guarantee
that we do indeed have an object in Pre+ ( (P)). We define a multiplication
and unit for M o F
m(x, y, a) = (x + y, a)
and e(a) = (0, a). Then condition c.) implies that these give natural transfor
mations of functors and yield an abelian object over F .
2.1.25 Lemma. The functor
() o F : Mod F (P)! Abpre+ ( (P))=F
is an equivalence of categories.
Proof. We write down the inverse functor. If G ! F is an abelian object, let
M 2 Pre+ (P) be defined by the split short exact sequence of A modules
0 ! M(P ) ! G( (P )) ! F ( (P )) ! 0
and, for f : (Q) ! (P ), let OEf be defined by the composite
M(P ) x F ( (P )) ~=G( (P )) G(f)!G( (Q)) ! M(Q).
Then the evident isomorphisms G( (P )) ! M(P ) x F ( (P )) assemble to give_
an isomorphism of abelian objects over F . __
We define ShMod F(P) to be the full subcategory of those modules M for
which M o F 2 Sh+ ( (P)).
The following result is now a moreorless evident consequence of Theorem
2.1.12 and Proposition 2.1.20.
2.1.26 Proposition. Fix an algebra 2 Alg . Then the Yoneda embedding
y* : Mod ! Mod y* ( (P))
defines an equivalence of categories from Mod to ShMod y* ( (P)).
51
2.2 Thetaalgebras and the padic Ktheory of
E1 ring spectra
In this section we define and discuss the concept of a thetaalgebra, which is
the algebraic model for the padic Ktheory of an E1 ring spectrum. We also
discuss the appropriate notion of modules over such rings. The key point for our
obstruction theory is that the padic Ktheory of E1 ring spectra can be made
algebraic in the following sense. There is a forgetful functor from thetaalgeb*
*ras
to (certain) continuous Zxpmodules, and it has a left adjoint S`. Furthermore,
if X is a cofibrant spectrum such that K*X is torsion free, and C is an operad
weakly equivalent to the commutative monoid operad, then the natural map
S`(K*X) ! K*(C(X))
is an isomorphism.
Let K denote the pcomplete Ktheory spectrum. If X is any spectrum, we
define the padic Ktheory of X by the equation
K*X = ss*LK(1)(K ^ X).
Under favorable circumstances, which will nearly always apply here,
K*X = limK*(X ^ M(pk))
where M(pk) is the evident Moore spectrum. Thus, we should really adorn K*
with some sort of completion symbol, but it is the only kind of Ktheory that
we will have, so we forebear.
Note that K*X is not really a homology theory: it does not take coproducts
to direct sums of abelian groups. However, it is the appropriate analog for
homology when discussing K(1)local spectra, where K(n) is the nthMorava
Ktheory. This sort of phenomenon discussed at length in [29] and we draw
freely from that source.
As with all 2periodic homology theories, we may either regard K*X as Z
graded or Z=2Zgraded. The latter is often more convenient, but the former can
be important, for example, when keeping track of behavior under suspension.
To talk about the structure of K*X, we first need a definition. Let L0 be
the zeroth derived functor of pcompletion. Then a Zpmodule is Lcomplete
if the natural map M ! L0M is an isomorphism. If M is torsion free, this is
equivalent to being pcomplete.
If X is any spectrum, K*X is Lcomplete. Furthermore, K*X has a contin
uous action by the group Zxpof units in the padics. If k 2 Zxpthe action of k
is by the kth Adams operation _k:
_k ^ X : LK(1)(K ^ X) ! LK(1)(K ^ X).
However, not every continuous action can arise as the K* homology of some
spectrum.
52
2.2.1 Definition. Let CK*K denote the category of Lcomplete Z=2Zgraded
Zxpmodules M with the property that the quotient Z=pZ M = M=pM is a
discrete Zxpmodule. We will call this the category of K*Morava modules or
simply Morava modules.
2.2.2 Proposition. The pcompleted Ktheory K*() = ss*LK(1)(K ^ ())
takes any spectrum to a Morava module.
Proof. This follows from the facts that K1K = 0 and
K0K = Hom cont(Zxp, Zp)
~=lim colimjHom(Zx =Uj, Z=pnZ)
n p
where Uj runs over a sequence of normal subgroups so that \Uj = {1}. ___
An elementary example of a Morava module we will use often is the following:
if u 2 K2 = [S2, K] = "K0(S2) is the Bott element, then
(2.2.1) _k(u) = ku.
2.2.3 Definition. A thetaalgebra is a Z=2Zgraded continuous commutative
Zpalgebra A so that
1.For i = 0, 1, the module Ai is a Moravamodule. Write the action of k on
Ai as Adams operations _k : Ai! Ai.
2.The Adams operations _k : A ! A are linear and
8
< _k(x)_k(y) x = 0 or y = 0
_k(xy) = :
1_k k
k_ (x)_ (y) x = 1 = y.
3.There is a continuous operation ` : Ai ! Ai so that `_k = _k` for all
k 2 Zxpand
8 P p11 p
< `(x) + `(y)  s=1_ps xsypsx = 0 = y;
`(x + y) = :
`(x) + `(y) x = 1 = y.
4.`(1) = 0, where 1 2 A0 is the multiplicative identity and
8
< `(x)yp + xp`(y) + p`(x)`(y)x = 0 or y = 0;
`(xy) = :
`(x)`(y) x = 1 = y.
Thetaalgebras form an obvious category Alg`.
53
The following result explains the origin of this definition; it is implied b*
*y the
work of McClure [12], Chapter IX.
2.2.4 Theorem. Suppose that X is an E1 ring spectrum. Then K*X is nat
urally a thetaalgebra.
2.2.5 Remark. If A is a thetaalgebra, define an operation _ : A ! A by the
equation
_(x) = xp + p`(x).
If the degree of x is 1, then _(x) = p`(x). The operation _ is a continuous, li*
*near
endomorphism of A that commutes with the Adams operations; furthermore
8
< _(xy), x = 0 ory = 0;
(2.2.2) _(x)_(y) = :
p_(xy), x = 1 = y.
The operation _ is also a lift of the Frobenius in the sense that _(x) = xp
modulo p. If A is torsion free, then the operation _ also determines `; indeed,
any lift of the Frobenius that commutes with the Adams operations and has the
multiplicative properties of Equation 2.2.2 will then determine an operation `
with desired properties.
2.2.6 Example. Suppose X is a finite CW complex. Let D(X+ ) = F (X+ , S0)
denote the SpanierWhitehead dual of X with a disjoint basepoint added. Then
D(X+ ) is naturally an E1 ring spectrum and there is a natural duality isomor
phism
o : K*D(X+ ) = K*(X) def=limK*(X, Z=pkZ)
given by applying homotopy to the homotopy inverse limit of the evident maps
M(pk) ^ K ^ F (X+ , S0) ! F (X+ , M(pk) ^ K).
Note that in degree 1 this defines an isomorphism
~= 1
o : K1D(X+ )! K (X).
The morphism o is an isomorphism of graded Zpalgebras that commutes with
Adams operations _k and
o(`(x)) = `p(o(x))
where `p is in the unstable cohomology operation so that _p(x) = xp + p`p(x).
This allows for the following easy, but crucial calculation: as a thetaalgebra
K*D(S1+) ~=Zp[ffl]
where ffl = 1, _k(ffl) = kffl and `(ffl) = ffl.
In fact, the element ffl is defined to be the element which goes to the Bott
element u under the isomorphisms
K1D(S1+) ~=K1 S1 ~="K0(S2).
and we can apply Equation 2.2.1.
54
We now come to the notion of a module over a thetaalgebra.
2.2.7 Definition. Let A be a thetaalgebra. Then an Amodule is a continuous
Z=2Zgraded module M over the commutative ring A equipped with continuous
homomorphisms _k : M ! M, k 2 Zxpand ` : M ! M so that M is a Morava
module and
1.if k 2 Zxp, a 2 A, and x 2 M, then
_k(ax) = _k(a)_k(x);
2.if a 2 A and x 2 M, then
8
< ap`(x) + p`(a)`(x)a = 0 orx = 0;
`(ax) = :
`(a)`(x) a = 1 = x.
If A is a thetaalgebra, there is an evident abelian category of Amodules.
2.2.8 Remark. Suppose that A is a thetaalgebra and that M is an Amodule.
Then we can define a new thetaalgebra M o A as follows. As a module, this
algebra is M A and we give it the usual infinitesimal multiplication:
(x, a)(y, b) = (ay + xb, ab).
Define _k(x, a) = (_k(x), _k(a)) and
`(x, a) = (`(x)  ap1x, `(a)).
One easily checks this yields a thetaalgebra. Furthermore, there is an evident
short exact sequence of modules
__q__//
0_____//M_____//M o Aoos__ A ____//_0
so that s and q are thetaalgebra maps, the inclusion M ! M o A commutes
with the Adams operations and `, and M2 = 0. We will call such a diagram a
split squarezero extension or split infinitesimal extension of thetaalgebras.
This process can be reversed. If q : B ! A is an abelian group object in the
category of thetaalgebras over A, then there is a split squarezero extension
__q__//
0 ____//_M____//_Boso_A _____//0
where M is the kernel of q. This diagram gives M the structure of a module
over the thetaalgebra A and defines an isomorphism B ~=M o A. Thus, the
functor M 7! M o A is an equivalence of categories between Amodules and
abelian thetaalgebras over A.
55
2.2.9 Example. If A is thetaalgebra, then A is not a module over itself, as
` : A ! A is not linear. However, one can define a new module A with
[ A]n = An+1.
If x 2 An+1, let us write fflx for the corresponding element in [ A]n. (If it's*
* not
clear already, see Equation 2.2.3 for a reason to choose this notation.) Then we
define the action of the Adams operations by
8
< kffl_k(x),x = 0;
_k(fflx) = :
ffl_k(x),x = 1;
and the action of ` by
8
< ffl_(x),x = 0;
`(fflx) = :
ffl`(x);x = 1.
Recall that _(x) = xp + p`(x) is linear in x. The action of A on A is the
obvious one:
a(fflx) = ffl(ax).
The resulting split squarezero extension can be written
A o A def=A[ffl]
where ffl = 1 and with _k(ffl) = kffl, `(ffl) = ffl, and ffl2 = 0. This mimi*
*cs K
theory: if X is an E1 ring spectrum, then there is a natural isomorphism of
thetaalgebras
(2.2.3) K*F (S1, X) ~=(K*X)[ffl].
Indeed, the natural pairing F (S1, S0) ^ X ! F (S1, X) defines the isomorphism
~= 1
K*(S1) ^K*X ! K*F (S , X).
Compare Example 2.2.6.
2.2.10 Example. The functor () can be extended to modules as well. If A
is a thetaalgebra and M is an Amodule, define M to be the shifted graded
Zp module with 8
< kffl_k(x),x = 0;
_k(fflx) = :
ffl_k(x),x = 1;
and 8
< fflp`(x),x = 0;
`(fflx) = :
ffl`(x);x = 1.
56
Of course, if a 2 A and x 2 M, then a(fflx) = ffl(ax).
This definition of () and the one given in the previous example dovetail
in the following way. There is a split short exact sequence of M o A modules
0_____// M _____// (M o A)____//_oAo___//_0
and the action of M o A on M factors through A.
We can iterate the functor to form a functor k. For example, if M is
an Amodule, then 2nM ~=M as an ordinary Z=2Zgraded Amodule, but we
write ffl2nx for x under this identification, then
_k(ffl2nx) = knffl2n_k(x) and `(ffl2nx) = pnffl2n`(x).
We now show that we have listed all the possible operations in the pcomplete
Ktheory of E1 ring spectra. As in Definition 2.2.1, let CK*K denote the cat
egory of Morava modules. The following also follows from results of McClure
in Chapter IX of [12]. Let C be any operad weakly equivalent to the commuta
tive monoid operad. Then C() is a model for the free E1 algebra functor on
spectra. (See Theorem 1.2.4.)
2.2.11 Theorem. The forgetful functor Alg` ! CK*K sending a thetaalgebra
to the underlying module over the Adams operations has a left adjoint S`. Fur
thermore, if X is a cofibrant spectrum so that K*X is torsion free, then the
natural map
S`(K*X)! K*(CX)
is an isomorphism.
2.2.12 Remark. It is possible to write down a formula for S`. There is a
category Alg0`of continuous graded Zpalgebras equipped with an operation `
satisfying such conditions that there is a forgetful functor Alg` ! Alg0`. The
forgetful functor from Alg 0`all the way down to continuous Zp modules has
a left adjoint which, by abuse of notation, we also call S`. The abuse is not
great: if M is a continuous Zpmodule the two obvious meanings of S`(M) in
Alg 0`agree up to natural isomorphism. Calculations can now be made using
two basic facts. First, there is a natural isomorphism
~=
S`(M1) ^S`(M2)! S`(M1 M2).
The source of this isomorphism is the completed tensor product. Second, if
M = Zp with generator x we have a completed polynomial algebra
S`(Zp) ~=Zp[x, `(x), `2(x), . .].^p
if M is concentrated in degree 0, and a completed exterior algebra
S`(Zp) ~= [x, `(x), `2(x), . .].^p
if M is concentrated in degree 1.
57
For our applications, we would like to write down a model category structure
on simplicial thetaalgebras so that the cofibrant objects are sfree on a set *
*of
objects of the form S`(M), where M is a free continuous Zpmodule. This can
be done using the arguments used in [19]. We will give an outline here.
2.2.13 Lemma. Let A = {Mff} be a set with one representative for each iso
morphism class of Morava modules which are free and finitely generated as Zp
modules. Then the elements of the set A generate the category CK*K of Morava
modules.
Proof. We reduce to a simpler case. There is an isomorphism of topological
groups Zxp~=G x Zp where G is a finite cyclic group. Let C0K*Kbe the category
of continuous modules over the profinite group ring Zp[[Zp]] modules M so that
M=pnM is discrete for all n. Then there is a forgetful functor CK*K ! C0K*K
with a left adjoint given by inducing up along G. We will show CK*K has a set
of generators {Nff} where with each element free and finitely generated as a Zp
module. Since our set A includes the classes of modules obtained by inducing
up the modules Nff, the result will follow.
By choosing a topological generator fl 2 Zp, we obtain an isomorphism
Zp[[t]] ! Zp[[Zp]] sending t to fl  1. (This is an old result of Serre, and ea*
*sy to
prove.) So we can translate our problem as follows. Let M be a Zp[[t]]module
with the property that every element in M=pM has a nontrivial annihilator
ideal in Fp[[t]]. Let x 2 M. Then we show there is a Zp[[t]]module N which is
free and finitely generated as a Zpmodule and a morphism N ! M of Zp[[t]]
modules so that x is in the image. Note that we may assume that M is cyclic
as a Zp[[t]]module and generated by x.
Let I Zp[[t]] be the annihilator ideal of x. Since the annihilator ideal of
x + pM 2 M=pM must be of the form (tn) Fp[[t]] for some n, 1 n < 1, I
is nontrivial; in fact, there is a sequence of surjections
I ! I=pI ! (tn).
In particular, there is an element g(t) 2 I so that g(t) is congruent to tn mod*
* p.
If we apply the Weierstrass preparation theorem to g(t), we see we may assume
that g(t) is the of the form
tn + an1tn1 + . .+.a1t + a0
where ai= 0 mod p. Then we set N = Zp[[t]]=(g(t)), and the result follows. __*
*_
2.2.14 Remark. From the previous proof it is easy to see that each of the
elements Mffof the set of generators A of CK*K is a cyclic Zp[[Zxp]]module with
a preferred generator xff. Define a diagram of these generators by specifying
a morphism of Morava modules Mfi! Mffif and only if xfi7! xff. Then we
immediately have that for all Morava modules M, evaluation at the generators
yields a natural isomorphism
~=
colimffHomCK*K (Mff, M)____//_M.
58
Note that every object in the set A of generators is small in CK*K . Then
the arguments in [19], xx3 and 4 immediately imply the following. Give sCK*K
the standard structure of a simplicial category: see [35], xII.2.
2.2.15 Proposition. The category sCK*K of simplicial Morava modules sup
ports the structure of a cofibrantly generated simplicial model category where
f : X ! Y is
1.a weak equivalence if ss*f is an isomorphism; and
2.a cofibration if it is a retract of a morphism which is sfree on set {Zn}
of Morava modules with each Zn a coproduct of objects in the generating
set A.
Furthermore, the cofibrations are generated by the set I of morphisms
Mff @ n ! Mff n
with n 0 and Mff2 A.
2.2.16 Remark. This model category is the localization of an auxiliary model
category created from the generators Mff. Compare Remark 2.5.1.
This result and the standard lifting lemmas (in [25], for example) imply
the result we want. Similar arguments appear in [19]. Again give sAlg `the
standard structure of a simplicial category.
2.2.17 Theorem. The category sAlg `of simplicial thetaalgebras supports the
structure of a cofibrantly generated simplicial model category where f : X ! Y
is
1.a weak equivalence if ss*f is an isomorphism; and
2.a cofibration if it is a retract of a morphism which is sfree on set {S`(*
*Zn)}
of Morava modules with each Zn a coproduct of objects in the generating
set A.
We can immediately write down the following consequence of the fact that
every object in the generating set is free as a continuous Zpmodule. Give the
category sAlg Zpof simplicial commutative continuous Zp algebras the usual
simplicial model category structure of [35] xII.4.
2.2.18 Corollary. The forgetful functor from the category sAlg `of simplicial
thetaalgebras to sAlg Zppreserves cofibrations.
2.3 Homotopy pushouts of simplicial algebras
The category of simplicial algebras over a simplicial operad is often not left
proper, and we seek to give a condition which serves as an acceptable substitut*
*e.
We will state this condition in Definition 2.3.3 and then show the condition is
59
satisfied when the operad is E1 or A1 . Mandell has related results in the E1
case. See [31].
Recall that the category of simplicial algebras over an operad supports in
the standard simplicial model category structure. Thus, we let C = Co be a
simplicial operad in Rmodules and sAlg C the category of simplicial algebras
over C. This is a simplicial category in the external simplicial structure; for
example, if K is a simplicial set and X 2 sAlg C then
a
(A K)n = An
Kn
with the coproduct in Cnalgebras. Also, among the morphisms of sAlg C we
single out the free maps: a morphism X ! Y is free if the underlying morphism
of degeneracy diagrams is isomorphic to a map of the form
X ! X t C(Z)
where Z is a sfree diagram on a free Rmodule. The definition of sfree is in
Definition 1.1.9.
The main theorem of [35] xII.4 immediately implies the following:
2.3.1 Proposition. The category sAlg C has the structure of a simplicial model
category with a morphism f : X ! Y
1.a weak equivalence if ss*f : ss*X ! ss*Y is an isomorphism;
2.a fibration if the induced map Nf : NX ! NY of normalized chain
complexes in Rmodules is surjective in positive degrees;
3.a cofibration if it is a retract of a free map.
Recall that a model category is left proper if whenever there is a pushout
square
f
A _____//B
j  
fflfflfflfflg
X _____//Y
with j a cofibration and f a weak equivalence, then g is a weak equivalence.
For example, the category of simplicial associative algebras is not left proper;
see Example 2.3.4. This lends teeth to the following example.
2.3.2 Example. Let R be a commutative ring. Then the category of simplicial
commutative Ralgebras is left proper. Suppose we are given a twosource B
A ! X with A ! X a cofibration. Then, by [35], xII.6, there is a spectral
sequence
Torss*A(ss*B, ss*X) ) ss*B A X.
Since B A X is the pushout in simplicial Ralgebras, the claim follows. Exactly
the same argument shows that the category sAlg `of simplicial thetaalgebras
is left proper.
60
2.3.3 Definition. Fix a commutative ring R and a simplicial operad C of R
modules. The model category sAlg C of simplicial Calgebras is relatively left
proper if
1.whenever W is cofibrant simplicial Rmodule and f : A ! B is a weak
equivalence between simplicial Calgebras which are cofibrant as simplicial
Rmodules, then
A t C(W ) ! B t C(W )
is a weak equivalence, and
2.any cofibrant X 2 sAlg C is cofibrant as a simplicial Rmodule.
2.3.4 Example. The category of simplicial associative algebras is not left
proper, but is relatively left proper. For if C is the free associative algebra
functor, W is an Rmodule and A any associative algebra, then
M
A t C(W ) ~= A W A . .A. W A
n 0
where W appears n times and A appear (n + 1) times in the nth summand.
This follows from the fact that A t C(W ) is the free algebra under A on the
Abimodule A W A.
In order to explore the implications of this relative notion of properness, *
*we
will use the following standard observation.
2.3.5 Lemma. Let X 2 s(sAlg C) be a simplicial object in the category of
simplicial Calgebras. Then the geometric realization of X is the diagonal:
X ' diagX = {Xn,n}.
The nomenclature "relatively left proper" is justified by the next result.
2.3.6 Lemma. Let
f
A _____//B
j  
fflfflfflfflg
X _____//Y
be a pushout square in sAlg C with j a cofibration and f a weak equivalence
between objects which are cofibrant as simplicial Rmodules. Then g is a weak
equivalence.
Proof. In the simplicial model category sMod R, define objects R ^ n=@ n,
n 0, by the pushout diagram
R ~=R * ____//_R n=@ n
 
