ALGEBRAIZATION OF E1 RING SPECTRA MICHAEL A. MANDELL Abstract.For a commutative ring k, the homotopy category of commutative Hk-algebras (strictly unital E1 ring spectra under the Eilenberg-Mac La* *ne spectrum Hk) is equivalent to the homotopy category of E1 differential g* *raded k-algebras. The functor from topology to algebra is a CW approximation a* *nd cellular chain functor; the inverse equivalence is constructed by Brown'* *s rep- resentability theorem. Introduction The goal of this paper is to construct a functor from the homotopy category of E1 ring spectra to some algebraic category, preserving as much structure as possible. Roughly, an E1 ring spectrum is a spectrum with a multiplication that is associative and commutative up to all higher coherence homotopies. In algebr* *a, there is an analogous notion of an E1 differential graded algebra, which is ro* *ughly a chain complex with a multiplication that is associative and commutative up to all higher coherence homotopies. In this paper, we prove the following theorem relating these notions. Theorem. (Algebraization) There is a functor A from the homotopy category of E1 ring spectra to the homotopy category of E1 differential graded algebras a* *nd a natural isomorphism of commutative rings H*(-; Z) ~=H*(A-). In order to explain the sense in which this functor preserves as much structu* *re as possible, we need to relativize over a base "S-algebra". Following [3], we * *say that an E1 ring spectrum R is a commutative S-algebra if it is an S-module [3, II.1.1], and that a commutative S-algebra A is a commutative R-algebra if there* * is a map of S-algebras R -! A. If k is a commutative ring, we can find a model for the Eilenberg-Mac Lane spectrum Hk of k that is a (cell) commutative S-algebra. The operadic smash product of an E1 ring spectrum with Hk provides a func- tor Hk ^L (-) from the homotopy category of E1 ring spectra to the homotopy category of commutative Hk-algebras, and there is a natural isomorphism of com- mutative k-algebras ss*(Hk ^L (-)) ~=H*(-; k): In particular, taking k = Z, we obtain a functor from the category of E1 ring spectra to the category of commutative HZ-algebras and a natural isomorphism ss*(HZ ^L (-)) ~= H*(-; Z). The category of commutative HZ-algebras (or its weak-unital minor variation) is philosophically the closest category to the cat* *egory of E1 ring spectra with the property that the Hurewicz homomorphism is an iso- morphism. Furthermore, the functor HZ ^L (-) preserves the E1 multiplication and all structure derived from it, for example, the primary power operations. T* *he ____________ 1991 Mathematics Subject Classification. Primary 55P42; Secondary 55P20. 1 2 MICHAEL A. MANDELL Algebraization Theorem is therefore best viewed as a corollary of following the* *orem in the special case k = Z. Main Theorem. For a commutative ring k, the homotopy category of commuta- tive Hk-algebras is equivalent to the homotopy category of E1 differential gra* *ded k-algebras. There is a natural isomorphism of commutative k-algebras between the homotopy ring of the commutative Hk-algebra and the homology ring of the corre- sponding E1 differential graded k-algebra. The bulk of the paper is devoted to the proof of the previous theorem. The fu* *nc- tor from commutative Hk-algebras to E1 differential graded k-algebras is made by constructing an appropriate kind of CW approximation functor and composing with a cellular chain functor C*. Most of the work is in describing what the ap* *pro- priate CW structure should be. In fact, we do not describe this until Section 9* *, but to make the arguments more concrete a closely related but more general structure that is easier to describe is defined in Section 2. Brown's representability th* *eorem provides the inverse equivalence. The Main Theorem asserts an equivalence on the homotopy category level, but in fact the functor from topology to algebra can be constructed on the point-set l* *evel. In other words, we construct a functor C* from the category of commutative Hk- algebras to the category of E1 differential graded k-algebras that converts we* *ak equivalences to quasi-isomorphisms and whose derived functor gives the equivale* *nce above. The point-set functor C* allows us to state a relative version of the Ma* *in Theorem. If A is a commutative Hk-algebra, we can consider the category of commutative A-algebras (the commutative Hk-algebras under A), and the category of E1 differential graded k-algebras under C*A. We prove the following relative theorem for the homotopy category of commutative A-algebras. Theorem A. Let A be a commutative Hk-algebra, cofibrant in the model struc- ture of [3, VII.4.10]. The derived functor of C* induces an equivalence between the homotopy category of