HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA
MARIA BASTERRA AND MICHAEL A. MANDELL
Abstract.We show that every homology or cohomology theory on a category
of E1 ring spectra is Topological Andr'e-Quillen homology or cohomology
with appropriate coefficients. We show that the cotangent complex of MU *
*is
MU ^ bu.
Introduction
Homology and cohomology theories in various contexts provide some of the most
effective tools in mathematics because of their computability, their usefulness*
* for
classification problems, and their close relationship to extensions and obstruc*
*tions.
In the context of homotopy theory, homology and cohomology typically refer to
theories that satisfy the Eilenberg-Steenrod (and Milnor) axioms. Although orig*
*i-
nally phrased for topological spaces, these axioms make sense in the more gener*
*al
context of homotopy theory of closed model categories.
The Eilenberg-Steenrod axioms involve a category of pairs. For a closed model
category C, an appropriate category of pairs is the category of arrows in C: A *
*pair
is a map A ! X in C and a map of pairs is a commutative square. Standard
notation is to write (X, A) for the pair f :A ! X (with f understood) and to wr*
*ite
X rather than (X, ;) for the pair ; ! X where ; is the initial object of C. A m*
*ap
of pairs (X, A) ! (Y, B) is called a weak equivalence when the maps A ! B and
X ! Y are weak equivalences in C. In this terminology, we understand cohomology
theories as follows:
Definition. Let C be a closed model category. A cohomology theory on C con-
sists of a contravariant functor h* from the category of pairs to the category *
*of
graded abelian groups together with natural transformations of abelian groups
ffin :hn(A) ! hn+1(X, A) for all n, satisfying the following axioms:
(i)(Homotopy) If (X, A) ! (Y, B) is a weak equivalence of pairs, then the
induced map h*(Y, B) ! h*(X, A) is an isomorphism of graded abelian
groups.
(ii)(Exactness) For any pair (X, A), the sequence
n
. .-.! hn(X, A) -! hn(X) -! hn(A) ffi-!hn+1(X, A) -! . . .
is exact.
(iii)(Excision) If A is cofibrant, A ! B and A ! X are cofibrations, and Y is
the pushout X [A B, then the map of pairs (X, A) ! (Y, B) induces an
isomorphism of graded abelian groups h*(Y, B) ! h*(X, A).
____________
Date: June 30, 2004.
1991 Mathematics Subject Classification. Primary 55P43; Secondary 55P48, 55U*
*35.
Corresponding author. Fax: (603) 862-4096; E-mail address: basterra@math.unh*
*.edu.
The second author was supported in part by NSF grant DMS-0203980.
1
2 MARIA BASTERRA AND MICHAEL A. MANDELL
(iv)(Product) If {Xff} is a set ofQcofibrant objects and X is the coproduct,
then the natural map h*(X) ! h*(Xff) is an isomorphism.
A map of cohomology theories OE: h* ! k* is a natural transformation of contrav*
*ari-
ant functors that makes the diagram
n
hn(A) __ffi//_hn+1(X, A)
OEAfflffl| OE(X,A)fflffl|
kn(A) __ffin//_kn+1(X, A)
commute for all (X, A), n.
A homology theory consists of a covariant functor h* together with natural tr*
*ans-
formations @n :hn+1(X, A) ! hn(A) satisfying completely analogous axioms, with
the Product Axiom replaced by the following Direct Sum Axiom:
(iv)0(Direct Sum) If {Xff}Lis a set of cofibrant objects and X is the coprod*
*uct,
then the natural map h*(Xff) ! h*(X) is an isomorphism.
As a matter of pure algebra, the axioms above imply the stronger property that
for A ! B ! X, the sequence
. .-.! hn(X, B) -! hn(X, A) -! hn(B, A) -! hn+1(X, B) -! . . .
is exact, where the last map is the composite hn(B, A) ! hn(B) ! hn+1(X,`B).`
Likewise,Qthey imply the stronger property that the natural map h*( Xff, Aff*
*) !
h*(Xff, Aff) is an isomorphism when each Xffand Affis cofibrant. As a matter
of pure homotopy theory, whether or not (h*, ffi*) or (h*, @*) satisfies the ax*
*ioms for
a particular model structure on C depends only on the weak equivalences and not
on the cofibrations (see Section 9 for details).
The purpose of this paper is to study homology and cohomology theories on
categories of E1 ring spectra, or equivalently, on the modern categories of EK*
*MM
commutative S-algebras [5], where the initial object is the sphere spectrum S a*
*nd
the coproduct is the modern symmetric monoidal smash product. Because the final
object in the category of commutative S-algebras is the trivial (one-point) spe*
*ctrum
*, if we take C above to be the model category of commutative S-algebras, then
there are no non-trivial cohomology theories: If A is any cofibrant commutative
S-algebra and C ! * is a cofibrant approximation, then A ^ C is contractible, a*
*nd
so by the Homotopy Axiom and the Exactness Axiom, h*(A ^ C, C) = 0, for any
cohomology theory h*. Then the Excision Axiom applied to the cofibrations S ! A
and S ! C implies that the map
h*(A) = h*(A, S) -! h*(A ^ C, C) = 0
is an isomorphism. Now it easily follows from the Homotopy Axiom and the Ex-
actness Axiom that h* is zero on any pair. In order to have non-trivial homology
and cohomology theories, we therefore need to consider categories of commutative
S-algebras with non-trivial final objects. We do this by considering over-categ*
*ories,
and we work more generally with categories of commutative R-algebras for a cofi-
brant commutative S-algebra R. For a commutative R-algebra B, let CR =B denote
the category of commutative R-algebras lying over B. This is a closed model cat-
egory with initial object R, final object B, and coproduct ^R , the smash produ*
*ct
over R.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 3
Topological Andr'e-Quillen cohomology with various coefficients provides exam-
ples of cohomology theories on CR =B. For a cofibrant commutative R-algebra A
and a cofibrant commutative A-algebra X, Basterra [1] constructs the cotangent
complex L AX as the derived commutative X-algebra indecomposables (of the
derived augmentation ideal) of X ^A X. The cotangent complex L AX is an
X-module, and restricting to maps A ! X in CR =B, we can regard
LAb BAX = B ^X L AX
as a functor from the homotopy category of CR =B to the homotopy category of B-
modules. Topological Andr'e-Quillen cohomology with coefficients in a B-module
M is defined by
D*R(X, A; M) = Ext*B(LAb BAX, M)
(where Ext is as in [5, IV.1.1]). Using the connecting homomorphism in the tran-
sitivity sequence [1, 4.4], D*R(-; M) becomes a cohomology theory on CR =B, fun*
*c-
torially in M in the homotopy category of B-modules. Our main result is the
following theorem, which says in particular that every cohomology theory on CR *
*=B
is isomorphic to D*R(-; M) for some B-module M.
Theorem 1. Topological Andr'e-Quillen cohomology, viewed as a functor from the
homotopy category of B-modules to the category of cohomology theories on CR =B,
is an equivalence of categories.
A characterization of the category of homology theories on CR =B is slightly
trickier because of the failure of Brown's Representability Theorem for homology
theories [3]. Given a homology theory h* on the category MB of B-modules, we
obtain a homology theory hD*on the category CR =B by setting
hD*(X, A) = h*(LAb BAX, *).
This describes a functor from the category of homology theories on MB to the ca*
*t-
egory of homology theories on CR =B. We prove that this functor is an equivalen*
*ce
of categories.
Theorem 2. The category of homology theories on CR =B is equivalent to the cat-
egory of homology theories on MB .
As always, there is a close relationship between homology and cohomology the-
ories and a category of "spectra" that we review in Section 1. For R = B, the
category CB =B is enriched over the category of based spaces, with tensors and
cotensors, and so in particular we have a suspension functor. We denote this su*
*spen-
sion functor by E to avoid confusion with suspension of the underlying B-module.
A CB =B-spectrum is a sequence of objects An, n 0, together with "structure
mapsö en :EAn ! An+1. A map of spectra is a collection of maps An ! A0nthat
commute with the structure maps. We define the homotopy groups of a spectrum
__A = {An} by
ßq__A = Colim~ßq+nAn,
where ~ß*A = Ker(ß*A ! ß*B). Standard techniques [6, 10] allow us to prove in
Section 7 that the category of CB =B-spectra forms a closed model category, with
weak equivalences the maps that induce isomorphisms on homotopy groups. The
resulting homotopy category is called the "stable categoryö f CB =B. In Sectio*
*n 3,
we prove the following theorem:
4 MARIA BASTERRA AND MICHAEL A. MANDELL
Theorem 3. Let B be a cofibrant commutative R-algebra. The stable category of
CB =B is equivalent to the homotopy category of B-modules.
In fact, as explained in Section 3, the equivalence of homotopy categories ar*
*ises
from a Quillen equivalence. Although the technical hypotheses do not quite appl*
*y,
this theorem is closely related to the title theorem of Schwede-Shipley [16] th*
*at
stable categories are categories of modules. Theorem 3 is in marked contrast to*
* the
corresponding situation for simplicial commutative algebras studied by Schwede
[15], where the stable category is equivalent to the homotopy category of modul*
*es
over a ring spectrum that is generally very different from the ground ring; see*
* also
Theorem 2.8 below.
For an object A in CR =B, B ^R A is naturally an object of CB =B and we have
an associated CB =B-spectrum E1 (B ^R A) = {En(B ^R A)} called the üs spension
spectrum". Closely related to the previous theorems is the following:
Theorem 4. Let B be a cofibrant commutative R-algebra, and A a cofibrant object
in CR =B. Under the equivalence of Theorem 3, the suspension spectrum E1 (B^R A)
corresponds to the B-module LAb BRA.
The suspension for commutative S-algebras turns out to be closely related to
delooping for spaces. When X is an E1 space, that is, a space with an action of*
* an
E1 operad, the work of May, Quinn, and Ray [13, IVx1] implies that the suspensi*
*on
spectrum 1 X+ is an E1 ring spectrum. We can make sense of the Andr'e-Quillen
Cohomology and the cotangent complex of E1 ring spectra. Up to equivalence,
these do not depend on the operad, and we can understand these in terms of an
equivalent commutative S-algebra. The work of May and Thomason [12] shows that
up to isomorphism in the stable category, there is a canonical spectrum associa*
*ted
to X whose zeroth space is the group completion of X; it is any spectrum output
by an "infinite loop space machine". In Section 6, we reinterpret part of the p*
*roof
of the previous theorem to prove the following delooping theorem:
Theorem 5. For an E1 space X, the cotangent complex of the E1 ring spec-
trum 1 X+ is the extended 1 X+ -module ( 1 X+ ) ^ Z, where Z is the spectrum
associated to X.
According to Lewis [8, xIX], the Thom spectrum M obtained from a map of
E1 spaces X ! BF naturally has the structure of an E1 ring spectrum (see also
Mahowald [9]), and the diagonal map
M -! M ^ X+
is a map of E1 ring spectra. The derived extension of scalars to E1 M-algebra*
*s,
M ^ M -! M ^ X+ ~=M ^ 1 X+
induces the Thom isomorphism and is a weak equivalence. The cotangent complex
commutes with extension of scalars [1, 4.5], and we obtain the following coroll*
*ary
of the previous theorem:
Corollary. Let ff: X ! BF be a map of E1 spaces, M the associated E1 ring
Thom spectrum, and Z the spectrum associated to X. Then the cotangent complex
L SM is the M-module M ^ Z.
As a special case, we obtain the following result. In it bu denotes the spect*
*rum
with zeroth E1 space BU; it is equivalent to 2ku, where ku is connective K-
theory.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 5
Corollary. The cotangent complex L SMU of MU is the MU-module MU ^ bu.
We have stated these results in terms of EKMM commutative S-algebras, but
because of the homotopy invariant nature of Theorems 1 and 2, they hold in the
context of commutative symmetric spectra and commutative orthogonal spectra
[10], or in any Quillen equivalent modern category of E1 ring spectra.
More generally, we have stated results in terms of commutative S-algebras, but
results analogous to Theorems 1-4 hold for any sort of operadic algebras in EKMM
S-modules with slight modifications of the arguments below; see Section 8 for d*
*e-
tails.
1. Reduced Theories
Although the most natural statement of Theorems 1 and 2 is in terms of coho-
mology and homology theories defined on pairs, the most natural proof is in ter*
*ms
of reduced cohomology and homology theories. The purpose of this section is to
record some basic facts about the relationship of cohomology theories on pairs,*
* re-
duced cohomology theories, Ö mega weak spectra" (see Definition 1.4 below), and
spectra. These results hold quite generally and their arguments depend very lit*
*tle
on the specifics of the category CR =B of commutative R-algebras over B. Since *
*the
arguments are familiar from other contexts, we omit many of the details.
We begin with the definition of reduced theories. These are defined for the
homotopy categories of öp inted closed model categories" (closed model categori*
*es
where the initial object is the final object); these categories have the extra *
*structure
of a üs spension functor" [14, Ix2] and of öc fiber sequences" [14, Ix3].