 
fflffl fflffl
0 ________//_R ^ n=@ n.
61
This forms a collection of cofibrant cogroup objects in sMR ; hence the objects
C(R ^ n=@ n) 2 sAlg C
form a set of cofibrant cogroup objects. Let
A ! Wo ! X
be a factorization of A ! X as a cofibration followed by a weak equivalence in
the resolution model category on s(sAlg C) determined by these objects. (The
bullet (o) here refers to the new, external, simplicial degree.) Then Wo ! X *
*is
a weak equivalence of cofibrant objects in the under category A=sAlg C. Since
every object of this under category is fibrant, this map is necessarily a homot*
*opy
equivalence under A. It follows that
B tA Wo ! B tA X ~=Y
is a homotopy equivalence. Thus we need only show
Wo ! B tA Wo ~=B tA Wo
is a weak equivalence. By the previous lemma, it is enough to show
Wn ! B tA Wn
is a weak equivalence. This is a retract of a morphism of the form
A t C(Z) ! B t C(Z)
L
where Z ~= R ^ nff=@ nff. This map is a weak equivalence by the definition
ff __
of relatively left proper. __
We now prove:
2.3.7 Proposition. Suppose sAlg C is relatively left proper and
A _____//B
j  
fflfflfflffl
X _____//Y
is a pushout diagram. If j is a cofibration and A, B are cofibrant in sMod R,
then Y is weakly equivalent to the homotopy pushout in sAlg C.
Proof. Choose a surjective weak equivalence A0 ! A with A0 cofibrant and, as
in the proof of Lemma 2.3.6, let
A0 ! Wo ! X
be a factorization of A0 ! X as a cofibration followed by a weak equivalence
in the resolution model category structure on s(sAlg C). By Lemma 2.3.6,
62
Wo ! X is a weak equivalence; furthermore A0 ! Wo is a cofibration.
Factor A0 ! B as A0 ! B0 ! B where the first map is a cofibration and the
second a weak equivalence  now simply in sAlg C. Then
B0 tA0 Wo ~=B0 tA0 Wo
is a model for the homotopy pushout.
There is a homotopy equivalence under A0
(2.3.1) Wn ' A0 t C(Zn)
where Zn is a sum of copies of R ^ k=@ k. From this it follows that there is
a homotopy equivalence under A
A tA0 Wn ' A t C(Zn)
and, hence,
Wn ! A tA0 Wn
is a weak equivalence. Thus, the natural map
A tA0 Wn ~=A tA0 Wn ! A tA0 X ~=X
is a weak equivalence. The last isomorphism uses that A ! A0 is surjective and
that we already have the map j : A ! X. Since A ! AtA0Wn is a cofibration
and every object of sAlg C is fibrant we have (as in Lemma 2.3.6) that
B tA0 Wo = B tA (A tA0 Wo) ! Y
is a weak equivalence. By (2.3.1) we have that for each n
B0 tA0 Wn ! B tA0 Wn
is a homotopy equivalent to
B0 t C(Zn) ! B t C(Zn).
This map is a weak equivalence and the result follows. ___
It is possible to prove that any cofibrant simplicial operad in Rmodules is
relatively left proper. We will make a remark on this below, but this result is
tangential to our project here, so we won't elaborate. More important is the
case of an E1 operad; we want any such operad to be relatively left proper.
This result can be obtained from [31], but we will give an outline here as well.
We begin a decomposition which we learned from Charles Rezk. For each
simplicial operad C, each Calgebra A, and each k 0, we claim there is a
R[ k] module DkCA so that D0TA = A and there is an isomorphism of simplicial
Rmodules
M
(2.3.2) A t C(W ) ~= DkCA R[ k]W k.
k
63
The isomorphism is natural in A, C, and W . We have D0TA ~=A.
To see this, first note that if A = C(A0) for some simplicial Rmodule A0,
then
A t C(W ) = k[ nC(n + k) R[ n]A0n ] R[ k]W k
which gives
DkCC(A0) = nC(n + k) R[ n]A0n .
For more general A, we write down a coequalizer diagram
(2.3.3) DkCC2(A) _____////_DkCC(A)_//DkCA.
The parallel arrows
nC(n + k) R[ n]C(A) n ____//_//_ nC(n + k) R[ n]A n
are given respectively by the evaluation C(A) ! A and the partial operad maps
(2.3.4) C(n + q) C(m1) . . .C(mn) ! C(m1 + . .m.n+ q).
2.3.8 Definition. Let R be a commutative ring. Then a simplicial E1 operad
over R is an augmented simplicial operad C ! Comm with the properties that
1.)The augmentation induces an isomorphism ss*C ! Comm ; and
2.)for all n, the simplicial R[ n]module C(n) is cofibrant.
The last requirement implies that C(n) is levelwise projective as a R[ n]
module.
2.3.9 Lemma. Let C be an E1 operad in simplicial Rmodules and let A be
any Calgebra. Then there is a natural zigzag of homotopy equivalences of
R[ k]modules between DkCA and C(k) A.
Proof. The operad multiplication
~ : C(2) C(n) C(k)! C(n + k)
supplies a weak equivalence between cofibrant simplicial R[ k]modules. As a
result, ~ has a homotopy inverse in this category. From this we obtain, for all
simplicial Rmodules A0, a homotopy equivalence of simplicial Rmodules
[ nC(2) C(n) C(k)] R[ n]A0n ! n C(n + k) R[ n]A0n = DkCC(A0).
The equalizer diagram in Equation 2.3.3  and the description below that equa
tion of the two maps to be equalized  now yields a homotopy equivalence of
simplicial Rmodules
C(2) C(k) A ! DkCA
for any A. Since this a morphism of simplicial R[ k]modules, it is a weak
equivalence of simplicial R[ k]modules. To complete the zigzag, take the
projection
C(2) C(k) A! R C(k) A.
Since C(2) C(k) ! C(k) is a weak equivalence of cofibrant simplicial R[_k]
modules, we obtain a homotopy equivalence. __
64
2.3.10 Remark. If C is a cofibrant simplicial operad a more delicate argument
using the language of trees analyzes DkCA and shows that sAlg C is also relativ*
*ely
left proper.
The following is immediately obvious from Lemma 2.3.9 and Equation 2.3.2.
2.3.11 Proposition. Let C be an E1 operad in simplicial Rmodules and A a
Calgebra. Then:
1.) If W is any simplicial Rmodule, there is a natural zigzag of weak
equivalences between the simplicial Rmodules A t C(W ) and A C(W ).
2.) Let B be a cofibrant Calgebra. Then there is a natural zigzag of weak
equivalences between the simplicial Rmodules A t B and A B.
3.) The model category sAlg C is relatively left proper.
Proof. For the first statement, the previous lemma supplies a natural zigzag of
homotopy equivalences between A t C(W ) and
[ kC(k) R[ k]W k] A.
For the second statement, take resolution Wo ! B of B is s(sAlg C) using
the objects C(R ^ n@ n) as the homotopy cogroup objects. (See the proof
of Lemma 2.3.6.) Then Wo ! B is a weak equivalence between cofibrant
Calgebras, hence a homotopy equivalence. Now part (1) supplies a homotopy
equivalence of simplicial Rmodules between A t Wo and A Wo. The third_
statement follows immediately from the first. __
2.3.12 Corollary. Let C be an E1 operad and suppose we are given a two
source
j f
X oo___ A ____//_B
in sAlg C with j a cofibration and A and B cofibrant as simplicial Rmodules.
Then there is a spectral sequence
Torss*Ap(ss*X, ss*B)q =) ssp+q(X tA B).
Proof. This follows immediately from Lemma 2.3.7, Proposition 2.3.11.2 and
the fact that we can use the bar construction to calculate the homotopy_push
out. __
Let f : A ! B a morphism of simplicial Rmodules. Define ss*(f) to be the
homotopy groups of the morphism. If f is a cofibration, then this is simply the
homotopy groups of the pair; more generally, it can be computed by replacing
f by a cofibration. As always, we will write ss*(B, A) when f is understood.
The following result is almost proved many places. See, for example, [6],
xI.C.4 or [43]. The wrinkle here is that we have may have a simplicial operad.
2.3.13 Theorem. Let sAlg F be either the category of simplicial algebras over a
simplicial E1 operad C, the category of simplicial thetaalgebras, or the cate*
*gory
65
of associative Ralgebras. Suppose we are given a homotopy pushout diagram
in sAlg C
f
A _____//X
j  
fflfflfflffl
B _____//Y
and, furthermore, that ssi(B, A) = 0 for i < m and ssi(X, A) = 0 for i < n.
Then
ssi(B, A)! ssi(Y, X)
is an isomorphism for i n + m  2 and onto for i = n + m  1.
Proof. The for the category of simplicial algebras over an E1 operad, we apply
the spectral sequence of Corollary 2.3.12; similarly, for simplicial thetaalge*
*bras,
apply the spectral sequence of Example 2.3.2. The case of simplicial associative
algebras is covered by [6], xI.C.4. Alternatively, we could use a bar complex_
argument and the decomposition result of Example 2.3.4. __
2.3.14 Remark. The previous result is actually true for an arbitrary simplicial
operad. This can be proved by adapting the methods of [7], Section 5. Indeed,
these methods make it clear that this result really follows from very general
considerations about functors from sets to itself.
2.3.15 Corollary. Let sAlg F be either the category of simplicial algebras over*
* a
simplicial E1 operad C, the category of simplicial thetaalgebras, or the cate*
*gory
of associative Ralgebras. Suppose we are given a pushout diagram in sAlg C
f
A _____//X
i 
fflfflfflffl
B _____//Y
and, furthermore, that ssi(B, A) = 0 for i < m and ssi(X, A) = 0 for i < n.
Then there is a partial long exact sequence
ssm+n2 (B) ssm+n2 (X) ! ssm+n2 (Y ) ! ssm+n3 (A) ! . .!.ss0(Y ) ! 0.
Proof. Given any commutative square (not necessarily a pushout), we can de
fine two modules Dn and Kn by the formulas
Dm = (j*)1Im (ssm (B, A) ! ssm (Y, X)) ssm Y
where j* : ssm (Y ) ! ssm (Y, X) is the natural map and
Km1 = ssm1 A=ffi(Ker(ssm (B, A) ! ssm (Y, X)).
where ffi : ssm (B, A) ! ssm1 A is connecting map. Note that D* and K* are
functors of the square; furthermore, Dm = ssm (Y ) if ssm (B, A) ! ssm (Y, X) is
66
onto and Km1 = ssm1 (A) if ssm (B, A) ! ssm (Y, X) is onetoone. A diagram
chase shows there is a long exact sequence
. .!.Dm+1 ! Km ! ssm (B) ssm (X) ! Dm ! Km1 ! . .!.D0 ! 0.
The result now follows easily. ___
2.4 Andr'eQuillen cohomology
If A is a commutative algebra over a commutative ring R, M an Amodule and
X ! A a morphism of Ralgebras, then the Andr'eQuillen cohomology of X
with coefficients in M is the nonabelian right derived functors of the functor
X 7! DerR(X, M)
which assigns to X the Amodule of Rderivations from X to M. This coho
mology has natural generalization to algebras over operads and their modules;
it also has a generalization to thetaalgebras and their modules. Indeed, much
of the formalism of Quillen's paper [34] goes through without difficulty  in t*
*he
thetaalgebra case the formalism is nearly identical. This section outlines the
details and gives an example of an application to the computation of the homo
topy type of the space of maps between between K(1)local E1 ring spectra.
2.4.1 Cohomology of algebras over operads
This first part is written algebraically. We fix a commutative ring R, possi
bly graded, and we consider Rmodules (again possibly graded), operads in
Rmodules, and so on. All tensor products will be over R. In our applications
R will be E* for some homotopy commutative ring spectrum E. Any omitted
details can be found in [19].
Let C be an operad in Rmodules and suppose A is a C algebra. We define
what it means for M to be an Amodule. Let (A, M) to be the graded R
module with
M
(A, M)n = A . . .A M A . . .A
i i
with each summand having nterms, M appearing once in each summand and
then in the ith slot. Note that (A, M)n has an obvious action of the symmetric
group n. Define
M M
C(A, M) = C(n) k n (A, M)n = C(n) R n1 A (n1) M.
n n
It is an exercise to show that there is a natural isomorphism of bifunctors
C(C(A), C(A, M)) ~=(C O C)(A, M)
67
where (.) O (.) is the composition of operads. The Rmodule M is an A
module over C (or simply an Amodule) if there is a morphism of kmodules
j : C(A, M) ! M which fits into a coequalizer diagram
d0 j
C(C(A), C(A, M)) ~=(C O C)(A, M) ' C(A, M) ! M
d1
where the maps d0 and d1 are induced by the operad multiplication of C, and
by j and the algebra structure on A respectively. Furthermore, the unit 1 ! C
defines a morphism of Rmodules M = 1(A, M) ! C(A, M) which is required
to be a section of j.
If A is a commutative Ralgebra, and M is an Amodule, we can can form a
new commutative algebra over A called M o A, which as an Rmodule is simply
M A, but with algebra multiplication
(x, a)(y, b) = (xb + ay, ab).
The algebra M o A is an infinitesimal extension and an abelian object in the
category of algebras over A; all abelian group objects in this category have th*
*is
form.
Now let k ! A be a morphism of commutative Ralgebras. Then A o M
represents the functor that assigns to an algebra over A the Amodule of k
derivations from A to M:
Derk(X, M) ~=(Alg k=A)(A, M o A)
where we write Alg k=A is the category of kalgebras over A; that is, Alg kis
the category of algebras over the commutative algebra operad over A and under
k.
These concepts easily generalize. If C is an operad, A a Calgebra and M
an Amodule, define a new Calgebra over A called M o A as follows: as a
Rmodule M o A is simply M A, but the Calgebra structure is defined by
noting that there is a natural decomposition
C(M A) ~=E(A, M) C(A, M) C(A)
where E(A, M) consists of those summands of C(M A) with more than one
M term. Since M is an Amodule we get a composition
C(M A) ! C(A, M) C(A) ! M A
which defines the Calgebra structure on M o A. Again we obtain an abelian
object in the category of algebras over A; again, all abelian objects have this
form. This last observation makes it possible to define the category of Amodul*
*es
over C to be the category of abelian Calgebras over A. For comparison, see
Remark 2.2.8.
Note that if we are in a graded setting and M is an Amodule, then the
graded object tM with
( tM)k = Mk+t
68
is also an Amodule. In this operadic setting, an obvious example of an A
module is A itself.
The object M oA in the category of Calgebras over A represents an abelian
group valued functor which we might as well call derivations. If k ! A is a
morphism of Calgebras and M is an Amodule, we define
(2.4.1) Derk(A, M) def=Algk=A(A, M o A).
Such a derivation is determined by a Rmodule homomorphism d : A ! M
which fits into an appropriate diagram which reduces to the usual definition of
derivation in the commutative or associative algebra case. We invite the reader
to fill in the details.
Note that the definition of derivations in Equation 2.4.1 depends on the
operad C; thus we might want C in the notation somewhere. But we hope that
C will always be implicit from the discussion, so we leave it out.
Cohomology in this context should be derived functors of derivations; for th*
*is
we need the model category structure on sAlg C discussed in Proposition 2.3.1.
We now allow ourselves the generality of a simplicial operad C in Rmodules.
If A 2 sAlg C then ss0A is a ss0Calgebra. If M is a ss0Amodule (over the
operad ss0C) then M is an Anmodule (over Cn) for all n 0. Then we can form
the simplicial module K(M, n) over A whose normalization NK(M, n) ~= M
concentrated in degree n. From this object we can form the simplicial Calgebra
KA (M, n) = K(M, n)oA over A. In following definition, we will use the notion
of relatively left proper, which appeared in Definition 2.3.3.
2.4.1 Definition. Suppose that C is a simplicial operad in Rmodules so that
the model category sAlg C is relatively left proper. Let k ! A be a morphism
of simplicial Calgebras. Let X be a Calgebra under k and over A. Then
Andr'eQuillen cohomology of X with coefficients in M is defined by
HnC(X=k, M) def=[X, KA (M, n)]sAlgk=A~=ss0map sAlgk=A(X, KA (M, n)).
Here we are writing sAlg k=A for simplicial Calgebras over A and under k
and in this formula we mean, as always, the derived mapping space. If C is
understood, we will write H*(X=k, M); if k is the initial object in sAlg C, we
may write simply H*(X, M).
We note immediately that there are natural isomorphisms
HniC(X=k, M) ~=ssimap sAlgk=A(X, KA (M, n))
and that, in fact, the collection of spaces map sAlgk=A(X, KA (M, n)), n 0,
assemble into a spectrum hom sAlgk=A(X, KA M) so that
HnC(X=k, M) ~=ssn hom sAlgk=A(X, KA M).
2.4.2 Remark. We have defined a relative Andr'eQuillen cohomology for a
morphism k ! A of simplicial Calgebras. At this level of generality, it may
69
actually be necessary to resolve the source k as well as the target A to get a
good theory. By this we mean that we ought to choose a cofibrant model k0! k
for k as a simplicial Calgebra and set
HnC(X=k, M) = ss0map sAlgk0=A(X, KA (M, n)),
again using the derived mapping space. Only in this way, for example, do
we get a transitivity sequence for this cohomology theory. (See Remark 2.4.3
below.) However, we have assumed that the category sAlg Cis relatively left
proper. Then if k is projective as Rmodule, Lemma 2.3.7, implies for all weak
equivalences f : k0! k restriction yields an adjoint pair
f* = k tk0()sAlg k0___//_sAlgok:of*_
which is part of a Quillen equivalence. Hence the more na"ive definition of
Andr'eQuillen cohomology given above agrees with the definition wherein one
also resolves k. The more general situation is discussed in [19].
2.4.3 Remark (Transitivity Sequence). As defined, there is a long exact
sequence, or transitivity sequence for Andr'eQuillen cohomology. Suppose we
have a sequence of Calgebras k ! X ! A and suppose that M is a ss0A
module. Then there is a homotopy pullback square
(2.4.2) map sAlgX=A(A, KA (M, n))____//mapsAlgk=A(A, KA (M, n))
 