commutative A-algebras and the homotopy category of E1 differential graded k-algebras under C*A. The functor C* takes its values in the category of algebras over the E1 operad C introduced in [5, V]. This category is a proper closed model category [9]. By* * [8], the following augmented version of the Main Theorem then follows from Theorem A as a categorical consequence. Alternatively, the following theorem can be proved using the same techniques developed here for the proof of the Main Theorem. Theorem B. The homotopy category of augmented commutative Hk-algebras is equivalent to the homotopy category of augmented E1 differential graded k-alge* *bras. As a final variation, we can consider connective ((-1)-connected) augmented commutative Hk-algebras. As an elementary consequence of the existence of closed model structures, the homotopy category of connective augmented commutative Hk-algebras is a full subcategory of the homotopy category of augmented commuta- tive Hk-algebras and the homotopy category of connective augmented E1 differen- tial graded k-algebras is a full subcategory of the homotopy category of augmen* *ted E1 differential graded k-algebras. As an immediate consequence of Theorem B, we obtain the following theorem. ALGEBRAIZATION OF E1 RING SPECTRA 3 Theorem C. The homotopy category of connective augmented commutative Hk- algebras is equivalent to the homotopy category of connective augmented E1 dif* *fer- ential graded k-algebras. The theory of Postnikov towers of E1 ring spectra and the study of E1 ring spectrum structures on the Brown-Peterson spectra BP in [4] required the previo* *us theorem and a heuristic argument for it was given there. For a commutative Hk-algebra A, [3] constructs a category of A-modules with a symmetric monoidal product ^A . Similarly, for an E1 differential graded k-alge* *bra E, [5] constructs a category of E-modules with a symmetric (weak) monoidal prod- uct A . For E = C*A, the theory developed in Sections 8-11 can be expanded to prove that the homotopy category of A-modules is symmetric monoidally equiv- alent to the homotopy category of E-modules. We intend to return to this in a future paper. An A1 ring spectrum algebraization theorem can also be proved, based on a noncommutative version of the Main Theorem, comparing the homotopy category of (noncommutative) Hk-algebras with the homotopy category of associative diffe* *r- ential graded k-algebras. Such a theorem can be proved with the same outline but with substantially simpler arguments than the proof of the commutative case giv* *en here. A proof along slightly different lines has appeared in [14]. A more struc* *tured theorem of a similar nature is proved in [13], where it is shown that the closed model category of Hk-algebras of Gamma rings and the closed model category of simplicial k-algebras are related by a Quillen equivalence. It is well-known that the homotopy category of associative differential grade* *d k- algebras is equivalent to the homotopy category of A1 differential graded k-alg* *ebras (cf. [5, V.1.7]). The E1 differential graded k-algebra associated to a commuta* *tive Hk-algebra A is equivalent in the homotopy category of A1 differential graded * *k- algebras to the associative differential graded k-algebra associated to A by re* *garding A as a (noncommutative) Hk-algebra. We leave the details to the interested read* *er. In this paper k denotes a fixed commutative ring. All constructions in algebra occur in the category of k-modules or differential graded k-modules, and denot* *es tensor product over k. For the operad C of differential graded k-modules, we fo* *llow the notational conventions of [9] rather than those of [5]; in particular, C de* *notes the free C-algebra functor and not the k-algebra C(1). 1. E1 Hk-Algebras As mentioned in the introduction, the functor that underlies the equivalence of the Main Theorem is the composition of a cellular chain functor with an ap- propriate CW approximation functor. Our approximations take place not in the category of commutative Hk-algebras, but in a category of E1 Hk-algebras for a topological E1 operad G closely related to the algebraic E1 operad C. We begin by introducing this operad. Definition 1.1.Let G be the E1 operad of spaces obtained from the linear isome- tries operad L [10, I.1.2] by geometric realization of the total singular compl* *ex. We use the operad G to define a category of E1 Hk-algebras, which provides a close topological analogue for the category of C-algebras. We define the monad G on the category of Hk-modules by W (n) GX = G(n)+ ^n X : n0 4 MICHAEL A. MANDELL Here X(n)denotes the n-th smash power of the Hk-module X, X ^Hk . .