Definition 1.1. Let C be a pointed closed model category. A reduced cohomology
theory on C consists of a contravariant functor h* from the homotopy category H*
*oC
to the category of graded abelian groups together with a natural isomorphisms of
abelian groups oe :hn(X) ! hn+1( X) (the suspension isomorphism) for all n,
satisfying the following axioms:
(i)(Exactness) If X ! Y ! Z is part of a cofibration sequence, then
hn(Z) -! hn(Y ) -! hn(X)
is exact for all n.
(ii)(Product) If {Xff} is aQset of objects and X is the coproduct in Ho C, *
*then
the natural map h*(X) ! h*(Xff) is an isomorphism.
A reduced homology theory consists of a covariant functor h* together with natu*
*ral
isomorphisms oe :hn+1( X) ! hn(X) satisfying an analogous exactness axiom and
the following Direct Sum Axiom:
(ii)0(Direct Sum) If {Xff}Lis a set of objects and X is the coproduct in Ho *
*C,
then the natural map h*(Xff) ! h*(X) is an isomorphism.
A map of reduced cohomology theories or of reduced homology theories is a natur*
*al
transformation that commutes with the suspension isomorphisms.
Whenever the final object in a closed model category is cofibrant, there is a
close relationship between cohomology theories and reduced cohomology theories
on the under-category of the final object. Writing B for the final object and C*
*\B
for the under-category of B (for C = CR =B, the under-category C\B is CB =B), a
cohomology theory h on C leads to a reduced cohomology theory ~h*on C\B with
6 MARIA BASTERRA AND MICHAEL A. MANDELL
~h*(X) = h*(X, B) and the suspension isomorphism (for X cofibrant) obtained from
the connecting homomorphism hn(X, B) ! hn+1(CX, X) and the inverse of the
excision isomorphism hn+1( X, B) ! hn+1(CX, X), where X ! CX is a Quillen
cone. Conversely, given a reduced cohomology theory ~h*on C\B, we obtain a
cohomology theory on C by setting hn(X, A) to be ~h*of the homotopy pushout
B [A X. In general, we have the following proposition:
Proposition 1.2. Let C be a closed model category with final object B, and let
B0! B be a cofibrant approximation (an acyclic fibration with B0 cofibrant). The
following categories are equivalent:
(i)The category of cohomology theories on C.
(ii)The category of cohomology theories on C=B0.
(iii)The category of reduced cohomology theories on (C=B0)\B0.
The analogous result holds for homology theories.
The homotopy category Ho[CB =B] of CB =B together with a skeleton of the full
subcategory of finite cell commutative B-algebras over B satisfy the hypotheses
of Brown [2, x2] for a öh motopy category". Brown's Abstract Representability
Theorem [2, 2.8] therefore applies to show that certain functors are representa*
*ble.
The following is the relevant special case; a standard homological algebra argu*
*ment
(the Barratt-Whitehead äl dderä rgument) reduces its hypotheses to those of the
Representability Theorem.
Proposition 1.3. Let h be a contravariant functor from Ho [CB =B] to abelian
groups that satisfies hypotheses (i) and (ii) in the definition of reduced coho*
*mology
theory. Then there exists an object Xh in Ho[CB =B] and a natural isomorphism of
functors h(-) ~=Ho[CB =B](-, Xh).
It follows that when h* is a reduced cohomology theory on CB =B, each functor
hn is representable by an object Xhn. If we write EL for the left derived funct*
*or
of E, the suspension functor on Ho[CB =B], and R for its right adjoint, the lo*
*op
functor on Ho[CB =B], then the suspension isomorphism
Ho[CB =B](-, Xhn) ~=hn(-) -! hn+1(EL-) ~=Ho[CB =B](EL-, Xhn+1)
~=Ho[CB =B](-, R Xhn+1)
induces (by the Yoneda Lemma) an isomorphism (in Ho[CB =B]),
Xhn -! R Xhn+1.
This leads to the following definition:
Definition 1.4. Let C be a pointed closed model category. An Omega weak spec-
trum in C consists of objects X0, X1, . .,.and isomorphisms ~oen:Xn ! Xn+1 in
Ho C, where denotes the (Quillen) loop functor on Ho C. A map of Omega weak
spectra from __X = {Xn} to __Y = {Yn} consists of maps Xn ! Yn in Ho C that
commute with the structure maps ~oen.
The rule hn(-; __X) = Ho[CB =B](-, Xn) defines a functor from the category of
Omega weak spectra to the category of reduced cohomology theories on CB =B. The
Yoneda Lemma implies that this functor is a full embedding, and the discussion
above implies that every reduced cohomology theory is isomorphic to one associa*
*ted
to an Omega weak spectrum. In summary, we have the following proposition:
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 7
Proposition 1.5. The category of reduced cohomology theories on CB =B is equiv-
alent to the category of Omega weak spectra in CB =B.
Finally, we need a general result about the relationship between Omega weak
spectra and spectra in a simplicial or topological pointed model category C. In
this context, when X is cofibrant, the tensor X ^ I+ is a cylinder object, and
so the tensor EX = X ^ S1 represents the (Quillen) suspension functor on the
homotopy category. Likewise, when Y is fibrant, the cotensor of Y with S1, Y ,
represents the (Quillen) loop functor on the homotopy category. A spectrum __X *
*is
defined as a sequence of objects X0, X1, . .a.nd maps oe :EXn ! Xn+1 in C. We
say that a spectrum __X is öc fibrant" if each Xn is cofibrant and each structu*
*re
map EXn ! Xn+1 is a cofibration. In the case of interest the cofibrant spectra
in CB =B are the cofibrant objects in the stable model structure (see Theorem 3*
*.1
below); quite generally, the cofibrant spectra are the cofibrant objects in some
model structure on the category of spectra, q.v. [6, 1.13-14].
We write ~oefor the adjoint of the structure map Xn ! Xn+1. A spectrum __X
is called an Omega spectrum when each Xn is fibrant and each adjoint structure
map is a weak equivalence. Then by neglect of structure (passing from C to Ho C*
*),
an Omega spectrum becomes an Omega weak spectrum. The following lemma is
the standard observation that every map of Omega weak spectra can be rectified
to a map of spectra (though typically not uniquely).
Lemma 1.6. Let C be a simplicial or topological pointed closed model category.
(i)Every Omega weak spectrum is (non-canonically) isomorphic to the under-
lying Omega weak spectrum of a cofibrant Omega spectrum.
(ii)Let __X and __Y be Omega spectra and suppose that __X is cofibrant. A*
*ny
map of Omega weak spectra f :__X ! __Y is represented by a map of spectra
(generally not uniquely).
Proof.Given an arbitrary Omega weak spectrum __X, each Xn is isomorphic in Ho C
to an object X0nthat is fibrant, and using these isomorphisms, we get an isomor*
*phic
Omega weak spectrum __X0. We choose a cofibrant Omega spectrum __X00as follows:
We choose X000to be a cofibrant approximation of X00. Then since X000is cofibra*
*nt
and X01is fibrant, we can choose a map X000! X01representing the composite
map in Ho C
X000-! X00-! X01.
We factor the adjoint map EX000! X01as a cofibration followed by an acyclic
fibration
EX000-! X001-! X01.
Continuing in this fashion constructs a cofibrant Omega spectrum __X00and an is*
*o-
morphism of Omega weak spectra __X00! __X0.
For (ii), since X0 is cofibrant and Y0 is fibrant, we can choose a map X0 ! Y0
representing f0. The hypothesis SM7 that C is a simplicial model category or
the analogous hypothesis that C is a topological model category implies that the
induced map of simplicial sets or of spaces
C(X1, Y1) -! C(EX0, Y1)
is a fibration and identifies the induced map on components as Ho C(X1, Y1) !
Ho C(EX0, Y1). Since the given map f1 in Ho C maps to the same component as
8 MARIA BASTERRA AND MICHAEL A. MANDELL
the map adjoint to the composite X0 ! Y0 ! Y1, there exists a map X1 ! Y1 in
C that represents f1 and makes the diagram
EX0D_____//X1
DD |
DDD |
DD""fflffl|
Y1
commute in C. Applying this argument inductively to each map Xn ! Yn con-
structs a compatible map Xn+1 ! Yn+1, and the map of spectra __X ! __Y .
2. Nucas, Indecomposables, and Stabilization
The functors L AX and LAb BAX mentioned in the introduction are somewhat
awkward to work with formally because they are composites of both left and right
derived functors. However, the technical trick introduced in [1] of working wi*
*th
non-unital commutative algebras (ün cas") alleviates this problem by providing a
point-set (left adjoint) functor whose left derived functor is a model for LAb *
*B. The
first half of this section consists of a brief overview of the theory of nucas *
*from [1].
Another technical advantage of the category of nucas is that every object is fi*
*brant,
and this makes the construction of a "stabilization" functor S from cofibrant s*
*pectra
to Omega spectra easier. The second half of this section constructs this funct*
*or
and explores its basic properties.
Definition 2.1. Let B be a commutative R-algebra. A non-unital commutative
B-algebra (or B-nuca) consists of a B-module N together with an associative and
commutative multiplication ~: N ^B N ! N. A map of B-nucas is a map of B-
modules N ! N0 that commutes with the multiplications. We write NB for the
category of B-nucas.
The category of B-nucas may also be described as the category of algebras in
B-modules over the operad Cofm with Cofm(k) = * for k > 0 and Cofm(0) empty.
We have a free B-nuca functor
`
NM = M(k)= k
k>0
for a B-module M (where M(k)= M ^B . .^.BM), and we have a functor K from
NB to CB =B defined by formally adding a unit: On the underlying B-modules,
KN = B _ N,
with the multiplication extended from N to KN by the usual multiplication on
B and the B-action maps of B on N. The functor K is the left adjoint of the
ä ugmentation ideal" functor I from CB =B to NB defined by setting I(A) to be
the (point-set) fiber of the augmentation A ! B. According to Basterra [1, 1.1,*
*2.2],
the adjunction (K, I) is a Quillen equivalence.
Proposition 2.2. The category NB is a topological pointed closed model category
with weak equivalences and fibrations the maps that are weak equivalences and f*
*i-
brations (resp.) of the underlying B-modules. The adjunction (K, I) is a Quillen
equivalence between NB and CB =B.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 9
Since K preserves all weak equivalences, it is harmless to use the same notat*
*ion
for the derived functor. We denote the right derived functor of I by IR . As in*
* any
Quillen adjunction, the derived functor IR preserves the Quillen loop functor, *
*but
since the derived functor K is an inverse equivalence, it also preserves the Qu*
*illen
loop functor, and we obtain the following proposition:
Proposition 2.3. The derived functors K and IR induce an equivalence between
the category of Omega weak spectra in NB and the category of Omega weak spectra
in CB =B.
Basterra [1, x3] constructs an "indecomposables" functor Q from NB to MB that
is left adjoint to the "zero multiplication" functor Z from MB to NB . Precisel*
*y,
the indecomposables functor QN is defined as the (point-set) pushout
QN = * [(N^BN) N
in the category of B-modules of the multiplication N ^B N ! N over the trivial
map N ^B N ! *. The functor Z simply assigns a B-module M the trivial map
M ^B M ! M for its multiplication. Since Z clearly preserves weak equivalences
and fibrations, we have the following observation of [1]:
Proposition 2.4. The functors (Q, Z) between NB and MB form a Quillen ad-
junction.
Again, since Z preserves all weak equivalences, it is harmless to denote its *
*derived
functor by the same notation. We write QL for the left derived functor of Q.
The derived functors QL and IR are needed in [1, 4.1] to defined the cotangent
complex. For a cofibrant commutative R-algebra A and a cofibration of commuta-
tive R-algebras A ! X, the cotangent complex of X relative to A is defined to be
the X-module
L AX = QLIR (X ^A X),
where IR and QL are understood in terms of CX =X and NX . When A is not
cofibrant or A ! X is not a cofibration, the cotangent complex is constructed by
choosing a cofibrant approximation A0! X0of A ! X, and then using the derived
extension of scalars functor X ^LX0(-):
L AX = X ^LX0L A0X0= X ^LX0(QLIR (X0^A0X0)).
Because the category this construction lands in depends on its inputs, a discus*
*sion of
functoriality would be somewhat involved, but (as mentioned in the introduction*
*),
for (X, A) a pair in CR =B, the construction we consider,
LAb BAX = B ^LX0L A0X0
assembles to a functor from the category of pairs in CR =B to the homotopy cate*
*gory
of B-modules, and this functoriality suffices for our work. The last fact we ne*
*ed
about the cotangent complex is the following version of [1, 4.4-5]:
Proposition 2.5. Assume that B is a cofibrant commutative R-algebra. If A is
cofibrant and A ! X is a cofibration in CR =B, then LAb BAX is isomorphic to
QLIR (B ^A X), naturally in A and cofibrations A ! X.