 
fflffl fflffl
{s} _______________//mapsAlgk=A(X, KA (M, n))
where s is composition X ! A ! KA (M, n) induced by the zerosection. Hence
there is a long exact sequence
. .!.HnC(A=X, M) ! HnC(A=k, M) ! HnC(X=k, M)
! Hn+1C(A=X, M) ! . ...
To get the fiber sequence 2.4.2, choose a commutative square
j
X0 _____//A0
'  '
fflffl fflffl
X _____//_A
where the vertical maps are weak equivalences, X0is cofibrant in sAlg kand j is
a cofibration in sAlg k. Then, because we have assumed that sAlg C is relatively
left proper, the induced map
X tX0 A0! A
70
is a weak equivalence and the source is cofibrant in sAlg X. Then we have a
pullback
map sAlgX=A(X tX0 A0, KA (M, n))___//mapsAlgk=A(A0, KA (M, n))
 
 
fflffl fflffl
{s} __________________//mapsAlgk=A(X0, KA (M, n))
as needed.
2.4.4 Remark. In all the applications we have in mind both k and A will be
constant simplicial Calgebras, or equivalently, k and A will be ss0Calgebras,
regarded as constant simplicial Calgebras. In this case, we have a natural
isomorphism
H0C(A=k, M) = Derk(A, M).
Also, Andr'eQuillen cohomology can be written down as the cohomology of a
chain complex.
To do this, suppose k ! A a morphism of constant Calgebras. Let M be a
A = ss0Amodule. Then for any simplicial Calgebra Y under k and over A, we
have abelian groups
Derk(Yn, M) = (Alg k=A)(Yn, M o A).
Furthermore, if OE : [n] ! [m] is a morphism in the ordinal number category, the
Yn is a Cm algebra by restriction of structure along OE* : Cm ! Cn and then
OE* : Ym ! Yn
is a morphism of Cm algebras. Hence we get a map
Derk(Yn, M)! Derk(Ym , M)
and, in fact, Derk(Y, M) becomes a cosimplicial abelian kmodule. Then, if
X 2 sAlg k=A, we have
(2.4.3) HnC(X=k, M) = HnN DerC(Y, M)
where Y is some cofibrant model for X and N is the normalization functor
from cosimplicial kmodules to cochain complexes of kmodules. This concept
is important enough that we will write
(2.4.4) DC (X=k, M) 2 Ho (Ch *k)
for the welldefined object in the derived category of cochain complexes defined
by N Derk(Y, M), with Y a cofibrant model for A.
71
2.4.5 Example (Cohomology of associative algebras). This discussion
applies to the case where k ! A is a morphism of associative algebras over our
ground ring R. If k is commutative and k is central in A, then H*(A=k, M) is, by
the results of [34], closely related to the Hochschild cohomology of the kalge*
*bra
A. In this case, an Amodule is an Abimodule and there are isomorphisms
Hs(A=k, M) ~=HHs+1(A=k, M), s 1
and an exact sequence
f 0 1
0 _____//Z(M)_____//M_____//H (A=k, M)____//_HH (A=k, M)_____//0
where
Z(M) = HH0(A=k, M) = {x 2 M  ax = xa for alla 2 A }
and f sends x 2 M to the derivation @x 2 Derk(A, M) = H0(A=k, M) given by
@x(a) = ax  xa.
2.4.6 Example (Cohomology over an E1 operad). Recall that we defined
an E1 operad to be a simplicial operad C of Rmodules so that each C(k) is
a cofibrant R[ k]module and so that there is a weak equivalence of operads
C ! Comm to the commutative algebra operad.
If A is a commutative Ralgebra and M is an Amodule, we can  by using
the augmentation  regard A as a constant Calgebra and M as an Amodule
over C. Hence we may form the Andr'eQuillen cohomology groups H*C(A=k, M)
for any morphism k ! A of commutative Ralgebras. These groups turn out to
be a independent of the choice of C, and are naturally isomorphic to almost any
other version of E1 algebra cohomology of A one might possible contrive. In
particular, by work of Mandell [32] H*C(A=k, M) is isomorphic to the topological
Andr'eQuillen cohomology of the EilenbergMacLane spectrum HA regarded
as an Hkalgebra and, combining this with work of Basterra and McCarthy
[5], H*C(A=k, M) is also isomorphic to the cohomology of the kalgebra A as
defined by Robinson and Whitehouse in [42].
2.4.2 Cohomology of algebras in comodules
In our applications we will have a homology theory E*(.) and R = E*. We will
also have a simplicial operad T  that is, a simplicial object in the category *
*O of
simplicial operads  so C = E*T and a typical Calgebra will be of the form E*X
where X 2 sAlg T. If E*E if flat over E*, this will imply that we are actually
working with operads, algebras and so forth in the category of E*Ecomodules,
rather than simply in the more basic category of E*modules. Under appropriate
hypotheses  for example, if E satisfies the Adams condition of Definition 1.4.1
 the E*Ecomodule version of Proposition 2.4.7 is true, and one can use this
to define Andr'eQuillen cohomology in the category of E*Ecomodules.
72
To do this requires a little care, as we are forced to resolve not only alge*
*bras,
but also the modules; the short reason for this technical difficulty is that not
every chain complex of comodules is fibrant. The same problem arose in [30]
and our solution is not much different.
To get started, fix a simplicial operad C in E*Ecomodules and a ss0C algebra
A, also all in E*Ecomodules.
To ease notation, let us abbreviate the extended comodule functor by
(M) = E*E E* M.
The functor also induces a right adjoint to the forgetful functor from A
modules in E*Ecomodules to Amodules. Indeed, if M is an Amodule, the
module structure on (A) is determined by the top split row of the diagram
(M) ____//_ (M) o A______//Aoo_
=  _A
fflffl fflffl fflffl
(M) ____//_ (M o A)____//o(A),o_
where the right square is a pullback and where _A is the comodule structure
map, which, by assumption, is a morphism of algebras. The functor () thus
becomes the functor of a triple on Amodules in E*Ecomodules.
Let k ! A be a morphism of ss0Calgebras in E*Ecomodules and let Y be a
simplicial Calgebra under k and over A in E*E comodules. Then we can form
the bicosimplicial E*module
Derk(Y, o(M))= {Derk(Yp, q+1(M))}
= {Alg k=A(Yq, q+1(M) n A)}.
where Alg k=A is the category of Calgebras under k and over A. If X is a
simplicial Calgebra in E*E comodules under k and over A, we now write
(2.4.5) DC=E*E (X=k, M) 2 Ho (Ch *E*E)
for the object in the derived category of comodules defined by taking Y to
be some cofibrant model for A in simplicial Calgebras under k and then tak
ing the total complex of the double normalization of the cosimplicial object
Der k(Y, o(M)). Then, still assuming that the sAlg C is relatively left proper,
we define the Andr'eQuillen cohomology by
(2.4.6) HnC=E*E(X=k, M) = HnDC=E*E (X=k, M).
However, with luck, one can reduce the calculation of the comodule coho
mology to the case of module cohomology. Here is the result we will use. The
definitions should make the following results plausible; the proof is in [19].
73
2.4.7 Proposition. Let C be a simplicial operad in E*E comodules and k ! A
a morphism of ss0Calgebra in E*Ecomodules. If M is a Amodule in E*
modules, then the extended comodule (M) = E*E E* M is an Amodule in
E*Ecomodules and there is a natural isomorphism
H*C=E*E(X=k, E*E E* M) ~=H*C(X=k, M).
A stronger assertion is true: there is an isomorphism
DC=E*E (X=k, E*E E* M) ~=DC (X=k, M)
in the derived category of E*modules.
2.4.8 Remark (Comodule transitivity sequence). In this setting there is
also a transitivity sequence identical to that of Remark 2.4.3. The argument
remains the same.
2.4.3 The cohomology of thetaalgebras
Another variant on the cohomology of a commutative algebras occurs in the
context of thetaalgebras and their modules. Here we use the model category
structure on simplicial theta algebras developed at the end of x2.2. See, in
particular, Theorem 2.2.17.
Let k be a thetaalgebra and let Algk`be the category of `algebras under k.
If A is an object in Algk`and M is a `module over A. In this case, we simply
define
Hn`(A=k, M) = ss0map sAlgk`=A(A, KA (M, n))
where, as always, we are taking the derived mapping space. So in particular,
for computations, we will have to choose a cofibrant replacement X ! A for A
as a simplicial object in Algk`. As before there is welldefined object
D`(A=k, M) 2 Ho (Ch *k)
whose cohomology is H*`(A=k, M). There is a mild wrinkle here: Ho (C*k) is
the derived category of continuous kmodules.
2.4.9 Remark. One example of a thetaalgebra is the algebra Zp = K*S0.
This is the initial object in the category of thetaalgebras and we will abbrev*
*iate
H*`(A=Zp, M) as H`(A, M).
2.4.10 Remark (Thetaalgebra transitivity sequence). The cohomology
of thetaalgebras also has a transitivity sequence. The proof in [34] goes thro*
*ugh
verbatim, but we could also use the arguments of Remark 2.4.3.
This example is very closely related to the standard Andr'eQuillen coho
mology of A as a commutative kalgebra. If k ! A is a morphism `algebras
and M is module over A, then we have a module Der`k(A, M) of continuous
74
kderivations @ : A ! M which commute with the Adams operations and so
that
8
< `(@x)  xp1dx x = 0
@`(x)= :
`(@x) x = 1.
This formula is obtained by viewing the natural isomorphism
Der`k(A, M) ~=Algk`=A(A, M o A)
In the end H*`(A=k, M) are the right derived functors of Der`k.
The functor Der`k(A, ) of Amodules is representable by the Amodule on
A=k of continuous Adifferentials. This inherits a natural structure as a `
module over A and the universal derivation d : A ! A=k is a derivation for
the thetaalgebra A. As always, one derives this functor by taking a cofibrant
resolution of X ! A as a simplicial `algebra under k and setting
L`(A=k) = A X X=k
where should be interpreted as a completed tensor product. Then there is a
composite functor spectral sequence
(2.4.7) RHom sMod`A(HtL`(A=k), M) =) Hs+t`(A=k, M)
where RHom denotes the derived functors of Hom in the category of `modules
over the thetaalgebra A. More is true. Since free `algebras are free commuta
tive Zpalgebras, there is a natural isomorphism
(2.4.8) H*L`(A=k) ~=H*LA=k
where LA=k is the ordinary cotangent complex of the the completed algebra A.
In particular, if A is smooth as complete graded kalgebra, then
8
< A=k t = 0
HtL`(A=k) = :
0 t > 0
regardless of the action of ` and the module A=k is projective as a continuous*
* A
module. (Although not a projective Amodule in the category of thetamodules.)
In particular, the spectral sequence of 2.4.7 collapses and we have
(2.4.9) RHom sMod`A( A=k, M) ~=Hs`(A=k, M).
If, in addition, M is an induced `module  which in this case means it is of t*
*he
form Hom c(Zxp, M0) where M0 is some continuous Amodule  then we have a
further reduction
(2.4.10) RHom sMod`A( A=k, M) ~=ExtsA[`]( A=k, M0)
where the target Extgroup is the derived functors of continuous homomorphisms
over the ring A[`]^p. Then, since A=k is a projective Amodule
(2.4.11) Hs(A=k, M) ~=ExtsA[`]( A=k, M0) = 0, s > 1.
75
2.4.4 Computing mapping spaces  the K(1)local case
In this part, we show how to construct a BousfieldKan spectral sequence for
the mapping space of E1 ring spectrum morphisms from an E1 ring spectrum
X to K(1)local E1 ring spectrum Y . A similar spectral sequence for simplicial
T algebras in another setting was constructed in [20].
In this subsection, our E1 ring spectra will be algebras over the commu
tative monoid operad  that is, we will work with commutative Salgebras (or
simply "Salgebras", for short). This is so we have a simple description of the
coproduct in this category. By Theorem 1.2.4, this is not a loss of generality.
We begin with some preliminary results. Let K be the padic complex K
theory spectrum. Note that for any spectrum Y there is a homotopy pairing
~ : K ^ LK(1)(K ^ Y ) ! LK(1)(K ^ Y )
obtained as the unique completion of the diagram
K ^ K ^ Y ___m____//K ^ Y____//_LK(1)(K3^3Yg)
g g
K^j  g g g g
fflfflgg g
K ^ LK(1)(K ^ Y )
obtained by from the multiplication m of K and the fact that K ^ j is a K(1)*
equivalence. This yields, for any two spectra X and Y , a K"unneth map
ss0map (X, LK(1)(K ^ Y )) ! Hom K*(K*X, K*Y )
sending a morphism f to the map obtained by applying homotopy to the com
posite
LK(1)(K ^ X)_K^f_//LK(1)(K ^ (LK(1)(K ^ Y_))~_//LK(1)(K ^ Y ).
Here is a continuous version of one of the key items in the definition of
Adams's condition on ring spectra. See Definition 1.4.1. Here and below we will
specify that K*Y be pcomplete. A priori K*Y is only Lcomplete. See the
material before Definition 2.2.1. However K*Y will be pcomplete if K(1)*Y is
in even degrees or even if K*Y is torsionfree.
2.4.11 Lemma. Let X be a finite CW complex with cells in even degrees and
let Y be any spectrum so that K*Y is pcomplete. Then the K"unneth map
ss0map (X, LK(1)(K ^ Y )) ! Hom K*(K*X, K*Y )
is an isomorphism.
Proof. The result is obvious if X is a sphere. Now induct over the number_of
cells. __
76
If X and Y are commutative Salgebras, then their coproduct as a commu
tative Salgebra is isomorphic to X ^ Y . (See [18], Proposition II.3.7; the pr*
*oof
there works in any of the models of spectra with a symmetric monoidal smash
product.) In particular, if Y is an Salgebra, so is K ^ Y . Also, if X is an
Salgebra, there is a model for LK(1)X, which is also an Salgebra. (See [18],
xVIII.2; again, the argument is very general.) Thus we may conclude that if Y is
an Salgebra, so is LK(1)(K ^ Y ). More than that, we can form the augmented
cobar construction
(2.4.12) Y ! LK(1)(K(.)^ Y )
obtained from the usual cobar construction by applying the localization functor;
this will be a cosimplicial Salgebra. We will use this cosimplicial Salgebra *
*to
build our spectral sequence.
Lemma 2.4.11 has the following obvious consequence. Let C be the free
commutative Salgebra functor. As a bit of notation, if Z is a commutative
Salgebra, write map Z(, ) for the (underived) space of Zalgebra maps. Sim
ilarly, write Hom K*Z(, ) for the set thetaalgebra maps under K*Z.
2.4.12 Proposition. Let Y be a K(1)local commutative Salgebra so that K*Y
is pcomplete. Let X be a finite CW spectrum concentrated in even (or in odd)
degrees. Then the natural map
ss0map Salg(C(X), Y ) ! Hom Alg`(K*C(X), K*Y )
is an isomorphism. More generally, let Z = _Zffbe any spectrum which is
wedge of spectra Zffwith cells in even (or odd) degrees Then
ss0map C(Z)(C(Z) q C(X), Y ) ! Hom K*C(Z)(K*(C(Z) q C(X)), K*Y )
is an isomorphism.
Proof. This is routine, using Lemma 2.4.11 and Theorem 2.2.11. ___
We also have the following convergence fact.
2.4.13 Lemma. Let Y be a K(1)local Salgebra so that K*Y is pcomplete.
Then the natural map
Y ! holim LK(1)(K(.)^ Y )
is a weak equivalence of commutative Salgebras.
Proof. The natural map is a morphism of Salgebras, so we need only show it is
a weak equivalence. Under the hypotheses listed, we have from [29] Proposition
7.10(e) that there is a natural weak equivalence
LK(1)(K(.)^ Y )_'__//holimn[(K(.)^ Y ) ^ M(pn)]
where M(pn) is the mod p Moore space. Now the arguments at the end of the __
proof Proposition 7.4 of [24] imply the result. __
77
Putting this all together, we have the following result.
2.4.14 Theorem. Let Z be a commutative Salgebras and let X be a commuta
tive Zalgebra. Let Y a K(1)local commutative Zalgebra with K*Y pcomplete.
Fix a morphism OE : X ! Y of Zalgebras. Then there is a second quadrant spec
tral sequence abutting to
ssts(map Z(X, Y ); OE)
with E2term
E0,02= Hom K*Z(K*X, K*Y )
and
Es,t2= Hs`(K*X=K*Z, tK*Y ), t > 0.
Proof. Since pcompleted Ktheory is Landweber exact, we can use the resolu
tion model category structure of Theorem 1.4.9, with T = C, the commutative
monoid operad. We use Lemma 1.4.15 to compute the effect of K* on cofibrant
objects.
In the category sAlg C, form a commutative diagram
j
Zcf _____//Xcf
'  '
fflffl fflffl
Z ______//_X
where ()cf denotes a simplicial Pcofibrant replacement and the morphism j
is a Pcofibration. Now form the cosimplicial space
Mo = diagmapZcf(Xcf, LK(1)(K(.)^ Y )).
The morphism OE : X ! Y supplies this with the basepoint. Since the geometric
realization of Zcf is weakly equivalent to Z, the geometric realization of Xcf
is weakly equivalent to X, and using Lemma 2.4.13, the total space of this
cosimplicial space will be weakly equivalent to map Z(X, Y ). We now identify
the E2term.
First, since ss0K*Xcf ~=K*X and ss0K*LK(1)(K(.)^Y ) ~=K*Y , Proposition
2.4.12 implies that
ss0ss0Mo = Hom K*Z(K*X, K*Y ).
For the rest of the E2term we use a bicomplex argument.
There is a spectral sequence converging to ssp+qsstMo with
Ep,q1= ssqsstmap Zcfp(Xcfp, LK(1)(K(.)^ Y )).
Since t > 0, Proposition 2.4.12 implies that
sstmap Zcfp(Xcfp,LK*K(1)(K(q+1)^ Y ))
~=DerK cf(K Xcf, tK L (K(q+1)^ Y ))
*Zp * p * K(1)
~=DerK*Z (K*Z K cf K Xcf, tK L (K(q+1)^ Y )).
*Zp * p * K(1)
78
The augmented cosimplicial K*Y module
tK*Y ! tK*LK(1)(K(q+1)^ Y )
has a cosimplicial retraction as K*Y modules and, thus, as K*Xcfpmodules. It
follows that
8 cf t
< DerK*Z(K*Z K*ZcfpK*Xp , K*Y ) q = 0
E1p,q= : .
0 q > 0
Since K*Z K*Zcf K*Xcf ! K*X is a cofibrant resolution of K*X as a K*Z __
algebra in thetaalgebras, the result follows. __
Bousfield's work [8] on obstructions in the total tower of a cosimplicial sp*
*ace,
implies the following result:
2.4.15 Corollary. Let Z be a commutative Salgebras and let X be a com
mutative Zalgebra. Let Y a K(1)local commutative Zalgebra with K*Y p
complete. Then there are successively defined obstructions to realizing a map
f 2 Hom K*Z(K*X, K*Y ) in the groups
Hs+1`(K*X=K*Z, sK*Y ) s 1.
In particular, if these groups are all zero, then the Hurewicz map
(2.4.13) ss0(map Z(X, Y )) ! Hom K*Z(K*X, K*Y )
is surjective. If, in addition, the groups
Hs`(K*X=K*Z, sK*Y ) = 0
for s 1, the Hurewicz map of Equation 2.4.13 is a bijection.
2.5 Postnikov systems for simplicial algebras
In this section we supply a detailed description of the Postnikov systems of a
simplicial algebra. We are particular interested in simplicial algebras in simp*
*li
cial comodules over some Adamstype Hopf algebroid (A, ); therefore, we will
concentrate on this case. However, the theory is very general and will apply,
for example, to the case of simplicial thetaalgebras, as discussed in x2.2. The
primary technical input in this case will be supplied by Lemma 2.2.13, Remark
2.2.14, and Theorem 2.2.17.
The discussion parallels section 5 of [7] very closely.
Let C be the category of comodules over our fixed Adamstype Hopf al
gebroid (A, ) and let {Cj} be an arbitrary, but fixed, generating system of
comodules. (See Definition 2.1.2.) Let D() be the duality functor on co
modules which are finitely generated and projective as Amodules. (See Lemma
79
2.1.5.) Since our Hopf algebroid and comodules will be graded, let us write
M[k] for the shifted comodule obtained from M withM[k]n = Mk+n. Thus, in
the language of Example 2.2.10, we might also write M[k] = kM; however,
the bracket notation is simpler for this section.
We now consider the category sC of simplicial objects in C . In [19] we
supplied the category sC with the structure of a simplicial model category so
that
1.a morphism f : X ! Y is a weak equivalence if ss*X ! ss*Y is an
isomorphism; and
2.a morphism f : X ! Y is a cofibration if it is in the class of morphisms
generated by the set of maps
DCj[k] @ n ! DCj[k] n.
for all j, all integers k and all positive integers n.
The fibrations are determined by the lifting property and a localization ar
gument. They are not easily otherwise described.2
2.5.1 Remark. More specifically, there is an auxiliary model category structure
on sC with the cofibrations above, but we specify that f : X ! Y is a weak
equivalence or fibration if for all j and k, the induced morphism of underived
simplicial mapping spaces
map sC(DCj[k], X)! mapsC (DCj[k], Y )
is a weak equivalence or fibration. Any such weak equivalence is automatically
induces an isomorphism ss*X ! ss*Y , and it is this auxiliary model category
that gets localized.
These technicalities not withstanding, we can ground the model category
structure on sC with the following comparison result. Give the category
sMod A of simplicial Amodules its standard simplicial model category structure
[35].
2.5.2 Lemma. 1.) The forgetful functor from sC to sMod A preserves weak
equivalences and cofibrations. The extended comodule functor
A () : sMod A! sC
preserves fibrations and weak equivalences.
2.) The forgetful functor from sC to sMod A preserves fibrations.
____________________________2
A similar, but perhaps more elegant model category structure could be obtain*
*ed using
the techniques of [27].
80
Proof. 1.) The statements about the forgetful functor follow from the definition
of weak equivalence and the fact that each of the Cj is a projective Amodule.
The statements about the extended comodule functor follow from the fact that
is flat over A and an adjointness argument.
2.) Let X ! Y be a fibration in sC . For each j and k, and each s and t,
the map of simplicial sets
mapsC (DCj[k] s, X)


