Topological Andreґ–Quillen Cohomology and EN

ARTICLE IN PRESS

Advances in Mathematics 177 (2003) 227–279 http://www.elsevier.com/locate/aim

Topological Andre´ –Quillen Cohomology and EN Andre´ –Quillen Cohomology

Michael A. Mandell1 Department of Mathematics, University of Chicago, 5734 S University Avenue, Chicago, IL 60637, USA

Received 10 January 2001; accepted 29 April 2002 Communicated by Mark Hovey

Abstract

We show that the Andre´ –Quillen cohomology of an EN simplicial algebra with arbitrary coefﬁcients and the topological Andre´ –Quillen cohomology of an EN ring spectrum with Eilenberg–Mac Lane coefﬁcients may be calculated as the Andre´ –Quillen cohomology of an associated EN differential graded algebra. r 2003 Elsevier Science (USA). All rights reserved.

MSC: 55P43 (primary); 18G55; 18D50 (secondary).

Keywords: Andre´ -Quillen cohomology; EN ring spectrum

0. Introduction

Andre´ [1] and Quillen [17] introduced a cohomology theory for commutative rings now called Andre´ –Quillen cohomology. For a commutative ring k; a commutative k- algebra A; and an A-module M; the Andre´ –Quillen cohomology of A relative to k with coefficients in M can be defined as a sort of derived functor of derivations. This functor satisfies the appropriate axioms for a cohomology theory and is closely related to ‘‘square zero’’ commutative k-algebra extensions of A: Andre´ –Quillen cohomology computations play an important role in the construction of the ring spectrum EO2 by Hopkins and Miller. The existence of EO2 and the closely related ring spectrum eo2 have already had a big impact on our

E-mail address: [email protected]. 1 Supported in part by NSF Postdoctoral Research Fellowship DMS 9804421.

228 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 understanding of the stable category and of the homotopy groups of spheres. Recent work of Goerss and Hopkins has shown that EO2 is actually a ‘‘commutative S- algebra’’ in the sense of [4], the analogue in the category of spectra of a commutative (differential graded) algebra. Their proof develops an obstruction theory for the existence of commutative S-algebra structures. The obstruction groups are algebraically defined in terms of the homology of the underlying ring spectrum, and are a generalization of Andre´ –Quillen cohomology to EN simplicial algebras in a category of comodules. In practice, these groups often reduce to the Andre´ –Quillen cohomology of an EN simplicial algebra. The category of commutative S-algebras has enough of the formal properties of the category of commutative algebras that many of the different definitions of Andre´ –Quillen cohomology also make sense in this category. Basterra [2] compares these definitions, and shows that they are equivalent. We therefore obtain a ‘‘Topological Andre´ –Quillen’’ cohomology theory for the category of commutative S-algebras. This version of Andre´ –Quillen cohomology is also closely related to square zero extensions, and it leads to a different obstruction theory for the existence of commutative S-algebra structures, and in addition, an obstruction theory for the existence of commutative S-algebra maps. This is because the Postnikov tower of a connective commutative S-algebra provides a canonical example of a sequence of square zero extensions. As explained in [2] (following [9]), the k-invariants of a commutative S-algebra lift to unique elements of topological Andre´ –Quillen cohomology in the following sense. Given an S-module (or spectrum) A; a commutative S-algebra structure on the nth stage of the Postnikov tower A½n can be lifted to the ðn þ 1Þth stage A½n þ 1 if and only if the k-invariant can be lifted to the nþ2 topological Andre´ –Quillen cohomology D ðA½n; Hpnþ1AÞ; and the set of homotopy classes of lifts of the commutative S-algebra structure is in one-to-one correspondence with the set of lifts of the k-invariant. This is the obstruction theory for the construction of commutative S-algebra structures. From here, obstruction theory for the construction of commutative S-algebra maps works just as the analogous theory works for spaces or spectra: a map of commutative S-algebras B-A½n lifts to A½n þ 1 if and only if the k-invariant pulls back to zero in nþ2 D ðB; Hpnþ1AÞ and the set of homotopy classes of lifts has a free transitive action nþ1 of D ðB; Hpnþ1AÞ: In this way, the topological Andre´ –Quillen cohomology of commutative S-algebras provides a natural and complete obstruction theory for connective commutative S-algebras. These generalizations of Andre´ –Quillen cohomology to very different looking categories have similar, interesting applications. A third theory is the Andre´ –Quillen cohomology of EN differential graded algebras. This last theory is less rigid than the theory for EN simplicial algebras and less technically demanding than the theory for commutative S-algebras. Moreover, a number of general tools exist for calculating the Andre´ –Quillen cohomology of EN differential graded algebras. In this paper we show that the Andre´ –Quillen cohomology of an EN simplicial algebra and the topological Andre´ –Quillen cohomology of a commutative S-algebra may be calculated as the Andre´ –Quillen cohomology of an associated EN differential graded algebra. In any of the three categories, Andre´ –Quillen ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 229 cohomology is defined for an algebra A with coefficients in an A-module M and is natural in both A and M as reviewed below. It is a cohomology theory in the algebra A and a homology theory in the module M: There is a ‘‘transitivity’’ long exact sequence associated to a map of algebras A-B (cf. [17, 5.1]), and a ‘‘coefficient’’ long exact sequence associated to a map of modules M-N: In the obstruction theory context above, the coefficients are of a simple form: In the simplicial context, the simplicial structure on M is constant and M is merely a p0A-module. In the topological context, M ¼ Hp for some p0A-module p; i.e. p * M is concentrated in degree zero, and the A-module structure is determined by the canonical map of commutative S-algebras from A to the Eilenberg–Mac Lane spectrum Hp0A: We call coefficients of this form ‘‘constant’’ in the simplicial context and ‘‘Eilenberg–Mac Lane’’ in the topological context. In either context, we form the corresponding coefficients for EN differential graded algebras using the p0A-module p0M (with zero differential). Our main results are then the following theorems. (See Corollaries 1.8 and 7.9 for generalizations to other coefficients.)

Theorem. Let k be a (graded) commutative ring. There is a functor N from EN simplicial k-algebras to EN differential graded k-algebras that induces an isomorphism of Andre´–Quillen cohomology in constant coefﬁcients, which preserves the transitivity and coefﬁcient long exact sequences.

Theorem. Let R be a connective cofibrant commutative S-algebra, and let k ¼ p0R: There is a functor X from commutative R-algebras to EN differential graded k- algebras that induces an isomorphism of Andre´–Quillen cohomology in Eilenberg– Mac Lane coefficients, which preserves the transitivity and coefficient long exact sequences.

In the simplicial context, the functor N is the normalization functor on the underlying simplicial k-module and we have a canonical isomorphism of D commutative k-algebras H * NA p * A: We give an outline of the argument and generalizations in Section 1. In the topological context, the functor X is the composite of a cellular approximation functor and a cellular chain functor, and D we have a canonical isomorphism of commutative k-algebras H * XA H * A; where H * A ¼ p * ðA4RHkÞ is the ‘‘ordinary’’ homology theory of [4, Section IV.2]. The condition that R be connective means that its homotopy groups are concentrated in non-negative degrees; the condition that R be coﬁbrant is a technical one. Both conditions are satisﬁed by the sphere spectrum S: We give an outline of the argument and generalizations in Section 7. These comparison theorems allow us to apply the tools developed in the differential graded context to the other two contexts. In a future paper, Maria Basterra and the author will describe these tools and use them to calculate topological Andre´ –Quillen cohomology and EN simplicial algebra Andre´ –Quillen cohomology in some interesting examples. ARTICLE IN PRESS

230 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

1. EN simplicial algebras and EN differential graded algebras

In this section, we state the main results for the comparison of EN simplicial algebras and EN differential graded algebras. We work relative to a base commutative ring k; in categories of k-modules exclusively, and the tensor product should be understood to be the tensor product over k: In the context of [6], k is the coefficient ring of a cohomology theory and so we must allow k to be graded. In this case, k-modules are graded and differential graded k-modules are bigraded. The internal degree plays no role in the theory below and for the most part can be ignored; in arguments below ‘‘degree’’ always refers to the differential or simplicial degree. Before stating in detail our comparison theorems, we begin with a review of the definition of Andre´ –Quillen cohomology. The same basic ingredients go into the definition for EN algebras as go into the definition for commutative algebras. Recall that for a commutative k-algebra A and a simplicial A-module M; we can define the ‘‘square-zero extension’’ ZM ¼ A"M to be the simplicial commutative A-algebra obtained by taking the product of any two elements in M to be zero. The projection ZM-A is a map of simplicial commutative algebras, and we can look at the set of maps from A to ZM in the ‘‘homotopy category’’ of simplicial commutative k-algebras lying over A: Since ZM is an abelian group object in this category, this set is actually an abelian group. When we take M ¼ SnN for an A-module N; the abelian groups we obtain are by definition the Andre´ –Quillen cohomology of A (as a commutative algebra) with coefficients in M: We give an entirely parallel treatment of the Andre´ –Quillen cohomology of EN simplicial and differential graded algebras. A basic principle of the philosophy of EN algebras is that the particular choice of EN operad used should not matter, and that homological invariants, correctly defined, should be invariant not just with respect to weak equivalence of algebras and modules, but also with respect to change of EN operad. As we explain in Section 6, Andre´ –Quillen cohomology is an invariant of this type, but working with particular operads allows us to avoid unpleasant technical hypotheses and give cleaner statements. In the differential graded context we use the EN operad of differential graded k-modules C from [8], and in the simplicial context, we use a closely related operad of simplicial k-modules that we denote as E: We describe these operads precisely in Definition 2.1. These operads are both closely related to and share many of the special properties of the linear isometries operad L exploited in [4]. For the whole of this paper, we understand the category of EN simplicial k-algebras to mean the category EE of E-algebras and the category of EN differential graded k-algebras to mean the category EC of C- algebras. For an EN simplicial or differential graded k-algebra A; we have a notion of A-module, explained for example in [5,8] and reviewed in Section 2. These categories of modules have suspension functors S that refine the suspension in the category of simplicial k-modules and in the category of differential graded k-modules (where suspension is the shift functor). In addition, Definition 2.7 describes a ‘‘square zero multiplication’’ functor Z from the category of A-modules to the category of EN ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 231 algebras lying over A; for an A-module M; the underlying simplicial or differential graded module of ZM is A"M: (We write ZA for Z when A is unclear.) In both the simplicial and differential graded contexts, we can deﬁne the Andre´ –Quillen cohomology groups as follows.

Definition 1.1. Let E denote either the category of EN simplicial k-algebras or the category of EN differential graded k-algebras. Let A be an object of E and let M be an A-module. We define the Andre´ –Quillen cohomology of A with coefficients in M as

DnðA; MÞ¼h%ðE=AÞðA; ZSnMÞ:

Here E=A denotes the over-category of A in E (the category of maps in E with codomain A), and h%ðE=AÞ denotes its homotopy category, the category obtained by formally inverting the quasi-isomorphisms.

In this definitions n ranges over all values that make sense: all integers in the differential graded context and the non-negative integers in the simplicial context. The functor Z preserves categorical products, and naturality in M gives DnðA; MÞ the structure of a graded k-module. Andre´ –Quillen cohomology is a functor of the EN algebra A in the following way. Let f : A-B be a map in E: Just as for modules over a ring, we can pull back a B- module structure along f to an A-module structure. In other words, we obtain a functor from the category of B-modules to the category of A-modules that is the identity functor on the underlying (simplicial or differential graded) k-modules; we denote this functor as f n when notation for it is required. Likewise we have a functor from E=B to E=A that performs the fiber product A ÂB ðÀÞ on the underlying k- modules; it is the pullback in the category E; and so in particular for any C in E=A; the canonical map C-A ÂB C is in E=A: We have a natural isomorphism ZAMDA ÂB ZBM in E=A refining the isomorphism of k-modules A ÂB ðB"MÞDA"M: The composite

% % % hðE=BÞðB; ZBMÞ-hðE=BÞðA; ZBMÞ-hðE=AÞðA; ZAMÞ defines a map DnðB; MÞ-DnðA; MÞ that makes Dn appropriately contravariantly functorial in the algebra variable. We can also define the relative Andre´ –Quillen cohomology groups for a map f : A-B in E: For this, we replace the over-category of A in the definition above with the ‘‘between-category’’ of f : We define Ej f to be the over-category of B in the under-category of A: An object of Ejf consists of maps A-C-B in E such that the composite map A-B is f ; a map is a map under A and over B: For a B-module M; we can regard ZM as an object of Ejf as follows: The inclusion of B in B"M is a map of EN algebras B-ZM; composing with f gives a map A-ZM such that the composite A-ZM-B is f : ARTICLE IN PRESS

232 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Definition 1.2. Let f : A-B be a map in E; and let M be a B-module. We define the Andre´ –Quillen cohomology of B relative to A (or better, ‘‘relative to f ’’) with coefficients in M as

DnðB\A; MÞ¼h%ðEjf ÞðB; ZSnMÞ:

Here h%ðEjf Þ denotes the homotopy category of Ej f ; the category obtained by formally inverting the quasi-isomorphisms.

In order to compare the Andre´ –Quillen cohomology of EN simplicial and differential graded algebras, we ﬁrst need to compare the categories of EN simplicial and differential graded algebras. We say an EN differential graded k-algebra is ‘‘non-negative’’ if its underlying differential graded k-module is non-negative, that is, concentrated in non-negative degrees. We deﬁne the category of non-negative EN differential graded k-modules to be the full subcategory of the category of EN þ differential graded k-algebras, and we denote it as EC: As we review in the next section, the category of EN simplicial k-algebras and the category of non- negative EN differential graded k-algebras are closed model categories with weak equivalences the quasi-isomorphisms. In Section 4, we prove the following theorem.

Theorem 1.3. The normalization functor N from simplicial k-modules to non-negative differential graded k-modules reﬁnes to a functor from EN simplicial k-algebras to non- negative EN differential graded k-algebras. The normalization functor N is the right adjoint of a Quillen equivalence:

(i) þ Between the categories EE and EC: (ii) þ Between the over-categories EE=A and EC=NA; for every EN simplicial k-algebra A: (iii) þ Between the between-categories EEj f and ECjNf ; for every map of EN simplicial k-algebras f : A-B:

A Quillen equivalence, in particular, induces an equivalence of homotopy categories but is much stronger. Quillen equivalence is the natural notion of weak equivalence of closed model categories and preserves all homotopy theoretic constructions; see for example [7]. We can consider the categories of non-negative modules over non-negative EN differential graded k-algebras; these again form a closed model category with weak equivalences the quasi-isomorphisms. For an EN simplicial k-algebra A; the normalization functor N then also reﬁnes to a functor from the category of A- modules to the category of non-negative NA-modules. In Section 4, we prove the following theorem.

Theorem 1.4. Let A be an EN simplicial k-algebra. The normalization functor from A- modules to non-negative NA-modules is the right adjoint of a Quillen equivalence. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 233

% % In order to understand the relationship between hðEE=AÞ and hðEC=NAÞ and to compare the Andre´ –Quillen cohomology theories, we next need to understand the relationship between the model category of EN differential graded k-algebras and the model category of non-negative EN differential graded k-algebras. We cannot expect a Quillen equivalence, since a general EN differential graded k-algebra can have homology in negative degrees, but we do have a Quillen adjunction. The inclusion of the subcategory of non-negative differential graded k-modules into the category of all differential graded k-modules has a right adjoint ðÀÞþ satisfying ( X þ HqM for q 0; HqðM Þ¼ 0 for qo0 for any differential graded k-module M: We explain the following theorem in Section 3.