Next we move on to the stabilization functor. In other contexts, this functor
is typically denoted by Q, but here we denote it by S to avoid confusion with t*
*he
indecomposables functor. We have the notion of a spectrum in the category of
10 MARIA BASTERRA AND MICHAEL A. MANDELL
B-nucas, as discussed in the previous section. The stabilization functor S turn*
*s out
to be a functor from spectra in NB to Omega spectra in NB .
Definition 2.6. For a NB -spectrum __X, we define a NB -spectrum S__X = {Sn__X}
as follows: Let
Sn__X = Telk n k-nXk,
the telescope over the adjoint structure maps. We have a map of telescopes
Telk n+1E k-nXk -! Telk n+1 k-(n+1)Xk
induced by sending E k-nXk to k-(n+1)Xk using the counit of the suspension,
loop adjunction applied to the innermost factor of , and we have a map of tele-
scopes
Telk nE k-nXk -! Telk n+1E k-nXk
induced by collapsing down the map EXn ! E Xn+1. We define the structure
map oe :ESn__X ! Sn+1__X to be the composite
E(Telk n k-nXk) ~=Telk nE k-nXk -!
Telk n+1E k-nXk -! Telk n+1 k-(n+1)Xk.
The construction S assembles in the obvious way to a functor from NB -spectra
to NB -spectra. Since the composite map
EXn E~oe--!E Xn+1 -! Xn+1
is the structure map oe, the inclusion of Xn into the telescope defining Sn__X *
*defines
a natural transformation __X ! S__X. The main fact we need about S is the follo*
*wing
proposition. In it, the homotopy groups of a spectrum __X are defined by
ßq__X = Colimßq+nXn.
Proposition 2.7. For any spectrum __X in NB , S__X is an Omega spectrum in NB
and the natural transformation __X ! S__X induces an isomorphism on homotopy
groups.
Proof.The telescope in the category of B-nucas is naturally weakly equivalent to
the telescope in the category of B-modules, and so the usual map
(Telk n+1 k-(n+1)Xk) -! Telk n+1 k-nXk
is a weak equivalence. The composite of the adjoint structure map and the map
above,
Telk n k-nXk -! (Telk n+1 k-(n+1)Xk) -! Telk n+1 k-nXk
is a homotopy equivalence, and so the adjoint structure map Sn__X ! Sn+1__X is*
* a
weak equivalence. Thus S__X is an Omega spectrum. We have
ßq+nSn__X ~=Colimk nßq+kXk,
and the map Xn ! Sn__X induces on homotopy groups the inclusion of ßq+nXn
into this colimit system. Under this identification, the map __X ! S__X induces*
* on
homotopy groups the map
Colimnßq+nXn -! ColimnColimk n ßq+kXk,
which is clearly an isomorphism.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 11
For a B-nuca N, let E1 N be the üs spension spectrum" which has as its n-th
object the n-fold suspension EnN and structure maps the identity map
E(EnN) -! En+1N.
In the case when N is free, i.e., N = NM for some B-module M, we have a canonic*
*al
isomorphism
EnNM ~=N nM,
and in particular, we have canonical maps nM ! SnE1 NM for all n. The fol-
lowing result on the suspension spectra of free B-nucas represents the fundamen*
*tal
difference between the context of commutative S-algebras and simplicial commu-
tative algebras.
Theorem 2.8. Assume that B is a cofibrant commutative R-algebra. If M is a cofi-
brant B-module, then the canonical maps nM ! SnE1 NM are weak equivalences
for all n.
Proof.The general case follows from the case n = 0, where we are studying the
map
ßqNM ~=ßq+k kNM -! ßq+kN kM.
The map kNM ! N kM takes the wedge summand kM(m)= m to the corre-
sponding wedge summand ( kM)(m)= m via the diagonal map on the sphere Sk.
The proposition is an immediate consequence of the following lemma.
Lemma 2.9. Let B be a cofibrant commutative R-algebra, let M be a cofibrant
B-module, and let x be an element of ßq(M(m)= m ) for some m > 1 and some
integer q. Then for some k, the composite map
ßq(M(m)= m ) ~=ßq+k k(M(m)= m ) -! ßq+k(( kM)(m)= m )
sends x to zero.
Proof.By [5, III.5.1], for any cofibrant B-module N (e.g., M, kM), the map
E m+ ^ m N(m) -! N(m)= m
is a weak equivalence. The cellular filtration of the m -CW complex E m in-
duces an increasing filtration on E m+ ^ m N(m) and on the homotopy groups of
N(m)= m ; clearly, this filtration is trivial (zero) below the zero level. The *
*map
ffi :E m+ ^ m M(m) -! E m+ ^ m ( M)(m)
(induced by the diagonal S1 ! (S1)(m)) preserves the filtration. Since m > 1, t*
*he
map M(m) ! ( M)m is null homotopic, and so ffi induces the zero map on the
E1-term of the homotopy group spectral sequence associated to the filtration. It
follows that the map on homotopy groups induced by ffi strictly lowers filtrati*
*on
level. The map ßqffik is the map
ßq(M(m)= m ) ~=ßq+k k(M(m)= m ) -! ßq+k(( kM)(m)= m )
in the statement, which therefore takes every element in filtration level n to *
*an
element of filtration level n - k. Taking k greater than the minimum filtration*
* level
of x, the map must send x to zero.
12 MARIA BASTERRA AND MICHAEL A. MANDELL
We can illustrate the previous lemma and theorem in the case R = S and
B = HF2 since in this case, ß*NM is easy to describe in terms of ß*M. Specifica*
*lly,
ß*NM is the polynomial F2-nuca on the free allowable Dyer-Lashof ("DL") module
on ß*M modulo the relation that the square (in the algebra structure) is equal
to the squaring operation (in the DL structure) on each element. The suspension
map oe :ß*NM ! ß*+1N M kills decomposables (in the algebra structure) and is
a map of DL-modules. An element of the form Qsx for x 2 ßqNM is therefore
killed by the map oek: ßqNM ! ßq+kN kM for k = s - q + 1 because oek-1(Qsx) =
Qs(oek-1x) = (oek-1x)2 is decomposable. From this it is easy to see directly th*
*at
the map ß*M ! Colimß*+kN kM is an isomorphism.
3.Proof of Theorems 3 and 4
In this section, we prove Theorems 3 and 4 of the introduction. The arguments
take advantage of the technical simplifications the category of B-nucas provide
and use Theorem 2.8 above as the key step. They also make use of the following
theorem, proved in Section 7:
Theorem 3.1. Let C be MB , NB , or CB =B. Then the category Sp(C) of C-spectra
is a topological closed model category with:
(i)Cofibrations the maps __X ! __Y with X0 ! Y0 a cofibration and each
EYn [EXn Xn+1 ! Yn+1 a cofibration,
(ii)Fibrations the maps __X ! __Y with each Xn ! Yn a fibration and each
Xn ! Yn x Yn+1 Xn+1 a weak equivalence, and
(iii)Weak equivalences the maps that induce an isomorphism on homotopy
groups.
In the statement, "Eä nd " " denote the (point-set) tensor and cotensor with
the based space S1 in the pointed topological category C, and öc fibration" mea*
*ns
a cofibration in the model structure on C (which was called a "q-cofibration" in
[1] and [5]). We have defined homotopy groups for CB =B-spectra and NB -spectra
above, and the definition is the same for MB -spectra:
ßq__X = Colimßq+nXn.
To avoid confusion, we use the term "stable equivalence" for weak equivalence in
the model structure on Sp(C) above, and we call the homotopy category of this
model structure the "stable categoryö f C.
We have the following easy consequence of the characterization of fibrations:
Proposition 3.2. An object is fibrant in one of the model structures in Theorem*
* 3.1
if and only if it is an Omega spectrum.
The following proposition is also clear:
Proposition 3.3. A map of Omega spectra __X ! __Y is a stable equivalence if and
only if it is a weak equivalence X0 ! Y0.
For any of the model categories C in Theorem 3.1, we have a Quillen adjunction
between C and the category Sp(C) of C-spectra with left adjoint the suspension
spectrum functor (that sends an object X to the suspension spectrum E1 X =
{EnX}) and the zeroth object functor (that sends a spectrum __X to the zeroth
object X0). The previous propositions applied to C = MB imply that this is a
Quillen equivalence.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 13
Proposition 3.4. Let C be one of the categories in Theorem 3.1. The suspension
spectrum functor and the zeroth object functor are a Quillen adjunction between
the model category C and the model category Sp(C) of C-spectra. In the case of
C = MB , this Quillen adjunction is a Quillen equivalence.
If we consider any adjunction that is enriched over the category of based spa*
*ces,
the left adjoint preserves tensors and the right adjoint preserves cotensors, a*
*nd
so both functors extend to functors between the categories of spectra. It is e*
*asy
to see that the induced functors on spectra remain adjoints. When in addition
the categories are ones considered in Theorem 3.1 and the adjunction is a Quill*
*en
adjunction, the characterization of the cofibrations and fibrations in the cate*
*gories
of spectra imply that the induced adjunction on spectra is a Quillen adjunction.
When we combine the previous proposition with the observations of the last
paragraph applied to the free, forgetful adjunction MB ø NB and to the K, I
adjunction NB ø CB =B, we have the following sequence of Quillen adjunctions:
___1_// __N_//_ ____K//_
(3.5) MB oo___Sp(MB ) oo___Sp(NB )oo___Sp(CB =B)
(-)0 _I
The first and last of these Quillen adjunctions are Quillen equivalences, and we
prove that the middle one is also a Quillen equivalence.
Lemma 3.6. Assume that B is a cofibrant commutative R-algebra. Then the free,
forgetful adjunction between MB -spectra and NB -spectra is a Quillen equivalen*
*ce.
Proof.Let M be a cofibrant B-module and let __X be a spectrum in NB that is
fibrant in the model structure above. It suffices to show that a map 1 M ! __X
is a stable equivalence if and only if the adjoint map E1 NM ! __X is a weak
equivalence. By Proposition 2.7 and Propositions 3.2 and 3.3 above, it suffices*
* to
show that M ! X0 is a weak equivalence if and only if S0E1 NM ! S0__X is a weak
equivalence. The diagram
M _________//X0
| |
| |
fflffl| fflffl|
S0E1 NM ____//_S0__X
in MB commutes, and the result follows from Theorem 2.8.
As an immediate consequence, we obtain a Quillen equivalence between the
category of B-modules and the category of CB =B-spectra. This is sufficient to
prove Theorem 3 as stated, but we would like to know that the equivalence of
homotopy categories is induced by the functor that sends a B-module M to the
CB =B-spectrum
ZM = {KZ nM}
that represents Andr'e-Quillen cohomology with coefficients in M. Also, for The-
orem 4, we need to know that the equivalence of homotopy categories takes the
suspension spectrum E1 (B ^R A) of a cofibrant object A of CR =B to the B-module
LAb BRA. Both of these are consequences of the following theorem.
Theorem 3.7. Let B be a cofibrant commutative R-algebra. Then the __Q, __Z ad-
junction between Sp(NB ) and Sp(MB ) is a Quillen equivalence.
14 MARIA BASTERRA AND MICHAEL A. MANDELL
Proof.Since the Quillen adjunction Q, Z between NB and Sp(MB ) is enriched over
the category of based spaces, as observed above, we obtain a Quillen adjunction
__Q, __Z between the categories of spectra. According to MMSS [10, A.2.(ii)], *
*for
this to be a Quillen equivalence, we just need to show that one of the derived
functors is an equivalence on the homotopy categories. If we write ___QL for th*
*e left
derived functor of __Q: Sp(NB ) ! Sp(MB ) and ___NL for the left derived functo*
*r of
__N: Sp(MB ) ! Sp(NB ), then the composite functor ___QL O___NL is naturally is*
*omorphic
to the derived functor of the composite and so is naturally isomorphic to the i*
*dentity
functor on Sp(MB ). By the previous lemma, ___NL is an equivalence, and it foll*
*ows
that ___QL is an equivalence. We conclude that __Q, __Z is a Quillen equivalenc*
*e.
In the course of the previous argument, we also proved the following result:
Proposition 3.8. If B is a cofibrant commutative R-algebra, then the derived
functors of __N and __Z are naturally isomorphic functors Ho Sp(MB ) ! Ho Sp(NB*
* ),
and the derived functors of __Q and the forgetful functor are naturally isomorp*
*hic
functors Ho Sp(NB ) ! Ho Sp(MB ).