fflffl
map sC (DCj[k] st, X) xmapsC (DCj[k] st,Ym)apsC(DCj[k] s, Y )
is an acyclic fibration. (Here we are not using the derived simplicial mapping
spaces, but the usual mapping spaces for a simplicial category.) If K is a fini*
*te
simplicial set, then there are natural isomorphisms
colimjmapsC (DCj[k] K, X)~=colimjmapsC (A[k] K, Cj A X)
~=map sC(A[k] K, A X)
~=map sMod A(A[k] K, X).
The filtered colimit of fibrations of simplicial sets is a fibration and_the re*
*sult
follows. __
We will be interested in various categories of algebras in comodules. Let
F be a triple on sC . We are thinking of the triple TE which arises from a
homotopically adapted operad T ; see Definition 1.4.16. In particular, we could
have either the free simplicial E1 algebra functor (for a general Hopf algebro*
*id)
or the prolonged free `algebra (for pcomplete Ktheory). Let sAlg F be the
category of F algebras and will assume that the forgetful functor
sAlg F! sC
creates a simplicial model category structure on sAlg F. This model category
will automatically be cofibrantly generated and the cofibrations will be gener
ated by
F (DCj[k] @ n) ! F (DCj[k] n).
2.5.3 Remark. In Remark 2.5.1 we noted that the model category structure
on sC is the localization of an auxiliary model category structure with fewer
weak equivalences. This auxiliary structure also lifts to an auxiliary model
category structure on sAlg F and again we have a localization, at least in all
our examples. Compare [19].
2.5.4 Lemma. Suppose the triple F is a lift of a triple F0 on sMod A, and sup
pose the forgetful functor sAlg F0! sMod A creates a simplicial model category
structure on sAlg F0. Then there is a forgetful functor
sAlg F! sAlg F0
81
which preserves cofibrations and weak equivalences.
Proof. This follows immediately from Lemma 2.5.2. ___
The hypotheses of this result are satisfied in both the examples we are in
terested in.
We now come to Postnikov towers.
2.5.5 Definition. Let X 2 sAlg F. Then an nth Postnikov section of X is a
morphism f : X ! Y in sAlg F so that sskY = 0 for k > n and f induces an
isomorphism sskX ~=sskY for k n. A Postnikov tower for X is a tower under
X
X ! . .!.Xn ! Xn1 ! . .X.1! X0
so that X ! Xn is an nth Postnikov section.
Note that in a Postnikov tower for X, Xn is an nth Postnikov section of Xk
for k n.
We will see below that functorial Postnikov towers and functorial kinvarian*
*ts
exist in sAlg F. We begin with the towers.
2.5.6 Proposition. The category sAlg F has functorial Postnikov towers: for
all X 2 sAlg F there is a natural tower under X
X ! . .!.PnX ! Pn1X ! . .!.P1X ! P0X
so that for all n, PnX is an nth Postnikov section for X.
Proof. The argument here is the standard one, but with the twist that we begin
with the auxiliary model category mentioned above in Remark 2.5.3. We will
say that X ! Yn is a Postnikov section if
ssk mapsC (DCj[k], X) ! ssk mapsC (DCj[k], Yn)
is an isomorphism for k n and if the target homotopy group is zero for k > n.
There is an associated notion of a Postnikov tower and we first claim that
functorial Postnikov towers exits. This is the standard argument:
PnX = colimiPn,iX
where Pn,iX = X for i n and, for i > n, Pn,iX fits into a pushout diagram
` i _____//
W F (DCj[k] @ ) Pn,i1X
 
 
` fflffl fflffl
W F (DCj[k] i)_____//_Pn,iX.
Here W is the set of all maps F (DCj[k] @ i) ! Pn,i1X. Then Corollary
2.3.15 and the fact that map sC (DCj[k], ) commutes with filtered colimits
82
implies that X ! PnX is a natural Postnikov section. There is an evident
inclusion PnX ! Pn1X induced from the inclusions Pn,iX ! Pn1,iX, and
we obtain the natural tower.
We would now like to claim that the same tower is actually a Postnikov
tower. This follows immediately from the formula
sskX = colimssk mapsC (DCj[*], X).
___
We next write down kinvariants. For this we will need our triple F on sC
to have an augmentation F ! to a triple on C . Here is the definition of that
concept.
2.5.7 Definition. Let F be triple on sC . Then an augmentation for F is a
triple on C equipped with a natural isomorphism
dX = d : ss0F X ! ss0X
so that there are commutative diagrams
ss0X___=__//_ss0X
j  jF
fflffl fflffl
ss0F X__d__// ss0X
and
p ss0F(d)
ss0F 2X_____//ss0F (ss0F_X)_//ss0F (ss0 X)
dFX dss0X
fflffl fflffl
(ss0X)____________d__________// 2(ss0X)
where p in induced by the augmentation F X ! ss0F X and
ss0F 2X__d__// ss0F X_d__// 2ss0X
ss0fflF ffl
fflffl fflffl
ss0F X________d_________//_ ss0X.
Here j and ffl are the unit and multiplication of the respective triples. As an
abuse of notation we may write that there is an augmentation of triples F ! .
This concept fits closely with all our major examples.
2.5.8 Example. If F is the triple induced by a simplicial operad sC then we
may take to be the triple induced by the operad ss0F . The augmentation is
then the observation that there is a natural isomorphism ss0F (X) ~=ss0F (X).
83
Indeed, the forgetful functor from F algebras to to sC creates reflexive co
equalizers.
In particular, if F is a simplicial E1 operad (see Definition 2.3.8), then *
*ss0F
is the commutative algebra operad. If F is the constant associative algebra
operad, then ss0F is simply the associative algebra operad.
The other case of interest in thetaalgebras. In this case, F is the free
thetaalgebra triple, prolonged to the simplicial setting and is also the free
thetaalgebra triple.
The following result is an exercise in diagrams.
2.5.9 Proposition. Suppose F ! is an augmentation from a triple on sC
to a triple on C . Then
1.if A is a algebra in C , then the constant simplicial comodule A is an
F algebra in sC with structure morphism
p d fflA
F A _____//ss0F_A___// A_____//A;
2.if X is an F algebra in sC , then ss0X is a algebra in C with structure
morphism
d1X ss0fflX
(ss0X)_____//ss0F_X___//ss0X;
3.the functor X 7! ss0X from F algebras to algebras is left adjoint to the
functor that assigns to any algebra A the constant simplicial F algebra
A.
The existence of an augmentation not only has implications for ss0, but for
the higher homotopy groups as well. In fact, if X is an F algebra, ssiX will
be a ss0X module. For any triple and any algebra A, an Amodule M is
determined by a split extension of algebras
(2.5.1) M ____//_B____//Aoo_
with the further additional property that B is an abelian algebra over A with
unit given by the splitting.
2.5.10 Proposition. Suppose F ! is an augmentation from a triple on sC
to a triple on C and suppose that X is an F algebra. Then for all i 1, ssiX
is a module over the algebra ss0X.
Proof. If K is a simplicial set and X 2 sC , let hom (K, X) denote the internal
exponential object in sC . Since the forgetful functor sAlg F ! sC creates
the simplicial model category structure on sAlg F, if X is a fibrant F algebra,
so is hom (K, X). If K is pointed, then let hom *(K, X) be defined by fiber at
0 of the morphism hom (K, X) ! hom (*, X) = X. To obtain the result, apply
ss0() to the split extension of F algebras
hom *( i=@ i, X) ____//_hom( i=@ i, X)____//Xoo_
and apply Proposition 2.5.9.2. ___
84
Finally, for our proofs, we are going to have to assume that pushouts in
the category sAlg F are quite regular. Thus, for the rest of this monograph,
we make the following assumption. It is satisfied for all our main examples by
Theorem 2.3.13 and, in fact, for many other examples as well. See Remark
2.3.14.
2.5.11 Assumptions. The category sAlg F satisfies the following Blakers
Massey Excision property: Suppose we are given a homotopy pushout diagram
in sAlg F
f
A _____//X
j  
fflfflfflffl
B _____//Y
and, furthermore, that ssi(B, A) = 0 for i < m and ssi(X, A) = 0 for i < n.
Then
ssi(B, A)! ssi(Y, X)
is an isomorphism for i n + m  2 and onto for i = n + m  1.
When this assumption is satisfied, there is a truncated MayerVietoris se
quence in homotopy, as in Corollary 2.3.15.
We can now introduce our EilenbergMacLane objects.
2.5.12 Definition. 1.) Let A be a algebra. Then X 2 sAlg F is of type KA
if ss0X ~=A and the augmentation X ! A is a weak equivalence of simplicial
F algebras. In particular ssiX = 0 for i > 0.
2.) Let M be an Amodule and let n 1. Then a morphism X ! Y in
sAlg F is of type KA (M, n) if X is of type KA , the morphism ss0X ! ss0Y is
an isomorphism and ae
ssiY ~= M0 ii=6n= n, i > 0
This isomorphism should be as Amodules. If the morphism X ! Y is under
stood, we will simply call Y an object of type KA (M, n).
Collectively, we will call the objects of type KA and KA (M, n) Eilenberg
MacLane objects. As would be expected such objects exist; indeed, A itself,
regarded as a constant object is of type KA and if M is an Amodule, the
twisted object
K(M, n) n A
yields a morphism of type KA (M, n). Here K(M, n) is the simplicial module
whose normalization is M is degree n; this is naturally a simplicial Amodule,
and K(M, n) n A is the simplicial infinitesimal extension.
In fact, Proposition 2.5.19 below says that the moduli space of all Eilenber*
*g
MacLane objects is a space of the form BG where G is a discrete group of
automorphisms. Before proving that however, we state and prove the result
about kinvariants and pullbacks.
85
Suppose we are given a morphism X ! Y in sAlg F for which ssiX ! ssiY
is an isomorphism for i < n, n 1. Write A for ss0X ~=ss0Y and let M = ssnF
where F is the homotopy fiber of X ! Y . Then let C be the homotopy pushout
of
Y oo___X _____//P0X.
Then Assumption 2.5.11 (or Corollary 2.3.15 in our main examples) implies that
P0X ! Pn+1C is of type KA (M, n + 1). A calculation of homotopy groups now
implies the following result.
2.5.13 Proposition. If Z is the homotopy fiber of X ! Y and ssiZ = 0 for
i 6= n, then the natural diagram
X ______//P0X
 