þ - þ Theorem 1.5. The functor ðÀÞ reﬁnes to a functor EC EC that is right adjoint to the inclusion functor, and which preserves ﬁbrations and weak equivalences. The counit of þ the adjunction A -A is a weak equivalence when HqA ¼ 0 for all qo0; the unit of the adjunction B-Bþ is always an isomorphism.

It follows that for any map of non-negative EN differential graded k-algebras f : A-B; the functors h%ðEþ=BÞ-h%ðE=BÞ and h%ðEþj f Þ-h%ðEj f Þ are full and faithful embeddings. As a consequence, we obtain the following corollary to Theorem 1.5.

Corollary1.6. For a non-negative EN differential graded k-algebra B and a non- negative B-module M; we have a natural isomorphism

n D % þ n D ðB; MÞ hðEC=BÞðB; ZS MÞ for nX0: For any map of non-negative EN differential graded k-algebras f : A-B; we have a natural isomorphism

n \ D % þ n D ðB A; MÞ hðECj f ÞðB; ZS MÞ

n for nX0: When HqM ¼ 0 for all q > 0; we have in addition that D ðB; MÞ¼0 and DnðB\A; MÞ¼0 for all no0:

The equivalence of virtually all properties of Andre´ –Quillen cohomology can be deduced from the following theorem together with the usual properties of the normalization functor N:

Theorem 1.7. Let A be an EN simplicial k-algebra, and let M be an A-module. The canonical isomorphism of differential graded k-modules NA"NM-NðA"MÞ is an isomorphism ZNANM-NðZAMÞ of EN differential graded k-algebras. ARTICLE IN PRESS

234 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

For A; M; as above, the natural quasi-isomorphism of differential graded k- modules SNM-NðSMÞ is a map of NA-modules. We therefore get the following corollary of Theorems 1.3, 1.5, and 1.7.

Corollary1.8. Let B be an EN simplicial k-algebra and let M be an B-module. There is a natural isomorphism,

DnðB; MÞDDnðNB; NMÞ for nX0: For a map of EN simplicial k-algebras f : A-B; there is a natural isomorphism

DnðB\A; MÞDDnðNB\NA; NMÞ

n for nX0: When pqM ¼ 0 for all q > 0; we have in addition that D ðNB; NMÞ¼0 and DnðNB\NA; NMÞ¼0 for all no0:

Finally, we say a few words about exact sequences in Andre´ –Quillen cohomology. In both the simplicial and differential graded contexts, we have associated to a map of EN algebras A-B the transitivity long exact sequence

?-DnðB\A; MÞ-DnðB; MÞ-DnðA; MÞ-Dnþ1ðB\A; MÞ-?; or more generally, for maps A-B-C;

?-DnðC\B; MÞ-DnðC\A; MÞ-DnðB\A; MÞ-Dnþ1ðC\B; MÞ-?:

(In the simplicial context, with only the definitions above, Dn is not defined for no0 and these sequences are half-exact at n ¼ 0 unless pqM ¼ 0 for q > 0:) As we review in Section 5, these exact sequences arise from cofiber sequences, which are preserved by the Quillen equivalences in Theorem 1.3. Likewise, we have associated to a short exact sequence of B-modules

0-M0-M-M00-0 the coefﬁcient long exact sequence

?-DnðB; M0Þ-DnðB; MÞ-DnðB; M00Þ-Dnþ1ðB; M0Þ-?; and similarly for DnðB\A; ÀÞ in the relative case. As we review in Section 5 these exact sequences arise from ﬁber sequences. We prove the following theorem in Section 5.

Theorem 1.9. The isomorphisms in Corollary 1.8 commute with the transitivity and coefﬁcient long exact sequences. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 235

2. Algebras, modules, and the operads E; C

In this section, we give a short review of the theory of EN simplicial algebras, EN differential graded algebras, and their modules in terms of the operads E and C: One advantage of using these operads is that, as shown in [8] (in the differential graded context), the theory of algebras over these operads and modules over these algebras can be reformulated to have a close formal resemblance to the theory of commutative algebras and modules over commutative algebras; we discuss the theory in these terms. We begin with the deﬁnition of the operads.

Deﬁnition 2.1. Let L be the linear isometries operad, the EN operad of spaces with N n N LðnÞ the space of linear isometries from ðR Þ to R [4, I.3.1]. Let SL denote the singular simplicial set of L; an EN operad of simplicial sets. Let E be the free k- module on SL; an EN operad of simplicial k-modules. Let C be the normalization of E; an EN operad of differential graded k-modules.

The operad structure on L is induced by composition: for n > 0; j1; y; jnX0; the operadic multiplication

LðnÞÂðLð j1ÞÂ? Â Lð jnÞÞ-Lð j1 þ ? þ jnÞ

N n N N ji N takes linear isometries f : ðR Þ -R ; and gi : ðR Þ -R ; i ¼ 1; y; n; to the linear isometry

N j1þ?þjn N f 3 ðg1"?"gnÞ : ðR Þ -R :

The collection of simplicial sets SL inherits its operad structure from L since the singular functor commutes with cartesian products. Likewise, the collection E inherits its operad structure from SL since the free k-module functor converts cartesian products to tensor products. The collection C inherits its operadic structure from E since the normalization functor is lax symmetric monoidal: the shufﬂe map gives an associative and symmetric natural transformation

CðnÞ#ðCð j1Þ#?#Cð jnÞÞ ¼ NEðnÞ#ðNEð j1Þ#?#NEð jnÞÞ

-NðEðnÞ#ðEð j1Þ#?#Eð jnÞÞÞ; see [14, p. 132] or [9, Section I.5] for details. As mentioned in the previous section, we work with the operad E in simplicial k-modules and C in differential graded k-modules, and we denote the categories of EN simplicial k-algebras, EN differential graded k-algebras, and non-negative EN þ differential graded k-algebras as EE; EC; and EC respectively. Following [9],we use the following version of EN simplicial k-modules and EN differential graded k-modules. ARTICLE IN PRESS

236 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Deﬁnition 2.2. The category ME of EN simplicial k-modules is the category of modules over the simplicial algebra Eð1Þ: The category MC of EN differential graded k-modules is the category of modules over the differential graded algebra Cð1Þ: The þ category MC of non-negative EN differential graded k-modules is the category of non-negative modules over the differential graded algebra Cð1Þ:

It is clear from this definition that the tensor product M#X of an EN simplicial k-module M and a simplicial k-module X is naturally an EN simplicial k-module. In particular, we have a suspension functor S on the category of EN simplicial k- modules; this functor shifts homotopy groups pnMDpnþ1SM: Likewise, the tensor product of an EN differential graded k-module and a differential graded k-module is naturally an EN differential graded k-module, and we have a suspension functor S on the category of EN differential graded k-modules that shifts homology groups HnNDHnþ1SN: The main results of Part V of [9] concern a symmetric ‘‘weak-monoidal’’ product 2 for EN differential graded k-modules analogous to the tensor product in the category of differential graded k-modules. Although [9] considered only Z-graded modules, it is easy to see that 2 restricts to a product on non-negative modules with the same formal properties. Essentially, the same construction defines an analogous product for EN simplicial k-modules: We define the bifunctor M2N for EN simplicial k-modules M and N by

M2N ¼ Eð2Þ # ðM#NÞ: Eð1Þ#Eð1Þ

The left action of Eð1Þ on Eð2Þ makes M2N naturally an EN simplicial k-module. We have a natural symmetry isomorphism M2NDN2M induced by the symmetry isomorphism M#NDN#M and the transposition isomorphism Eð2ÞDEð2Þ induced by the right S2-action on Eð2Þ: We also have unit a map l : k2N-N induced by the operad multiplication Eð2Þ#ðk#Eð1ÞÞ ¼ Eð2Þ#ðEð0Þ#Eð1ÞÞ-Eð1Þ:

It is important to note that l is generally not an isomorphism. Just as in the differential graded case [9, Section V.1], the product 2 is coherently associative, and the natural transformations satisfy all the properties of a symmetric monoidal product except that the unit map is not an isomorphism. We summarize these in the following deﬁnition and proposition.

Deﬁnition 2.3. Let E be a category. A symmetric weak-monoidal product on E consists of a functor 2 : E Â E-E; a unit object k of E; an associativity natural isomorphism

a : ðX2YÞ2ZDX2ðY2ZÞ ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 237 a symmetry natural isomorphism

t : X2YDY2X and a unit map

l : k2X-X satisfying Mac Lane’s coherence conditions [10, Section VII.1] except that l is not required to be an isomorphism.

2 þ Proposition 2.4. The products on ME; MC; and MC are symmetric weak-monoidal.

The proof is the same in all important details as the one in [9]. The main point is that the operad E has the special property that the maps

Eð2Þ # ðEðiÞ#Eð jÞÞ-Eði þ jÞ Eð1Þ#Eð1Þ induced by the operadic multiplication are isomorphisms for all i; j > 0 (‘‘Hopkins’ Lemma’’). This induces the associativity isomorphism. As mentioned above, the symmetry isomorphism is induced by the S2 action on Eð2Þ together with the symmetry isomorphism for the tensor product. The unit map is induced by the operadic multiplication: Eð2Þ#k ¼ Eð2Þ#Eð0Þ-Eð1Þ: A commutative monoid in a symmetric weak-monoidal category is deﬁned just as in a symmetric monoidal category: it is an object A together with maps i : k-A and m : A2A-A such that the following associativity, commutativity, and unit diagrams commute:

For the reader unfamiliar with EN algebras, we offer the following definition. A proof of the equivalence of this definition and the traditional definition of EN differential graded algebra (over C) can be found in [9, Section V.3] and the equivalence in the simplicial context follows the same argument.

Deﬁnition 2.5. The category EE of EN simplicial k-algebras is the category of 2 commutative monoids for in ME: The category EC of EN differential graded k- 2 algebras is the category of commutative monoids for in MC: ARTICLE IN PRESS

238 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

A left A-object is an object M in ME together with a map x : A2M-M such that the following associativity and unit diagrams commute:

For the reader unfamiliar with modules over EN algebras, we offer the following definition. A proof of the equivalence of this definition and the traditional definition can be found in [9, Section V.3] and the equivalence in the simplicial context follows the same argument.

Deﬁnition 2.6. For an EN simplicial k-algebra A; the category of A-modules MA is the category of left A-objects for 2 in ME: For an EN differential graded k-algebra B; the category of B-modules MB is the category of left B-objects for 2 in MC:

Finally, we can describe the functor Z: Let A be an EN simplicial or differential graded k-algebra and let M be an A-module. We deﬁne a square zero multiplication m : ZM2ZM-ZM on ZM ¼ A"M as follows. Since the functor 2 commutes with direct sums, we have a canonical isomorphism

ZM2ZMDðA2AÞ"ðA2MÞ"ðM2AÞ"ðM2MÞ:

We deﬁne m to be the map induced by the multiplication of A on the summand A2A; the action of A on M on the summands A2M and M2A; and the zero map on the summand M2M: This makes ZM an EN simplicial or differential graded k- algebra, and the projection map ZM-A is a map of EN simplicial or differential graded k-algebras.

Deﬁnition 2.7. For an EN simplicial or differential graded k-algebra A; deﬁne the functor Z : MA-E=A by ZM ¼ A"M; an object of E=A via the square zero multiplication and the projection map A"M-A:

3. The normalization functor

The purpose of this section is to review the background needed to prove Theorems 1.3–1.5 and 1.7. The arguments for the last two theorems are relatively straightforward and are given in this section; the proofs of Theorems 1.3 and 1.4 require the construction of left adjoint functors and are given in the next section. The basis for these theorems is the reﬁnement of the normalization functor to a functor of EN algebras and EN modules. We begin with the following observation. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 239

Theorem 3.1. The normalization functor reﬁnes to a functor from the category ME of EN simplicial k-modules to the category MC of EN differential graded k-modules and is lax symmetric monoidal for the 2 product.

Proof. Since ME is the category of simplicial Eð1Þ-modules, MC is the category of differential graded Cð1Þ-modules, and Cð1Þ¼NðEð1ÞÞ; the ﬁrst observation is classical: if M is a simplicial Eð1Þ-module, the shufﬂe map

Cð1Þ#NM ¼ NðEð1ÞÞ#NM-NðEð1Þ#MÞ composed with the normalization of the Eð1Þ-action map Eð1Þ#M-M deﬁnes the Cð1Þ-action map on NM: For EN simplicial k-modules M1; M2; the shufﬂe map Cð2Þ#NðM1Þ#NðM2Þ-NðEð2Þ#M1#M2Þ induces the natural transformation NðM1Þ2NðM2Þ-NðM12M2Þ: It is straightforward to check that this natural transformation is suitably associative, commutative, and unital. &

If A is an EN simplicial k-algebra, the natural map NA2NA-NðA2AÞ induces an associative, commutative, and unital multiplication on NA: Likewise, if M is an A-module, the natural map NA2NM-NðA2MÞ induces an associative and unital NA action on NM: We obtain the following immediate corollary of the previous theorem.

Corollary3.2. The normalization functor reﬁnes to a functor from the category EE of EN simplicial k-algebras to the category EC of EN differential graded k-algebras. For an EN simplicial k-algebra A; the normalization functor reﬁnes to a functor from the category MA of A-modules to the category MNA of NA-modules.

As another consequence, we get the proof of Theorem 1.7.

Proof of Theorem 1.7. The inclusions of A and M in A"M induce an isomorphism

NA"NM-NðA"MÞ; i.e. an isomorphism of EN differential graded k-modules ZðNMÞ-NðZMÞ: The composite of the isomorphism

ðNA2NAÞ"ðNA2NMÞ"ðNA2NMÞ"ðNM2NMÞ

DNðZMÞ2NðZMÞ with the multiplication NðZMÞ2NðZMÞ-NðZMÞ is the multiplication of NA on the ﬁrst factor, the action maps on the middle two factors, and the zero map on the last factor. This therefore coincides with the multiplication on ZðNMÞ under the isomorphism above. & ARTICLE IN PRESS

240 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Before moving on to Theorems 1.3 and 1.4, we need to review the closed model structures. In general, categories of simplicial algebras over simplicial operads of k- modules admit a right proper simplicial closed model structures [15, 3.3.6]. The following proposition is a special case. Left properness is an immediate consequence of [13, 7.1–2].

Proposition 3.3. The category EE of EN simplicial k-algebras admits a proper simplicial closed model structure with the weak equivalences and ﬁbrations the weak equivalences and ﬁbrations of the underlying simplicial sets.

Since EN simplicial k-algebras are in particular simplicial abelian groups, a map between them is a weak equivalence if and only if it is a quasi-isomorphism after normalization. Likewise, a map between them is a ﬁbration if and only if it is a surjection in positive degrees after normalization. The model structure described in the previous proposition therefore appears very similar to the model structure described in the following proposition. The ﬁrst part is the main result of [13]; the second part is an easy consequence of the same arguments.

Proposition 3.4. The category EE of EN differential graded k-algebras admits a proper closed model structure with the weak equivalences the quasi-isomorphisms and the þ ﬁbrations the surjections. The category EE of non-negative EN differential graded k- algebras admits a proper closed model structure with the weak equivalences the quasi- isomorphisms and the ﬁbrations the maps that are surjections in positive degrees.

Comparing the two closed model structures in the previous propositions, we obtain the following portion of Theorem 1.3. As mentioned above, the remainder of the proof is in the next section.