We can be more explicit about these natural isomorphisms. The first is induced
by the natural transformation of point-set functors N ! Z that arises from the
universal property of the free functor. When __M is a cofibrant object in Sp(MB*
* ),
the map __N__M ! __Z__M is a stable equivalence because the composite map
__M -! __N__M -! __Z__M
in Sp(MB ) is the identity and the map __M ! __N__M is a stable equivalence by
Lemma 3.6. The other natural transformation is induced by the unit of the Q, Z
adjunction, specifically, the natural transformation Id! __Q in Sp(MB ). This m*
*ap
is hard to study directly; instead, the argument implicit in the proof of Theor*
*em 3.7
studies the (solid) zigzag
__X0____//C_N__X0//___Q__N__X0
CC _____ |
CCC _____ |
CC!!fflffl___fflffl|
__X______//______________Q__X
where __X is a cofibrant object in Sp(NB ), and __X0 ! __X is a cofibrant appro*
*xi-
mation in Sp(MB ). The diagonal solid arrow is therefore a stable equivalence by
assumption, and the composite horizontal map __X0 ! __Q__N__X0 is the isomorphi*
*sm
(explicit) in the proof of the theorem. The remaining solid arrows are also sta*
*ble
equivalences: The top one __X0! __N__X0 is a stable equivalence by the lemma an*
*d the
right-hand one is a stable equivalence because, as a Quillen left adjoint, __Q *
*preserves
stable equivalences between cofibrant objects. This is the concrete argument th*
*at
shows that the natural map __X ! __Q__X in Sp(MB ) is a stable equivalence for *
*__X
cofibrant in Sp(NB ).
Proof of Theorems 3 and 4.The equivalence of the stable category of CB =B and
the homotopy category of B-modules we need for Theorem 3 follows from the three
Quillen equivalences in (3.5), or better, by the outer two in (3.5)and the one *
*in
Theorem 3.7. Theorem 4 then follows by Proposition 2.5.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 15
4. Proof of Theorem 1
In this section, we prove Theorem 1 from the introduction. By Propositions 1.*
*2,
1.5, and 2.3, it suffices to consider the case when B is a cofibrant commutative
R-algebra and show that the functor that sends a B-module M to the Omega weak
spectrum in NB
__Z 1 M = {Z nM}
gives an equivalence between the homotopy category of B-modules and the category
of Omega weak spectra in NB . We write W (NB ) for the category of Omega weak
spectra in NB .
The work of the previous section suggests that we should be able to use the
forgetful functor on the zeroth object as the inverse equivalence W (NB ) ! Ho *
*MB .
The composite functor Ho MB ! Ho MB is the identity. The composite functor
W (NB ) ! W (NB ) sends __X to __Z 1 X0; the proof of Theorem 1 is completed by
showing that this functor is naturally isomorphic to the identity.
To construct an isomorphism in W (NB ) between __X and __Z 1 X0, we have to
take a detour through the category of spectra in NB . To avoid confusion, we wr*
*ite
W __A for the underlying Omega weak spectrum of an Omega spectrum __A. First
we choose a cofibrant Omega spectrum __A__Xand an isomorphism of Omega weak
spectra W __A__X! __X as in Lemma 1.6.(i). We write A0__Xfor the zeroth object
of __A__X, and we choose a cofibrant approximation M__X! A0__Xin the category of
B-modules. Then we have the following chain of stable equivalences in Sp(NB ):
__A__X- __N 1 M__X-! __Z 1 M__X-! __Z 1 A0__X.
(See the explanation following Proposition 3.8 for a proof that the first two m*
*aps
are stable equivalences.) Using the functor S, we then have the following chain*
* of
stable equivalences of Omega spectra:
__A__X-! S__A__X- S(__N 1 M__X) -! S(__Z 1 A0__X) - __Z 1 A0__X.
Finally, applying W , we have the following chain of isomorphisms of Omega weak
spectra:
__X ~=W __A__X-! W S__A__X- W S(__N 1 M__X) -!
W S(__Z 1 A0__X) - W __Z 1 A0__X~=_Z 1 X0.
Let OE__X:_X ! __Z 1 X0 be the composite isomorphism.
The isomorphism OE__Xappears to depend on the choices made above, and it is
far from obvious that OE is a natural transformation. Let f :__X ! __Y be a map*
* of
Omega weak spectra; we must show that the diagram
OE_X 1
__X_________//__Z X0
f fflffl| |f0fflffl
__Y_____OE_Y_//_Z 1 Y0
commutes in the category of Omega weak spectra. According to Lemma 1.6.(ii),
we can choose a map of spectra __A__X! __A__Ywhose underlying map of Omega weak
spectra is the composite
f ~
__A__X~=_X -! __Y = __A__Y.
16 MARIA BASTERRA AND MICHAEL A. MANDELL
Likewise, since M__Y! A0__Yis an acyclic fibration, we can choose a map of B-
modules M__X! M__Ymaking the diagram
M__X____//_A0__X
fflffl| fflffl||
M__Y_____//A0__Y
commute in MB . Then the following diagram commutes in W (NB ):
__XoW__A_Xo_//_WS__A_XWS(_No1oM_X)_//_WS(_Z 1oA0__X)W_Zo1_A0__X//__Z 1 X0
ffflffl||fflffl|fflffl| fflffl| fflffl|| fflffl|| ff0flffl||
__YoW__A_Yo_//_WS__A_YWS(_No1oM_Y)_//_WS(_Z 1oA0__Y)W_Zo1_A0__Y//__Z 1 Y0
The required naturality now follows. (The same argument applied to the identity
map on __X shows that OE__Xis in fact independent of the choices.) This complet*
*es
the proof of Theorem 1.
5. Proof of Theorem 2
In this section we prove Theorem 2. By Proposition 1.2, it suffices to consid*
*er
the case when B is a cofibrant commutative R-algebra, and by Proposition 2.2
it suffices to prove the analogous theorem for reduced homology theories on the
category NB . The work of Section 3 reduces this to proving the following theor*
*em:
Theorem 5.1. The category of reduced homology theories on NB is equivalent to
the category of reduced homology theories on Sp(NB ).
In this equivalence, the functor in one direction sends the reduced homology
theory k* in Sp(NB ) to the theory kE*on NB defined by
kE*(N) = k*(EL1 N).
where EL1 N denotes the left derived functor of the suspension spectrum functor
E1 . On the other hand, for a reduced homology theory h* on NB , we define a
functor hc*from Sp(NB ) to graded abelian groups by
hcq(__X) = Colimhq+nXn.
Lemma 5.2. For any __X in Sp(NB ), the map hc*(__X) ! hc*(S__X) is an isomorphi*
*sm.
Proof.The map hcq_X ! hcqS__X is the map
Colimnhq+nXn -! Colimnhq+n(Telj n j-nXj)
~=ColimnColimj nhq+n( j-nXj).
To see that this is an isomorphism, it suffices to show that the map
Colimnhq+nXn -! Colimnhq+n( iXn+i)
is an isomorphism for all i. The maps
hq+n( iXn+i) ~=hq+n+i(ELi iXn+i) -! hq+n+iXn+i
induce a map
Colimnhq+n( iXn+i) -! Colimnhq+nXn
inverse to the map above.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 17
It follows that hc*sends stable equivalences to isomorphisms and therefore in*
*duces
a functor from the stable category Ho Sp(NB ) to the category of graded abelian
groups. The suspension isomorphism for h* induces a suspension isomorphism for
hc*, and the Direct Sum Axiom for h* implies the Direct Sum Axiom for hc*. This
then defines a functor from the category of homology theories on NB to the cate*
*gory
of homology theories on Sp(NB )
In order to prove Theorem 5.1, it suffices to prove that these functors (-)c *
*and
(-)E are inverse equivalences. It is clear that when we start with a homology t*
*heory
h* on NB , we have a natural isomorphism between h* and the composite functor
hcE*. To produce the natural isomorphism between the other composite and the
identity, we use the following construction:
Definition 5.3. For a spectrum __X, let Tn__X be the spectrum with j-th object *
*Xj
for j n and Ej-nXn for j > n.
We have compatible natural maps Tn__X ! Tn+1__X, with __X the colimit. More
usefully for the work below, the map TelTn__X ! __X is an objectwise weak equiv-
alence (a weak equivalence on each object); this implies the following less pre*
*cise
result:
Proposition 5.4. The natural map TelTn__X ! __X is a stable equivalence.
If we write FnXn for the spectrum that has j-th object the initial object * f*
*or
j < n and Ej-nXn for j n, then we have natural maps
EnTn__X - EnFnXn -! E1 Xn
that are isomorphisms on j-th objects for j n and are therefore stable equiv-
alences. Applying k* and the suspension isomorphism, we obtain the following
proposition:
Proposition 5.5. For __X cofibrant, kqTn__X ~=kq+nE1 Xn .
The previous two propositions give us a natural (in both __X and k*) isomorph*
*ism
kEcq(__X) = Colimkq+n(E1 Xn ) -! kq(__X).
Since this natural isomorphism commutes with the suspension isomorphisms, it is
a natural isomorphism of homology theories. This completes the proof of Theo-
rem 5.1.
6. Proof of Theorem 5
In this section we prove Theorem 5 that interprets the cotangent complex of t*
*he
suspension spectra of E1 spaces in terms of the associated spectra. As indicat*
*ed
in the introduction, up to equivalence, the definition of cotangent complex sho*
*uld
not depend on the E1 operad involved. In order to take advantage of the work in
previous sections, we work with the linear isometries operad L, and consider the
category of L-spaces, the E1 spaces for the E1 operad L. If X is an L-space,
then 1 X+ is an "L-spectrum", an E1 ring spectrum for the E1 operad L. The
category of L-spectra is closely related to the category of commutative S-algeb*
*ras:
The functor S ^L (-) studied in EKMM [5, Ix8] converts L-spectra to weakly
equivalent commutative S-algebras. We study the functor that takes an L-space
X to the commutative S-algebra 1S+X = S ^L 1 X+ and we prove the following
theorem.
18 MARIA BASTERRA AND MICHAEL A. MANDELL
Theorem 6.1. For an L-space X, the S-module LAb SS 1S+X ~=QLIR 1S+X is
naturally isomorphic in the stable category to the spectrum associated to X.
In the theorem, we are regarding 1S+X as augmented over S = 1S+*via
the map induced by the trivial map X ! *, and LAb SSis as in Section 2 (with
A = B = S). Since the cotangent complex of 1S+X is weakly equivalent to
the extended 1S+X-module ( 1S+X) ^ LAb SS 1S+X, Theorem 5 is an immediate
consequence.
We analyze LAb SS 1S+X using the results of Section 3. Since the work of that
section is phrased in terms of model structures and Quillen adjunctions, it is *
*con-
venient to make a number of model category observations for L-spaces. Since the
category of L-spaces is the category of algebras for a continuous monad, Quille*
*n's
small object argument and standard techniques prove the following proposition.
Proposition 6.2. The category of L-spaces is a topological closed model category
with weak equivalences and fibrations the weak equivalences and (Serre) fibrati*
*ons
of the underlying spaces.
May, Quinn, and Ray [13, IV.1.8] observe that the functor 1 (-)+ from L-
spaces to L-spectra is left adjoint to the zero-th space functor 1 . Since the
functor 1 preserves weak equivalences and fibrations, this is in fact a Quill*
*en
adjunction. Since the functor S ^L (-) from L-spectra to commutative S-algebras
is a Quillen left adjoint, we obtain the following result.
Proposition 6.3. The functor 1S+from the category of L-spaces to the category
of commutative S-algebras is a Quillen left adjoint.
The previous propositions in particular give us a notion of cofibrant L-space
and prove that the suspension spectrum functor 1S+takes cofibrant L-spaces to
cofibrant commutative S-algebras. Since in L-spaces, the one-point L-space is b*
*oth
the initial and final object, the category of L-spaces is enriched over the cat*
*egory of
based spaces. When we regard 1S+as a functor into the category of commutative
S-algebras lying over S, the functor 1S+is enriched over based spaces. As a fo*
*rmal
consequence we obtain the following proposition.
Proposition 6.4. The functor 1S+from the category of L-spaces to the category *
*of
commutative S-algebras over S preserves the tensor with based spaces. In partic*
*ular,
it converts the suspension functor B in the category of L-spaces to the suspens*
*ion
functor E in the category of commutative S-algebras over S.
We use the notation B for the suspension in L-spaces because of the following
theorem, which is well-known to experts. The theorem is closely related to the
uniqueness theorem of May and Thomason [12] and the relationship between the
homotopy category of L-spaces and the homotopy category of connective spectra.
For the statement, note that since the category of L-spaces may be defined as t*
*he
category of algebras for a continuous monad in based spaces, the loop space X *
*is
the underlying based space of cotensor of an L-space X with the based space S1.
Theorem 6.5. The derived functor of B is a (one-fold) delooping functor: If X is
a cofibrant L-space, then the unit of the suspension, loops adjunction,
X -! BX
is group completion.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 19
Since no proof of this theorem has appeared in the literature, we outline a p*
*roof
at the end of the section.
The canonical maps BnX ! Bn+1X make {BnX} a spectrum in the category
of spaces, or in the terminology of Lewis-May [8], an "indexed prespectrum". Wh*
*en
X is cofibrant, this has the further property that the adjoint structure map Bn*
*X !