 
fflffl fflffl
Y _____//Pn+1C
is a homotopy pullback diagram.
In other words, we have a natural formulation of the fact that there is a
homotopy cartesian square
(2.5.2) PnX ___________//_KA
 
 
fflffl fflffl
Pn1X _____//KA (ssnX, n + 1).
2.5.14 Remark. The construction we used in Proposition 2.5.13 will be re
peated throughout later sections, so we will give it a name. Given a morphism
f : X ! Y , let us write
ffin(f) : P0X ! Pn+1C
for the resulting morphism, and call it the nth difference construction. It is
natural in the morphism f.
2.5.15 Remark (The relative version). In our applications we will need to
consider the relative case where we have fixed a morphism k ! A of algebras.
In order to do this, we will assume that the category sAlg F is relatively left
proper (as in Definition 2.3.3) and that k is projective as an Rmodule. This
is to avoid the question of whether or not we have to resolve the algebra k or
not. See Remark 2.4.2. Then all of the constructions we have made so far are
valid not simply in sAlg F, but in the relative category sAlg kof simplicial F 
algebras under k. Thus we have Postnikov towers under A, for example, and we
can require that our EilenbergMacLane objects KA and KA (M, n) be objects in
sAlg kas well. The difference construction and Proposition 2.5.13 also remains
valid in sAlg kas homotopy pullbacks in sAlg kare created in sAlg F. Keeping
this in mind, we will work, for the rest of this section in this relative case.*
* Note
that a simplicial kalgebra will be an object in sAlg F under k.
86
Proposition 2.5.13 has a continuous version that is phrased in terms of modu*
*li
spaces. Let k ! A be a morphism of algebras and let Y be a simplicial k
algebra. Suppose ssiY = 0 for i > n. Let M be a ss0Y = A module and
write M(Y (M, n)) for the moduli space of all simplicial kalgebras so that
Pn1X ' Y and ssnX ~=M as an Amodule. (Neither the weak equivalence nor
the isomorphism are part of the data.) The notation using the arrows # was
defined in Example 1.1.19.
Note that we might write M(Y (M, n)) as Mk(Y (M, n)) if we want to
emphasize the role of k; however, we hope that k normally remains clear from
the context.
2.5.16 Theorem. The difference construction defines a natural weak equiva
lence
M(Y (M, n))'!M(Y # KA (M, n + 1)" KA ).
Proof. The difference construction is natural and provides a functor from the
category whose nerve defines M(Y (M, n)) to the category whose nerve defines
M(Y # KA (M, n + 1)" KA ). A natural version of homotopy pullback defines
the functor back. Then Proposition 2.5.13  which remains true in the relative
case  supplies the natural transformations needed to make these functors_into
an equivalence on nerves. __
There is a variant of these results which can be used to analyze Eilenberg
MacLane objects. Let k ! A be a morphism of algebras and M an Amodule.
If X ! Y is a morphism of type KA (M, n), then the difference construction
and Proposition 2.5.13 supplies a homotopy cartesian diagram in sAlg k
Y _______//P0Y
 
 
fflffl fflffl
P0Y _____//Pn+1C
and the morphism P0Y ! Pn+1Y is of type KA (M, n + 1). Write MA=k(M, n)
for the moduli space of all morphisms of type KA (M, n) in sAlg k.
2.5.17 Lemma. Let n 1. The assignment
{f : X ! Y } 7! {ffin(f) : P0Y ! Pn+1C}
yields a weak equivalence of moduli spaces
MA=k(M, n)'!MA=k(M, n + 1).
Proof. The functor back takes sends a morphism f : X ! Y of type KA (M, n+
1) to the homotopy pullback of the twosink
f f
X _____//Yoo___X.
___
87
We now analyze the uniqueness of EilenbergMacLane objects. We assuming
we have an augmented triple F ! and we are keeping mind the results of
Propositions 2.5.9 and 2.5.10. Anything labeled "EilenbergMacLane" should
represent cohomology and that is indeed the case here. Let k ! A be a mor
phism of algebras and M an A module. Let X be a simplicial F algebra
under k equipped with an augmentation X ! A. Recall that we can define the
Andr'eQuillen cohomology of X with coefficients in M by the formula
(2.5.3) HnF(X=k, M) = ss0map sAlgk=A(X, KA (M, n))
~=sstmap sAlg (X, K (M, n + t)).
k=A A
Here sAlg k=A is the category of simplicial F algebras under k and over A. The
following is now immediately obvious.
2.5.18 Lemma. Let k ! A be a morphism of algebras and M an A module.
Let X be a simplicial F algebra under k. Then there is a natural isomorphism
a
ss0map sAlgk(X, KA (M, n)) = HnF(X=k, M).
f:ss0X!A
Only slightly more complicated is the following result. If A is an algebra a*
*nd
M is an Amodule, the group Aut(A, M) of automorphisms of the pair (A, M)
is defined to be the group of automorphisms in the category of algebras of the
diagram
M n A _____//A.oo_
For example, if A is a commutative algebra, this is equivalent to specifying
an algebra automorphism f : A ! A and an isomorphism of abelian groups
OE : M ! M so that OE(ax) = f(a)OE(x) for all a 2 A and x 2 M.
In the following result, recall that sAlg kis the category of F algebras un*
*der
a fixed F algebra k.
2.5.19 Proposition. 1.) Let k ! A be a morphism of algebras and Autk(A)
the group of automorphisms of A under k as a algebra. If MA is the moduli
space of all objects in sAlg kof type KA , then there is a weak equivalence
MA ' B Autk(A).
2.) Let k ! A be a morphism of algebras and M an Amodule. Let
Aut k(A, M) denote the group of automorphisms of the pair (A, M) under k. If
MA=k(M, n) is the moduli space of all morphisms in sAlg kof type KA (M, n),
then there is a weak equivalence
MA=k(M, n) ' B Autk(A, M).
In particular, this moduli space is connected and any object of MA=k(M, n)
represents Andr'eQuillen cohomology.
88
Proof. The first claim follows immediately from the definition of type KA ,
Proposition 2.5.9.3, and the case M = 0 of the previous lemma.
For the second claim, let us write MA=k(M, n) = Mn for n 1. Because
of Lemma 2.5.17 we need only calculate M1. Let M0 be the moduli space of
pairs of the form KMnA AE KA ; that is diagrams of the form
Y AE X
of simplicial F algebras under k so that Y and X have trivial higher homotopy
there is an isomorphism of algebras from ss0Y AE ss0X to M n A AE A. An
easy calculation shows M0 ' B Autk(A, M). We establish a weak equivalence
M1 ' M0.
If X ! Y is a morphism of type KA (M, 1), we take the homotopy pullback
of X ! Y X to get a morphism Y 0! X  with the evident section  of
the form KMnA AE KA . This gives the map M1 ! M0. To get map back let
Y AE X be a morphism of the form KMnA AE KA and let M0 be the kernel of
ss0Y ! ss0X and form
Kss0X! K(M0, 1) n Kss0X
That these two functors have natural transformations to the identity is an ex *
* __
ercise left to the reader. Or see the proof of Proposition 6.5 of [7]. *
* __
2.5.20 Remark. This last result provides an equivalence of moduli spaces
(2.5.4) MA=k(M, n)'!M(KA (M, n)" KA ).
In particular, MA=k(M, n) is connected and any morphism of kalgebras of type
KA (M, n) is weakly equivalent (although not canonically) to KA ! KA (M, n).
Combining this statement with the pullback diagram of Proposition 2.5.13 and
the isomorphism of Lemma 2.5.18 we have the following statement: if X is
a simplicial F algebra under the constant simplicial F algebra k, then the k
invariants of the Postnikov tower of X lie in
Hn+1F(PnX=k, ssnX).
By Proposition 2.5.10 we know that ssnX is, in fact, a ss0Xmodule.
We record the following result for later use. Recall that all the moduli spa*
*ces
we are considering are built in the category sAlg kof F algebras under k.
2.5.21 Lemma. Let k ! A be a morphism of algebras, let M be an Amodule
and let m 1. Then there is a commutative square with horizontal maps weak
equivalences
M(KA #KA (M, n)) ___'___//_M(KA #KA (M, n + 1))
 
 
fflffl ' fflffl
M(KA (M, n)) ______//M(KA #KA (M, n + 1)" KA ).
89
The left vertical map sends X ! Y to Y and the right vertical map sends a
morphism X ! Y to X ! Y X.
Proof. This is a combination of Theorem 2.5.16, Lemma 2.5.17, and the equiv_
alence 2.5.4. __
We now investigate the homotopy type of the one of the spaces that arises
here.
Let k ! A be a morphism algebras, M an Amodule and B a simplicial
F algebra under k.
2.5.22 Proposition. There is a homotopy fiber sequence
`
HnF(B=k, Mf) _____//M(B# K (M, n)" A)__p__//M(B) x B Aut(A, M)
f A
where f : ss0B ! A runs over all algebra isomorphisms under k and Mf
indicates the ss0Bmodule induced by f
Proof. We will identify the fiber of the arrow p as
map sAlgk(X, KA (M, n))
and then apply Proposition 2.5.22.
As in the proof of Proposition 1.1.17, the fiber of the morphism p is the
nerve of the category of diagrams
KA (M, n)oo____A
' '
' fflffl fflffl
B oo___ U ________//_Voo______W
However, the functor that takes such a diagram to the diagram
B oo'__U _____//Voo'__KA (M, n)
induces an equivalence of categories and the result follows from Example_1.1.15.
__
Finally, we specialize to the case where B = A. The following is an easy
consequence of the previous result, the fact that M(A) = B Autk(A), and the
fact that Autk(A) acts freely on ss0map 0(A, KA (M, n)). Recall that
^HnF(A=k, M) = E Aut(A, M) xAut(A,M)Hn(A, M)
is the Borel construction of the natural action of Autk(A, M) on the Andr'e
Quillen cohomology space.
90
2.5.23 Corollary. There is a homotopy fiber sequence
HnF(A=k, M) _____//M(A# KA (M, n)" A)__p__//B Autk(A, M)
and the induced action of Autk(A, M) on HnF(A=k, M) is the natural action on
the Andr'eQuillen cohomology space. Furthermore, there is a weak equivalence
M(A# KA (M, n)" A) ' ^HnF(A=k, M).
91
Chapter 3
Decompositions of Moduli
Spaces
3.1 The spiral exact sequence
The spiral exact sequence displays the relationship between two different sets
of homotopy groups that can be defined on a simplicial T algebra in spectra.
The existence of this exact sequence and its properties are discussed in [17]
and [7] and this section is an amalgamation of those two papers. The added
value here and the whole reason for running through these ideas once again
is so that we can prove Corollary 3.1.18, which displays a localized version of
the more traditional spiral exact sequence. This version is at the heart of our
computations.
3.1.1 Natural homotopy groups and the exact sequence
We give ourselves a model category C and a set P of small projectives, in the
sense of Bousfield  all as discussed in section 1.4. We also assume enough
that we get the Presolution model category on sC; this is a simplicial model
category. Given P 2 P, there are two notions of homotopy groups for objects
in sC. First, if X 2 sC, we can form the simplicial abelian group [P, X], where
[, ] denotes the morphisms in the homotopy category of C. We can then take
the homotopy groups of this simplicial abelian group:
ssi[P, X] def=ssissP (X).
These are the homotopy groups used to define the weak equivalences in the
Pmodel category structure. On the other hand, we can form the simplicial
mapping space map (P, X), where we now regard P as a constant simplicial
object in sC and, as always, we either assume that X is Pfibrant or we take
the derived mapping space. Because the objects of P are homotopy cogroup
92
objects, this mapping space has a basepoint given by the morphism
P ! OE ! X
where OE is the initial object. Define the natural homotopy groups by
ssi,PX def=ssimap (P, X).
These natural homotopy groups are representable. If K is any pointed simplicial
set and P 2 P, define P ^ K by the pushout diagram
P *_____//P K
 
 
fflffl fflffl
OE *____//P ^ K.
Then there is a natural isomorphism
ssi,PX ~=[P ^ i=@ i, X]P
where the symbol [, ]P means homotopy classes of maps in the Presolution
model category structure. In contrast, the homotopy groups ss1ssP (X) do not
seem to be representable. (The groups ssissP (X) are representable if i 6= 1. S*
*ee
[17].)
The representability of ssi,P() suggests a construction. Let K be a pointed
simplicial set and let C=OE be the arrow category of objects in C equipped with
an augmentation Z ! OE to the initial object. Then we have defined a functor
() ^ K : C=OE! sC.
This functor has a right adjoint CK (). Indeed the functor from C to sC which
assigns Z K to Z has a right adjoint given by the zeroth object in the expo
nential object
(3.1.1) MK X def=hom(K, M)0.
If * is "onepoint" simplicial set, then CK X is defined by the pullback diagr*
*am
(3.1.2) CK X _______//_MK X
 
 
fflffl fflffl
OE______//M*X = X0.
The construction of CK X is natural in K; in other words, we have a bifunctor
C()() : sSetsop*x sC ! C=OE.
93
where sSets*is the category of pointed simplicial sets. Note also that if K ! L
is a cofibration of simplicial sets, then there is pullback diagram
(3.1.3) CL=K X ______//MLX
 