- þ Proposition 3.5. The normalization functor EE EC preserves ﬁbrations and weak equivalences.

Proposition 3.4 also gives us what we need to prove Theorem 1.5.

Proof of Theorem 1.5. Recall that the functor ðÀÞþ takes a differential graded k- module M to the differential graded k-module Mþ that is the same as M in positive degrees, the zero cycles of M in degree zero, and zero in negative degrees. For any differential graded k-module M; we therefore obtain a natural map

Cð2Þ#Mþ#Mþ-Cð2Þ#M#M;

þ þ and hence a natural map M 2M -M2M: For an EN differential graded k-algebra A; composing the map Aþ2Aþ-A2A with the multiplication, we obtain a map

Aþ2Aþ-A ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 241 that must factor (uniquely) through Aþ since Aþ2Aþ is non-negative. This þ þ multiplication makes A an EN differential graded k-algebra and the map A -A a þ map of EN differential graded k-algebras. Clearly, ðÀÞ is right adjoint to the inclusion þ- & EC EC; and it preserves weak equivalences and ﬁbrations by Proposition 3.4.

We now turn to the module categories. For an EN differential graded k-algebra B; Kriz and May [8, V.4.3] describes a free functor FB from differential graded k- modules to B-modules. For an EN simplicial k-algebra A; we can give an entirely similar description of the free functor FA from simplicial k-modules to A-modules. For a simplicial k-module X; the following diagram is a pushout:

Of course, Eð1Þ#X is the free EN k-module on X: If the 2 product were strictly unital, A2ðEð1Þ#XÞ would be the free A-module on X; the pushout along the unit map l corrects for the fact that l is not an isomorphism. It is convenient to record here the following observation. It is proved in [8, V.4.6; 13, 6.1] in the differential graded case; the simplicial case is an immediate consequence of [13, 7.1–2].

Proposition 3.6. (i) Let A be an EN simplicial k-algebra. There is a natural homotopy equivalence of simplicial k-modules FAX-A#X: (ii) Let B be an EN differential graded k-algebra. There is a natural homotopy equivalence of differential graded k-modules FBX-B#X:

The closed model structures on the module categories follow easily from Quillen’s small objects argument.

Proposition 3.7. Let A be an EN simplicial k-algebra. The category MA of A-modules is a proper simplicial closed model category with weak equivalences and ﬁbrations the weak equivalences and ﬁbrations of the underlying simplicial sets.

Proposition 3.8. Let B be an EN differential graded k-algebra. The category MB of B- modules is a proper closed model category with weak equivalences the quasi- þ isomorphisms and ﬁbrations the surjections. If B is non-negative, then the category MB of non-negative B-modules is a proper closed model category with weak equivalences the quasi-isomorphisms and ﬁbrations the maps that are surjections in positive degrees.

We get the following portion of Theorem 1.4. The remainder is proved in the next section.

Proposition 3.9. Let A be an EN simplicial k-algebra. The normalization functor - þ MA MNA preserves ﬁbrations and weak equivalences. ARTICLE IN PRESS

242 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

4. The proofs of Theorems 1.3 and 1.4

In this section, we prove Theorems 1.3 and 1.4. We concentrate the discussion on the argument for Theorem 1.3(i) for definiteness. The argument for Theorem 1.4 has a similar outline but is easier in technical details. The remaining parts (ii) and (iii) of Theorem 1.3 are deduced from part (i) and model category arguments. The proof of Theorem 1.3(i) begins with the construction of the left adjoint functor L to the normalization functor N: Proposition 3.5 tells us that N preserves fibrations and all weak equivalences; this means that ðL; NÞ is then a Quillen adjoint pair. For the Quillen adjunction ðL; NÞ to be a Quillen equivalence means that for þ - any cofibrant object A in EC and any fibrant object B in EE; a map LA B in EE is a - þ weak equivalence if and only if the adjoint map A NB in EC is a quasi- isomorphism. (Note that all objects of EE are fibrant.) The argument is slightly simplified by the fact that N not only preserves weak equivalences but also reflects weak equivalences: A map f in EE is a weak equivalence if and only if its þ normalization Nf is a quasi-isomorphism in EC: This reduces the argument to showing that the unit of the adjunction A-NLA is a quasi-isomorphism for every þ - cofibrant object A in EC: This is because for a map LA B in EE; the adjoint map - þ A NB in EC is the composite A-NLA-NB:

Thus, Theorem 1.3(i) is a consequence of the following theorem proved in this section.

- þ Theorem 4.1. The normalization functor N : EE EC has a left adjoint L: IfAisa þ - coﬁbrant object in EC; then the unit of the adjunction A NLA is a quasi-isomorphism.

Before constructing the left adjoint L; we need some additional notation. Recall that the normalization functor from simplicial k-modules to non-negative differential graded k-modules is an equivalence of categories; we denote its inverse as G: We denote the free functor from simplicial k-modules to EN simplicial k- algebras as E and the free functor from differential graded k-modules to EN differential graded k-algebras as C: We observed in the last section that the normalization of an EN simplicial k-algebra is an EN differential graded k-algebra; this gives us a natural map CNE-NE: Composing with the unit map CN-CNE; gives a natural map CN-NE: Applying this to a simplicial module of the form GðÀÞ; and then applying G again gives a natural transformation GCNG-GNEG: Finally, using the natural isomorphisms GNDIdDNG; we obtain a natural transformation GC-EG:

Construction 4.2. Let B be a non-negative EN differential graded k-module. We construct LB as the following coequalizer in the category of simplicial k-modules:

EGðCBÞ4EGB-LB: ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 243

Here one map is induced by the action map CB-B; and the other is the composite EGðCBÞ-EEGB-EGB of the map induced by the natural transformation GC-EG and the multiplication EE-E:

Proposition 4.3. For any non-negative EN differential graded k-module B; LB has the þ- natural structure of an EN simplicial k-algebra. The functor L : EC EE is left adjoint - þ to N : EE EC

Proof. An elementary category theory argument [4, II.6.6] shows that LB is an EN simplicial k-algebra and the diagram deﬁning it is a coequalizer in the category þ- of EN simplicial k-algebras; this gives L as a functor EC EE: Unraveling the adjunctions and using the universal property of the coequalizer gives D þ a natural bijection EEðLB; AÞ ECðB; NAÞ; which identiﬁes L as the left adjoint of N: &

Finally, before proving Theorem 4.1, we recall the deﬁnition of ‘‘cell algebra’’ [13, 3.1]. In the following deﬁnition, we follow [9] and denote the coproduct and pushout of EN differential graded k-algebras by the symbol ‘‘ ’’.

Deﬁnition 4.4. A cell non-negative EN differential graded k-algebra is a non- negative EN differential graded k-algebra B ¼ Colim Bn with B0 ¼ k and Bnþ1 ¼ y Bn CXn CCXn where X0; X1; are degreewise free non-negative differential graded k-modules with zero differential. Here CXn denotes the cone on Xn; the contractible differential graded k-module that comes with a conical inclusion Xn-CXn such that the quotient CXn=Xn is canonically isomorphic to SXn:

Proof of Theorem 4.1. It remains to show that when B is cofibrant, the unit map B-NLB is a quasi-isomorphism. By a standard argument (q.v. [13, 3.2]), every cofibrant non-negative EN differential graded k-algebra is the retract of a cell non- negative EN differential graded k-algebra, and so it suffices to consider the case when B is a cell non-negative EN differential graded k-algebra. Let Bn and Xn be as above. The underlying module of a sequential colimit of EN algebras is the sequential colimit of the underlying modules. It follows that NLBDColim NLBn; and so it suffices to show that the unit map Bn-NLBn is a quasi-isomorphism for each n: Clearly it is for n ¼ 0; since in that case B0 ¼ k and NLB0 ¼ k and the unit map is an isomorphism. So we may assume by induction that the unit map Bn-NLBn is a quasi-isomorphism. For the inductive step, choose a splitting of the underlying graded module of CXn as Xn"SXn; and let Y denote the graded submodule of Bnþ1 consisting of the (isomorphic) image of SXn; then, ignoring differentials, the underlying graded module of Bnþ1 is isomorphic to the underlying graded module of Bn CY: In fact, if we define a filtration on Bnþ1 by giving elements of Bn degree zero, elements of Y degree 1; and products (indexed by C) the sum of degrees, then the associated graded is Bn CSXn: We have an analogous filtration on LBnþ1; whose associated graded is ARTICLE IN PRESS

244 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

LBn EGSXn: We obtain a ﬁltration on NLBnþ1; and the unit map preserves the ﬁltration. The map of associated gradeds is a quasi-isomorphism by [13, 2.12] (or [12, 14.1]) and its analogue in the simplicial context. &

The proof of Theorem 1.4 follows essentially the same outline.

Proof of Theorem 1.4. Just as above, it suffices to construct a left adjoint functor þ - - L : MNA MA and to show that the unit of the adjunction M NLM is a quasi- isomorphism when M is a cofibrant NA-module. A coequalizer just like the one in 4.2 constructs L: We have an analogous definition of cell NA-algebra from [9, III.1.1] (or [13, 2.3]): M is a cell NA-module if M ¼ Colim Mn; where M0 ¼ 0 and Mnþ1 ¼ Mn,FNAXn FNACXn where each Xi is a degreewise free non-negative differential graded k-module with zero differential. Every cofibrant NA-module is a retract of a cell NA-module, and so it suffices to show that the unit M-NLM is a quasi-isomorphism when M is a cell NA-module. In this case, we have that D LMnþ1 LMn,FAXn FACXn:

Since in the module context colimits are formed in the underlying category of simplicial or differential graded k-modules, N (as well as L) preserves all colimits. It follows that the maps - D NLMn NLMnþ1 NLMn ,NFAXn NFACXn are injections. Examining the induced long exact sequences on homology, by induction and the ﬁve-lemma, we see that each map Mn-NLMn is a quasi- isomorphism. Passing to the colimit, we conclude that the map M-NLM is a quasi- isomorphism. &

We close the section by deducing parts (ii) and (iii) of Theorem 1.3 from part (i). Part (ii) is a consequence of the following general fact about closed model categories. It is an immediate consequence of the deﬁnition of Quillen equivalence.

Proposition 4.5. Let A; B be closed model categories, and let N : B-A be the right adjoint of a Quillen equivalence. Then for any ﬁbrant object B of B; the functor N : B=B-A=NB is the right adjoint of a Quillen equivalence.

Part (iii) of Theorem 1.3 is a consequence of the following general fact about left þ proper closed model categories, applied to the over-categories EC=NB and EE=B: In the statement B\A denotes the under-category of an object A of B:

Proposition 4.6. Let A; B be left proper closed model categories, and let N : B-A be the right adjoint of a Quillen equivalence. Then for any ﬁbrant object A of B; the functor N : B\A-A\NA is the right adjoint of a Quillen equivalence. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 245

Proof. Let L denote the left adjoint to N : B-A; and let LA denote the left adjoint to N : B\A-A\NA; then LA can be described as the composite of L with the pushout ANLNAðÀÞ: Since by hypothesis N preserves fibrations and weak equivalences between fibrant objects in B; it preserves fibrations and weak equivalences between fibrant objects in B\A; and so ðLA; NÞ is a Quillen adjunction. We need to see that whenever X is cofibrant in A\NA and B is a fibrant in B\A; a map LAX-B is a weak equivalence if and only if the adjoint map X-NB is a weak equivalence. The argument would be easy if we knew that the map LNA-A were a weak equivalence, but it may not be in general. To fix this, let A0-NA be a cofibrant approximation in A (i.e. A0 is cofibrant and A0-NA is a weak equivalence), and let X 0-X be a cofibrant approximation in A\A0: Then since A is left proper, we have 0 that the map ðNAÞNA0 X -X is a weak equivalence. Since LA is the left adjoint of a Quillen adjoint pair, it preserves weak equivalences between cofibrant objects, and so 0 the map LAððNAÞNA0 X Þ-LAX is a weak equivalence. Note that 0 0 LAððNAÞNA0 X ÞDANLA0 LX : Since L is the left adjoint of a Quillen equivalence, we have that LA0-A is a weak equivalence, and since B is left proper, we have that 0 0 the map LX -ANLA0 LX is a weak equivalence. We conclude that the map 0 LX -LAX is a weak equivalence. Since ðL; NÞ is a Quillen equivalence, a map LX 0-B is a weak equivalence if and only if the adjoint map X 0-NB is a weak equivalence. Now, using the work of the last paragraph and the two-out-of-three property of weak equivalences, we see that a 0 map LAX-B is a weak equivalence if and only if the composite map LX -B is a weak equivalence if and only if the adjoint map X 0-NB is a weak equivalence if and only if the adjoint map X-NB is a weak equivalence. Thus, ðLA; NÞ is a Quillen equivalence. &

5. The transitivityand coefﬁcient sequences

In this section we discuss the transitivity and coefficient long exact sequences and prove Theorem 1.9. The reader already familiar with the construction of these long exact sequences should see that Theorem 1.9 is an easy consequence of Theorems 1.3 and 1.4. We spend the major part of this section reviewing the construction of these long exact sequences. As mentioned in Section 1, the transitivity long exact sequence arises from a cofiber sequence and the coefficient long exact sequence arises from a fiber sequence. In order to follow standard references and the model category literature, we reformulate everything in terms of pointed model categories, i.e. model categories where the initial object is isomorphic to the final object. The following proposition is the main fact that allows us to do this. In it ‘‘ ’’ denotes the coproduct in the category of EN simplicial or differential graded k-algebras.

Proposition 5.1. Let A be an EN simplicial or differential graded k-algebra, let f : A-B be a coﬁbration of EN k-algebras, let A-C be a map of EN k-algebras, and ARTICLE IN PRESS

246 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 let M be a ðC ABÞ-module. Then the natural map

n n D ðC AB\C; MÞ-D ðB\A; MÞ is an isomorphism.

Proof. Let i : C-C AB be the obvious map. The functor C AðÀÞ : Ej f -Eji is the - left adjoint to the pullback functor B ÂC AB ðÀÞ : Eji Ejf : Since fibrations and weak equivalences of EN k-algebras are fibrations and weak equivalences of the underlying (simplicial or differential graded) modules, the pullback functor is easily seen to preserves fibrations and weak equivalences between fibrant objects. It follows that the adjoint pair is a Quillen adjunction. Since by hypothesis B is cofibrant in Ejf ; we get a natural bijection

% D % hðEjiÞðC AB; RÞ hðEj f ÞðB; B ÂC AB RÞ

for any ﬁbrant R in Eji: Applying this to ZC ABM; and identifying the map completes the proof. &

Correctly deﬁned, the Andre´ –Quillen cohomology groups DnðB\A; MÞ should preserve weak equivalences in all three variables A; B; and M: It is clear from the deﬁnition that it preserves weak equivalences in B and M; and the following proposition shows that it preserves weak equivalences in A:

A0 - B0 Proposition 5.2. Let kkbe a commutative diagram of EN simplicial or A - B differential graded k-algebras. If the vertical maps A0-A and B0-B are weak equivalences, then for every B-module M; the natural map DnðB\A; MÞ-DnðB0\A0; MÞ is an isomorphism.