Bn+1X is a weak equivalence for n > 0 and is group completion for n = 0. Also
when X is cofibrant, the structure maps are cofibrations, and so
Z = ColimS-n ^ BnX,
is a Lewis-May spectrum whose zeroth space is a group completion of X. Although
this construction does not constitute an infinite loop space machine on L-space*
*s,
we have the following proposition; see Remark 6.10 below for further discussion.
Proposition 6.6. If X is cofibrant, Z is a model for the spectrum associated to
X.
The Lewis-May spectrum Z is naturally isomorphic in the stable category to the
S-module
ZS = ColimS-nS^ BnX.
This is the composite of the "free L-spectrum" functor of EKMM applied to Z,
LZ = L(1) n Z, and the functor S ^L (-) from L-spectra to S-modules. For the
proof of Theorem 5, it is useful to reframe this in the context of the spectra *
*in the
category of S-modules, i.e., the MS-spectra of Section 3. We have a MS-spectrum
__Z defined by
Zn = SS ^ BnX,
and we have a canonical natural map of MS-spectra from __Z to the suspension MS-
spectrum of ZS induced by the inclusion of SS ^ BnX ~= n(S-nS^ BnX) in the
colimit system defining ZS. An easy colimit argument shows that this map is a
stable equivalence.
The MS-spectrum __Z is one MS-spectrum associated to {BnX}, but we also have
a different one that takes into account the action of the topological monoid L(*
*1) on
the based spaces BnX. The (Lewis-May) suspension spectrum functor 1 takes
based L(1)-spaces to L-spectra; we write 1Sfor the composite functor S ^L 1 (*
*-)
that lands in S-modules. The purpose for introducing this construction is that *
*we
have a canonical natural isomorphism of S-modules
1S(X+ ) ~= 1S+X.
We have a MS-spectrum __B defined by
Bn = 1SBnX.
The identity isomorphism of 1 BnX in the category of Lewis-May spectra induces
a map of L-spectra
(LS) ^ (BnX) ~=L( 1 BnX) -! 1 BnX
and a map of S-modules
SS ^ BnX = S ^L LS ^ BnX -! S ^L 1 BnX = 1SBnX,
that induces a map of MS-spectra __Z ! __B. Since each map displayed above is a
weak equivalence, the map __Z ! __B stable equivalence. We now have everything
needed for the proof of Theorem 6.1.
20 MARIA BASTERRA AND MICHAEL A. MANDELL
Proof of Theorem 6.1.It suffices to consider the case when X is a cofibrant L-s*
*pace
and show that QLIR 1S+X is naturally isomorphic to ZS in the homotopy cate-
gory of S-modules. Applying Propositions 3.4 and 3.8, and combining with the
work above, it suffices to show that the underlying MS-spectrum of the derived
suspension spectrum EL1 IR 1S+X is naturally isomorphic to __B in the homotopy
category of Sp(MS). Since I is part of a Quillen equivalence, its derived funct*
*or pre-
serves suspension and EL1 IR 1S+X is naturally isomorphic to _IR E1 1S+X in t*
*he
homotopy category of Sp(MS). Since the underlying MS-spectrum of _IR E1 1S+X
is the homotopy fiber of the augmentation, it is naturally weak equivalent to t*
*he
MS-spectrum cofiber of the unit map:
E1 1S+X [E1 SCE1 S ~-!_IR E1 1S+X.
Since the unit map is a cofibration, the cofiber is equivalent to the quotient,
(E1 1S+X)=E1 S. Using the natural isomorphisms of S-modules
(En 1S+X)=S ~=( 1S+BnX )=S ~= 1S(BnX+ =S0)~= 1SBnX ,
we obtain our chain of natural isomorphisms in Ho Sp(MS) between EL1 IR 1S+X
and __B as the zigzag
~ 1 1 1 ~ n 1 ~ 1 n
_IR E1 1S+X - E S+ X [E1 SCE S -! (E S+ X)=S = S B X = __B.
We now go on to the proof of Theorem 6.5. We write L for the monad on based
spaces associated to the operad L. For a based space T , LT is the quotient of *
*the
disjoint union of L(n) x n T n(cartesian power of T ) by an equivalence relatio*
*n in
terms of the basepoint and the operad degeneracies (operadic multiplications wi*
*th
L(0) = *), described in detail in [11, 2.4].
We study the suspension in L-spaces in terms of geometric realization. For an
L-space X and a based simplicial set To, let X bTo be the simplicial L-space wh*
*ich
in degree n is the tensor of X with the based set Tn; this is the coproduct of *
*copies
of X indexed on the non-basepoint simplexes of Tn. Writing S1ofor the simplicial
model of the based circle with one vertex and one non-degenerate 1-simplex, the
following lemma implies in particular that BX is the geometric realization of t*
*he
simplicial L-space X b S1o.
Lemma 6.7. The geometric realization of a simplicial L-space is naturally an L-
space. For any L-space X and any based simplicial set To, the map X b |To| !
|X b To| induced by the universal property of the tensor is an isomorphism of L-
spaces.
Proof.The first statement is [11, 12.2]: Since cartesian products and colimits *
*of
spaces commute with geometric realization, for any simplicial based space Yo, we
have a natural isomorphism L|Yo| ~= |LYo|. It is straight-forward to check that
the composite of this isomorphism and the geometric realization of the L-space
structure map |LYo| ! |Yo| provides an L-space structure map for |Yo|.
For the statement about tensors, consider first the free L-space LX. The uni-
versal property of the free functor and the coproduct induce an isomorphism of
simplicial L-spaces
L(X ^ To) ~=(LX) b To,
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 21
and applying the isomorphism of the previous paragraph, we get an isomorphism
(LX) b |To| ~=L(X ^ |To|) ~=L|X ^ To| ~=|L(X ^ To)| ~=|(LX) b To|,
which is the statement for LX. In the general case, the tensor X b|To| is const*
*ructed
as the reflexive coequalizer
L(LX ^ |To|)____////_L(X ^ |To|)//_X b |To|.
Commuting geometric realization with L and the coequalizer, we see that the co-
equalizer displayed above is the geometric realization of the reflexive coequal*
*izer
L(LX ^ To) _____////_L(X ^_To)//_X b To
describing the tensor of the L-space X with the based set To.
The tensor of the L-space X with a finite based set defines a functor from fi*
*nite
based sets to based spaces that takes the trivial based set * to the trivial ba*
*sed set
*. This constructs a -space associated to X, and the previous lemma identifies*
* the
suspension BX as the classifying space of this -space. Segal [17] proved that *
*when
a -space is "special", the loop space of the classifying space is a group comp*
*letion.
In this case, special means that the map from X q . .q.X ! X x . .x.X is a weak
equivalence. Theorem 6.5 is therefore an immediate consequence of the following
lemma.
Lemma 6.8. If X and Y are cofibrant L-spaces, the map from the coproduct X qY
to the cartesian product X x Y is a weak equivalence.
We prove this using a shortcut. Recall from [11, 9.6], the two-sided monadic *
*bar
construction B(L, L, X) which is the geometric realization of the simplicial L-*
*space
Bn = L L_._.L.-z_"X.
n
The iterated structure map L . .L.X ! X induces a map of L-spaces B(L, L, X) !
X. When X is cofibrant, we can choose a map of L-spaces X ! B(L, L, X) and a
homotopy from the composite X ! X to the identity through maps of L-spaces.
Choosing such a map for cofibrant Y as well, we obtain a diagram
X q Y _____//B(L, L, X) q B(L, L,_Y_)//_X q Y
| | |
| | |
fflffl| fflffl| fflffl|
X x Y _____//B(L, L, X) x B(L, L,_Y_)//_X x Y
where the horizontal composites are homotopic to the identity (through maps of
L-spaces). The lemma therefore reduces to showing that the map
B(L, L, X) q B(L, L, Y ) -! B(L, L, X) x B(L, L, Y )
is a weak equivalence. Both the coproduct of L-spaces and the cartesian product
commute with geometric realization. Since this bar construction is the geometric
realization of a proper simplicial space, we are reduced to showing that the map
Bn(L, L, X) q Bn(L, L, Y ) -! Bn(L, L, X) x Bn(L, L, Y )
is a weak equivalence for all n. The following lemma therefore completes the pr*
*oof
of Lemma 6.8.
22 MARIA BASTERRA AND MICHAEL A. MANDELL
Lemma 6.9. If T and U are nondegenerately based, then the map L(T _ U) !
LT x LU is a homotopy equivalence.
As always, ön ndegenerately based" means that the inclusion of the base point
is an unbased h-cofibration. The proof of Lemma 6.9 involves studying the double
filtration on L(T _ U) and LT x LU of homogeneous degree in T and U: Let
F m,nL(T _U) L(T _U) be the image of L(m+n)xT mxUn , and let F m,n(LT x
LU) LT x LU be F mLT x F nLU where F mLT LT is the image of L(m) x T m,
and similarly for F nLU. The map in Lemma 6.9 preserves this double filtration.
Let W m,ndenote the subspace of T m x Un consisting of those points where at
most m + n - 1 coordinates are not the basepoint (equivalently when m, n > 0,
the subset where at least one coordinate is the basepoint). Since we have assum*
*ed
that T and U are nondegenerately based, the inclusion of W m,nin T mx Um is a
m x n-equivariant h-cofibration; moreover, F m,nL(T _ U) is formed from
F m-1,nL(T _ U) [Fm-1,n-1L(T_U)F m,n-1L(T _ U)
as the pushout over the map
L(m + n) x m x nW m,n-! L(m + n) x m x n(T mx Un ).
Likewise, F m,n(LT x LU) is formed from
F m-1,n(LT x LU) [Fm-1,n-1(LTxLU)F m,n-1(LT x LU)
as the pushout over the map
(L(m) x L(n)) x m x nW m,n-! (L(m) x L(n)) x m x n(T mx Un ).
The maps
L(m + n) x m x nW m,n-! (L(m) x L(n)) x m x nW m,n
and
L(m + n) x m x n(T mx Un ) -! (L(m) x L(n)) x m x n(T mx Un )
are homotopy equivalences since the maps L(m+n) ! L(m)xL(n) are equivariant
homotopy equivalences. Since the map
F 0,0L(X _ Y ) -! F 0,0(LT x LU)
is the isomorphism L(0) ! L(0) x L(0), an easy double induction shows that we
have a homotopy equivalence on F m,nfor all m, n. Passing to the colimit, we see
that the map L(T _ U) ! LT x LU is a homotopy equivalence. This completes the
proof of Lemma 6.9.
Remark 6.10. Let X be a cofibrant L-space, and let Z be as above. This remark
explains in the terminology of May and Thomason [12], why Z is equivalent to the
output of an "infinite loop space machine". Let E be any infinite loop space ma*
*chine
for L-spaces. Writing n for the finite based set {0, . .,.n} (with 0 as basepoi*
*nt),
the collection {X bn} forms a -space in the category of L-spaces, or an FL-spa*
*ce
[12, 3.1]. Applying the machine E, a "whiskering functor", if necessary, and Se*
*gal's
machine, we obtain a bispectrum [12, 3.9ff], that is equivalent to Z in one dir*
*ection
and EX in the other.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 23
7. Proof of Theorem 3.1
This section is devoted to the proof of Theorem 3.1, the topological closed m*
*odel
structure on the categories of spectra in CB =B, NB , and MB . For convenience *
*we
repeat the definition of the cofibrations, fibrations, and weak equivalences; w*
*e say
that a map of spectra __X ! __Y is:
(i)A cofibration if X0 ! Y0 is a cofibration and each EYn[EXn Xn+1 ! Yn+1
is a cofibration,
(ii)A fibration if each Xn ! Yn is a fibration and each Xn ! Ynx Yn+1 Xn+1
is a weak equivalence, and
(iii)A stable equivalence if it induces an isomorphism of homotopy groups
ß*__X ! ß*__Y , where ßq__X = Colimnß~q+nXn for ~ß*X = ß*X in NB and
MB , and ~ß*X = Ker(ß*X ! ß*B) in CB =B.
It is clear that the categories of spectra in CB =B, NB , and MB have all sma*
*ll
limits and colimits. Likewise, it is clear from the definitions above that cofi*
*brations,
fibration, and weak equivalences in these spectra are closed under retracts and*
* that
weak equivalences have the two-out-of-three property. The proof of Theorem 3.1
therefore amounts to proving the factorization and lifting properties, and prov*
*ing
the topological version of SM7. The arguments are identical for all of the cate*
*gories,
and we use C to denote any of the categories CB =B, NB , and MB in what follows.
We begin with an alternative characterization of the acyclic fibrations.
Lemma 7.1. A map __X ! __Y is an acyclic fibration in Sp(C) if and only if each
map Xn ! Yn is an acyclic fibration in C.