^.HkX, where we understand X(0)= Hk. The unit map X -! GX is induced by the inclusion of the element 1 in G(1), and the multiplication GGX -! GX is induced by the operadic multiplication of G. Definition 1.2.We define the category G of G-algebras to be the category of algebras over the monad G. In Section 12, we verify that G satisfies the conditions [3, VII.4.9] that gi* *ve the category of G-algebras a closed model structure with the usual weak equivalence* *s. In more detail, we prove the following proposition. Proposition 1.3.The category of G-algebras is a closed model category with weak equivalences and fibrations the weak equivalences and Serre fibrations of the u* *nderly- ing Hk-modules and cofibrations the retracts of relative cell inclusions [3, VI* *I.4.11]. The augmentation of G, the map that sends each space G(n) to a point, induces a natural map of Hk-modules W (n) W (n) GX = G(n)+ ^n X -! X =n = PX: n0 n0 This natural transformation provides a map of monads G -! P, and therefore induces a forgetful functor from the category of commutative Hk-algebras to the category of G-algebras. In Section 12, we prove the following theorem. Theorem 1.4. The forgetful functor from the category of commutative Hk-algebras to the category of G-algebras is the right adjoint of a Quillen equivalence, an* * adjoint pair that satisfies the hypotheses of [2, 9.7.(i-ii)] and therefore induces an * *equivalence of homotopy categories. Thus, it suffices to define the functor C* on the category of G-algebras. This actually provides a significant reduction, since we shall see that G retains mo* *st of the convenient formal properties of the operad L exploited in [3], but in addition,* * G has a CW structure: By construction, each space G(m) is a free n-CW complex, and the operadic multiplication of G is given by cellular maps. In particular, appl* *ying the cellular chain functor with coefficients in k, C*G is an operad of differen* *tial graded k-modules. Since the operad C is defined to be the operad of differenti* *al graded k-modules obtained by applying the singular chain complex functor with coefficients in k to the operad L, we obtain the following proposition. Proposition 1.5.The operad C is canonically isomorphic to the operad C*G. More generally, we have the following proposition to motivate the shift in fo* *cus from commutative Hk-algebras to G-algebras. Proposition 1.6.Let M be a CW Hk-module. The cellular chain complex of GM, C*GM, is naturally a C-algebra, canonically isomorphic to the free C-algebra on C*M, CC*M. Technically, for a CW Hk-module M, GM is not a CW Hk-module since Hk is not a CW Hk-module [3, III.2.5]. The non-unit summands of GM are all CW Hk-modules, and so we can make sense of the cellular chain complex of GM nev- ertheless. We say more about this in the next section. ALGEBRAIZATION OF E1 RING SPECTRA 5 2.CW G-Algebras and the Cellular Chain Functor In this section we describe a category of algebras on which we can apply a cellular chain functor. As we have seen in the previous section, when M is a CW Hk-module, the free G-algebra GM is not quite a CW Hk-module, but is close enough that we can make sense of its cellular chain complex. We begin by making precise the structure on an Hk-module needed to make sense of the cellular chain functor. Definition 2.1.A skeletal filtration on an Hk-module M is a filtration by cofi- brations * . . .M-n . .M.0 M1 . .M.n . . .M such that ss*(Mn =Mn-1) is concentrated in dimension n and ssm Mn = 0 for m > n. A map M -! N between skeletally filtered Hk-modules is cellular if it preserves skeletal filtrations. We say that M is an almost CW Hk-module if M ^Hk SHk is a CW Hk-module with skeletal filtration Mm ^Hk SHk , where SHk is the (cell) sphere Hk-module [3, II.2]. The cellular chain functor makes sense on any skeletally filtered Hk-module, * *and is natural in cellular maps. Definition 2.2.Define the cellular chain functor by CnM = ssn(Mn =Mn-1) with differential dn :Cn+1M -! CnM given by the composite Cn+1M = ssn+1(Mn+1=Mn ) -! ssn+1((Mn =Mn-1)) ~=ssn(Mn =Mn-1) = CnM where the map ssn+1(Mn+1=Mn ) -! ssn+1((Mn =Mn-1)) is the map in the Puppe sequence for the cofibration Mn =Mn-1 -! Mn+1=Mn-1 together with a sign determined by the convention that on Hk^I+ ^Sn, the boundary takes the standard generator of ssn+1(Hk^Sn+1) = Cn+1(Hk^I+ ^Sn) to the image of the fundamental class of ssn(Hk^Sn) ~=ssn(Hk^{1}+ ^Sn) minus the image of the fundamental class of ssn(Hk^Sn) ~=ssn(Hk^{0}+ ^Sn) in ssn(Hk^{0; 1}+ ^Sn) = Cn(Hk^I+ ^Sn). Of course the definition of C* just given makes sense for an arbitrary filtra* *tion by cofibrations, but the force of the definition of a skeletal filtration is th* *e following proposition. Proposition 2.3.For a skeletally filtered Hk-module M, there is a canonical iso- morphism ss*M ~=H*(C*M), natural in cellular maps. The usefulness of