 
fflffl fflffl
OE_______//_MK X.
An important aspect of this construction is the following:
3.1.1 Lemma. 1.) Let A ! B be an acyclic cofibration in C and K ! L a
fibration of simplicial sets. Then
A L tA K B K ! B L
and
A ^ L tA^K B ^ K ! B ^ L
are acyclic Reedy cofibrations.
2.) Suppose X 2 sC is Reedy fibrant and K ! L is a fibration of pointed
simplicial sets. Then the morphism
MLX ! MK X
is a fibration in C and the morphism
CLX ! CK X
is a fibration in C=OE with the fiber at OE ! CK X naturally isomorphic to CK=L*
* X.
Proof. The first statement is simply a matter of inspection. The second state
ment follows from an adjointness argument, using the first statement. Alterna_
tively, combine the diagrams 3.1.2 and 3.1.3. __
To shorten notation, we define
CnX def=C n= n0X
ZnX def=C n=@ n X.
Then the morphism
d0 : n1=@ n1 ! n= n0
and Lemma 3.1.1 define  at least for X Reedy fibrant  a fibration sequence in
C=OE
(3.1.4) ZnX ____//_CnX_d0__//Zn1X.
The following lemma starts the calculations. If A is a simplicial abelian
group, let NA be its normalized chain complex.
94
3.1.2 Lemma. Let X be a Reedy fibrant object in sC.
(1)For all P 2 P projective, there is a natural isomorphism [P, CnX] ~=
Nn[P, X];
(2)If P 2 P is a projective, then there is a natural exact sequence
[P, Cn+1X]_d0_//_[P, ZnX]___//ssn map(P, X) ! 0
Proof. The cofiber sequence
n0! n ! n= n0
of simplicial sets yields, using Lemma 3.1.1, a fibration sequence
(3.1.5) CnX ! Xn ! M n0X.
Furthermore
[P, M n0X] ! M n0[P, X]
is an isomorphism, by the standard induction argument. (See [21], VIII.1.8, for
the cosimplicial analog.) The fibration sequence of 3.1.5 yields a short exact
sequence
0 ! [P, CnX] ! [P, Xn] ! [P, MnX] ! 0
and part (1) now follows.
For (2), note that the adjoint isomorphism
Hom C(P, ZnX) ! Hom sC(P ^ n=@ n , X)
and Lemma 3.1.1.1 yields a well defined map
[P, ZnX]! ssn map(P, X).
Since any element in ssn map(P, X) is represented by an element P ^ n=@ n !
X, this morphism is onto. If P ^ n=@ n ! X represents the zero object in __
ssn map(P, X), then it automatically extends over P ^ n+1= n+10. __
3.1.3 Corollary. There is a natural isomorphism
~=
ss0ssP (X) ! ss0,PX
Proof. This is case n = 0 of Lemma 3.1.2.2. ___
We now get a set of long exact sequences
. .!.[ P, Zn1X] ! [P, ZnX] ! [P, CnX] ! [P, Zn1X]
95
which can be spliced together into an exact couple
(3.1.6) [ q+1P, Zn1X] ` ` ` ` ` ` `//[ q+1P, ZnX]
ffMMM s
MMM sss
MMM sss
M yysss
[ q+1P, CnX]
Using Lemma 3.1.2 we immediately see that the first derived long exact se
quences of this exact couple yield the spiral exact sequence:
3.1.4 Proposition. For all P 2 P and all Reedy fibrant X in sC there is a long
exact sequence
. .!.ssi+1ssP (X) ! ssi1, PX! ssi,PX ! ssissP X !
. .!.ss0, PX ! ss1,PX ! ss1ssP X ! 0.
For the rest of the section, it is convenient to write
ss*(X; Pd)ef=ss*ssP (X)
ss"*(X; Pd)ef=ss*,P(X)
in order to avoid very complicated subscripts.
The long exact sequences of Proposition 3.1.4 can be spliced together to give
a spectral sequence
(3.1.7) ssp(X; qP ) =) colimkss"k(X; p+qkP ).
using the triangles
(3.1.8) ss"p1(X; q+1P_)_________//_ss"p(X; qP )
eeK
K K www
K www
K www
ssp(X; qP )
as the basis for an exact couple. Here and below the dotted arrow means a
morphism of degree 1. In the basic case when C = S is the category of spectra
and P = PE is the set of projective arising from an Adamstype homology
theory (see Definition 1.4.2), this is actually a very familiar spectral sequen*
*ce
in disguise, as we now explain.
So let us assume we are working with spectra and simplicial spectra and that
P = PE .
We may assume that X is Reedy cofibrant spectrum, and let sknX denote
the nth skeleton of X as a simplicial spectrum. Then geometric realization
96
makes {sknX} into a filtration of X and the standard spectral sequence of
the geometric realization of a simplicial spectrum is gotten by splicing the to
gether the long exact sequences obtained by apply the functor [ p+qP, ] to the
cofibration sequence
skp1X! skpX! p(Xp=LpX).
If we let
[ p+qP, skpX](1)= Im{[ p+qP, skpX]! [ p+qP, skp+1X]}
then the first derived long exact sequence of this exact couple is
(3.1.9) [ p+qP, skp1X](1)________//_[ p+qP, skpX](1)
ffMM r
M rrr
M rrr
M yyrrr
ssp[ qP, X]
and we obtain the usual spectral sequences
(3.1.10) ssp(X; qP ) = ssp[ qP, X] =) [ p+qP, X].
Thus the two spectral sequences have isomorphic E2terms. More is true. The
next result says that the two exact couples obtained from the triangles of 3.1.8
and 3.1.9 are isomorphic; hence, we have isomorphic spectral sequences and we
can assert that geometric realization induces an isomorphism
~= p+q
colimkss"k(X; p+qkP ) ! [ P, X].
3.1.5 Lemma. Geometric realization induces as isomorphism between the spiral
exact sequence
. .!.ss"p1(X; q+1P ) ! ss"p(X; qP ) ! ssp(X; qP ) ! . . .
and the derived exact sequence
. .!.[ p+qP, skp1X](1)! [ p+qP, skpX](1)! ssp[ qP, X] ! . . .
Proof. We construct a map between the exact sequences which induces an iso
morphism ssp(X; qP ) ~=ssp[ qP, X]. Once that is in place, the five lemma and
an induction argument show that we must have an isomorphism. To do this,
we write down the map
Hom S(Z, CK X) ~=Hom sS(Z ^ K, X)! Hom S(Z ^ K, X).
This does not induce a map out of the triangle of 3.1.6; however, after taking *
*first
derived triangles, we get a morphism from the triangle of 3.1.8 to the triangle_
3.1.9, as required. __
97
3.1.6 Remark. Lemma 3.1.5 implies that we have a spectral sequence
ssp[ qDEff, X] =) [ p+qDEff, X]
where the Effare the finite cellular spectra so that colimEff' E. Taking the
colimit of ff, as in Remark 1.4.8 we get a spectral sequence
sspEq(X) =) Ep+qX.
Lemma 3.1.5 implies that this is the usual homology spectral sequence of a
simplicial spectrum.
3.1.2 The module structure
The spiral exact sequence is natural in X and P and the naturality in P leads
to the module structure of the exact sequence. To be concrete, we will limit
ourselves to the situation which will arise here, but there are possibilities f*
*or
almost infinite generalization. Thus in our basic case we will work with spectra
and P = PE as in Definition 1.4.2.
Thus we will have a simplicial operad T that is homologically adapted to E*
and so that the resulting triple TE on E*E has an augmentation TE ! . The
notion of homologically adapted was defined in Definitions 1.4.13 and 1.4.16.
The notion of an augmented triple was defined in Definition 2.5.7. In particula*
*r,
we have a triple on E*Ecomodules so that if X is a T algebra, then ss0E*T
is a algebra. See Example 2.5.8 and Propositions 2.5.9 and 2.5.10.
3.1.7 Example. Here are the main examples:
1.In the case where T is the constant simplicial commutative monoid operad
(so that a T algebra is a simplicial E1 ring spectrum) and E* = K* (p
completed Ktheory), then is the free `algebra functor.
2.In the case when T is a simplicial E1 operad and E* is arbitrary, then
is simply the graded commutative algebra functor. Recall that T is a
simplicial E1 operad if for all k the space T (k) is contractible and if *
*the
action of k on T (k) is levelwise free.
3.In the case when T is the constant simplicial associative monoid operad
(so that T algebras are simplicial A1 ring spectra), we can take to be
the associative algebra operad.
Now let T (P) be the category with objects the simplicial T algebras T (P ),
P 2 P (regarded as constant objects) and morphisms all classes of morphisms
of T algebras in the Presolution homotopy category obtained from Theorem
1.4.9. Let Pre+ (T (P)) be the product preserving presheaves of sets on T (P)
(there are no sheaves).
3.1.8 Example. The main example we have of an object in Pre+ (T (P)) is
T (P ) 7! ss0map T(T (P ), X) ~=ss0,PX
98
when X is a (fibrant) simplicial T algebra. Let ss0,*X denote this object in
Pre+ (T (P))
If we let P stand (by abuse of notation) for the category with objects P and
morphisms all homotopy classes in spectra. There is a forgetful functor
Pre+ (T (P))! Pre+ (P).
given by restricting along the functor T : P ! T (P). In particular, we see that
for each P 2 P and each object F 2 Pre+ (T (P)), the set F (T (P )) is actually
an abelian group. However, not every transition function F (T (P )) ! F (T (Q))
need be a homomorphism of abelian groups.
We would like to regard the objects of Pre+ (T (P)) as algebras of a certain
sort. In Section 2.1.2 we showed that there was an equivalence of categories
y* : AlgE*E ! Sh+ ( (E*P))
where Sh+ ( (E*P)) Pre+ ( (E*P)) was a fullsubcategory satisfying a de
scent (or sheaf) condition. The functor y* is the Yoneda embedding
A 7! Hom (, A).
Less formally, the left adjoint to this equivalence was given by
y*G = colimffG(E* *DEff).
See Lemma 2.1.21 for an exact statement. This functor extends to a functor
y* : Pre+ ( (P)) ! AlgE*E .
The functor
ss0E*() : T (P)! (E*P)
guaranteed by our assumptions defines a restriction functor
Pre+ ( (E*P)) ! Pre+ (T (P))
which has a left adjoint given by left Kan extension. This yields a composable
pair of functors
*
Pre+ (T (P))LKan//_Pre+( (P))_y__//AlgE*E
By abuse of notation we write y* : Pre+ (T (P)) ! AlgE*E for this composite
functor as well; it is left adjoint to the functor
A 7! Hom (ss0E*(), A).
3.1.9 Lemma. This composite functor y* : Pre+ (T (P)) ! AlgE*E is isomor
phic to the functor
F 7! colimffF (T ( *DEff)).
99
Proof. Let us drop the suspensions from the notation. After dissecting the
definitions, we find that the composite is given by the coend1
Z T(P)
F (T (P )) (E*P ).
Since (E*P ) ~=ss0E*T (P ), we can write
(E*P )~= colimiss0map (DEi, T (P ))
~= colimiss0map T(T (DEi), T (P )).
Thus, evaluation gives a map
Z T(P)
ffl : F (T (P )) (E*P ) ! colimiF (T (DEi)).
The claim is that this natural map is an isomorphism. It clearly is if F is a
representable of the form
F () = ss0map T(, T (P )).
Since the coend and the colimit commute all colimits in F , this implies that
ffl is an isomorphism if F is a coproduct of representables. The general case
follows, since every F is the coequalizer of a pair of maps between coproducts_
of representables. __
Modules over algebras can be defined as abelian objects in an over category
and a similar definition applies to the objects in Pre+ (T (P)); see Proposition
3.1.11 below. However, we can offer a more concrete definition exactly as in
Definition 2.1.23. Only the base category on which our contravariant functors
has changed.
3.1.10 Definition. Let F 2 Pre+ (T (P)). Then we specify an F module M by
the following data:
1.)an object M 2 Pre+ (P); and
2.)for each f : T (Q) ! T (P ) a map of sets
OEf : M(P ) x F (T (P ))! M(Q)
subject to the conditions that
a.)if f = T (f0), then OEf(x, a) = M(f0)x;
b.)for any composable pair of arrows in T (P),
OEgf(x, a) = OEf(OEg(x, a), F (g)a);
____________________________1 `
If X is set and A is any category with coproducts, then X A = X A.
100
c.)for all a 2 F (T (P )), the function OEf(, a) is a homomorphim of abelian
groups.
The F modules form a category Mod F(P) in the obvious way.
If M is an F module, we form a new object M o F of Pre+ (T (P)) exactly
as in Remark 2.1.24 and then we have the analog of Proposition 2.1.25. The
proof remains the same.
3.1.11 Proposition. The functor
() n F : Mod F (P)! Abpre+ (T (P))=F
is an equivalence of categories.
3.1.12 Remark. If M is an F module, then then there is a split projection of
algebras
y*(M n F )_____//y*Foo_
which defines a module y*M over the algebra y*F . In our examples, this will
actually be an ordinary module over the ring y*F , perhaps with some additional
structure if the operation ` is present. Lemma 3.1.9 implies that the module
y*M has a simple formula
y*M = colimiM( *DEi).
3.1.13 Example. Let F 2 Pre+ (T (P)). Then F is not a module over itself,
but there are modules nF , for n 1 and these modules play a very important
part in this discussion. For any spectrum X set n+X denote the spectrum
Sn+^ X, where Sn+is the topological nsphere with a disjoint basepoint. If
P 2 P, then n+P 2 P. Then, if F 2 Pre+ (T (P)), we define a new object
n+F 2 Pre+ (T (P)) by the formula
n+F (T (P)) = F (T ( n+P )).
The evident split short exact sequence
0_____// nF_____// n+F____//Foo__//_0
defines nF and its module structure over F . Note that as an abelian group
nF (P ) = F (T ( nP )).
If M is an E*Ecomodule we can define the shifted E*E comodule nM by
the formula [ nM]k = Mn+k. (In Section 2.5 we called this module M[n].) If
M is a module over the algebra A, then so it nM. Now one easily checks
that
y* nF ~= n(y*F )
as a module over algebra y*F .
101
3.1.14 Example. 1.) If X is a simplicial T algebra, then
T (P ) 7! ssn mapT (T (P ), X) = ssn,PX
is a ss0,*X module which we will call ssn,*X. In fact, the natural split cofibr*
*ation
sequence of simplicial T algebras
T (P )____//To(Po_ n=@ n_)___//T (P ^ n=@ n )
yields the abelian object over ss0,*X necessary to display ssn,*X as a module:
ssn,*X____//_ss0map T(T (* ^ n=@ n ),_X)//_ss0,*Xoo_
An immediate consequence of these observations is that y*ssn,*X = ssn,*EX has
a natural structure over the algebra y*ss0,*X = ss0E*X.
2.) Slightly less obvious is that ssnss*X = ssn[, X] is also a module over
ss0,*X ~= ss0ss*X, for n > 0. To see this, let T (P)Reedydenote the category
with objects T (P ), P 2 P and morphisms the Reedy homotopy classes of maps
in simplicial T algebras. Then C0[, X] 2 Pre+ (T (P)Reedy) and Lemma 3.1.2
implies that the functor Cn[, X] is an object in Mod C0[,X](T (P)Reedy). The
projection functor T (P)Reedy!T (P) gives a restriction functor
Pre+ (T (P))! Pre+ (T (P)Reedy)
and this gives ss0ss*X the structure of an object in Pre+ (T (P)Reedy). The
fact that the categories of modules have kernels and cokernels now imply that
ssnss*X is an object in Mod ss0ss*X(T (P)Reedy). We now have to argue that it
actually descends to an object in Mod ss0ss*X(T (P)). Because the morphisms
f : T (Q) ! T (P ) in T (P)Reedy(or T (P) for that matter) form an abelian grou*
*p,
it is sufficient to show that if f descends to the trivial morphism T (Q) ! T (*
**) !
T (P ) in T (P), then the induced morphism on ssnss*X is trivial. But we have a
factoring
0
T (Q ^ n=@ n )__d__//T (Q n+1= n+10)
f^ n=@ n  
fflffl fflffl
T (P n=@ n )______//T (P n=@ n )0
where ()0means some functorial fibrant replacement. The claim follows.
An immediate consequence of these observations is that y*ssnss*X = ssnE*X
has a natural structure as a module over the algebra ss0E*X.
The main result on module structures is the following:
3.1.15 Theorem. Let X 2 sAlg T be a fibrant simplicial T algebra. Then the
isomorphism
ss0,*X ! ss0ss*X
is an isomorphism of objects in Pre+ (T (P)) and the the spiral exact sequence
is naturally an exact sequence of ss0,*Xmodules.
102
The proof is exactly the same as for Proposition 7.13 of [7]. Since it is te*
*dious
we won't give it here.
We now come to the main result. In order to state it, we need a bit of
notation.
3.1.16 Definition. If X is a simplicial spectrum and E* is a homology theory
with representing spectrum E, form the new simplicial spectrum EX = E ^ X
and define its bigraded homotopy groups by the equation
ssp,qEX = sspmap sS(Sq, EX)
The mapping space here is the external mapping space defined using the
standard simplicial structure on a category of simplicial objects and we derived
the mapping space, if necessary, using the resolution model category structure
based on the set of projectives {Sq}, q 2 Z. See Theorem 1.4.6.
3.1.17 Example. From Example 3.1.14 we immediately have that ssp,*EX and
sspE*(X) are modules over the algebra ss0,*EX = ss0E*X.
The following now immediately follows from Theorem 3.1.15 by applying the
functor y*; that is, by passing to a colimit.
3.1.18 Corollary. . Let X 2 sAlg T be a fibrant simplicial T algebra. Then
the isomorphism
ss0,*EX ~=ss0E*X
is an isomorphism in AlgE*E and the spiral exact sequence
. .!. ssn1,*EX ! ssn,*EX ! ssnE*X !
ssn2,*EX ! . .!.ss1,*EX ! ss1E*X ! 0
is an exact sequence of ss0,*EXmodules.
3.2 Postnikov systems for simplicial algebras in
spectra
This section sets up a theory of Postnikov towers for simplicial T algebras, w*
*here
T is one of our simplicial operads. The important correspondence to the theory
for simplicial algebras constructed in Section 2.5 is provided by the kinvaria*
*nts
and the EilenbergMacLane objects, which will represent Andr'eQuillen coho
mology. In order to make this correspondence explicit, we must make some
assumptions. The following holds for this rest of this monograph, and we note
that most of this has come up before. The notion of homologically adapted was
defined in Definitions 1.4.13 and 1.4.16. The notion of an augmented triple was
defined in Definition 2.5.7.
103
3.2.1 Assumptions. Let T be a simplicial operad and sAlg T the category of
simplicial algebras in spectra. Fix an Adamstype homology theory E* and give
sAlg T the PE = Presolution model category structure. Furthermore
1.The simplicial operad T is homotopically adapted to E*;
2.the resulting triple TE on simplicial E*Ecomodules has an augmentation
TE ! . In particular, we have a triple on E*Ecomodules so that if
X is a T algebra, then ss0E*X is a algebra.
3.the zeroth simplicial set T (0) of the simplicial operad T is a point; in
particular, the sphere spectrum is the initial object in sAlg T;
4.the category sAlg TE satisfies BlakersMassey Excision, as in 2.5.11.
3.2.2 Example. There are three examples we have in mind. The following
statement collect the results of Example 2.5.8, Propositions 2.5.9 and 2.5.10,
and Theorem 2.3.13.
1.Let T be the associative monoid operad, regarded as a constant simplicial
operad. The sAlg T is the category of simplicial associative algebras in
spectra  that is, simplicial A1 ring spectra. We can let F and be the
associative algebra triple on E*Ecomodules.
2.Let T be a simplicial E1 operad. Then we can let F = E*T regarded as
triple and we can let be the commutative algebra triple.
3.For this example, we specialize to the case of E* = K*, pcompleted K
theory. Then we can let T be constant commutative monoid operad, so
that sAlg T is the category of simplicial commutative algebras in spectra
 that is, simplicial E1 ring spectra. Then we can let F = be the free
thetaalgebra triple.
The question of whether these operads are relatively left proper and satis
fied BlakersMassey excision was settled in Example 2.3.2, Example 2.3.4, and
Proposition 2.3.11.
3.2.3 Remark (Notation for Andr'eQuillen Cohomology). In the rest of
this paper were are going to work with Andr'eQuillen cohomology of simplicial
E*algebras.
Suppose k is algebra and Y is a simplicial T algebra equipped with a
weak equivalence of E*T algebras E*Y ! k. Equivalently, we could require
that ssnE*Y = 0 for n > 0 and ss0E*Y ~= k as algebras. (In the context of
the three examples just given, we are thinking of the example where Y is the
constant simplicial algebra on some E1 ring spectrum.) Suppose we are given
a morphism of k ! A of algebras and an Amodule M. Now let Y ! X be
a morphism of simplicial T algebras so that X is equipped with a morphism of
algebras ss0X ! A so that the composite
k ~=ss0E*Y ! ss0E*X ! A
104
is our chosen morphism k ! A. Then we will be concerned with the Andr'e
Quillen cohomology groups
HnTE=E*E(E*X=k, M).
This is a bit of a mouthful, so we will write Hn(E*X=k, M) for these groups,
or even Hn(E*X, M) if k = E*S = E* with the algebra structure obtained
from Assumptions 3.2.1.
We now get down to our construction of Postnikov towers. Recall that we
have two homotopy theories on simplicial T algebras. First, there is the P
resolution model category structure where P is a fixed set of finite CWspectra
closed under coproducts and containing the spectra kDEi. This simplicial
model category structure was defined and discussed in Section 1.4 and figured
in the Assumptions 3.2.1. Second, there is the localization of this category,
where we define a morphism f : X ! Y to be an ss*E*()equivalence if
ss*E*X ! ss*E*Y
is an isomorphism. This yielded only a semimodel category (See Definition
1.1.6.); the cofibrations remained the same as in Presolution model category.
While the latter is the one that is ultimately important, the former is the key
to constructions, and we will take care to keep them straight.
3.2.4 Definition. Let X 2 sAlg T be a simplicial T algebra in spectra. Then an
nth Postnikov section for X is a morphism of simplicial T algebras q : X ! Y
so that there is an isomorphism
~=
f* : ssi,PX ! ssi,PY, i n
for all P 2 P and so that ssi,PY = 0 for i > n. More succinctly, we will say
that f* : ssi,*X ! ssi,*Y is an isomorphism for i n and that ssi,*Y = 0 for
i > n. The asterisk (*) is a placeholder for P 2 P. A Postnikov tower for X is
a tower of simplicial T algebras under X
X ! . .!.Xn ! Xn1 ! . .!.X0
so that X ! Xn is an nth Postnikov section.
The reader will have noticed that this definition depends on P and, perhaps,
that P should be included in the notation at some point. However, since P will
be fixed throughout, we forebear.
3.2.5 Lemma. Let X be a simplicial T algebra in spectra. Then there exists a
natural Postnikov tower for X
X ! . .!.PnX ! Pn1X ! . .!.P0X
105
Proof. The only wrinkle on the standard construction is that not every object
in sAlg T is Reedy fibrant. We let X ! X0 denote some functorial acyclic
cofibration from X to a fibrant object. Then PnX = colimPn,tX where Pn,0X =
X0 and Pn,t+1X = Y 0with Y defined by the pushout diagram
` ` k k
P,k>n f:P^ k=@ k!Pn,tXP ^ =@ ______//_Pn,tX
 