Proof.Since preserves fibrations and acyclic fibrations (even on non-fibrant *
*ob-
jects), when each Xn ! Yn is an acyclic fibration in C, the map __X ! __Y is a
fibration and stable equivalence in Sp(C). Conversely, assume __X ! __Y is a fi*
*bration
and stable equivalence; then each map Xn ! Yn is a fibration, and so it suffices
to show that each map Xn ! Yn is a weak equivalence. Let ___W be the fiber (for
C = CB =B, this means Wn = B xYnXn); then ___W is an Omega spectrum. Since the
sequential colimit (of abelian groups) is an exact functor, the levelwise long *
*exact
sequences of homotopy groups for Wn ! Xn ! Yn induce a long exact sequence
of homotopy groups for ___W ! __X ! __Y . It follows that ß*W = 0 and therefore
that ~ß*Wn = 0 for all n, since ___W is an Omega spectrum. We see from the long
exact sequence of homotopy groups for Wn ! Xn ! Yn that Xn ! Yn is a weak
equivalence.
The previous lemma allows us to prove the lifting property for cofibrations a*
*nd
acyclic fibrations: If the solid rectangle
__A___//_fflffl_X;;w
| w ~|
fflffl|wfflfflfflffl|
__B___//__Y
is a commutative diagram with the left-hand map a cofibration and the right-hand
map an acyclic fibration of C-spectra, we can construct the dashed arrow making
the diagram commute as follows. We have that A0 ! B0 is a cofibration and
X0 ! Y0 is an acyclic fibration, and so we use the lifting property in C to con*
*struct
the required map B0 ! X0. Inductively, having constructed Bn ! Xn, since we
have that EBn qEAn An+1 ! Bn+1 is a cofibration and Xn+1 ! Yn+1 is an acyclic
24 MARIA BASTERRA AND MICHAEL A. MANDELL
fibration, we can use the lifting property in C to construct a map Bn+1 ! Xn+1,
compatible with the map EBn ! EXn ! Xn+1 and making the required diagram
commute. This proves the following proposition.
Proposition 7.2. In Sp(C), cofibrations have the left lifting property with res*
*pect
to acyclic fibrations.
We could construct the factorization for cofibrations and acyclic fibrations *
*anal-
ogously, but in order to analyze the topological version of SM7, it is useful to
construct them instead using Quillen's small object argument. For this, we reca*
*ll
the set IC of "generating cofibrationsä nd JC of "generating acyclic cofibrati*
*ons"
in C, the definition of which is implicit in the construction of the model stru*
*cture
on C in EKMM [5, VIIx4]. For C = MB , I is simply the set of "cells"
IMB = {SmB- ! CSmB| m 2 Z},
where SmB denotes the cofibrant sphere B-module [5, IIIx2] and J is the set of
cylinders of spheres,
JMB = {SmB- ! SmB^ I+ | m 2 Z}.
For C = NB , the sets I and J are
INB = NIMB = {Ni | i 2 IMB }
JNB = NJMB = {Nj | j 2 JMB }.
For C = CB =B, the sets I and J are only slightly more complicated: I is the se*
*t of
diagrams of commutative B-algebras
PSmB ____________//KPCSmB
KKK rrrrr
K%%Kxxrr
B
where P denotes the free commutative B-algebra, and the map SmB ! CSmB is
always the usual inclusion. The set J has an entirely analogous description usi*
*ng
the inclusion SmB! SmB^ I+ . The fundamental property of the sets IC and JC is
the following:
Proposition 7.3. A map in C is an acyclic fibration if and only if it has the r*
*ight
lifting property with respect to the maps in IC; it is a fibration if and only *
*if it has
the right lifting property with respect to the maps in JC.
In order to describe sets of generating cofibrations and generating acyclic c*
*ofi-
brations for Sp(C), we need one more piece of notation. For an object X in C, w*
*e let
FnX denote the C-spectrum with (FnX)j the initial object for j < n and Ej-nX
for j n; maps of C-spectra from FnX into a C-spectrum __Y are in one-to-one
correspondence with maps in C from X to Yn. We set ISp to be the set of maps
ISp = {Fnf | f 2 IC, n = 0, 1, 2, . .}..
We have the following analogue of the first part of the previous proposition.
Lemma 7.4. A map in Sp(C) is an acyclic fibration if and only if it has the rig*
*ht
lifting property with respect to ISp.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 25
The description of the set JSp is slightly more complicated. Certainly JSp sh*
*ould
contain the maps Fnf for f 2 JC, but these maps only generate the cofibrations
that are levelwise weak equivalences. Another general sort of stable equivalen*
*ce
occurs in the following way: If f :S ! T is a map in IC, then for any map S ! X*
*n,
we could attach the cell Fnf at the n-th level or attach the cell Fn+1Ef at the
(n + 1)-st level, and the map
__X qFn+1(ES)Fn+1(ET ) -! __X q(FnS)FnT
is a stable equivalence. This map is not a cofibration, but we can make it a co*
*fi-
bration using a cylinder: If we denote by (-) I the tensor in C of an object *
*with
the (unbased) unit interval, the map
__X qFn+1(ES)Fn+1(ES I) qFn+1(ES)Fn+1(ET )
-! (__X qFnS FnT ) qFn+1(ET)Fn+1(ET I)
gives a version of the previous map that is a cofibration. With this as motivat*
*ion,
for f :S ! T in IC, let
__Sn;f= FnS qFn+1(ES)Fn+1(ES I) qFn+1(ES)Fn(ET ),
__T n;f= FnT qFn+1(ET)Fn+1(ET I),
and let ~n;f:__Sn;f! __T n;fbe the map induced by f. Then on j-th objects, ~n;f*
*is
the identity map (on *) for j < n, is the map S ! T (from IC) for j = n, and is
the map
Ej-nS I qEj-nSEj-nT -! Ej-nT I,
for j > n. This last map is easily seen to be the inclusion of a deformation re*
*traction.
(In fact, it is isomorphic to a map in JC.)
In particular, we have that ~n;fis a cofibration and for j > n is a weak equi*
*va-
lence on j-th objects. It follows that ~n;fis an acyclic cofibration. We set JS*
*p to
be the set of maps
JSp = {Fng | g 2 JC, n = 0, 1, 2, . .}.[ {~n;f| f 2 IC, n = 0, 1, 2, . .}.
In studying these maps, it is convenient to use the following notation: Let
Mn__X = Xn x Xn+1 ( Xn+1)I,
where (-)I denotes the cotensor in C with the unbased interval. The two endpoint
of the interval induce two maps ( Xn+1)I ! Xn+1. The construction of Mn__X
uses one of these maps; the other gives us a map Mn__X ! Xn+1. Unwinding the
definition of of the maps ~n;fand the universal property of tensors and cotenso*
*rs
leads to the following proposition.
Proposition 7.5. Let f :S ! T be a map in IC and let __X be a C-spectrum. Then
maps in Sp(C) from __T n;fto __X are in one-to-one correspondence with maps in C
from T to Mn__X. Maps in Sp(C) from __Sn;fto __X are in one-to-one corresponden*
*ce
with commutative diagrams in C,
S ______//Mn__X
f || ||
fflffl| |fflffl
T _____// Xn+1.
26 MARIA BASTERRA AND MICHAEL A. MANDELL
We need one more observation about the map Mn__X ! Xn+1 before moving
on to the analogue for JSp of Lemma 7.4.
Lemma 7.6. Let __X ! __Y be a map in Sp(C) and assume that the maps Xn ! Yn
are fibrations in C for all n. Then the map
Mn__X -! Mn__Y x Yn+1 Xn+1
is a fibration in C.
Proof.Abbreviate Mn__X to M and Mn__Y x Yn+1 Xn+1 to N; we have a commu-
tative cube where the double-headed arrows are known to be fibrations and the
dotted arrow is the map we want to show is a fibration.
M ___________________//____( Xn+1)I
| ________ | NNNN
| _______ | NNNN
| _______ | NN''N
| _$$___ | ''
| N _______________________//( Yn+1)I
| | | |
| | | |
fflfflfflffl|||| fflfflfflffl|| ||
Xn x Xn+1 _____|_____//_ Xn+1 x Xn+1 |
KKK | OOO |
KKKK | OOOOO |
KK%%fflfflfflffl||%%K OO'''fflfflfflffl||'O
Yn x Xn+1 ________________// Yn+1 x Yn+1
The front and back (rectangular) faces are pullbacks, and the map
( Xn+1)I -! ( Yn+1)I x( Yn+1x Yn+1) Xn+1 x Xn+1
is a fibration. Since fibrations in C are characterized by the right lifting pr*
*operty
with respect to IC, it follows that M ! N is a fibration.
Lemma 7.7. A map in Sp(C) is a fibration if and only if it has the right lifting
property with respect to JSp.
Proof.Given h: __X ! __Y , it follows from Proposition 7.3 that the maps Xn ! Yn
are fibrations for all n if and only if h has the right lifting property with r*
*espect to
the set {Fng | g 2 JC, n = 0, 1, 2, . .}.. We can therefore restrict to the cas*
*e when
Xn ! Yn is a fibration for all n and prove that the map Xn ! Yn x Yn+1 Xn+1
is a weak equivalence if and only if h has the right lifting property with resp*
*ect
to {~n;f| f 2 IC, n = 0, 1, 2, . .}.. Proposition 7.5 implies that h having the*
* right
lifting property with respect to the maps ~n;ffor f 2 IC (for fixed n) is equiv*
*alent
to
Mn__X -! Mn__Y x Yn+1 Xn+1
having the right lifting property with respect to IC, which is equivalent to it*
* being
an acyclic fibration (by Proposition 7.3). By the previous lemma, the displayed
map is a fibration, so being an acyclic fibration is equivalent to being a weak
equivalence.
Recall that for a set of maps A (e.g., A = ISp or A = JSp), a relative A-comp*
*lex
is a map __X ! Colim__Xn where __X0 = __X and each __Xn+1 is formed from __Xn as
the pushout over a coproduct of maps in A. Lemmas 7.4 and 7.7 therefore give
us the right lifting property of acyclic fibrations and fibrations with respect*
* to
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 27
relative ISp-complexes and relative JSp-complexes (respectively). We have alrea*
*dy
observed that the maps in ISp and JSp are cofibrations and it follows that rela*
*tive
ISp-complexes and relative JSp-complexes are cofibrations. The remainder of the
proof of following lemma requires a compactness argument that we give at the end
of the section.
Lemma 7.8. A relative ISp-complex is a cofibration. A relative JSp-complex is an
acyclic cofibration.
The following lemma constructs factorizations:
Lemma 7.9. Let A = ISp or JSp. Any map __X ! __Y can be factored as a relative
A-complex __X ! __Z and a map __Z ! __Y that has the right lifting property wit*
*h respect
to the maps in A.
Proof.Proposition 7.5 (for A = JSp) and the characterization of maps out of Fn
(for A = ISp) show that for __A the domain or codomain of a map in A, the set of
maps of C-spectra out of __A commutes with sequential colimits,
Colim Sp(C)(__A, __Xn) ~=Sp(C)(__A, Colim__Xn),
when the maps __Xn ! __Xn+1 are cofibrations. We can now apply Quillen's small
object argument to construct the required factorizations.
The usual retract argument (factoring using the previous lemma and applying t*
*he
lifting property of Proposition 7.2) then proves the following lemma, the conve*
*rse
of Lemma 7.8.
Lemma 7.10. If a map in Sp(C) is a cofibration, then it is a retract of a relat*
*ive
ISp-complex; if it is an acyclic cofibration, then it is a retract of a relativ*
*e JSp-
complex.
We have now assembled everything we need for the proof of the theorem.
Proof of Theorem 3.1.As observed above, the proof that classes of maps defined *
*in
the statement form a closed model structure is completed by proving the required
factorization and lifting properties. Using the characterization of the acyclic*
* cofibra-
tions from 7.10, the lifting properties follow from Proposition 7.2 and Lemma 7*
*.7.
The factorization properties follow from Lemma 7.8 and Lemma 7.9.
It remains to prove the topological version of SM7: We need to show that when
i: __A ! __B is a cofibration and p: __X ! __Y is a fibration, the map of spaces
Sp(C)(__B, __X) -! Sp(C)(__B, __Y ) xSp(C)(__A,__YS)p(C)(__A, __X)
is a (Serre) fibration and is a weak equivalence if either i or p is a stable e*
*quivalence.
To show that the map is a fibration, it suffices to consider the case when i *
*is a
relative ISp-complex by Lemma 7.10, and for this, it suffices to consider the c*
*ase
when i is a map in ISp. Then i is a map Fnf :FnS ! FnT for some n and some f
in IC, and we can identify the map in question with the map of spaces
C(T, Xn) -! C(T, Yn) xC(S,Yn)C(S, Xn).
This is a fibration by the topological version of SM7 for C. An entirely simil*
*ar
argument proves that this map is an acyclic fibration when p is an acyclic fibr*
*ation.