almost CW Hk-modules over general skeletally filtered Hk- modules is their closure properties given in the following proposition. Proposition 2.4.The category of almost CW Hk-modules and cellular maps has the following closure properties: (i)Hk is an almost CW Hk-module. (ii)If M is a CW Hk-module then M is an almost CW Hk-module. (iii)If M is an almost CW Hk-module and X is a based CW space or X = Z+ for an unbased CW space Z, then X ^ M is an almost CW Hk-module, and C*(X ^ M) is canonically isomorphic to "C*(X; k) C*M, where "C*denotes the reduced cellular chain complex of a based CW space. (iv)The smash product ^Hk is a symmetric monoidal functor on the category of almost CW Hk-modules and cellular maps and C* is a symmetric monoidal natural transformation, i.e. for almost CW Hk-modules M and N, M ^Hk N 6 MICHAEL A. MANDELL is naturally an almost CW Hk-module and there is an associative, commuta- tive, and unital natural isomorphism C*(M ^Hk N) ~=C*M C*N. With the sign convention we have chosen for the differential on the cellular * *chain complex, the following diagram of natural isomorphisms commutes for almost CW Hk-modules M and N. ss*M ss*N ________________________________//ss*(M ^Hk N) | | | | fflffl| fflffl| (H*C*M) (H*C*N) _____//H*(C*M C*N) oo___H*(C*(M ^Hk N)) Compare this with [6, VIII.1.3] and its proof. Using the last part of the previous proposition, we can now define the catego* *ry of CW G-algebras. Definition 2.5.A CW G-algebra is a G-algebra A with an almost CW structure on its underlying Hk-module such that the multiplication maps G(n)+ ^ A(n)-! A are cellular for n 0. A cellular map of CW G-algebras is a map of G-algebras t* *hat is a cellular map of the underlying almost CW Hk-modules. Clearly GM is a CW G-algebra when M is a CW Hk-module. These CW G- algebras are free in the following sense. Proposition 2.6.Let M be a CW Hk-module and let A be a CW G-algebra. A map GM -! A is a cellular map of CW G-algebras if and only if the restriction M -! A is a cellular map of almost CW Hk-modules. Proof.The inclusion M -! GM is a cellular map of almost CW Hk-modules, so a cellular map GM - ! A restricts to a cellular map M - ! A. Conversely, if M -! A is cellular, the induced map W (n) W (n) G(n)+ ^ M -! G(n)+ ^ A -! A n0 n0 is cellular and factors through the quotient complex GM. The map GM -! A is_ then also cellular. |__| The main motivation for the introduction of the category of CW G-algebras is the following proposition. Proposition 2.7.The cellular chain complex of a CW G-algebra is a C-algebra, naturally in cellular maps. Proof.The structure maps are the composites C(n) (C*A)(n)~=C*(G(n)+ ^ A(n)) -! C*A: That these maps satisfy the associativity, unity, and equivariance diagrams fol* *lows_ from applying C* to the corresponding diagrams for A. |__| ALGEBRAIZATION OF E1 RING SPECTRA 7 3. Proof of the Main Theorem In this section, we prove the Main Theorem, assuming a representability theo- rem proved in Section 5 and the following CW approximation theorem proved in Section 6. Theorem 3.1. (CW Approximation) There exist functors and and natural transformations , , and O satisfying the following properties. (i) is a functor from the category of G-algebras to the category of CW G-alge* *bras and cellular maps. (ii) is a functor from the category of CW G-algebras and cellular maps to itse* *lf. (iii) is a natural transformation from to the identity functor in G and is always a weak equivalence. (iv) is a natural transformation from to the identity functor in the category of CW G-algebras and cellular maps and is always a weak equivalence. (v) O is a natural transformation -! in the category of CW G-algebras and cellular maps. (vi)Viewed as maps in G , O = O and so in particular O is always a weak equivalence. The following diagram summarizes the natural weak equivalences of Theorem 3.1 for a CW G-algebra A. The solid arrows denote cellular maps of CW G-algebras; the dotted arrow denotes a map of G-algebras that may not be cellular. O A B___________//_BA_ ___ BBB _________ BB!!B""_______ A It follows in particular from the previous theorem that the forgetful functor* * from the category of CW G-algebras and cellular maps to the category of G-algebras induces an equivalence of homotopy categories. The inverse is given by the deri* *ved functor of ; here is a natural isomorphism from to the identity in the homot* *opy category of G-algebras and O O-1 is a natural isomorphism from to the identity in the homotopy category of CW G-algebras and cellular maps. Corollary 3.2.The forgetful functor induces an equivalence between the homotopy category of CW G-algebras and cellular maps and the homotopy category of G- algebras. The other result we need to assume for now is the following representability theorem, proved in Section 5. For this, let hG denote the homotopy category of * *G- algebras, and let hC denote the homotopy category of C-algebras. Since preserv* *es weak equivalences and C* converts weak equivalences to quasi-isomorphisms, we c* *an regard C* as a functor from hG to hC . For an object E of hC , we can define a functor E from hG to sets to be the mapping set E (-) = hC (C*(-); E): This is in fact a set by [9, 1.1]. Theorem 3.3. (Representability) For every object E in hC , the functor E is representable in hG . 