Proof. We can assume, without loss of generality, that the horizontal maps A-B and A0-B0 are coﬁbrations (by factoring and renaming if necessary). Then, since the model category of EN simplicial or differential graded k-algebras is left proper, the 0 map A A0 B -B is a weak equivalence, and so induces an isomorphism n n 0 D ðB\A; MÞ-D ðA A0 B \A; MÞ: The statement now follows from the previous proposition. &

For the long exact sequences in Andre´ –Quillen cohomology we work in pointed model categories of the form EjB ¼ EjidB; categories of objects over and under an EN algebra B: If

0-M0-M-M00-0 ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 247 is a short exact sequence of B-modules, then for every n;

?-ZOSnM00-ZSnM0-ZSnM-ZSnM00 is a fiber sequence in EjB: This is because limits in EjB are limits of the underlying modules lying over B: Note also that the natural map M-OSM is an isomorphism, and so we could have written ZOSnM00 as ZSnÀ1M00 above. We can therefore patch together these fiber sequences for all n and continue them infinitely to the right. If we apply the functor h%ðEjBÞðB B; ÀÞ; we get a long exact sequence of abelian groups; when B is cofibrant, Proposition 5.1 identifies these abelian groups as

?-DnÀ1ðB; M00Þ-DnðB; M0Þ-DnðB; MÞ-DnðB; M00Þ-?: ð5:3Þ

% Likewise, for a map f : A-B; we can apply hðEjBÞðB AB; ÀÞ to get a long exact sequence of abelian groups, and when f is a coﬁbration, Proposition 5.1 identiﬁes these abelian groups as

?-DnÀ1ðB\A; M00Þ-DnðB\A; M0Þ-DnðB\A; MÞ-DnðB\A; M00Þ-?: ð5:4Þ

This leads to the coefﬁcient long exact sequence.

Definition 5.5. For a cofibrant EN simplicial or differential graded k-algebra B and a short exact sequence of B-modules M0-M-M00; the coefficient long exact sequence is the long exact sequence (5.3). For a cofibration of EN algebras A-B the relative coefficient sequence is the long exact sequence (5.4). For an arbitrary EN algebra B and an arbitrary map of EN algebras A-B the coefficient sequence and relative coefficient sequence are constructed using a cofibrant approximation of B0-B in E and using a cofibrant approximation of B0-B in E\A; respectively.

For the transitivity sequence, let A-B-C be coﬁbrations of EN simplicial or differential graded k-modules, and let M be a C-module. We have a canonical map C AC-C and the composite with either of the inclusions C-C AC is the identity map. This allows us to regard M as a ðC ACÞ-module such that the pullback C- module structure is the given original one. Consider the following commutative diagram in EjC: ARTICLE IN PRESS

248 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

This diagram identiﬁes

C AB-C AC-C BC as the start of a coﬁber sequence in EjC: The sequence continues (in h%ðEjCÞ) with

C BC-RCðC ABÞ-RCðC ACÞ-RCðC BCÞ-?; % where RC denotes the (Quillen) suspension functor on hðEjCÞ: Since ZC preserve ﬁbrations and limits, it preserves the Quillen loop functor, and the following proposition is a formal consequence.

Proposition 5.6. There is a canonical isomorphism

% nþ1 % nþ1 % n hðEjCÞðRCD; ZCS MÞDhðEjCÞðD; ZCOS MÞDhðEjCÞðD; ZBS MÞ % % natural in MAhMC and DAhðEjCÞ:

n Finally, looking at maps from the coﬁber sequence above into ZCS M; we get a long exact sequence, which Propositions 5.1 and 5.6 allow us to identify as

?-DnðC\B; MÞ-DnðC\A; MÞ-DnðB\A; MÞ-Dnþ1ðC\B; MÞ-?: ð5:7Þ

Definition 5.8. For cofibrations A-B and B-C and a C-module M; the transitivity sequence is the long exact sequence (5.7). For arbitrary maps A-B and B-C; the transitivity sequence is the long exact sequence obtained from a cofibrant approximation B0-B in E\A and a cofibrant approximation C0-C in E\B0:

Theorem 1.9 follows from the observation [16] that Quillen equivalences preserve ﬁber and coﬁber sequences.

6. Change of operads

Our review of Andre´ –Quillen cohomology in Sections 1 and 2 took advantage of the special properties of the operads C and E to eliminate technical hypotheses and to cast the theory in a form precisely paralleling the definition in the case of commutative (simplicial) algebras. In this section, we review the definition of Andre´ – Quillen cohomology for algebras over an arbitrary EN operad. We show that all EN operads lead to equivalent theories. For definiteness we concentrate on the differential graded context; the arguments in the simplicial context are for the most part identical, and we indicate when they differ. Although we are mainly interested in EN operads and EN algebras, the definitions and arguments in this section apply more generally to ‘‘flat’’ operads. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 249

Definition 6.1. For a ring R; we say that a differential graded right R-module M is right R-flat if the functor M #RðÀÞ preserves quasi-isomorphisms in the category of differential graded left R-modules. We say that an operad of differential graded k- modules F is flat if for each n; FðnÞ is right k½Sn-flat.

The main reason for restricting to flat operads is that, as explained in [12, Section 2], the category of algebras over a flat operad has a nice homotopy theory, and a structure nearly that of a model category. (Here the simplicial context is simpler: The category of algebras over an arbitrary operad forms a model category [15, 3.3.6].) In particular, when F is a flat operad, the homotopy category of F- % % algebras, hEF has small Hom-sets, as does the homotopy category hðEF=AÞ for an F-algebra A: Given an F-algebra A; recall that an A-module is a differential graded k-module M together with structure maps

FðnÞ#ðAðnÀ1Þ#MÞ-M satisfying certain unit, associativity, and equivariance requirements detailed in [8, I.4.1]. If we write ZM ¼ ZAM ¼ ZF;AM for A"M; then

# ðnÞD " # ðnÀjÞ# ð jÞ FðnÞ Sn ðZMÞ FðnÞ SnÀj ÂSj ðA M Þ: jX0

The composite of the projection onto the j ¼ 0; 1 summands

" # ðnÀjÞ# ð jÞ - # ðnÞ" # ðnÀ1Þ# FðnÞ SnÀj ÂSj ðA M Þ FðnÞ Sn A FðnÞ SnÀ1 ðA MÞ jX0 and the map

# ðnÞ" # ðnÀ1Þ# - " FðnÞ Sn A FðnÞ SnÀ1 ðA MÞ A M ¼ ZM obtained from the F-algebra structure map of A (on the ‘‘j ¼ 0’’ summand) and the A-module structure map of M (on the ‘‘j ¼ 1’’ summand) describe a sequence of maps # ðnÞ- FðnÞ Sn ðZMÞ ZM:

We leave it to the interested reader to verify that these maps deﬁne an F-algebra structure on ZM; natural in M; A; and F: The inclusion A-ZM and the projection ZM-A are maps of F-algebras. We deﬁne Andre´ –Quillen cohomology as follows.

Definition 6.2. Let F be a flat operad. Let A be an F-algebra, and let M be an A-module. We define the Andre´ –Quillen cohomology of A with coefficients ARTICLE IN PRESS

250 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 in M as

n n % n D ðA; MÞ¼DFðA; MÞ¼hðEF=AÞðA; ZS MÞ:

In order to define the relative Andre´ –Quillen cohomology groups, we need to be a little more careful than we had to be for the operad C: The key point is the invariance of DnðB\A; MÞ under quasi-isomorphisms of A: Proposition 5.2 uses in an essential way the left properness of the category of C-algebras. For a general flat (or even EN) operad F; there is no reason to expect that the category of F-algebras has the analogous property. This problem is alleviated by assuming that A is a ‘‘cell’’ or ‘‘cofibrant’’ F-algebra. The definition of cell F-algebra is completely analogous to the definition of cell C-algebra in Definition 4.4, and so we do not repeat it here. An F-algebra is a cofibrant if and only if it is a retract of a cell F-algebra. Even though the category of F-algebras may not be a model category, there is an appropriate notion of Quillen adjunction and Quillen equivalence given by [12, 2.14]. The argument of [11, 3.2] applies to give the following proposition.

Proposition 6.3. Let F be a ﬂat operad. Let X and Y be coﬁbrant F-algebras, let g : X-Y be a quasi-isomorphism, and let f : Y-Z be a map. Then the functor n g : EFj f -EFjðf 3 gÞ is a Quillen equivalence and induces an equivalence of homotopy categories.

As a consequence, we obtain the following proposition.

Proposition 6.4. Let F be a ﬂat operad. Let f : A-B be a map of F-algebras, and let C and D be objects of EFjf : Let q : X-A and r : Y-A be quasi-isomorphisms with X and Y coﬁbrant. There is a canonical bijection % D % hðEFjðf 3 qÞÞðC; DÞ hðEFjðf 3 rÞÞðC; DÞ:

Proof. This is an easy ‘‘left homotopy’’ argument. By Proposition 6.3 and factoring, we can assume, without loss of generality, that the maps X-A and Y-A are acyclic fibrations. The lifting property of cofibrant objects [12, Section 2] implies that we can factor the map X-A through the map Y-A; i.e. find a dotted arrow g making the triangle on the left commute.

Note that g is then a quasi-isomorphism, and so Proposition 6.3 gives us a n % - % bijection g : hðEFjðf 3 rÞÞðC; DÞ hðEFjðf 3 qÞÞðC; DÞ: To see that this bijection is ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 251

‘‘canonical’’, we need to see that any other factorization gives the same bijection. Let g0; g1 be two factorizations. Factor the codiagonal map XNX-X as a cofibration XNX-IX followed by an acyclic fibration s : IX-X; we write @0;@1 for the two maps X-IX: The lifting property for cofibrations implies that we can factor the map XNX-Y (given by the sum of g0 and g1) through a map G : IA-Y; making n n n the square on the right above commute. Then we have gi ¼ @i 3 G for i ¼ 0; 1: On n the other hand, we have that s 3 @i ¼ idX and so @i must be the inverse bijection to n n n n n s : In particular, we have @0 ¼ @1 ; and so g0 ¼ g1: &

The following deﬁnition now make sense; Proposition 5.2 implies that it agrees with Deﬁnition 1.2 when F ¼ C:

Definition 6.5. Let F be a flat operad. Let f : A-B be a map in EF; and let M be a B-module. We define Andre´ –Quillen cohomology of B relative to A with coefficients in M as

n \ n \ % 3 n D ðB A; MÞ¼DFðB A; MÞ¼hðEjðf qÞÞðB; ZS MÞ; where q : A0-A is a quasi-isomorphism and A0 is a coﬁbrant F-algebra.

n \ The following theorem, the main theorem in this section, says that DFðB A; MÞ preserves quasi-isomorphisms in F; A; B; and M: In it, a map of operads F0-F is a quasi-isomorphism means that each F0ðnÞ-FðnÞ is a quasi-isomorphism. For a map of operads f : F0-F; each F-algebra B becomes an F0-algebra with structure maps the composite maps

F0ðnÞ#BðnÞ-FðnÞ#BðnÞ-B:

- n This describes a functor EF EF0 ; we denote this functor as f when notation for it is required. Likewise, when M is a module over an F-algebra B; it becomes a module over the F0-algebra B: We therefore obtain a natural map

n - n DFðB; MÞ DF0 ðB; MÞ; and for a map of F-algebras A-B; a natural map

n \ - n \ DFðB A; MÞ DF0 ðB A; MÞ:

Theorem 6.6. Let F be a ﬂat operad, let A-B be a map of F-algebras, and let M be aB-module. Let F0 be a ﬂat operad, A0-B0 a map of F0-algebras, and M0 aB- module. Let F0-F be a quasi-isomorphism of operads, let A0-A and B0-Bbe quasi-isomorphisms of F0-algebras making the middle square commute, and let ARTICLE IN PRESS

252 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

M0-M be a quasi-isomorphism of B0-modules.

Then the natural maps

n - n 0 0 n \ - n 0\ 0 0 DFðB; MÞ DF0 ðB ; M Þ and DFðB A; MÞ DF0 ðB A ; M Þ are isomorphisms.

The previous theorem is an immediate consequence of the following theorem.

0- n - Theorem 6.7. Let f : F F be a map of ﬂat operads. Then the functor f : EF EF0 is the right adjoint of a Quillen adjunction. This adjunction is a Quillen equivalence if and only if f is a quasi-isomorphism.

The construction of the left adjoint f is similar to Construction 4.2 but simpler in * that it does not involve the ðG; NÞ adjunction. Since the right adjoint fn preserves fibrations and quasi-isomorphisms, the adjunction is a Quillen adjunction. When f is a quasi-isomorphism, to show that the adjunction is a Quillen equivalence, we just n need to check that whenever A0 is a cofibrant F0-algebra, the unit map A0-f f A0 * is a quasi-isomorphism (cf. Section 4). This is a special case of Lemma 6.10 at the end of the section. On the other hand, when f is not a quasi-isomorphism, it is easy to see that the adjunction cannot induce an equivalence of homotopy categories. Since a Quillen adjunction induces an adjunction on homotopy categories, and an adjunction is an equivalence if and only if the unit and counit are natural isomorphisms, it suffices to find an example of an F0-algebra for which the unit of the derived adjunction is not % 0 - an isomorphism in hEF0 : Suppose fðnÞ : F ðnÞ FðnÞ is not a quasi-isomorphism, and let A and A0 be the free F-algebra and F0-algebras on n generators in degree zero. Then A0 contains as a direct summand a differential graded module isomorphic to F0ðnÞ; A contains as a direct summand a differential graded module isomorphic to FðnÞ and the map of F0-algebras A0-A sending each generator of A0 to the corresponding generator of A induces the map fðnÞ between these submodules. This map is not a quasi-isomorphism but is adjoint to an isomorphism. It follows that the 0 n L 0 % unit of the derived adjunction A -f f A is not an isomorphism in hE 0 : * F Finally, we compare coefficients. Given an F0-algebra A0 and an A0-module M0; we would like to know that there is a quasi-isomorphic A ¼ f A0-module M: As * explained in [8, Section I.4], for an F-algebra A; the category of A-modules is the category of modules over a differential graded algebra UA; called the enveloping algebra of A; since UA depends on F as well as A; we shall denote it as UFA when ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 253 clarity is required. A map of F-algebras A-B induces a map of differential graded algebras UA-UB and the induced functor MUB-MUA coincides with the induced 0 functor MB-MA: Similarly, a map of operads F -F induces a map of differential graded algebras U 0 A-U A and the induced functor - coincides F F MUFA MUF0 A with the functor from F-algebra A-modules to F0-algebra A-modules described above. It is clear from this formulation that for an F0-algebra A0; the category of A0- modules is Quillen equivalent to the category of A-modules when the map of 0- differential graded algebras UF0 A UFA is a quasi-isomorphism.

Theorem 6.8. Let f : F0-F be a quasi-isomorphism of ﬂat operads, let A0 be a 0 0 coﬁbrant F -algebra, and let A be the F-algebra f A : Then the functor M -M 0 is * A A the right adjoint of a Quillen equivalence.

0 The proof is an induction and ﬁltration argument, using the ﬁltration on UF0 A and UFA described in [12, 13.7], which we now review.

i Notation 6.9. For an F-algebra A; deﬁne UFA to be the differential graded k½Si- module that makes the following diagram a coequalizer:

" # ð jÞ4 " # ð jÞ - i Fð j þ iÞ k½Sj ðFAÞ Fð j þ iÞ k½Sj A UFA: jX0 jX0

One map is induced by the F-algebra structure maps of A and the other is induced by the operadic multiplication of F:

0 1 Note that UFA is canonically isomorphic to A; and UFA is easily identiﬁed as UFA: Although, we do not use it, it is interesting to note that the collection fUðnÞ¼ n \ UFAg assembles into an operad whose category of algebras is isomorphic to EF A: As a particular case, when A is the initial F-algebra Fð0Þ; it is easy to check that i UFA ¼ FðiÞ: The following lemma in the case i ¼ 1 proves Theorem 6.8 and in the case i ¼ 0 proves the remaining part of Theorem 6.7.