Finally, we need to show that the map is an acyclic fibration when i is an
acyclic cofibration. As in the previous paragraph, this reduces to the case wh*
*en
28 MARIA BASTERRA AND MICHAEL A. MANDELL
i in in JSp. When i is Fng for some g in JC, the argument reduces to C just as
in the previous paragraph. Now consider the other case, when i = ~n;ffor some
f in IC. The argument is (as always) to go back over the proof of the lifting
property of Lemma 7.7 taking into account the topology of the mapping spaces.
Proposition 7.5 was stated in terms of a bijection of sets, but the argument re*
*fines
to give an isomorphism of spaces; this allows us to identify the map in question
with the map of spaces
C(T, M) -! C(T, N) xC(S,N)C(S, N),
where we have used the notation in the proof of Lemma 7.6. Lemma 7.7 shows
that when __X ! __Y is a fibration, M ! N is an acyclic fibration. It now follo*
*ws
from the topological version of SM7 in C that the map displayed above is an acy*
*clic
fibration.
Finally, we complete the proof of Lemma 7.8 by proving that a relative JSp-
complex is a stable equivalence. For this, it suffices to see that a pushout
a
__Y = __X q(` _Snff;fff)( __T nff;fff)
over a coproduct of maps ~nff;fffis a stable equivalence. This is clear for a f*
*inite
coproduct (since then for n large, the map on n-th objects is a deformation ret*
*ract).
The proof for the general case follows from the finite case, provided we can id*
*entify
the homotopy groups ß*__Y as the filtered colimit of the homotopy groups ß*__YA
where A ranges over the finite subsets of the index sets. For this it is suffi*
*cient
to identify the homotopy groups of the n-th object of __Y as the filtered colim*
*it of
the homotopy groups of the n-th objects. As observed above, for each ff, ~nff;f*
*ff
is on n-th objects either an isomorphism (if n < nff), the map ffffrom IC (when
n = nff), or the inclusion of a deformation retraction (if n > nff). The argume*
*nt is
therefore completed by the following lemma.
Lemma 7.11. Let {fff:Sff! Tff} be a set of maps in IC, let X be an object in C
and let a
Y = X q(` Sff)( Tff).
Any element of ~ßqY is represented in ~ßqYff1,...,ffm,
Yff1,...,ffm= X q(Sff1q...qSffm)(Tff1q . .q.Tffm),
for some finite subset of indexes ff1, . .,.ffm . If some element of ~ßqYff1,..*
*.,ffmis zero
in ~ßqY , then it is zero in ~ßqYff1,...,ffpfor some subset of indexes ff1, . .*
*,.ffp.
Proof.In order to fix notation, we treat the case C = CB =B, but the arguments *
*for
the other cases are similar. (We continue to write q instead of ^ for the copro*
*duct,
however, to avoid confusion for the other cases.) Since ~ßqis a subset of ßq, *
*it
suffices to prove the analogous lemma for ßq, and for this, it suffices to work*
* in CB ,
the category of commutative B-algebras. In this context, each map fffis just a
map PSmffB! PCSmffBfor some integer mff. For a finite subset A = {ff1, . .,.ffn*
*},
we set YA = Yff1,...,ffn(or YA = X when A is empty), and we set
`
MA = SmffB,
ff=2A
so that Y ~=YA qPMA P(CMA ). We construct various filtrations (of B-modules) on
Y using the bar construction, which we denote as "fio". Let fio(YA , PMA , P*) *
*be
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 29
the simplicial object in CB , with
fin(YA , PMA , P*) = YA q PMA_q_._.q.PMA_-z______"q P*,
n
with faces induced by the maps MA ! Y , MA ! *, and the codiagonal maps, and
with degeneracy maps induced by inserting an extra summand of PMA . We write
fi[A] for the geometric realization and we use the notation
YA = fi[A]0 -! fi[A]1 -! . .-.! fi[A]n -! . .,. fi[A] = Colimfi[A]n,
for the filtration arising from the geometric realization. Since the degeneracy*
* maps
are inclusions of wedge summands of B-modules, this filtration is a filtration *
*by
h-cofibrations of B-modules. The geometric realization of a simplicial object i*
*n CB
is an object in CB , and we have isomorphisms
fi[A] = |fio(YA , PMA , P*)| ~=YA qPMA |fio(PMA , PMA , PMA )| qPMA P*
~=YA qPMA P(MA ^ I) qPMA P* ~=YA qPMA P(CMA ) ~=Y
(see [5, VII.3.2]). For A A0, the map fi[A] ! fi[A0] covering the identity ma*
*p of
Y is not induced by a simplicial map but does preserve the filtrations above.
Now given an element x of ßqY , we show that x is the image of an element of
ßqYA for some finite subset of indexes A. Let A0 be the empty set. Using the
isomorphisms
ßqY ~=ßq(fi[A0]) ~=Colimßq(fi[A0]n),
we can represent x as the image of an element x0 of ßq(fi[A0]n) for some n. Now
suppose by induction, we have constructed a finite set Ak and found an element *
*xk
of ßq(fi[Ak]n-k) whose image in ßqY is x. The quotient fi[Ak]n-k=fi[Ak]n-k-1 is*
* the
quotient of n-kfin-k(YAk, PMAk, P*) by the degeneracies. Since the degeneracies
are inclusions of wedge summands, this quotient is still a wedge sum of B-modul*
*es,
each summand of which involves only finitely many of the indexes ff. The image
of xk factors through ßq of a finite wedge sum of these summands; let Ak+1 be t*
*he
union of Ak and the finite set of indexes involved in these summands. Since the
map fi[Ak] ! fi[Ak+1] is compatible with the filtration, and by construction, t*
*he
image of xk in ßq(fi[Ak+1]n-k=fi[Ak+1]n-k-1) is zero, we can find an element xk*
*+1
in ßq(fi[Ak+1]n-k-1) whose image in ßq(fi[Ak+1]n-k) is the image of xk. It foll*
*ows
that the image of xk+1 in ßqY is x. Continuing in this way, we get a finite sub*
*set
of indexes An and an element xn of ßq(fi[An]0) = ßqYAn whose image in ßqY is x.
The argument for a relation is similar, using a relative class in ßq+1 and st*
*arting
with A0 = {ff1, . .,.ffm }.
8. Cohomology Theories for Operadic Algebras
We have stated and proved the results in this paper in terms of the special c*
*ase
of particular interest, the category of algebras over the operad Com . Most of *
*these
results hold quite generally for the categories of algebras over other operads.*
* The
purpose of this section is to give precise statements of these general results *
*and to
indicate how to adapt the arguments in the earlier sections to the more general
case. We prove the following theorem.
30 MARIA BASTERRA AND MICHAEL A. MANDELL
Theorem 8.1. Let G be an operad of (unbased) spaces with each G(n) of the ho-
motopy type of a n-CW complex. Let B be a cofibrant G-algebra in EKMM R-
modules, and let UB be its universal enveloping algebra. Let GR=B be the catego*
*ry
of G-algebras of EKMM R-modules lying over B.
1.Topological Quillen Cohomology with coefficients in a left UB-module in-
duces an equivalence from the homotopy category of left UB-modules to
the category of cohomology theories on GR=B .
2.The category of homology theories on GR=B is equivalent to the category
of homology theories on the category of left UB-modules.
3.The stable category of G-algebras over and under B is equivalent to the
category of left UB-modules.
4.The equivalence in the previous statement takes the suspension spectrum
of an G-algebra A over B to the UB-module of infinitesimal UB-deforma-
tions.
We review the general definition of the universal enveloping algebra UB and
of Topological Quillen Cohomology below; the proof of the theorem essentially
amounts to formulating the definitions in a framework parallel to the case for *
*G =
Com . In the case G = Com , UB is just B, the category of left UB-modules is
the category of B-modules, and Topological Quillen Cohomology is Topological
Andr'e-Quillen Cohomology, as in the theorems in the introduction.
In general, unlike the case of the operad Com , a weak equivalence of G-algeb*
*ras
does not necessarily induce a weak equivalence of enveloping algebras, and this
is why we need to assume that B is cofibrant from the outset in the theorem
above. The theorem combined with Proposition 1.2 implies that for general B, the
category of cohomology theories on GR=B is equivalent to the homotopy category
of left UB0-modules and the category of homology theories on GR=B is equivalent
to the category of homology theories on the category of left UB0-modules, where
B0! B is a cofibrant approximation.
For G = Ass, the operad for associative algebras, UA is A ^R Aop, and so the
category of left UA-modules is the category of A-bimodules. When A is cofibrant,
one typically writes Ae for A ^R Aop. More generally, Ae denotes A0^R A0op,
for some fixed choice of cofibrant approximation A0 ! A. Lazarev [7] identifies
Topological Quillen Cohomology in terms of Topological Hochschild Cohomology,
and identifies the module of infinitesimal deformations of an associative algeb*
*ra A
as the homotopy fiber of the multiplication map Ae ! A. Part 1 of the previous
theorem then has the following corollary.
Corollary 8.2. Let B be an associative R-algebra. Every cohomology theory on
the category of associative R-algebras lying over B is of the form
h*(X, A) = ß-*F ib(T HHR (X, M) -! T HHR (A, M))
for some left Be-module M.
We now return to the general case of Theorem 8.1. We begin by describing the
universal enveloping algebra UB. At the same time, we describe the ü niversal
enveloping operadÜ B.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 31
Definition 8.3. The universal enveloping operad UB is the operad in R-modules
that has n-th object UB(n) defined by the coequalizer
W (k)____//W (k)____//
kG(n + k) ^ k (GB) _____//kG(n + k) ^ k B UB(n)
where G denotes the free G-algebra functor, one map is the G-action map of B
and the other is induced by the operadic multiplication of G. The operadic mul-
tiplication on UB is induced by the operadic multiplication of G. The universal
enveloping algebra UB is the R-algebra UB(1).
The fundamental property of the universal enveloping operad is given by the
following proposition, which is an easy consequence of the definitions.
Proposition 8.4. The category of G-algebras lying under B is equivalent to the
category of UB-algebras.
In particular, we have that UB(0) = B. Let ~UB be the operad with
(
U~B(n) = * n = 0
UB(n) n > 0
The category of ~UB-algebras plays the role for G-algebras under and over B that
the category of nucas plays for commutative algebras.
Definition 8.5. Let GB=B denote the category of UB-algebras lying over B, and
let NU~Bdenote the category of ~UB-algebras. Let K :NU~B! GB=B denote the
functor that takes a ~UB-algebra N to B _ N. Let I :GB=B ! NU~Bdenote the
functor that takes A to the (point-set) fiber of the augmentation A ! B.
The functors K and I are adjoint, and the following proposition holds just as*
* in
the commutative algebra case.
Theorem 8.6. The categories GB=B and NU~Bare topological closed model cate-
gories with weak equivalences the weak equivalences of the underlying R-modules.
The adjunction (K, I) is a Quillen equivalence.
Proof.The topological closed model structures and easy consequences of the gen-
eral theory in EKMM [5, VIIx4]. When we take T to be the monad associated
to the operad UB or U~B, or indeed any operad in the category of R-modules,
the proof of the öC fibration Hypothesisö f [5, VIIx4] for T follows just like *
*the
proof for associative and commutative algebras in [5, VIIx3]. The key observati*
*on
is that since colimits and smash products of R-modules commute with geometric
realization of simplicial R-modules, the monad T commutes with geometric real-
ization. This gives the geometric realization of a simplicial T-algebra a T-alg*
*ebra
structure and also proves that geometric realization commutes with colimits of *
*T-
algebras. As a consequence, we obtain the analogue of [5, VII.3.7]: For any maps
of T-algebras A ! A0and A ! A00, we can identify the geometric realization of t*
*he
äb r construction"
fin(A0, A, A00) = (A, TM, T*) = A0q A_q_._.q.A-z____"q A00,
n
as the double pushout in T-algebras
A0qA (A I) qA A00,
32 MARIA BASTERRA AND MICHAEL A. MANDELL
where A I is the tensor with the (unbased) interval (see for example the argume*
*nt
for Lemma 6.7). In the special case when A00= T* and A = TM for some R-module
M, the degeneracy maps are the inclusion of wedge summands, and so the filtrati*
*on
on the geometric realization is a filtration by h-cofibrations. In particular, *
*in this
case, the inclusion of the lowest filtration level A0 = A0q T* in the geometric
realization, A0qTM T(CM), is an h-cofibration; this is the Cofibration Hypothes*
*is.
Since I preserves fibrations an acyclic fibrations, (K, I) is a Quillen adjun*
*ction.
Finally, given any ~UB-algebra N, and any fibrant UB-algebra X over B, it is cl*
*ear
from the effect of K and I on homotopy groups that a map N ! IX is a weak
equivalence if and only if the adjoint map KN ! X is a weak equivalence, and so
(K, I) is a Quillen equivalence.