 
` ` fflffl fflffl
k+1 ______//_
P,k>n f:P^ k=@ k!Pn,tXP ^ k+1= 0 Y.
___
Recall that Pre+ (T (P)) is the category of functors
F : T (P)op! Sets
which preserve products.
3.2.6 Definition. 1.) Let F 2 Pre+ (T (P)). Then we say that a simplicial
T algebra is of type BF if ss0,*X ~=F and ssi,*X = 0 for i > 0.
2.) Suppose further that M is an F module. Then we say a morphism
X ! Y is of T algebras is of type BF (M, n), n 1, if X is of type BF , the
morphism
ss0,*X ! ss0,*Y
is an isomorphism, ssn,*Y ~=F as an F module, and ssi,*Y = 0 if i 6= 0 or n. As
a shorthand, we may say Y is of type BF (M, n), leaving the morphism X ! Y
understood.
Note that X ! Y is of type BF (M, n), then the composition
X ! Y ! P0Y
is a weak equivalence. This observation, the spiral exact sequence, and Theorem
3.1.15 immediately imply the following lemma.
3.2.7 Lemma. 1.) Let X be of type BF . Then ss0ss*X ~= F , ssiss*F = 0 if
i 6= 0, 2 and
ss2ss*X ~= F
as an F module.
2.) Let X ! Y be of type BF (M, n). Then there is an isomorphism
8
< M i = n;
ssiss*Y ~=ssiss*X x : M i = n + 2;
0 otherwise.
If i 1, this is an isomorphism of F modules.
106
3.2.8 Example. Let A 2 Alg and N an Amodule. Recall that the triple
on E*Ecomodules is built into our Assumptions 3.2.1. Then we have the
associated object F 2 Pre+ (T (P))
F () = Hom Alg (ss*E*(), A)
and the F module M
M() = Hom E*E(E*(), N).
The previous result and a colimit argument as in Remark 1.4.8 show that if X
is of type BF , then 8
< A i = 0;
ssiE*X ~=: A i = 2;
0 otherwise
and, by Corollary 3.1.18 this is an isomorphism of Amodules for i 1. Fur
thermore, if X ! Y is of type BF (M, n), then
8
< M i = n;
ssiE*Y ~=ssiE*X x : M i = n + 2;
0 otherwise.
Again this is an isomorphism of Amodules in positive degrees. Note, in partic
ular, that E*Y is not of type KA (M, n). Compare Definition 2.5.12.
We now come to a functorial construction of kinvariants. Let f : X ! Y
be any morphism in sAlg T and let C be the pushout of the twosource
Y 0 X0! (P0X)0
where use the symbol ()0to denote some functorial construction to replace X
be a Pcofibrant simplicial algebra and the two maps by Pcofibrations. Then,
applying the Postnikov section functor of Lemma 3.2.5, we obtain a commutative
diagram
(3.2.1) X oo'__ X0 ____//_(P0X)0
f   ffin(f)
fflffl'fflffl fflffl
Y oo___ Y 0____//Pn+1C.
We will refer to the morphism ffin(f) as the difference construction applied to*
* f.
3.2.9 Proposition. Let f : X ! Y be a morphism of simplicial T algebras and
suppose there is an n 1 so that
1.f* : ssi,*EX ! ssi,*EY is an isomorphism for i < n, and
2.f* : ssn,*EX ! ssn,*EY is surjective.
107
Let M = ssn+1,*(EY, EX). Then M is naturally an A = ss0,*EX ~= ss0E*X
module and there in an ss*E*()equivalence from ffin(f) to a morphism of type
BA (M, n + 1). If ssi,*(Y, X) = 0 for i 6= n + 1, then the right hand square of
3.2.1 induces an ss*E*()equivalence
X0 ! holim{Y 0! Pn+1C (P0X)}.
Proof. There is a homotopy pushout in simplicial E*T algebras
E*X0 _____//E*(P0X)0
 
 
fflffl fflffl
E*Y 0_______//E*C.
This is because the functor E*() : sAlg T ! sAlg E*T preserves cofibrations,
weak equivalences, and pushouts along free cofibrations. By the fivelemma
and the spiral exact sequence, we have that
ssiE*X ! ssiE*Y
is a surjection for i n and an isomorphism for i < n. Furthermore,
ssn+1E*(Y, X) ~=M
as an Amodule. Then, Corollary 2.3.15 implies that ssiE*(C, P0X) = 0 for
i n and
ssiE*(C, P0X) ~=M
as Amodules. This and using the spiral exact sequence in reverse proves that *
* __
ffin(f) is as claimed. It is then straightforward to check the final claim. *
* __
3.2.10 Remark. There is a stronger result than the one we just proved. Indeed,
let f : X ! Y be a morphism of simplicial T algebras and suppose there is an
n 1 so that
1.f* : ssi,*X ! fi,*Y is an isomorphism for i < n, and
2.f* : ssn,*X ! ssn,*Y is a pointwise surjective
Let M = ssn+1,*(Y, X). The M is naturally a F = ss0,*X module and ffin(f) is a
morphism of type BF (M, n + 1). If ssi,*(Y, X) = 0 for i 6= n + 1, then the rig*
*ht
hand square of 3.2.1 is a homotopy pullback square.
This can be proved exactly as the comparable result in section 8 of [7].
However, this would mean developing the homotopy theory of Pre+ (T (P)) and
we haven't done that. Since this is not relevant for our main applications, we
will be content with the previous result.
The next question is whether EilenbergMacLane objects exist. Again we
concentrate on the case where A is the kind of algebra which can arise as ss0E**
*X,
where X is a simplicial T algebra. Thus we will have a simplicial operad T that
is homologically adapted to E* and so that the resulting triple TE on E*E has
an augmentation TE ! . See Assumptions 3.2.1 and Examples 3.2.2.
108
3.2.11 Proposition. Let A be a algebra and M a module over A. Then
there is a simplicial T algebra of type BA and for each n 1 there is a morph*
*ism
of simplicial T algebras of type BA (M, n). Furthermore, for X 2 sAlg T there
are natural isomorphisms
ss0map (X, BA )~=ss0map sAlgE*T(E*X, A)
~=Hom Alg (ss0E*X, A).
and
ss0map (X, BA (M, n))~=ss0map sAlgE*T(E*X, KA (M, n))
~= a Hn(E*X, M).
ss0E*X!A
Proof. This can be done by a generator and relations argument. (See [7].)
Alternatively, we could use a Brown representability argument. (See [22].) We
need to show certain functors are representable  namely, the targets of the
isomorphisms listed in the statement of the result. The argument given in [44]
certainly works, where we use as our spheres the objects T (P k=@ k )._We_
leave the details to the reader. __
It is worth recording immediately that the EilenbergMacLane object BA
constructed in this result has a strong homotopy discreteness property.
3.2.12 Lemma. Let BA be an EilenbergMacLane object so that
ss0map (X, BA ) ~=Hom Alg (ss0E*X, A)
for all simplicial T algebras A. Then all of the components of map (X, BA ) are
contractible.
Proof. This follows from the fact that if * ! k=@ k is the inclusion of the
basepoint, then the induced map
X ~=X * ! X k=@ k
induces an isomorphism on ss0E*(). ___
We next turn to the project of identifying the homotopy type of the mapping
space map (X, BA (M, n)).
By taking the class of the identity in ss0map (BA (M, n), BA (M, n)) and usi*
*ng
the isomorphism supplied by the second part of Proposition 3.2.11, we have a
universal morphism u : E*BA (M, n)! KA (M, n) and a diagram
(3.2.2) E*BA (M, n)_u___//KA (M, n)
 
 
fflffl fflffl
E*BA ___________//A.
109
Now Example 3.2.8 implies that if X ! Y is of type BA (M, n), then
ffin(E*f) : P0alg(E*X) ! PnalgC
is of type KA (M, n). Here Pnalgdenote the algebraic Postnikov section of Propo
sition 2.5.6 (there simply called Pn) and C is the homotopy pushout in sAlg E*T
of
P0algE*X oo___E*X _____//E*Y.
Applying this observation to the universal morphism u we get a diagram
E*BA (M, n)_____//Pnalg+1Cv_//KA (M, n)
  
  
fflffl fflffl fflffl
E*BA __________//A____=_____//_A.
3.2.13 Lemma. The induced map
v : Pnalg+1C ! KA (M, n)
is a weak equivalence of simplicial TE algebras in E*Ecomodules.
Proof. Let X = T (P ^ n=@ n ). Then we get, by examining the definition of
u, a commutative diagram
~= n n
ssn,PBA (M, n)____//_ss0map sCE*E(E*P3^3 =@ , KA (M, n)).
ggggg
~= ggggggggg
fflfflggggg
ssnssP BA (M, n)
The horizontal map is an isomorphism by construction and the vertical map is
an isomorphism by the spiral exact sequence. In the end, we get an isomorphism
~= n n
ssnssP BA (M, n)! ss0map sCE*E(E*P ^ =@ , KA (M, n)).
Letting P = kDEi, taking the colimit over i and letting k vary gives an
isomorphism ~
ssnE*BA (M, n)=!ssnKA (M, n).
The result follows. ___
We now give a continuous version of the statement that EilenbergMacLane
objects represent cohomology, and we also take a moment to present a relative
version. If M is some Amodule, let
Hn(E*X=k, M) = map sAlgTE=A(E*X, KA (M, n))
110
denote the Andr'eQuillen cohomology space. This is the derived space of maps
of simplicial TE algebras over A. Of course,
ssiHn(E*X=k, M) ~=Hni(E*X=k, M).
We should really write HnTE=E*E(E*X=k, M), but in keeping with Remark 3.2.3
we shorten the notation. If k = E*, we write will continue to write H*(E*X, M)
for H*(E*X=E*, M).
First we have an absolute result.
3.2.14 Proposition. Let A be a algebra, M an Amodule and let BA (M, n)
be an EilenbergMacLane object which represents AndreQuillen cohomology as
in 3.2.11.2. Let n 2. Then functor which sends
X U ! V ! BA (M, n)
to
E*X E*U ! E*V ! E*BA (M, n) u!KA (M, n)
defines a natural weak equivalence
fX : map sAlgT=BA(X, BA (M, n)) ! Hn(E*X, M).
Proof. In this proof we will write
mapsAlgT=BA(, ) = map BA(, )
to make some of our more cluttered calculations easier on the eye. The morphism
fX is a morphism of Hspaces, so it is sufficient to show that fX induces an
isomorphism on homotopy groups. We choose as basepoint of the mapping space
map BA (X, BA (M, n)) the "constant" map
X ! BA ! BA (M, n).
This maps to the corresponding constant map
E*X ! A ! KA (M, n).
We have an isomorphism on ss0 by Proposition 3.2.11.
To examine what happens in higher homotopy groups, we make a construc
tion. Let C be any simplicial category. If K is a simplicial set and Y is in C
let hom (K, Y ) be the internal mapping (or exponential) object. We may fix an
object U and consider the category C=U of objects over U. If K is a simplicial
set and Y ! U is in C=U, we define the mapping object hom U(K, Y ) by the
pullback diagram
hom U(K, Y )_____//_hom(K, Y )
 
 
fflffl fflffl
U = hom (*, U)____//hom(K, U).
111
If Y ! U has a section and K is pointed, we may define the pointed mapping
object by making a further pullback
hom U*(K, Y_)_____//homU(K, Y )
 
 
fflffl fflffl
U _________//_homU(*, Y ) = Y.
Note that the section on Y induces a section U ! hom U*(K, Y ). One now checks
that we have a commutative square
~=
sspmap BA (X, BA (M, n))___//ss0map BA (X, homBA*( p=@ p , BA (M, n)))
 