Lemma 6.10. Let f : F0-F be a quasi-isomorphism of ﬂat operads, let A0 be a 0 0 i 0 i coﬁbrant F -algebra, and let A ¼ f A : Then the canonical map U 0 A -U Aisa * F F quasi-isomorphism for all iX0:

Proof. Since every cofibrant F0-algebra is a retract of a cell F0-algebra, the usual retraction argument shows that it suffices to consider the case when A0 is a cell F0- 0 0 0 0 0 algebra. Then A ¼ Colim An; where A0 is the initial F -algebra F ð0Þ; and each 0 0 0 Anþ1 is formed from An by attaching F -algebra ‘‘cells’’. Then A ¼ Colim An where 0 An ¼ f A ; and since sequential colimits commute with homology, it suffices to * n i 0 i show that the map U 0 A -U A is a quasi-isomorphism for each n: F n F n ARTICLE IN PRESS

254 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

i 0 0 The argument is by induction. For the base case, we note that U 0 A i ; F 0 ¼ F ð Þ i UFA0 ¼ FðiÞ; and the map in question is fðiÞ; this is a quasi-isomorphism by i 0 i assumption. Now assume by induction that U 0 A -U A is a quasi- F nÀ1 F nÀ1 i 0 i isomorphism for all i; and we show that U 0 A -U A is a quasi-isomorphism F n F n 0 0 0 for all i: We have that An ¼ AnNF0X F CX for some degreewise free differential graded k-module X with zero differential. Then An ¼ AnÀ1NFX FCX: We choose a splitting of the underlying graded k-module of CX as X"SX; and write Y for the submodule of CX isomorphic to SX to avoid confusion. We define a filtration on i 0 i 0 U 0 A by declaring elements of U 0 A to have degree zero, elements of Y to have F n F nÀ1 degree one, and products the sum of degrees. Just as in [12, 13.10], we see that the associated graded (‘‘E0’’) of this filtration is 0 i 0 " iþj 0 # ð jÞ E ðUF0 AnÞ¼ U 0 AnÀ1 Sj ðCX=XÞ : jX0 F

i We have an entirely analogous filtration on UFA; and the map in question preserves the filtration. Since by induction the map U iþjA0 -U iþjA is a quasi-isomorphism F0 nÀ1 F for all i; j; and since by [12, 13.6] (and [12, 13.13]), U iþjA0 and U iþjA are both F0 nÀ1 F nÀ1 right k½Sj-flat, we see that the map of associated gradeds is a quasi-isomorphism. It i 0 i follows that the map U 0 A -U A is a quasi-isomorphism for all i: & F n F n

7. Commutative R-algebras and Andre´ –Quillen cohomology

Now we turn to topology and begin our comparison of the topological Andre´ – Quillen cohomology of commutative S-algebras with the Andre´ –Quillen cohomology of differential graded k-algebras. In this section, we state in detail the main results and outline the strategy of argument. For the remainder of the paper, we work relative to a commutative S-algebra R: For our comparison of Andre´ –Quillen cohomology, we need to assume that R is connective and that R is cofibrant as a commutative S-algebra. In topology, we work in the category of R-modules, and 4 denotes the smash product over R: In algebra, we fix k ¼ p0R: The first problem is in producing an appropriate functor from commutative R- algebras to EN differential graded k-algebras. Since we have assumed that R is connective, the category of R-modules has an ‘‘ordinary homology theory’’ H * M that can be calculated as the composite of a CW approximation and a cellular chain functor [4, Section IV.2]. The basic strategy is to extend and refine this theory in some way to produce a functor from the category of commutative R-algebras to the category of EN differential graded k-algebras. In attempting to construct some commutative R-algebra version of the cellular chain functor, a difficulty immediately presents itself: There is no obvious cell structure on the free commutative R-algebra on a CW R-module X; _ ð jÞ PX ¼ X =Sj: ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 255

n ðpÞ This problem is intrinsic. For example, ðSRÞ =Sp is the quotient of a single cell and yet it has homology in inﬁnitely many degrees when p is not invertible in k [4, III.5.1]. The solution to this difﬁculty is to work with a category of EN R- algebras; we use the following operad G; closely related to the operad C we use in algebra.

Deﬁnition 7.1. Let G be the operad of spaces obtained from the operad of simplicial sets SL by geometric realization.

In the topological setting we work exclusively with the operad G: We have the evident notion of an action of the operad G on an R-module: An action of G on an R-module A consists of suitably associative and equivariant maps

ðnÞ- GðnÞþ4A A:

In the next section, we set up a theory of EN R-modules and a symmetric weak- monoidal product 2 parallel to the algebraic theory and product set up in [8, V] and reviewed in Section 2. We study the commutative monoids for the 2 product. The same formal argument as in algebra shows that the category of commutative monoids for 2 coincides with the category of algebras over G in R-modules. We call this category the category of EN R-algebras and denote it as EG: The trivial maps GðnÞ-* induce a map of operads from G to the commutative operad, and we obtain a forgetful functor from the category of commutative R- algebras to the category of EN R-algebras. We prove the following theorem in the next section.

Theorem 7.2. The category EG of EN R-algebras is a proper closed model category with weak equivalences and ﬁbrations the weak equivalences and ﬁbrations of the underlying R-modules. The forgetful functor from the category of commutative R- algebras to the category of EN R-algebras is a Quillen equivalence.

Recall that the functor G defined by _ ð jÞ GX ¼ Gð jÞþ4Sj X is a monad on the category of R-modules with monadic multiplication induced by the operadic multiplication on G: Then G becomes the free functor from R-modules to EN R-algebras. When X is a CW R-module, the free EN R-algebra GX does have a filtration suitable for calculating its homology; it is an example of a CW EN R- algebra, which we define in Section 10. The idea is that a CW EN R-algebra is an EN n n R-algebra built by inductively attaching EN R-algebra ‘‘cells’’ ðGCS ; GS Þ in a way that preserves a ‘‘skeletal’’ (or geometric) filtration. The main difficulty is in showing that the resulting EN R-algebra then inherits an appropriate skeletal filtration. These verifications are performed (and precise definitions are given) in Sections 9 and 10. Along the way, we prove the following theorem. ARTICLE IN PRESS

256 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Theorem 7.3. The cellular chain functor C * extends to a functor from the category of CW R-algebras to the category of EN differential graded k-algebras.

In order for this to provide a useful comparison functor for the category of EN R-algebras, we need to understand the relationship between the category of EN R-algebras and CW EN R-algebras; it is analogous to the relationship between the category of spaces and the category of CW spaces (with the cellular maps). In Section 10, we describe a CW approximation functor G and a natural weak equivalence g : G-Id: We prove the following version of the Whitehead theorem for % EN R-algebras. Here CWG denotes the category of CW EN R-algebras and hCWG denotes the category obtained from CWG by formally inverting the weak equivalences.

Theorem 7.4. There exists a CW approximation functor G : EG-CWG and natural weak equivalence g : G-Id with the following properties:

(i) The forgetful functor and the CW approximation functor induce inverse % % equivalences between hCWG and hEG: (ii) For any EN R-algebra A; the forgetful functor and CW approximation functor % % induce inverse equivalences between hðCWG=GAÞ and hðEG=AÞ: (iii) For any map of EN R-algebras f : A-B; the CW approximation functor induces % % an equivalence between hðEGjf Þ and hðCWGjGf Þ:

In the last part, just as in Section 1, EGjf denotes the ‘‘between category’’ of f ; the category whose objects are maps A-C-B where the composite A-B is f and whose maps are the maps of C under A and over B: The inverse functor is constructed as part of the proof in Section 10.

- 3 Deﬁnition 7.5. Let X : EG EC be the composite functor X ¼ C * G:

In order to compare Andre´ –Quillen cohomology, we need to discuss module categories. In Section 8, we use the special properties of the operad G to give a definition of modules over an EN R-algebra analogous to the description of modules over an EN differential graded algebra in Section 2 (from [8, Section V]). We denote the category of modules over an EN R-algebra A as MA; it is a proper closed model category with weak equivalences and fibrations the weak equivalences and fibrations of the underlying R-modules. Note that when A is an EN R-algebra by virtue of being a commutative R-algebra, MA differs from the category of modules described in [4]; we denote this latter category as MCom;A to distinguish it. We prove the following comparison theorem in Section 8.

Theorem 7.6. Let A be a commutative R-algebra. There is a forgetful functor MCom;A-MA that is the right adjoint of a Quillen equivalence of closed model categories. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 257

In both the commutative R-algebra and EN R-algebra contexts, we have a square zero multiplication functor ZCom;A or ZE;A from the category of A-modules to the category of commutative or EN R-algebras lying over A; see [2, Section 3] or Deﬁnition 8.5. The underlying R-module of ZCom;AM or ZE;AM is A3M and A3M-A is the map induced by the trivial map M-*: We deﬁne the topological Andre´ –Quillen cohomology of EN R-algebras as follows.

Deﬁnition 7.7. Let B be an EN R-algebra, and let M be an B-module. Deﬁne

n % n D ðB; MÞ¼hðEG=BÞðB; ZE;BS MÞ for nAZ: For a map of EN R-algebras f : A-B; deﬁne

n % n D ðB\A; MÞ¼hðEGj f ÞðB; ZE;BS MÞ for nAZ:

This is clearly a functor of M; and it becomes a functor of B; analogously as in algebra or as described in the commutative case in [2]: For a map of EN R-algebras - - R A B; the functor A ÂB ðÀÞ : EG=B EG=A has a right derived functor A ÂB % - % R D ðÀÞ : hðEG=BÞ hðEG=AÞ: We have a natural isomorphism A ÂB ZE;BM ZE;AM; and hence a map % % % hðEG=BÞðB; ZE;BMÞ-hðEG=BÞðA; ZE;BMÞ-hðEG=AÞðA; ZE;AMÞ that induces a map DnðB; MÞ-DnðA; MÞ: In [2], Deﬁnition 4.1 gives an analogous description of the topological Andre´ – Quillen cohomology of a commutative R-algebra. When A is a commutative R- algebra and M is an A-module in the commutative sense, we can form both ZCom;AM and ZE;AM: These are then two EN R-algebras lying over A whose underlying R- modules over A are the same. The identity map is a map of EN R-algebras ZE;AM-ZCom;AM; and it follows that the topological Andre´ –Quillen cohomology n DComðA; MÞ of A as a commutative R-algebra is canonically isomorphic to the n topological Andre´ –Quillen cohomology D ðA; MÞ of A as an EN R-algebra. Similar observations apply in the relative case. To compare with Andre´ –Quillen cohomology in algebra, we need a comparison functor on the module level. We describe an appropriate version of the CW approximation functor and cellular chain functor for modules, and we can use it to deﬁne a functor from the homotopy category of A-modules to the homotopy category of XA-modules (in algebra) but this functor goes in the wrong direction. We need its right adjoint. In Section 11, we prove the following theorem.

% % Theorem 7.8. Let B be an EN R-algebra. There is a functor R : hMXB-hMB with the following properties: (i) There is a natural isomorphism RSDSR: ARTICLE IN PRESS

258 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

A D (ii) For M MXB; there is an isomorphism of graded k-modules H * M p * RM; natural in M: (iii) For CAE=B; MAMXB; there is an isomorphism % % hðE=BÞðC; ZBRMÞDhðE=XBÞðXC; ZXBMÞ;

natural in C and M: (iv) For a map of EN R-algebras f : A-B; CAEjf ; MAMXB; there is an isomorphism % % hðEjf ÞðC; ZBRMÞDhðEjXf ÞðXC; ZXBMÞ;

natural in C and M:

We obtain the following immediate consequence.

Corollary7.9. Let A-B be a map of EN R-algebras, and let M be an XB-module. There are natural isomorphisms

DnðB; RMÞDDnðXB; MÞ and DnðB\A; RMÞDDnðXB\XA; MÞ:

Without being more explicit about the construction of the functor R or about the EN differential graded k-algebra XB; it might at first seem that Corollary 7.9 does not give any real information. However, from the point of view of obstruction theory, DnðB; NÞ is most interesting when the coefficients N have homotopy groups concentrated in degree zero. In this case, when B is connective, the p0B-module structure on p0N then uniquely identifies N up to isomorphism % in hMB; usually N is denoted Hp; where p ¼ p0N: On the other hand, by the Hurewicz theorem [4, IV.3.6], we have that H0XBDp0B; and there exists an % M in hMXB; unique up to isomorphism, with H * M ¼ p * N; that is, HqM ¼ 0 for qa0 and H0M ¼ p: We therefore have that NDRM: We summarize this as follows.

Corollary7.10. Let A be a connective EN R-algebra, and let p be a p0A-module. Then n n D ðA; HpÞDD ðXA; MÞ for any M with H0MDp and HqM ¼ 0 for qa0:

Since the comparison functor X is not part of a Quillen equivalence, we do not immediately obtain a comparison of the formal properties of topological Andre´ – Quillen cohomology of EN R-algebras with those of the Andre´ –Quillen cohomology of EN differential graded k-algebras. We discuss some further relations in the last two sections. In particular, in Section 13, we show that the isomorphism of Andre´ – Quillen cohomology preserves the fundamental cohomological exact sequence (the transitivity sequence) and in Section 14 we compare the operations in the topological Andre´ –Quillen cohomology of EN R-algebras with the operations in the Andre´ – Quillen cohomology of EN differential graded k-algebras. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 259

Finally, we have the following analogue of [4, IV.2.4], repeated and proved at the end of Section 12 (q.v. Theorem 12.5). In it, Hk denotes a coﬁbrant commutative S- algebra model of the Eilenberg–Mac Lane ring spectrum.

Theorem 7.11. Let R ¼ Hk: % - % (i) The functor X : hEG hEC is an equivalence of categories. % % (ii) For any EN Hk-algebra B the functor X : hðEG=BÞ-hðEC=BÞ is an equivalence of categories. % % (iii) For any map of EN Hk-algebras A-B; the functor X : hðEGjf Þ-hðECjXf Þ is an equivalence of categories. % % (iv) For any EN Hk-algebra B the functor R : hMXB-hMB is an equivalence of categories.

In general, up to natural isomorphism, the functors X and R for R factor through the equivalences X and R for Hk; see Theorem 12.4 for a precise statement. Using the methods of [18], the equivalence of Theorem 7.11 can be beefed up to a Quillen equivalence of model categories through a zigzag of adjoint pairs. Theorem 7.11 was ﬁrst proved in the author’s 1997 University of Chicago PhD Thesis but has never previously appeared in published form.

8. EN R-modules and the products 2; v ; and

The purpose of this section is to define modules over EN R-algebras and to prove the largely formal Theorems 7.2 and 7.6. In parallel to Section 2, we define the 2- product of EN R-modules, which drastically simplifies the technical details of working with EN R-algebras. For work in this section and the next, we need in addition the more complicated products v and described below. We begin by defining the category of EN R-modules, parallel to the category of EN k-modules of Definition 2.2. The operadic multiplication of G makes Gð1Þ a topological monoid and Gð1Þþ4R an associative R-algebra, weakly equivalent to R: A left module over Gð1Þþ4R is exactly an R-module M together with an associative - and unital map x : Gð1Þþ4M M:

Deﬁnition 8.1. The category MG of EN R-modules is the category of left modules 2 - over Gð1Þþ4R: We deﬁne the bifunctor : MG Â MG MG by 2 M N ¼ Gð2Þþ4Gð1ÞÂGð1ÞðM4NÞ with Gð1Þþ4R action induced by the left Gð1Þ action on the space Gð2Þ:

Just as in [8] and Section 2, the 2 product is coherently associative, commutative, and weakly unital. We have the following propositions. ARTICLE IN PRESS

260 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Proposition 8.2. The product 2 on MG is symmetric weak-monoidal.