The proof of the previous theorem used the free functor from R-modules to
~UB-algebras, but there is in addition a free functor NU~B from left UB-modules
to ~UB-algebras, left adjoint to the forgetful functor from ~UB-algebras to lef*
*t UB-
modules, defined by
`
NU~BM = (UB(n) ^UB(n)M(n))= n,
n 1
for a left UB-module M. We also have a zero multiplication functor Z that gives
M a ~UB-action where
~UB(n) ^UB(n)M(n)-! M
is the left UB-action map for n = 1 and the trivial map for n > 1. This functor
has a left adjoint Q defined by the coequalizer
NU~BX _____////_X__//QX,
where one map is the ~UB-algebra action map on X, and the other map is the unit*
* on
the ~UB(1) ^UB X summand and the trivial map on the summands (UB(n) ^UB(n)
M(n))= n for n > 1. Since the zero multiplication functor Z preserves fibrations
and weak equivalences, we obtain the following proposition.
Proposition 8.7. The (Q, Z) adjunction is a Quillen adjunction.
As a consequence of the previous two propositions, for a cofibrant G-algebra A
over B and any left UB-module M, we obtain bijections of sets (in fact, isomor-
phisms of abelian groups)
Ho GR=B (A, B _ ZM) ~=Ho GB=B (B q A, B _ ZM)
~=Ho GB=B (IR (B q A), ZM) ~=Ho MUB (QLIR (B q A), M),
where L and R denote left and right derived functors. This leads to the followi*
*ng
definition.
Definition 8.8. For a ~UB-algebra N, the module of infinitesimal UB-deformations
is the left UB-module QLN. For a cofibrant G-algebra A over B, the UB-module
of infinitesimal UB-deformations of A is the left UB-module QLIR (B q A), and
for a cofibration of G-algebras A ! X over B, the UB-module of infinitesimal UB-
deformations of X relative to A is the left UB-module QLIR (B qA X). For a left
UB-module M, we define the Topological Quillen Cohomology of X relative to A
with coefficients in M by
D*G(X, A; M) = Ext*UB(QLIR (B qA X), M).
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 33
For general A and a general map A ! X of G-algebras over B, the Topological
Quillen Cohomology is defined using a cofibration A0 ! X0 covering A ! X for
cofibrant approximations A0! A and X0! X.
Since (I, K) is a Quillen equivalence, IR preserves coproducts and cofibrati*
*on
sequences. Since Q is a Quillen left adjoint, QL also preserves coproducts and
cofibration sequences. From this, it is easy to see that for any left UB-module*
* M,
Topological Quillen Cohomology with coefficients in M forms a cohomology theory,
with the connecting maps ffi induced by Ext.
We now have the underlying theory for G-algebras parallel to the theory for
commutative algebras, and the argument for Theorem 8.1 parallels the proof of
Theorems 1-4. The Brown's Representability argument in Section 1 applies gen-
erally to any category to which the arguments of EKMM [5, VIIx4] apply. The
only argument in Sections 1-5 and 7 that does not immediately generalize in this
framework is the appeal to [5, III.5.1] in Lemma 2.9. (For the proof of Lemma 7*
*.11,
noteWthat the coproduct of UB-algebras X q PUB M has as its underlying module
(UX(n) ^R M(n))= n by the analogue of Proposition 8.4 for X.) The following
lemma fills this gap.
Lemma 8.9. Let M be a cofibrant left UB-module. The natural map
E m+ ^ m (UB(m) ^UB(m) M(m)) -! (UB(m) ^UB(m) M(m))= m
is a weak equivalence.
The proof depends strongly on the hypothesis that the spaces of G have equiva*
*ri-
ant CW homotopy types and that B is a cofibrant G-algebra. This latter hypothes*
*is
implies that B is a retract of a "cell G-algebra" [5, VII.4.11]. If B ! B0 ! B *
*is
such a retraction (with B ! B the identity), then we get a retraction of operads
UB ! UB0 ! UB. The analogue of Lemma 8.9 for B0, then implies the lemma
as stated for B. Thus, it suffices to consider the case when B is a cell G-alge*
*bra.
Specifically, this means that we can write B as ColimBn where B0 = G(0)+ ^ R,
and Bn+1 = Bn qGWn G(CWn), where W is a wedge of sphere R-modules SmR.
The filtration of B allows us to get a better hold on the m -equivariant R-
modules UB(m). For example, UB0(m) = G(m)+ ^ R, and
`
UB1(m) = G(m + k) ^ k ( W0)(k).
k 0
More generally, we have the following lemma.
Lemma 8.10. For each n 0, UBn+1(m) has a filtration by m -equivariant h-
cofibrations UBn+1 = ColimkUBn+1(m)k, with UBn+1(m)0 = UBn(m), and the
filtration quotients
UBn+1(m)k=UBn+1(m)k-1 ~=UBn(m + k) ^ k ( Wn)(k).
Proof.We set UBn+1(m)0 = UBn(m) as required. The idea is that UBn+1(m)k
is the image in UBn+1(m) of UBn(m + k) ^ k (CWn)(k). Precisely, consider the
k-equivariant filtration
Wn(k)= F 0CWn -! . .-.! F k-1CWn -! F kCWn = (CWn)(k)
34 MARIA BASTERRA AND MICHAEL A. MANDELL
obtained as the smash power of the filtration on CWn that has Wn in level zero
and CWn in level one. For k 1, define UBn+1(m)k as the pushout
UBn(m + k) ^ k F k-1CWn ____//_UBn(m + k) ^ k (CWn)(k)
| |
| |
fflffl| fflffl|
UBn+1(m)k-1 ________________//_UBn+1(m)k.
Then it is clear that the map UBn+1(m)k-1 ! UBn+1(m)k is an h-cofibration,
and the quotient is as indicated in the statement. The identification of UBn+1 *
*with
ColimkUBn+1(m)k follows from the universal properties and the fact that the map
UBn(m + k) ^1x k-1(Wn ^ (CWn)(k-1)) -! UBn(m + k) ^ k F k-1CWn
is a categorical epimorphism.
Proof of Lemma 8.9.By the usual retract argument, it suffices to prove the lemma
when M is a cell left UB-module, and the usual filtration argument then reduces
to the case when M = UB ^R SqR, that is, to proving that the map
E m+ ^ m (UB(m) ^R (SqR)(m)) -! (UB(m) ^R (SqR)(m))= m
is a weak equivalence. This now follows from previous lemma using the argument
of [1, x9] (which generalizes the argument of [5, III.5.1]).
9. Weak Equivalences and Excision in Model Categories
The axioms listed in the introduction for a homology or cohomology theory on a
closed model category C patently depend on the cofibrations. The reduced version
of these axioms in Section 1 implicitly depends on the cofibrations in terms of*
* the
definition of cofibration sequences. The purpose of this section is to prove t*
*he
following theorem.
Theorem 9.1. Let C1 and C2 be closed model structures on the same category with
the same weak equivalences. Let h* be a contravariant functor from the category*
* of
pairs to the category of graded abelian groups and let ffin :hn(A) ! hn+1(X, A)*
* be
natural transformations of abelian groups. Then (h*, ffi) is a cohomology theor*
*y on
C1 if and only if it is a cohomology theory on C2. Likewise, for a covariant fu*
*nctor
and natural transformations, (h*, @) is a homology theory on C1 if and only if *
*it is
a homology theory on C2.
As a technical point, with the Product Axiom and Direct Sum Axiom as stated in
the introduction, we need the standard assumption that the closed model category
has all small colimits. See Remark 9.5 below for further discussion of the case*
* when
this assumption does not hold.
The theorem above does not appear to be well known, and we offer it here for *
*its
intrinsic interest; it has not been used in the previous sections. The basic id*
*ea is that
although cofibrations determine the notion of excision in a category, an approp*
*riate
notion of "weak excision" defined in terms of öh motopy cocartesian" diagrams
depends only on the weak equivalences. The following definition is standard.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 35
Definition 9.2. A commutative diagram
A _____//X
| |
| |
fflffl|fflffl|
B _____//Y
in a closed model category is homotopy cocartesian means that there exists a co*
*m-
mutative diagram
X0 oo___A0o//__//_oB0
|~| |~| |~|
fflffl| fflffl| fflffl|
X oo____A _____//_B
with A0cofibrant, the top horizontal arrows cofibrations, and vertical arrows w*
*eak
equivalences, such that the induced map Y 0= X0[A0B0! Y is a weak equivalence.
More concisely, the diagram is homotopy cocartesian if the canonical map in t*
*he
homotopy category from the homotopy pushout to Y is an isomorphism. One class
of examples of homotopy cocartesian squares is given by the squares where A is
cofibrant, A ! X and A ! B are cofibrations, and Y is the pushout X [A B.
Another class of examples is given by the squares where A ! B and X ! Y are
weak equivalences. With these examples in mind, the following proposition is cl*
*ear
from the definition.
Proposition 9.3. Let C be a closed model category and let h be a contravariant
functor from the category of pairs to the category of abelian groups. The follo*
*wing
are equivalent:
(a)h satisfies the Homotopy Axiom and the Excision Axiom:
(i)(Homotopy) If (X, A) ! (Y, B) is a weak equivalence of pairs, then
h(Y, B) ! h(X, A) is an isomorphism.
(iii)(Excision) If A is cofibrant, A ! B and A ! X are cofibrations, and
Y is the pushout X [A B, then the map of pairs (X, A) ! (Y, B)
induces an isomorphism h(Y, B) ! h(X, A).
(b)h satisfies the following Weak Excision Axiom: Whenever
A _____//X
fflffl|fflffl|
B _____//Y
is homotopy cocartesian, the map of pairs (X, A) ! (Y, B) induces an
isomorphism h(Y, B) ! h(X, A).
The Weak Excision Axiom does not depend on the cofibrations but only on the
weak equivalences:
Lemma 9.4. Let C1 and C2 be closed model structures on the same category C with
the same weak equivalences. Then a commutative diagram
A _____//X
fflffl|fflffl|
B _____//Y
is homotopy cocartesian in C1 if and only if it is homotopy cocartesian in C2.
36 MARIA BASTERRA AND MICHAEL A. MANDELL
Proof.Assume the diagram is homotopy cocartesian in C1. Using the factorization
properties of C2 (starting with A), we can find a commutative diagram
X2 oo2__A2oo//2_//B2
f~flffl|f~flffl|f~flffl|
X oo____A _____//_B
with A2 cofibrant in C2, the top horizontal arrows cofibrations in C2, and the *
*vertical
arrows weak equivalences. Likewise, using the factorization properties of C1 a*
*nd
C2, we can extend this to a commutative diagram
X0 oo2__A0oo//2_//B0
f~flffl|f~flffl|f~flffl|
X1 oo1__A1oo//1_//B1
f~flffl|f~flffl|f~flffl|
X2 oo2__A2oo//2_//B2
f~flffl|f~flffl|f~flffl|
X oo____A _____//_B
where A1 is cofibrant in C1, A0is cofibrant in C2, the horizontal arrows labele*
*d with
the number i are cofibrations in Ci, and all the vertical arrows are weak equiv*
*alences.
Taking Yi= Xi[Ai Bi and Y 0= X0[A0B0, the previous diagram induces maps
Y 0-! Y1 -! Y2 -! Y.
Clearly the map Y 0! Y2 is a weak equivalence (see, for example, the characteri*
*za-
tion of homotopy pushouts by Dwyer and Spalinski [4, 10.7]). The hypothesis that
the original diagram is homotopy cocartesian in C1 implies that the map Y1 ! Y
is a weak equivalence. It follows that the map Y 0! Y is a weak equivalence, and
so the original diagram is homotopy cocartesian in C2.
Since coproducts of cofibrant objects represent the coproduct in the homotopy
category, it is clear that the Product Axiom of the introduction is equivalent *
*to the
following axiom.
(iv)wIf {Xff} is a set of objects andQX is the coproduct in the homotopy cate*
*gory,
then the natural map h*(X) ! h*(Xff) is an isomorphism.
The analogous observation holds for the Direct Sum Axiom. Since the homotopy
category depends only on the weak equivalences in the model structure and not t*
*he
cofibrations, this completes the proof of Theorem 9.1.
Remark 9.5. It is sometimes useful to consider model categories that do not have
all small colimits but (as in the original definition in [14]) are only assumed*
* to
have finite colimits. For these categories, it appears unlikely that the versi*
*on of
the Product Axiom above is equivalent to the one in the introduction, and it de-
pends on the application which axiom, if either, is the "rightö ne. Typically,*
* the
most useful version of the Product Axiom in this case is one where we assume the
isomorphism only for index sets of certain fixed cardinalities; when coproducts*
* of
the given cardinalities always exist in the point-set category, then the two ve*
*rsions
of the axioms are again equivalent. The version of the Product Axiom for finite
cardinalities follows from the other axioms.
HOMOLOGY AND COHOMOLOGY OF E1 RING SPECTRA 37
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Department of Mathematics, University of New Hampshire, Durham, NH
E-mail address: basterra@math.unh.edu
Department of Mathematics, University of Chicago, Chicago, IL
E-mail address: mandell@math.uchicago.edu