 
fflffl fflffl
sspmap A(E*X, KA (M, n))~=//_ss0map A(E*X, homKA*( p=@ p , KA (M, n))).
The result follows once one checks that BA ! hom BA*( k=@ k , BA (M, n)) and
A ! hom A*( k=@ k , KA (M, n)) are of type BA (M, n  k) and KA (M, n  k)
respectively, and that
E*hom BA*( k=@ k , BA (M, n)) ! hom A*( k=@ k , KA (M, n))
is a model for the universal morphism. This is easy and left to the reader. *
*___
We are now going to prove two results about the homotopy types of various
moduli spaces of EilenbergMacLane objects. It is important for the next section
that we have a relative version of the results here. Choose a morphism k ! A
of algebras and suppose we have an E1 ring spectrum Y so that E*Y ~=k as
algebras. We may regard Y as a constant object in sAlg T and then choose
a Pequivalence Yc ! Y with Yc cofibrant in the Presolution model category.
In particular, ssnE*Yc = 0 for n > 0 and we have an isomorphism of algebras,
ss0E*Yc ~=k. Corollary 1.4.12 implies that the induced map Yc ! Y is an E*
equivalence.
In this section and the next we are going to be working with the category
sAlg Ycof simplicial T algebras under Yc. Because of our assumptions 3.2.1, th*
*is
category is independent of the choice of Yc; specifically, we have the following
result.
3.2.15 Lemma. Suppose f : Y0 ! Y1 is a ss*E*() equivalence of Pcofibrant
objects in sAlg T. Then the adjoint pair
f* = Y1 tY0()sAlg Y0____//sAlgoY1:of*_
is a Quillen equivalence of semimodel categories.
112
Proof. Recall that we are using the ss*E*() isomorphisms as our weak equiv
alences. The functor f* sends a morphism Y1 ! X to the composition
f
Y0 _____//Y1____//X.
The functor f* preserves all ss*E*()equivalences and fibrations; the functor
f* preserves cofibrations for formal reasons and ss*E*()equivalences between
cofibrant objects by Lemma 1.5.10  or, more exactly, by the argument given
for the second part of the proof of that result.
The lemma here now follows as any two Pcofibrant replacements can be __
connected by a chain ss*E*()equivalences. __
Now select the model for an EilenbergMacLane object of type BA con
structed in Proposition 3.2.11. The morphism k ! A of algebras yields a
unique homotopy class of T algebra maps Yc! BA ; by fixing a representative,
we may assume that BA is a T algebra under Yc. Similarly, we may construct
EilenbergMacLane objects of type BA (M, n) under Yc.
3.2.16 Proposition. Let k ! A be a morphism of algebras, Y an E1 ring
spectrum so that E*Y ~= k as algebras and Yc ! Y a Pcofibrant model for
Y in simplicial T algebras. Let BA and BA (M, n) be the EilenbergMacLane
objects of 3.2.11.
1.Evaluation at ss0E*() defines a natural isomorphism
ss0map Yc(X, BA ) ~=Hom k(ss0E*X, A)
where map Ycis the derived space of morphisms of simplicial T algebras
under Yc and Hom k means homomorphisms of algebras under k. In
addition, the components of map Yc(X, BA ) are contractible.
2.If n 2, the universal element u : E*BA (M, n) ! KA (M, n) defines a
natural weak equivalence
mapYc=BA(X, BA (M, n) ' Hn(E*X=k, M)
where map Yc=BAdenotes the derived space of morphisms of simplicial T 
algebras under Yc and over BA .
Proof. The first statement follows from a pullback argument using Proposition
3.2.11.1 and Lemma 3.2.12. The second statement follows from a pullback __
argument, Proposition 3.2.14, and Remarks 2.4.3, 2.4.8, and 2.4.10. __
All our moduli spaces will be formed in the category sAlg Yc. In order to
specify these moduli spaces we need to specify a class of weak equivalences.
In both Proposition 3.2.17 and Proposition 3.2.16 we will mean the ss*E*()
equivalences of simplicial T algebras. Recall that M(KA #KA (M, n)) is the
moduli morphisms of simplicial E*T algebras which induce an isomorphism in
ss0. This is exactly the moduli space of all algebraic EilenbergMacLane objects
113
of type KA (M, n). See Definition 2.5.12 and Proposition 2.5.19. Even alge
braically, we are still working in a relative situation; for example, KA will be
an object in the category sAlg kof simplicial F algebras under k and M(KA )
is formed in sAlg k.
3.2.17 Proposition. Let k ! A be a morphism of algebras and M a 
module over A. Furthermore, let Y be an E1 ring spectrum so that E*Y ~= k
as algebras and suppose Yc! Y is a Pcofibrant model for Y as a simplicial
T algebra.
1.Let M(A) be moduli space of all simplicial T algebras of type BA under
Yc. Then the functor X 7! P0algE*X defines a weak equivalence
M(A) ! M(KA ) ' B Autk(A).
2.Let MA (M, n) be the moduli space of all morphisms of type BA (M, n) in
simplicial T algebras under Yc. Then the functor f 7! ffin1(E*) defines a
weak equivalence
MA (M, n) ! M(KA #KA (M, n)) ' B Autk(A, M)
In particular, these spaces are connected and any EilenbergMacLane object
in sAlg T will represent Andr'eQuillen cohomology.
Proof. Both of these statements follow from examining the functor that the
object in question represents. We begin with first point. Choose a fixed bifi
brant simplicial T algebra Z under Ycwhich represents Hom k(ss0E*(), A). See
Proposition 3.2.16. Then if X is any simplicial T algebra of type BA under Yc,
the isomorphism ss0E*X ! A defines a morphism X ! Z under Yc which is
Pequivalence. Thus M(A) ~=BAut (X). Now an easy calculation shows that
ss0Aut(X) ~=Autk(A).
via f 7! ss0E*f. To complete the argument, use Proposition 3.2.16 to show that
Aut (X) is homotopically discrete.
The second point is proved similarly. Choose a bifibrant model Z for BA
and a cofibration g : Z ! W of type BA (M, n) so that W represents
a
X 7! Hn(E*X=k, M).
ss0E*X!A
Then if we have any morphism f of type BA (M, n), there is an evident map
E*f ! ffin1E*f ~= E*g, which  using the strong representability result of
Proposition 3.2.16  defines an E*equivalence from f to g. This shows that
M(A, n) is connected, and now we need only show that Aut(g) is homotopically
discrete. But this is a simple calculation. Compare the corresponding_result_in
[7]. __
114
3.2.18 Remark. Combining Proposition 3.2.9 with Proposition 3.2.16 we can
identify where kinvariants for simplicial T algebras live. Indeed, when X 2
sAlg Ycis a simplicial T algebra under Yc and ss0X ~= A as algebras, the
Postnikov tower becomes a tower under Yc and the nth kinvariant determines
an equivalence class of elements in the group
Hn+1(E*Pn1X=k, ssnE*X).
3.3 The decomposition of the moduli spaces
Let us recall the basic setup. We have a simplicial operad T so that the as
sumptions of 1.4.16 and 3.2.1 hold. In particular, there is a fixed homology
theory E* and a triple on E*Ecomodules so that for all simplicial T algebras
X, ss0E*X is naturally a algebra. In our two main examples, is the free
commutative algebra functor or the free thetaalgebra functor.
The arguments and ideas of this section also apply to the case of associative
algebras. These are considerably easier, and left to the reader.
Note that if Y is simply an E1 ring spectra, then Y may be regarded as a
constant object in sAlg T; hence, our assumptions imply that E*X is algebra.
In the case where is the free commutative algebra functor, this amounts
to regarding E*Y simply as a commutative algebra and forgetting any other
structure that might be present  for example, any DyerLashof operations.
If A is a algebra in E*Ecomodules, then we have a moduli space T M (A)
of realizations of A. This is the nerve of the category R(A) with objects the
commutative ring spectra X so that E*X ~=A as algebras; the morphisms are
E*equivalences. The DwyerKan decomposition theorem of Proposition 1.1.12
gives a weak equivalence
a
T M (A) ' BAut (X)
[X]
where [X] runs over the E*equivalence class of objects in R(A), and Aut(X) is
the (derived) space of selfequivalences of X in the E*local category of E1 r*
*ing
spectra. The point of this section is give a decomposition of T M (A) in terms
of algebraic data.
We will actually work out a more general relative case. Fix a cofibrant E1 
ring spectrum Y and let k = E*Y . Choose a algebra morphism k ! A and let
T M (A=k) be the moduli space of realizations of the algebra A under k. This
is the nerve of the category R(A=k) with objects the morphisms of commutative
ring spectra Y ! X so that there is an isomorphism from E*Y ! E*X to the
chosen morphism k ! A. The morphisms in R(A=k) are morphisms under Y
which induce an isomorphism on E*. Again there is a decomposition
a
T M (A=k) ' BAut Y(X)
[X]
115
where [X] runs over the E*equivalence class of objects in R(A), and AutY (X)
is the (derived) space of selfequivalences of X under Y in the E*local catego*
*ry
of E1 ring spectra.
For our decomposition results, we will work with the ss*E*() localization of
the Presolution model category structure on simplicial T algebras in spectra,
where P is the fixed set of projectives defined in Definition 1.4.2. For a simp*
*licial
spectrum X, we are writing ssi,*X = {ssi,PX} where P runs over the elements
of P.
Regard our fixed E1 ring spectrum Y as a constant object in the category
sAlg T of simplicial T algebras, and choose a Pequivalence Yc ! Y so that
Yc is Pcofibrant; thus Yc is a Presolution of Y . The reason for making this
replacement is so that we can apply Lemma 3.2.15, which will imply that any
moduli space we construct out of the category sAlg Ycwill be independent of
the choice of Yc.
3.3.1 Definition. Let Y be an E1 ring spectrum and let k = E*Y be the
resulting algebra in E*Ecomodules. Let A be a algebra under k in E*E
comodules. A potential nstage for A is a simplicial T algebra X under Yc so
that the following three conditions hold
1.ss0E*X ~=A as algebra under k;
2.ssi,*X = 0 for i > n; and
3.ssiE*X = 0 for 1 i n + 1.
The partial moduli space T M n(A=k) is defined to be the moduli space of all
simplicial T algebras under Yc which are potential nstages for A. Morphisms
are the ss*E*() equivalences under Yc.
It follows from the spiral exact sequence that if X is a potential nstage f*
*or
A, then
8
< A i = 0;
(3.3.1) ssiE*X ~=: n+1A i = n + 2;
0 otherwise.
Furthermore, the structure of ssn+2E*X as a ss0E*Xmodule is the standard
one. See Examples 2.2.10 and 3.1.13.
Definition 3.3.1 makes sense for n = 1. If X is a potential 1stage for A,
then ae
ssiE*X ~= A0 ii=60;= 0.
Let T M 1 (A=k) be the resulting moduli space.
Here are two preliminary decomposition results.
3.3.2 Theorem. The geometric realization functor induces a weak equivalence
T M 1(A=k) ! T M (A=k).
116
Proof. The spaces T M 1 (A=k) and T M (A=k) are the nerves of categories
R1 (A=k) and R(A=k) respectively. Therefore, it is sufficient to define func
tors F : R1 (A=k) ! R(A=k) and G : R(A=k) ! R1 (A=k) so that the two
composites F G and GF are connected to the respective identity functors by
chains of natural transformations which are ss*E*() equivalences. The functor
G is easy: given a morphism Y ! X we may regard X as a constant T algebra
under Y and, hence, under Yc; this is a tautological potential 1stage.
We now define the functor F . If Yc ! X is a potential 1stage, form a
functorial factorization
p
Yc __i__//X0____//X
where X0 is a Pcofibrant simplicial T algebras and the morphisms p is a
ss*E*()equivalence. Apply geometric realization to the top map in this di
agram and form the pushout in E1 algebras
Yc___i____//X0
ffl 
fflffl fflffl0
Y _____//_Y tYcX .
We now apply Corollary 1.4.12 to the top row and use that Yc ! Y is an E*
equivalence between cofibrant E1 algebras to conclude that the bottom row is
in R(A=k). Then
F (Y ! X) = Y ! Y tYcX0.
We leave it to the reader to connect F G and GF by weak equivalences to the_
respective identities. __
3.3.3 Theorem. The nthPostnikov stage functor Pn induces a map of moduli
spaces
Pn : T M k(A=k)! T M n(A=k), n k 1
and the resulting map
T M 1(A=k)! holimn<1TnM(A=k)
is a weak equivalence.
Proof. This follows from [14], x4.6. ___
Because of the these results, we next address the homotopy type of the space
T M n(A=k).
3.3.4 Theorem. The functor ss0E*() induces a natural weak equivalence
T M 0(A=k) ' B Autk(A)
where Autk(A) is the group of automorphisms of the algebra A over k. In
particular, T M 0(A=k) is nonempty and connected.
117
Proof. A potential 0stage for A is nothing more nor less than a simplicial T *
*__
algebra of type BA under Yc. The result now follows from Proposition 3.2.17. *
*__
The main theorem of this section and, indeed, of this paper now identifies
how to pass up the layers of the tower. If A is algebra and M is an Amodule,
then we have defined
Hn(A=k, M) def=HkTE=E*E(A=k, M) = map sAlgk=A(A, KA (M, n))
and
H^n(A=k, M) def=^HnTE=E*E(A=k, M) = E Aut (A, M) xAut(A,M)Hn(A=k, M).
See Remark 3.2.3 for more on this notation.
3.3.5 Theorem. Let n 1, then there is a natural homotopy pullback diagram
T M n(A=k) ______//B Aut (A=k, nA)
Pn1 
fflffl fflffl
T M n1(A=k) _____//_^Hn+2(A=k, nA).
The proof will occupy the rest of the section. We begin with an analysis of
how to pass from potential (n  1)stages to nstages.
Suppose that X is a potential nstage for A. Then ssnE*X ~= nA as an
Amodule, by the spiral exact sequence. Then Z = Pn1X is a potential (n1)
stage for A and Proposition 3.2.9 implies that there is a homotopy pullback
square in the E*local category under Yc
(3.3.2) X __________//_BA
p q
fflfflf fflffl
Z _____//BA ( nA, n + 1).
Note that all the maps in this diagram induce an isomorphism on ss0E*(). The
next result shows how to reverse this process. Recall from Proposition 3.2.16
that the simplicial T algebra BA (M, n) represents Andr'eQuillen cohomology;
that is,
(3.3.3) ss0map sAlgYc(Z, BA (M, n)) ~=ss0map k(E*Z, KA (M, n)).
3.3.6 Proposition. Suppose that Z is a potential n  1stage for A and that
n 1. Suppose further that X lies in a homotopy fiber square of the form
displayed in 3.3.2. Then X is a potential nstage if and only if the map
g : E*Z ! KA ( nA, n + 1)
induced by f is a weak equivalence of simplicial TE algebras.
118
Proof. This is a simple calculation, using that there is a MayerVietoris se
quence in ss*,*()  and hence in ss*E()  for homotopy pullbacks. Compare_
Proposition 9.11 of [7]. __
3.3.7 Remark (Obstructions to realization). Given a potential (n  1)
stage Z for A, then E*Z, as an F algebra, has exactly two nonvanishing homo
topy groups; thus, taking algebraic Postnikov sections, we obtain a homotopy
pullback square in F algebras under k
E*Z ___________//_A
 
 
fflfflO fflffl
A _____//_KA ( nA, n + 2).
The previous result implies that there exists a potential nstage X so that
Pn1X ' Z if and only if
0 = O 2 Hn+2(A=k, nA).
Thus we see the obstructions to realizing A as elements of Andr'eQuillen coho
mology. The next result extends this observation to a statement about moduli
spaces.
We continue to work in the category sAlg Ycof simplicial T algebras under
Yc. If X and Z are two T algebras under Yc, then recall from Example 1.1.19
that M(X# Z) means the moduli space of all arrows X ! Z which induce
an isomorphism on nonzero homotopy groups. In this case, we would have
ssm E*(X) ! ssm E*(Z) is an isomorphism when both source and target are non
zero. If Z is a potential (n  1)stage for A, let M(Z ( nA, n)) denote the
moduli space of potential nstages X for A under Yc so that there is some ss*E*
weak equivalence Pn1X ! Z. This weak equivalence is not part of the data,
we are simply assuming we can find one.
3.3.8 Remark (Labeling of moduli spaces). In the rest of this section we
will be working with relative moduli spaces; that is, moduli spaces built from
objects either under Yc (on the topological side) or under E*Yc (on the algebra*
*ic
side). We could adorn our space to indicate this; for example, in the previous
paragraph we could have written MYc(X# Z) and in the statement of the next
result we could write ME*Yc(E*Z) for the moduli space of the object E*Z under
E*Yc. However, since this will be completely universal, we won't add this extra
bit of notation, but leave it understood.
3.3.9 Proposition. Suppose that Z is potential (n  1)stage for A under Yc
and that n 1. Then there is a natural homotopy fiber square
M(Z ( nA, n)) _____//M(E*Z# KA ( nA, n + 1)" KA )
Pn1 
fflffl fflffl
M(Z) __________E*________//M(E*Z).
119
Note that the space M(Z ( nA, n)) may be empty. By Proposition 3.3.6
this will happen if and only if there is no weak equivalence
E*Z ! KA ( nA, n + 1)
under E*Yc. In this case, the space M(E*Z# KA ( nA, n + 1)" KA ) will also
be empty.
Proof. Let M = nA. The difference construction supplies a map
M(Z (M, n)) ! M(Z ! BA (M, n + 1)" BA )
where the symbol in the target means morphisms Z ! BA (M, n + 1) under
Yc which correspond to weak equivalences E*Z ! KA (M, n + 1) under E*Yc.
See Proposition 3.2.16. Then Proposition 3.3.6 implies that this map is a weak
equivalence; thus we have a homotopy pullback square
M(Z (M, n)) _____//M(Z ! BA (M, n + 1)" BA )
Pn1 
fflffl = fflffl
M(Z) __________________//_M(Z).
Now applying homology and composing with the universal map of Diagram 3.2.2
u : E*BA (M, n + 1) ! KA (M, n + 1)
supplies a commutative diagram
M(Z ! BA (M, n + 1)" BA )____//M(E*Z# KA (M, n + 1)" KA )
 
 
fflffl fflffl
M(Z) ____________E*__________//M(E*Y ).
To complete the proof, we show that this is a homotopy pullback square. To
do this, note that Proposition 3.2.17 yields a weak equivalence
M(BA (M, n + 1)" BA )! M(KA (M, n + 1)" KA ).
Therefore it is sufficient to prove that there is a homotopy pullback square
M(Z ! BA (M, n + 1)" BA )_________//M(E*Z# KA (M, n + 1)" KA )
 
 
fflffl fflffl
M(Z) x M(BA (M, n + 1)" BA )_____//M(E*Z) x M(KA (M, n + 1)" KA ).
Note that the two spaces at the bottom of this diagram are connected. The
induced map on fibers is
map wYc(Z, BA (M, n + 1)) ! map wE*Yc(E*Z, KA (M, n + 1)).
120
Here the superscript w means, on the right, the subspace of the space of all ma*
*ps
which are weak equivalences and, on the left, those maps which correspond to
weak equivalences. Then Proposition 3.2.16 shows this morphism is a weak__
equivalence. The result follows. __
We can now supply the proof of our core result.
3.3.10 Proof of the Theorem 3.3.5. For any morphism k ! A of algebras,
any Amodule M, and any m 1, there is a commutative square
(3.3.4) M(KA (M, m)" KA ) ____'___//M(KA (M, m + 1)" KA )
 
 
fflffl ' fflffl
M(KA (M, m)) ______////_M(KA #KA (M, m + 1)" KA ).
(Recall that these are all moduli spaces of morphisms under E*Yc and that
E*Yc is weakly equivalent to k.) As indicated the horizontal maps are weak
equivalences, as demonstrated by the analysis of Postnikov sections given in
Proposition 2.5.16. In particular, we have a pullback square. If Y is a potent*
*ial
(n  1)stage for A, we take M = nA and m = n + 1. Then M(E*Z) is one
component of M(KA (M, m)). There are two cases.
The first case is that there is no weak equivalence of simplicial algebras
E*Z ! KA ( nA, M) under E*Yc. With that assumption Proposition 3.3.6
shows that M(Y ( nA, n)) is empty. We also have that the component
M(E*Y ) is not in the image of
M(KA (M, m)" KA ) ! M(KA (M, m)).
Together with the pullback 3.3.4, these facts imply that
(3.3.5) M(Z ( nA, n)) ________//M(KA (M, m + 1)" KA )
 
 
fflffl fflffl
M(Z) __________//M(KA #KA (M, m + 1)" KA )
is a pullback square  rather trivially, in fact.
For the second case we assume that there is some weak equivalence of sim
plicial algebras E*Z ! KA ( nA, M). Then we assert that there is is a weak
equivalence
(3.3.6) f : M(KA (M, m)" KA ) ! M(E*Z# KA ( nA, m)" KA ).
To see this recall that source and target are given by nerves of categories of
arrows. The morphism f sends U V to
U=! U  V ;
121
the homotopy inverse sends W ! U V to U V . Then Proposition 3.3.9
implies that the square of 3.3.5 is a homotopy pullback square in this case al*
*so.
Finally taking the coproduct over all weak equivalence classes of potential
(n  1)stages Z yields a pullback square
T M n(A=k) ________//_M(KA (M, m + 1)" KA )
 
 
fflffl fflffl
T M n1(A=k) ____//_M(KA #KA (M, m + 1)" KA ).
and the result follows. Indeed, the identification
M(KA (M, m + 1)" KA ) ' B Aut(A, nA)
follows from Proposition 2.5.19 and the identification
M(KA #KA (M, m + 1)" KA ) ' ^Hn+2(A=k, nA)
follows from Corollary 2.5.23.
122
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Department of Mathematics, Northwestern University, Evanston IL 60208
pgoerss@math.northwestern.edu
Department of Mathematics, MIT, Cambridge MA, 02139
mjh@math.mit.edu
126