Proposition 8.3. The category EG of EN R-algebras is exactly the category of 2 commutative monoids for in MG:

Using the previous proposition, we can deﬁne the category of modules over an EN R-algebra as follows.

Deﬁnition 8.4. Let A be an EN R-algebra. The category MA of EN A-modules is the category of left A-objects for 2 in MG:

Likewise, we can now deﬁne the square zero multiplication functor. Given an EN R-algebra A; and an EN A-module M; let ZM ¼ A3M; and let

m : ZM2ZMDðA2AÞ3ðA2MÞ3ðM2AÞ3ðM2MÞ-A3M ¼ ZM be the map that is the multiplication A2A-A on the ﬁrst wedge summand, is the action map A2M-M and M2ADA2M-M on the middle wedge summands, and is the trivial map M2M-* on the last wedge summand.

Deﬁnition 8.5. For an EN R-algebra A; deﬁne the functor ZE;A : MA-EG=A by ZE;AM ¼ ZM ¼ A3M with the square zero multiplication m and the projection map A3M-A:

We now move onto the comparison Theorems 7.2 and 7.6. The proofs of these theorems are complicated slightly by the fact that the product 2 is only weakly unital and so does not share all the formal properties of a symmetric monoidal product. For example, A2B is not the coproduct of EN R-algebras A and B; and A2M is not the free A-module on an EN R-module M: For the correct constructions we need ‘‘unital’’ variants given by the and v products introduced in [4, XIII; 9].

u Deﬁnition 8.6. The category MG of unital EN R-modules is the category of EN R- modules under R: For unital EN R-modules K and L and an EN R-module M; we deﬁne the products K v M and K L as the following pushouts:

We deﬁne M x L symmetrically.

The following associativity relations can be proved just as their analogues in [8]. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 261

Proposition 8.7. The bifunctor is a symmetric monoidal product on the category of unital EN R-modules. There are canonical isomorphisms

K v ðM2NÞDðK v MÞ2N; K v ðL v MÞDðK LÞ v M natural in the unital EN R-modules K; L and the EN R-modules M; N:

These associativity relations lead immediately to the following observations. The ﬁrst shows that our previous use of the symbol to denote the coproduct of EN algebras is consistent with the new deﬁnition.

Proposition 8.8. Let A and B be EN R-algebras. Then A BisanEN R-algebra and is the coproduct of A and B in the category of EN R-algebras.

Proposition 8.9. Let A be an EN R-algebra. Then A v ðÀÞ is the free functor from EN R-modules to EN A-modules.

We can now begin the proofs of Theorems 7.2 and 7.6, which are based on a comparison of the 2 product with the smash product of R-modules. We have a forgetful functor from EN R-modules to R-modules, which amounts to pulling back - the action of Gð1Þþ4R along the unit map R Gð1Þþ4R; this functor has as left adjoint the free functor Gð1Þþ4ðÀÞ: We could compare the smash product of R- modules with the 2 product of EN R-modules using these functors on the homotopy category, but on the point-set category another pair of adjoint functors works a little better. The trivial map of monoids Gð1Þ-* induces a map of - associative R-algebras Gð1Þþ4R R; and so we obtain a ‘‘trivial action’’ functor from R-modules to EN R-modules. The trivial action functor is a right adjoint; its left adjoint is the ‘‘indecomposibles’’ functor R ðÀÞ: The category of EN R- 4Gð1Þþ4R modules is a proper closed model category with weak equivalences and ﬁbrations the weak equivalences and ﬁbrations of the underlying R-modules by [4, VII.4.7]. The following comparison result is elementary.

Proposition 8.10. The free, forgetful adjunction and the trivial action, indecomposibles adjunction are both Quillen equivalence pairs. The indecomposibles functor is strong symmetric monoidal and the trivial action functor is lax symmetric monoidal. The derived functor of each of these functors is a symmetric monoidal equivalence of homotopy categories.

Since the trivial action functor from R-modules to EN R-modules is lax symmetric monoidal, it takes a left A-object for 4 to a left A-object for 2; that is, it reﬁnes to a functor MCom;A-MA: This is the ‘‘forgetful functor’’ of Theorem 7.6.

Proof of Theorem 7.6. We have that the category of EN A-modules is the category of algebras for a continuous monad on the category of R-modules, namely the monad v A ðGð1Þþ4ðÀÞÞ: We obtain the model structure from [4, VII.4.10]. Now suppose ARTICLE IN PRESS

262 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 that A is a commutative R-algebra. The trivial action functor preserves all weak equivalences, and so to see that we have a Quillen equivalence, we just need to see that the unit map

M-R M 4Gð1Þþ4R is a weak equivalence for all cell A-modules R: Passing to sequential colimits and using the usual long exact sequences of homotopy groups, it sufﬁces to treat the case when M is the free EN A-module on a sphere R-module [4, III.2], M ¼ v n A ðGð1Þþ4SRÞ: v v n Unwinding the deﬁnition of ; we see that A ðGð1Þþ4SRÞ is the pushout

and the unit map is the quotient of the left action of Gð1Þ on Gð2Þ: Since Gð2Þ is isomorphic to Gð1Þ as a left Gð1Þ-space (cf. [4, I.6.1]), it follows that the v n n indecomposibles functor takes A ðGð1Þþ4SRÞ to A4SR: In addition, since the singular simplicial set functor and geometric realization preserve homotopies, the homotopies in [13, 7.1–2] (in the case m ¼ 1) show that the right vertical arrow in the diagram above is a homotopy equivalence of R-modules. The map n - n ðGð2Þ=ðGð1ÞÂ*ÞÞþ4ðA4SRÞ A4SR is a homotopy equivalence of R-modules since Gð2Þ=ðGð1ÞÂ*Þ is contractible (by [13, 7.1], for example). It follows that the unit map v n - n A ðGð1Þþ4SRÞ A4SR is a homotopy equivalence of R-modules and therefore a weak equivalence of EN A- modules. &

As a consequence of the last observation of the proof, that for any EN R-algebra v n - n A; the natural map A ðGð1Þþ4SRÞ A4SR is a homotopy equivalence of R-modules, we also obtain the following corollary.

Corollary8.11. Let f : A-B be a map of EN R-algebras. The induced functor n f : MB-MA is the right adjoint of a Quillen adjoint pair. If f is a weak equivalence, then the Quillen adjoint pair is a Quillen equivalence.

Finally, we close the section with the proof of Theorem 7.2.

Proof of Theorem 7.2. The closed model structure is again an easy application of [4, VII.4.9]. Left properness follows from [13, 7.1–2]. For the Quillen equivalence, deﬁne ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 263 the functor L : EG-Com by the following coequalizer:

PGA4PA-LA:

Here, one of the left-hand arrows is the G action map on A and the other is the composite of the natural map PG-PP (induced by the map G-P) and the monadic multiplication PP-P: A check of universal properties reveals that L is left adjoint to the forgetful functor Com-EG: Since the weak equivalences and fibrations in Com are also the weak equivalences and fibrations of the underlying R- modules, a map in Com is a weak equivalence or fibration if and only if the forgetful functor takes it to a weak equivalence or fibration in EG: It follows that L and the forgetful functor are a Quillen adjunction. To see that this is a Quillen equivalence, we just have to see that the unit map is a weak equivalence for every cell G-algebra, i.e. for every cell G-algebra [4, VII.4.11]. The unit B-LB is a weak equivalence for B ¼ GX for any CW R-module X by [4, III.5.1; VII.6.7] (here is where we use the assumption that R is cofibrant). The remainder of the argument is follows by induction. &

9. The cellular chain functor

As explained in Section 7, our strategy for constructing the functor from topology to algebra is to use a CW approximation functor composed with a cellular chain functor. In this section, we make some preliminary observations on the cellular chain functor and on the sorts of filtrations preserved by the constructions we use to build CW EN R-algebras in the next section. We begin by reviewing the cellular chain functor on ‘‘CW R-modules’’ as described in [4]. We then describe a class of filtered EN R-algebras called ‘‘quasi-CW’’ EN R-algebras, to which we generalize the cellular chain functor. This class is a convenient one in which to state results, but is too general to be closed under many constructions. We therefore introduce the class of ‘‘E-CW’’ algebras, whose underlying R-modules are built out of extended power ‘‘cells’’. This class of algebras is closed under algebra cell attachment and provides the foundation for the definition of CW EN R-algebras, which is given in the next section. We begin with a quick review of CW R-modules. Recall that a cell R-module is an R-module M ¼ Colim Mn such that M0 ¼ *; and each Mnþ1 is formed from Mn as - q the cofiber of a map Fn Mn; where Fn is a wedge of sphere R-modules SR [4, III.2] (of varying dimensions q). The filtration Mn is called the sequential filtration. The only role of the sequential filtration is to allow inductive arguments; it is unrelated to any of the dimensions of the sphere R-modules the cells are attached along. A CW R- m module is a cell R-module M ¼ Colim Mn that has a ‘‘skeletal filtration’’ Mn ; mAZ; such that the attaching maps Fn-Mn are filtered, and the skeletal filtration on Mnþ1 is the obvious one induced on the cofiber; for example, for an attaching map q - q f : SR Mn; we understand that f must factor through Mn and the filtration on the m m m cofiber Cf is Cf ¼ Mn for moq þ 1; and Cf ¼ Cf for mXq þ 1: The skeletal ARTICLE IN PRESS

264 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

m m filtration on M; M ¼ Colim Mn ; is then closely tied to the formal dimensions of the cells out of which that M is built. A cellular map of CW R-modules is a map of R-modules that preserves the skeletal filtration. An EN R-algebra A has a unit map R-A: Technically, R is not a cell R-module and this makes a cell R-module structure on A inconvenient as well as unlikely; nevertheless, we can understand R as a cell of dimension zero and define a unital variant of CW R-modules. We define a CW unital R-module in the same way as a CW R-module except that we start with M0 ¼ R concentrated in skeletal filtration m m level zero (i.e. M0 ¼ * for mo0; M0 ¼ R for mX0). A cellular map of CW unital R-modules is a map of R-modules under R that preserves the skeletal filtration. For defining the cellular chain functor, only two properties of the skeletal filtration m mÀ1 m m are needed: M =M is weakly equivalent to a wedge of SR ’s and H * M is zero m mÀ1 above degree m: We define the cellular chains by C * M ¼ pmðM =M Þ with differential induced by the connecting homomorphism of the cofiber sequence

MmÀ1=MmÀ2-Mm=MmÀ2-Mm=MmÀ1 with an appropriate sign (namely the sign sm; where s ¼ 71 is the sign of commuting the suspension isomorphism with the connecting homomorphism in one’s preferred sign convention). These two properties then imply that D H * M H * ðC * MÞ: This leads to the following deﬁnition.

Definition 9.1. A skeletally filtered R-module is an R-module M together with a filtration Mm-M by h-cofibrations MmÀ1-Mm for mAZ such that Mm=MmÀ1 is m m weakly equivalent to a wedge of SR ’s and H * M is zero above degree m: A cellular map of skeletally filtered R-modules is a map that preserves the skeletal filtration.

Proposition 9.2. The cellular chain functor C * extends to a functor from the category of skeletally ﬁltered R-modules and cellular maps to the category of differential graded D k-modules. There is a natural isomorphism H * M H * C * M:

If X is a CW space and M is a CW unital R-module, then Xþ4M is not a CW unital R-module, but it is a skeletally filtered R-module. We have introduced the definition of skeletally filtered R-modules for just this reason, to give us a language to talk about such objects and their cellular chains. Skeletally filtered R-modules are not closed under enough of the usual constructions in homotopy theory to be useful for much else. In the EN R-algebra context, it is convenient to have a category of objects for which we can make sense of the cellular chain functor, but which is more general than the category of CW EN R-algebras.

Deﬁnition 9.3. A quasi-CW (unital) EN R-module is a (unital) EN R-module M that is also a CW (unital) R-module for which the action map Gþ4M-M is cellular. A quasi-CW EN R-algebra is an EN R-algebra A that is also a quasi-CW unital EN R- - module for which the multiplication Gð2Þþ4ðA4AÞ A is cellular. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 265

We understand a map between quasi-CW (unital) EN R-modules to be cellular if it is a map of EN R-modules (under R) and a cellular map of CW (unital) R-modules. Likewise a map between quasi-CW EN R-algebras is cellular if it is a map of EN R- algebras and a cellular map of CW unital R-modules. When we refer to the category of quasi-CW (unital) EN R-modules or quasi-CW R-algebras, we understand the maps to be the cellular maps.

Theorem 9.4. The cellular chain functor extends to functors:

(i) From the category of quasi-CW EN R-modules to the category of EN differential graded k-modules. (ii) From the category of quasi-CW unital EN R-modules to the category of unital EN differential graded k-modules. (iii) From the category of quasi-CW EN R-algebras to the category of EN differential graded k-algebras.

- Proof. The ﬁrst two statements are clear. The map Gð3Þþ4ðA4A4AÞ A is - cellular since the map Gð2Þþ4ððGð2Þþ4A4AÞ4AÞ A is, and so the multiplication # # - Cð2Þ C * ðAÞ C * ðAÞ C * ðAÞ is associative. It is easy to see that it is also & commutative and unital, and so C * A is an EN differential graded k-algebra.

If N is a quasi-CW (unital) EN R-module, we say that M is a subcomplex if it is a subcomplex of the underlying CW (unital) R-module and the inclusion is a map of EN R-modules (under R). We have the following closure properties for quasi-CW unital EN R-modules. They are immediate consequences of analogous closure properties for CW unital R-modules

Proposition 9.5. Let N be a quasi-CW unital EN R-module and let M be a subcomplex of N: If P is a quasi-CW unital EN R-module and M-P is a cellular map, then P ,M N is a quasi-CW unital EN R-module with P as a subcomplex. In addition, it is the pushout in the category of quasi-CW EN R-modules, and the universal map C P ,C M C N-C ðP ,M NÞ is an isomorphism. * * * *

Proposition 9.6. Let M0-M1-? be a sequence of subcomplexes of quasi-CW unital EN R-modules. Then Colim Mn is a quasi-CW unital EN R-module with each Mn as a subcomplex. In addition, it is the colimit in the category of quasi-CW unital R-modules, - and the universal map Colim C * ðMnÞ C * ðColim MnÞ is an isomorphism.

Similar statements hold for quasi-CW EN R-modules. The analogue of Proposition 9.6 also holds for quasi-CW EN R-algebras, but we need to include more structure before we can have a category of skeletally ﬁltered EN R-algebras that satisﬁes the analogue of Proposition 9.5. The extra structure we add is a cell ðnÞ structure based on extended powers GðnÞþ4Sn M for CW R-modules M; and with subcomplexes based on the map induced by an inclusion of a subcomplex MCN: ARTICLE IN PRESS

266 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Deﬁnition 9.7. We deﬁne an E-cell to be a quasi-CW R-module of the form

Bðn; XÞ¼GðnÞþ4Sn X for some n > 0; where X is a ﬁnite CW R-module with a cellular Sn-action. We say that Bðn; XÞ is a subcell of Bðn; YÞ when X is a subcomplex of Y and the inclusion X-Y is Sn-equivariant.

Deﬁnition 9.8. An E-CW (unital) module is a quasi-CW (unital) EN R-module M ¼

Colim Mn such that M0 ¼ * ðM0 ¼ RÞ; and Mnþ1 ¼ Mn ,Bn Cn where Bn; Cn are wedges of E-cells, Bn-Cn is a wedge of subcell inclusions, and Bn-Mn is a cellular map.

An E-CW (unital) module is then by neglect of structure a quasi-CW (unital) EN R-module by the previous two propositions and their variants. We have the notion of a subcomplex M of an E-CW module N; where the wedges of cells BnCCn out of which M is constructed are wedge summands of the wedges of cells out of which N is constructed. The analogues of Propositions 9.5 and 9.6 clearly hold for E-CW modules. In addition, we have the following result.

Theorem 9.9. Let M and N be E-CW unital modules. Then M N is an E-CW unital module with M ,R N as a subcomplex. There is a canonical isomorphism D D C * ðMÞ C * ðNÞ C * ðM NÞ: The commutativity isomorphism M N N Mis cellular, and for an E-CW unital module P; the associativity isomorphism

M ðN PÞDðM NÞ P is cellular. If L is a subcomplex of M; then L N is a subcomplex of M N:

Proof. We have an isomorphism EN R-modules 2 D Bðn; XÞ Bðq; YÞ Bðp þ q; Spþqþ4SpÂSq ðX4YÞÞ:

The construction of the E-CW unital module structure and the veriﬁcation that the maps are cellular are obtained by induction up the sequential ﬁltrations. &

Analogous theorems hold for the 2 product of E-CW modules and the v product of an E-CW unital module and an E-CW module. We can now deﬁne an E- CW algebra to be an E-CW unital module A with a cellular EN R-algebra multiplication A A-A: It is easy to see that this is just the same thing as a quasi- CW EN R-algebra that is an E-CW unital module.

Deﬁnition 9.10. An E-CW algebra is a unital E-CW module A together with a cellular map A A-A that is associative, commutative, and unital.

The following theorem is the motivation for introducing E-CW modules. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 267

Theorem 9.11. Let Y be a wedge of ﬁnite CW R-modules and let X be a subcomplex. Let A be an E-CW algebra, and let X-A be a cellular map. Then the pushout of EN R-algebras A GX GY is an E-CW algebra that contains A as a subcomplex. In addition, it is the pushout in the category of quasi-CW EN R-algebras, and the universal map

C A CC X CC Y-C ðA GX GYÞ * * * * is an isomorphism.

Proof. We can write A GX GY as a sequential colimit of pushouts of modules, A ¼ A; A ¼ A v A v ð ð1Þ YÞ; and in general 0 1 0 ,A ðGð1Þþ4XÞ G þ4

ðnÞ A ¼ A v A v ð ðnÞ Y Þ; n nÀ1 ,A ðGðnÞþ4Sn ZnÞ G þ4Sn

ðnÞ ðnÀ1Þ where Zn is the subcomplex of Y that is the union of the subcomplexes X4Y ; Y4X4Y ðnÀ2Þ; y; Y ðnÀ1Þ4X: &

10. CW EN R-algebras

The purpose of this section is to prove Theorem 7.4. We begin with the definition of a CW EN R-algebra. The definition of a CW EN R-algebra is analogous to the definition of CW R- module reviewed in the last section. We have the more basic notion of a cell EN R- algebra [4, VII.4.11]: A is a cell EN R-algebra when A ¼ Colim An; where A0 ¼ R (the initial EN R-algebra), and inductively Anþ1 is formed from An as an EN R- algebra cofiber of a wedge of spheres along attaching maps,

Anþ1 ¼ An GXnþ1 GCXnþ1:

If inductively An is an E-CW algebra (Deﬁnition 9.10), and the attaching map Xnþ1-An is a cellular map, then Anþ1 becomes an E-CW algebra by Theorem 9.11. In this case, the colimit A ¼ Colim An is both a cell EN R-algebra and an E-CW algebra.

Deﬁnition 10.1. A CW EN R-algebra is a cell EN R-algebra for which the attaching maps are cellular. Deﬁne the category CWG of CW EN R-algebras to have objects the CW EN R-algebras and maps the cellular maps.

We have the evident notion of a subcomplex algebra. A CW EN R-algebra A is a 0 subcomplex algebra of a CW EN R-algebra A when the cells ðGCXn; GXnÞ used to 0 0 0 form A consist of some of the cells ðGCXn; GXnÞ used to form A ; that is, when each 0 Xn is a wedge summand of Xn: Theorem 9.11 applied inductively gives the following proposition. ARTICLE IN PRESS

268 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Proposition 10.2. Let A; B; CbeCWEN R-algebras. If A is a subcomplex algebra of B and A-C is a cellular map, then B AC is a CW EN R-algebra containing C as a subcomplex algebra. In addition it is the pushout in the category of CW EN R- algebras, and the universal map C B C AC C-C ðB ACÞ is an isomorphism. * * * *

The usual process constructs a CW approximation functor.

Construction 10.3. We construct the functor G : EG-CWG and the natural transformation g : G-Id in EG as follows. For an EN R-algebra A; let G0A ¼ R - and let g0 : G0A A be the initial map. Inductively, having constructed the CW EN - R-algebra GnA and the map gn : GnA A; let Dnþ1 be the set of commutative diagrams

where the map f is cellular, and m ranges over all integers. Let _ ma Xnþ1 ¼ SR ; aADnþ1

and let Gnþ1A ¼ GnA GXnþ1 GCXnþ1; attached by the maps f in the diagrams in Dnþ1: - We deﬁne gnþ1 : Gnþ1A A using gn and the maps g in the diagrams in Dnþ1: Let GA ¼ Colim GnA; and let g ¼ Colim gn:

As an easy consequence of [4, III.2.9], we see that g1 induces a surjection on homotopy groups, and that the kernel of p * gn (in p * ðGnAÞ) maps to zero in D p * ðGnþ1AÞ: Since p * ðGAÞ Colim p * ðGnAÞ; we obtain the following proposition.

Proposition 10.4. For any EN R-algebra A; the natural map g is a weak equivalence.

When A is a quasi-CW EN R-algebra, there is no reason to expect the map g to be a cellular map. In this context we can deﬁne a functor L by attaching only those cells from diagrams ðf ; gÞ where g is also cellular. The CW EN R-algebra LA is a subcomplex algebra of GA; and the composite LA-GA-A is cellular. Using [4, III.2.9] again, we obtain the following proposition.

Proposition 10.5. Let A be a quasi-CW EN R-algebra. Then the natural transformation LA-A is a weak equivalence.

Finally, when A is a CW EN R-algebra, we have the variant LA : CWG\A-CWG\A deﬁned by starting with A in place of R: Then for any cellular map A-B; LA contains A as a subcomplex algebra and the natural ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 269 transformation LAB-B is a cellular map and a weak equivalence. We can now prove Theorem 7.4.

Proof of Theorem 7.4. For the purposes of the proof, write F for the forgetful functor. For parts (i) and (ii), the natural transformation g gives a natural % % isomorphism in hEG and in hðEG=BÞ between the identity functor and F 3 G; and the zigzag

B B Id ’ L - G 3 F

% deﬁnes a natural isomorphism between the identity functor and G 3 F in hCWG and % hðCWG=GBÞ: - For part (iii), we need to describe the inverse functor. Let G : CWGjGf EGjf be deﬁned by

GðCÞ¼A GAðLGACÞ:

Since GA is a subcell algebra of LGAC; the map GA-LGAC is a coﬁbration of EN R- algebras, and it follows that G preserves weak equivalences, and so induces a functor % - % hðCWGjGf Þ hðEGjf Þ: When C ¼ GD for D in EGjf ; the universal property of the pushout induces a map GGðDÞ-D in EGjf that is easily seen to be natural and a weak equivalence. This gives a natural isomorphism in h%ðEjf Þ from G 3 G to the identity. % Finally, we need to produce a natural isomorphism in hðCWG j Gf Þ between G 3 G and the identity. Deﬁne H : CWGjGf -CWGjG f by

HðCÞ¼GA LGAðLLGACÞ:

Since GA is a subcomplex algebra of LGAC; LGA is a subcomplex algebra of LLGAC; and it follows that this pushout exists in CWG by Proposition 10.2. The universal property of the pushout induces cellular maps

HðCÞ-C and HðCÞ-GðGðCÞÞ that are weak equivalences since the map LGA-GA is a weak equivalence. &

When A is a CW EN R-algebra, we define a CW A-module to be a cell A-module where the attaching maps are cellular. We define the category CWMA to be the category of CW A-modules and cellular maps. We can define a CW approximation functor from A-modules to CW A-modules and prove the following theorem.

Theorem 10.6. Let A be a CW EN R-algebra. The forgetful functor and CW % % approximation functor induce inverse equivalences between hMA and hCWMA: ARTICLE IN PRESS

270 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

11. The proof of Theorem 7.8

In this section, we use the results on CW objects and CW approximation from the previous section to prove the main theorem, Theorem 7.8. The argument is an application of the abstract form of Brown’s representability theorem [3]. % We apply Brown’s theorem to the homotopy category of CW B-modules hCWMB for a CW EN R-algebra B and to the homotopy category of the between-category of % CW EN R-algebras hðCWGjf Þ for a cellular map of CW EN R-algebras f : A-B: Proposition 9.5 implies that the cellular chain functor preserves pushouts along inclusions of subcomplexes of CW A-modules. Similarly, Proposition 10.2 implies that the cellular chain functor preserves pushouts along inclusions of subcomplex algebras. The cellular chain functor also preserves sequential colimits of inclusions of subcomplexes. These properties are all that are needed in this context to apply Brown’s representability theorem to the functors % % hðECjC f ÞðC ðÀÞ; CÞ and hMC BðC ðÀÞ; MÞ * * * * for an EN differential graded k-algebra C and a C * B-module M: Choosing representing objects FðCÞ and CðMÞ deﬁnes functors % % % % F : hðECjC f Þ-hðCWGjf Þ and C : hMC B-hCWMB: * * The deﬁning property of these functors give natural bijections % D % hðCWGj f ÞðÀ; FðCÞÞ hðECjC * f ÞðC * ðÀÞ; CÞ and % % hCWMBðÀ; CðMÞÞDhMC BðC ðÀÞ; MÞ: * * In other words, F and C give right adjoint functors to the cellular chain functor. We can now prove Theorem 7.8.

% % Proof of Theorem 7.8. Let R be the composite of C : hMXB-hCWMGB and the equivalences % % % hCWMGBChMGBChMB:

Since C * commutes with suspension, it follows that R commutes with the right adjoint of suspension; since suspension is an equivalence on the homotopy categories, it follows that R commutes with suspension. This proves (i). n Applying the adjunction above to the free CW GB-module on SR; we get a bijection HnMDpnCMDpnRM; and by naturality, it is an isomorphism of k- modules. This proves (ii). Moving on, we note that part (iii) is the special case of part (iv) where A is the initial EN R-algebra, R: We therefore concentrate on (iv) and work in the context of ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 271 a map of EN R-algebras f : A-B: If N is a CW GB-module, then by inspection we have an isomorphism

D C * ZGBN ZXBC * N in ECjXf : Applying this to CM and using the C * ; F adjunction, we obtain a map

- - ZGBCM FðZXBC * CMÞ FðZXBMÞ: ð11:1Þ

n We can make GA GSR an object of CWGjGf using the inclusion of GA in the n - coproduct for the map in and the map induced by trivial map SR GB for the map % n out. Looking at the set of maps in hðCWGjGf Þ out of GA GSR; we identify the map

D % n - % n D pnðCMÞ hðCWGjGf ÞðGA GSR; ZGBCMÞ hðECjXf ÞðXA CSR; ZXBMÞ HnðMÞ as the isomorphism pnCM-HnM: It follows that the map (11.1) is a weak % C % equivalence. Via the equivalence hðCWGjGf Þ hðEGjf Þ; the map (11.1) induces a natural transformation

% % hðEGjf ÞðÀ; ZBRMÞ-hðECjXf ÞðXðÀÞ; ZXBMÞ; which is therefore a natural isomorphism. &

12. Eilenberg–Mac Lane commutative S-algebras

In this section we discuss the stronger results of Theorem 7.11 we obtain when we use the Eilenberg–Mac Lane spectrum Hk for the base commutative S-algebra. We discuss the relationship between the functor X we obtain when we use Hk as the base commutative S-algebra and the functor X we obtain when we use a more general connective commutative S-algebra R (as always with p0R ¼ k) as the base. Throughout this section, to avoid confusion, we attach an additional subscript R or Hk to denote the category in which each construction is made, that is, whether we take R or Hk as the base commutative S-algebra. The ﬁrst basic fact is the following proposition, which is clear from [4, III.4.1] and the constructions.

Proposition 12.1. The change of base rings functor Hk4ðÀÞ and the forgetful functor - MG;Hk MG;R are the left and right adjoint of a Quillen pair. The functor Hk4ðÀÞ is strong symmetric monoidal and the forgetful functor is lax symmetric monoidal.

As a consequence, we also get the following proposition.

Proposition 12.2. The change of base rings functor Hk4ðÀÞ and the forgetful functor - EG;Hk EG;R are the left and right adjoint of a Quillen pair. ARTICLE IN PRESS

272 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

The previous proposition in particular gives us a natural map of EN R-algebras A-Hk4A that allows us to regard ðHk4AÞ-modules as A-modules. Since the forgetful functor takes the EN Hk-algebra ZHk4A;HkM to the EN R-algebra ZHk4A;RM; we obtain the following proposition.

Proposition 12.3. Let A be an EN R-algebra, and let M be an ðHk4AÞ-module. There is a natural isomorphism

n D n DRðA; MÞ DHkðHk4A; MÞ:

The change of rings functor Hk4ðÀÞ takes sphere R-modules to sphere Hk- modules and so it converts CW (unital) R-modules to CW (unital) Hk-modules. Moreover, it induces an isomorphism of the cellular chain functors. It follows that in the homotopy category, XR factors through XHk up to natural isomorphism. Likewise, RR factors through RHk up to natural isomorphism. In fact, we can prove the following more precise theorem.

Theorem 12.4. There is a natural map of CW EN Hk-algebras

Hk4GRðÀÞ-GHkðHk4ðÀÞÞ such that the diagram

commutes. The induced map XRðÀÞ-XHkðHk4ðÀÞÞ on cellular chains induces a map on Andre´–Quillen cohomology that makes the following diagram commute for any XHkðHk4AÞ-module M:

Proof. We have Hk4ðGRÞ0ðÀÞ ¼ Hk ¼ðGHkÞ0ðÀÞ as a base case, and by induction, - we construct a natural cellular map Hk4ðGRÞnðÀÞ ðGHkÞnðHk4ðÀÞÞ making the obvious diagram with id4gn and gn commute. Then smashing with Hk takes a diagram in ðDnþ1ÞR to a diagram in ðDnþ1ÞHk; and tells us how to construct the map ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 273

- Hk4ðGRÞnþ1ðÀÞ ðGHkÞnþ1ðHk4ðÀÞÞ: In the colimit we obtain the desired map. The second diagram above commutes by an easy adjunction argument. &

We close with the proof of Theorem 7.11, which we restate here for convenience. It sufﬁces to consider only parts (iii) and (iv) of the original formulation: Part (ii) is the special case of part (iii) where A is the initial EN Hk-algebra Hk; and part (i) is the special case of part (ii) where B is the ﬁnal EN Hk-algebra *:

Theorem 12.5. Let R ¼ Hk; and let f : A-B be a map of EN Hk-algebras. % % % % Then the functors X : hðEGj f Þ-hðECjf Þ and R : hMXB-hMB are equivalences of categories.

Proof. Since the functor X is the composite of the CW approximation functor G and the cellular chain functor C * on CW algebras, and since R is the composite of an equivalence with the right adjoint to the cellular chain functor C * on CW modules, % - % we just need to show that the functors C * : hðCWGjf Þ hðECjC * f Þ and % % C : hCWMA-hMC A are equivalences for a CW EN Hk-algebra A: Using the * * adjunctions F and C; the equivalence is easily deduced from the fact that the ordinary homology groups H * are naturally isomorphic to the homotopy groups p * for Hk-modules, cf. [4, IV.2.4]. &

13. The transitivitysequence

Corollary 7.9 provides a natural isomorphism of functors from the topological Andre´ –Quillen cohomology of an EN R-algebra to the Andre´ –Quillen cohomology of an EN differential graded k-algebra, but we would like to have an isomorphism of cohomology theories. In this section, we show that the isomorphism is cohomological in the algebra variable by showing that it commutes with the fundamental long exact sequence, the transitivity sequence. The transitivity sequence for EN R-algebras can be constructed in the same way as the transitivity sequence for commutative R-algebras [2, 4.3] and the transitivity sequence for EN differential graded k-algebras in Deﬁnition 5.8.

Proposition 13.1 (The transitivity sequence). Let A-B be a map of EN R-algebras, and let M be a B-module. There is a long exact sequence

?-DnðB\A; MÞ-DnðB; MÞ-DnðA; MÞ-Dnþ1ðB\A; MÞ-?:

We prove the following theorem, comparing the transitivity sequence for EN R-algebras to the transitivity sequence for EN differential graded k-algebras. ARTICLE IN PRESS

274 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Theorem 13.2. Let A-B be a map of EN R-algebras and let M be a XB-module. The natural isomorphisms of Corollary 7.9 make the following diagram commute:

Just as in Section 5, it is convenient to work in a ‘‘pointed’’ category, where the initial and final objects are isomorphic. In this case it is the category EGjB ¼ EGjidB of objects under and over B: It is clear from the construction of the transitivity sequence that to prove Theorem 13.2, we just need to see that X takes % % cofiber sequences in hðEGjBÞ to cofiber sequences in hðECjXBÞ and preserves the isomorphism in Proposition 5.6. The following two lemmas give precise statements.

- Lemma 13.3. Let f : X Y be a cofibration in EGjB and let Cf be the cofiber. Then Xf - is a cofibration in ECjXB; and the universal map CXf XCf is a quasi-isomorphism.

Proof. An easy induction argument shows that G converts categorical monomorph- isms into coﬁbrations. Since EG is left proper, the map of pushouts

CGf ¼ GY GX GB-Y X B ¼ Cf is a weak equivalence, and so the universal map CGf -GCf (induced by the map GY-GCf ) is a weak equivalence. Proposition 10.2 shows that this latter map is also cellular and identiﬁes the induced map on cellular chains as the universal map CXf -XCf : &

Passing to the homotopy category, the previous lemma in particular gives us a D % natural isomorphism RXBX XRB in hðECjXBÞ; where RB denotes the Quillen suspension functor as in Section 5. By adjunction, we obtain a natural map - % XXB XXBX in hðEGjXBÞ; where XB denotes the (Quillen) loop functor, the right adjoint to RB: On the other hand, we used a natural isomorphism SRMDRSM to construct the natural isomorphism of Andre´ –Quillen cohomology. We have the following consistency lemma.

% Lemma 13.4. Let M be an XB-module. The following diagram in hðECjXBÞ commutes:

We close with the proof of Theorem 13.2. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 275

Proof of Theorem 13.2. We can assume, without loss of generality, that A is cofibrant and the map A-B is a cofibration. Then as above, the map XA-XB is a cofibration (and XA is cofibrant). The first two squares commute by naturality. It follows from Lemma 13.3 that the following square commutes:

Combining this with Lemma 13.4 shows that the ﬁnal square commutes. &

14. Operations

In this section we compare the cohomology operations in the topological Andre´ – Quillen cohomology of EN R-algebras with those in the Andre´ –Quillen cohomology of EN k-algebras. As we explain, the local nature of the coefficients lead to characteristic classes or ‘‘constants’’ in Andre´ –Quillen and topological Andre´ – Quillen cohomology, which lead to ‘‘constant’’ operations. We see below that every operation can be written uniquely as a constant operation plus a ‘‘based’’ operation. The constants in topological Andre´ –Quillen cohomology generally differ from the constants in Andre´ –Quillen cohomology, but we show that the isomorphism in Corollary 7.9 induces an isomorphism of the based operations. The based operations include all the additive, linear, and multi-linear operations. Although we could argue in the full generality of Corollary 7.9, the motivation for the definition and the usefulness of operations is better described when we restrict to the example of main interest: where the coefficient module has only a single non-trivial homotopy group (Corollary 7.10). We begin by explaining what we mean by operations in topological Andre´ –Quillen cohomology. For the entirety of this section, let a denote a fixed commutative k- algebra (with no grading). We denote by Ha a fixed EN R-algebra with pqHa ¼ 0 for qa0 and with a fixed isomorphism p0Ha ¼ a: (Any other choice would be % isomorphic in hEG by a unique isomorphism.) For any a-module p (with no grading), we choose and fix an Ha-module Hp with pqHp ¼ 0 for qa0 and with a fixed % isomorphism p0Hp ¼ p: (Again, any other choice would be isomorphic in hMHa by a unique isomorphism.) A map of EN R-algebras A-Ha allows us to regard Hp as an A-module, and therefore allows us to define DnðA; HpÞ: Thus, it is natural to regard n % D ðÀ; HpÞ as a functor on hðEG=HaÞ: In fact, this is the correct generality, since for connective EN R-algebras A; the set of isomorphism classes of A-module structures % on Hp (isomorphism classes in hMA that map to the isomorphism class of Hp in % hMR) are in one-to-one correspondence with p0A-module structures on p: Moreover, factorizations p0A-a-End p are in one-to-one correspondence with isomorphism % % classes in hðEG=HaÞ that map to the isomorphism class of A in hEG: We therefore make the following definition. ARTICLE IN PRESS

276 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279

Deﬁnition 14.1. An operation

Dm1 ðA; HpÞÂ? Â Dmr ðA; HpÞ-DnðA; Hp0Þ

% - is a natural transformation of functors hðEG=HaÞ Set: An operation is based if it is a natural transformation of based sets.

The base point is of course the zero element in the k-module structure. We can also deﬁne additive, linear, and multi-linear operations, but these play no special role in what follows. To save space, we write that the operation displayed above is an 0 operation ‘‘from ðm1; y; mr; pÞ to ðn; p Þ’’. We deﬁne operations and based operations in the Andre´ –Quillen cohomology of EN differential graded k-algebras similarly. Since H * Ha is connective and H0Ha ¼ a; we can choose a map of EN differential graded k-algebras % XHa-a; and in hEC; this map is the unique one that on homology is inverse to the isomorphism a ¼ p0Ha-H0Ha: For each A; we can identify Hp as Rp: Then 0 any operation from ðm1; y; mr; pÞ to ðn; p Þ in the Andre´ –Quillen cohomology of EN 0 k-algebras induces an operation from ðm1; y; mr; pÞ to ðn; p Þ in the topological Andre´ –Quillen cohomology of EN R-algebras via the isomorphism of Corollary 7.9. Of course, when the operation in Andre´ –Quillen cohomology is based (or additive, linear, or multi-linear), the corresponding operation in topological Andre´ –Quillen cohomology is as well. The following is the main result of this section.

Theorem 14.2. The based operations in the Andre´–Quillen cohomology of EN differential graded k-algebras are in one-to-one correspondence with the based operations in the topological Andre´–Quillen cohomology of EN R-algebras.

To prove this, we need to understand the relationship between based operations and all operations. For this, note that the structure map e : A-Ha induces a map of topological Andre´ –Quillen cohomology en : DnðHa; Hp0Þ-DnðA; Hp0Þ: Each ele- n 0 0 ment xAD ðHa; Hp Þ induces an operation kx from ðm1; y; mr; pÞ to ðn; p Þ (for any ðm1; y; mr; pÞ) that sends every element of

Dm1 ðA; HpÞÂ? Â Dmr ðA; HpÞ to enxADnðA; Hp0Þ: We call elements of the form enx constants and the operations obtained in this way constant operations. The analogous observation and construction gives us constants and constant operations in the Andre´ –Quillen cohomology of EN differential graded k-algebras.

Proposition 14.3. Every operation in the topological Andre´–Quillen cohomology of EN R-algebras or the Andre´–Quillen cohomology of EN differential graded k-algebras can be written uniquely as a based operation plus a constant operation. ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 277

0 Proof. Let f be an operation from ðm1; y; mr; pÞ to ðn; p Þ and let x be the image of ð0; y; 0Þ in DnðHa; Hp0Þ or in Dða; p0Þ: Then by naturality, f sends ð0; y; 0Þ to enx n 0 n 0 in D ðA; Hp Þ or in D ðA; p Þ for all A: Thus, f À kx is based. &

We now turn to the proof of Theorem 14.2, which is essentially a Yoneda Lemma argument. The functor

Dm1 ðÀ; HpÞÂ? Â Dmr ðÀ; HpÞ

% on hðEG=HaÞ is a representable functor, represented by ZHaM where

M ¼ Sm1 Hp Â ? Â Smr Hp:

0 Thus, the set of operations from ðm1; y; mr; pÞ to ðn; p Þ is in one-to-one correspondence with the set

n 0 n 0 D ðZHaM; Hp ÞDD ðXZHaM; p Þ:

The following lemma picks out the based operations.

Lemma 14.4. The map

n 0 n 0 D ðZHaM\Ha; Hp Þ-D ðZHaM; Hp Þ is an injection and its image is the set of based operations.

Proof. We use the transitivity sequence (Proposition 13.1). The map

n 0 n 0 D ðZHaM; Hp Þ-D ðHa; Hp Þ in the transitivity sequence associated to Ha-ZHaRM is easily identiﬁed as the one that sends an operation f to f ð0; y; 0ÞADnðHa; Hp0Þ: By Proposition 14.3, this map is a surjection for every n; and its kernel is the set of based operations. &

At this point it is convenient to work entirely in the topological context. As in Section 12, we subscript with R or Hk for the category in which the constructions n D n are formed. The natural isomorphism DRðÀ; HpÞ DHkðHk4ðÀÞ; HpÞ allows us to associate an operation in the topological Andre´ –Quillen cohomology of EN R-algebras to each operation in the topological Andre´ –Quillen cohomology of EN Hk-algebras. The equivalence of Theorem 7.11 gives us a one-to-one correspondence between the operations for EN differential graded k-algebras and the operations for EN Hk-algebras. The second diagram in Theorem 12.4 implies that the operation of EN R-algebras associated to a given operation of EN differential graded k-algebras agrees with the operation associated to the ARTICLE IN PRESS

278 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 corresponding operation of EN Hk-algebras. Thus, Theorem 14.2 is equivalent to the following theorem.

Theorem 14.5. The based operations in the topological Andre´–Quillen cohomology of EN Hk-algebras are in one-to-one correspondence with the based operations in the topological Andre´–Quillen cohomology of EN R-algebras.

Proof. To avoid confusing notation, we assume without loss of generality, that the EN Hk-algebra model Ha is sent by the forgetful functor to the EN R-algebra model Ha; and likewise for the Ha-modules Hp: The association of an EN Hk-algebra operation to an EN R-algebra operation is the map

n 0 % 0 DHkðZHaM; Hp Þ¼hðEG;Hk=HaÞðZHaM; ZHaHp Þ - % 0 n 0 hðEG;R=HaÞðZHaM; ZHaHp Þ¼DRðZHaM; Hp Þ induced by the forgetful functor from EN Hk-algebras to EN R-algebras. It follows that the inclusion of the based operations is the map

n \ 0 % 0 DHkðZHaM Ha; Hp Þ¼hðEG;HkjHaÞðZHaM; ZHaHp Þ - % 0 n \ 0 hðEG;RjHaÞðZHaM; ZHaHp Þ¼DRðZHaM Ha; Hp Þ induced by the forgetful functor from the category of EN Hk-algebras to the category of EN R-algebras. The following proposition implies that this map is an isomorphism. &

Proposition 14.6. The forgetful functor from the category of EN Hk-algebras to the category of EN R-algebras induces an equivalence of homotopy categories % % hðEG;HkjHaÞ-hðEG;RjHaÞ:

Proof. We can assume, without loss of generality, that Ha is a commutative Hk- algebra and the map Hk-Ha is a coﬁbration. Then the forgetful functor from commutative algebras to EN algebras induces equivalences

% - % % - % hðComjHaÞ hðEG;HkjHaÞ and hðComjHaÞ hðEG;RjHaÞ making the following diagram commute: ARTICLE IN PRESS

M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 279

References

[1] M. Andre´ ,Me´ thode Simpliciale en Alge` bre Homologique et Alge` bre Commutative, in: Lecture Notes in Mathematics, Vol. 32, Springer, Berlin, 1967. [2] M. Basterra, Andre´ –Quillen cohomology of commutative S-Algebras, JPAA 144 (1999) 111–143. [3] E.H. Brown, Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965) 79–85. [4] A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, Rings, Modules, and Algebras in Stable Homotopy Theory, Amer. Math. Soc. Surveys & Monographs, Vol. 47, Amer. Math. Soc., Providence, RI, 1996 (Errata http://www.math.uchicago.edu/~mandell/ekmmerr.dvi). [5] V. Ginzburg, M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1) (1994) 203–272. [6] P.G. Goerss, M.J. Hopkins, Realizing commutative ring spectra as EN ring spectra, preprint. [7] M. Hovey, Model Categories, Amer. Math. Soc. Surveys & Monographs, Vol. 63, Amer. Math. Soc., Providence, RI, 1999. [8] I. Kriz, J.P. May, Operads, algebras, modules, and motives, Asterisque 233 (1995). [9] I. Kriz, Towers of EN ring spectra, preprint. [10] S. MacLane, Categories for the Working Mathematician, Springer, Berlin, 1971. [11] M.A. Mandell, Equivalence of simplicial localizations of closed model categories, JPAA 142 (2) (1999) 131–152. [12] M.A. Mandell, EN algebras and p-adic homotopy theory, Topology 40 (1) (2001) 43–94. [13] M.A. Mandell, Flatness for the EN tensor product, Homotopy Methods in Algebraic Topology (Boulder, CO, 1999), pp. 285–309, Contemporary Mathematics, Vol. 271, Amer. Math. Soc., Providence, RI, 2001. [14] J.P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, Princeton, NJ, 1967. [15] C.W. Rezk, Spaces of algebra structures and cohomology of operads, Thesis, M.I.T., 1996. [16] D.G. Quillen, Homotopical Algebra, in: Lecture Notes, Vol. 43, Springer, Berlin, 1967. [17] D.G. Quillen, On the (co-)homology of commutative rings, in: Applications of Categorical Algebra (Proceedings of Symposium on Pure Mathematics, Vol. XVII, New York, 1968), Amer. Math. Soc., Providence, RI, 1970, pp. 65–87. [18] S. Schwede, B. Shipley, Equivalences of monoidal model categories, preprint.