8 MICHAEL A. MANDELL Writing (E) for the representing object, it follows from Yoneda's lemma, that (-) defines a functor from hC to hG . Then from Theorem 3.1 and Theorem 3.3, we deduce the following corollary by arguing that gives an inverse equivalence* * to the derived functor of C*. Corollary 3.4.The functor C* induces an equivalence between the homotopy category of G-algebras and the homotopy category of C-algebras. Proof.We have a natural bijection hG (A; E) ~=hC (C*A; E), and therefore an adjunction between C* and . The map on ssn induced by the unit A -! C*A can be identified with the map induced by C* hG (GSnHk; A) -! hC (C*GSnHk; C*A): For an arbitrary C-algebra E, the quasi-isomorphisms C*GSnHk-O C*GSnHk-! C*GSnHk~=Ck[n] induce a natural isomorphism between ssn(E) = hG (GSnHk; E) and Hn(E) = hC (Ck[n]; E). For E = C*A, the identity = O allows us to identify the composite isomorphism ssn(C*A) ~=Hn(C*A) ~=ssn(A) as the inverse of the isomorphism ssn :ssn(A) -! ssnA. It follows that the unit induces an isomorphism on ssn, and is therefore an isomorphism in hG . We have seen in the previous paragraph that the functor Hn(-) is naturally isomorphic to the functor ssn(-). It follows that a map f :E0- ! E is an isomor- phism in hC if and only if (f): E0- ! E is an isomorphism in hG . It follows from the adjunction that the composite of the unit and of the counit E -! C*E -! E is the identity. The counit is therefore an isomorphism in hC . We conclude_that C* and are inverse isomorphisms. |__| The Main Theorem is a direct consequence of the preceding corollary. 4. Properties of Special CW G-Algebras We have defined CW G-algebras in simple terms to specify concretely the G- algebras to which we can apply the cellular chain functor to obtain a C-algebra. The drawback is that the category of CW G-algebras is too general for us to per- form many constructions in; for example, for general CW G-algebras, we cannot identify the coproduct as a CW G-algebra. We overcome this problem by studying a class of CW G-algebras with a certain combinatorial structure defined in Sec- tion 9. These are the special CW G-algebras, and they form a full subcategory of the category of CW G-algebras and cellular maps. In this section, we describe the formal properties that special CW G-algebras have; these are sufficient to * *prove the CW Approximation Theorem (Theorem 3.1) and the Representability Theorem (Theorem 3.3) in the next two sections. We begin with an example of a special CW G-algebra. The following proposition will be immediate from the definition of a special CW G-algebra in Section 9. Proposition 4.1.If M is an arbitrary wedge sum of finite CW Hk-modules, then GM is a special CW G-algebra. ALGEBRAIZATION OF E1 RING SPECTRA 9 Much of the reason for introducing the special CW G-algebras is that we can analyze the coproduct. The following theorem is proved in Section 9. Theorem 4.2. Let A and B be special CW G-algebras. The coproduct of A and B exists in the category of CW G-algebras and cellular maps as a special CW G-alg* *ebra and its underlying G-algebra is the coproduct of the underlying G-algebras. The first part of the theorem stated in other words asserts that the coproduc* *t of G-algebras A q B can be given the structure of a special CW G-algebra such that the maps A -! A q B and B -! A q B are cellular and a G-algebra map from A q B to a CW G-algebra C is cellular if and only if the restrictions A -! C and B -! C are cellular. Because of Theorem 4.2, it now makes sense to ask how the cellular chain comp* *lex of the coproduct A q B relates to the cellular chain complexes of A and B. Reca* *ll from [5, V.3.4] that is the coproduct of C-algebras. The maps of C-algebras C*A -! C*(A q B) and C*B -! C*(A q B) induce a canonical map of C-algebras C*AC*B -! C*(AqB). We prove in Section 9 that this map is an isomorphism. Theorem 4.3. The canonical map C*A C*B -! C*(A q B) is an isomorphism. The previous identification of the cellular chain complex of a coproduct as t* *he point-set coproduct of the cellular chain complexes is indispensable in our lat* *er arguments, but we would also like to know when it represents the derived coprod- uct. In [9], we introduced the term -flat C(1)-module to mean a unital C(1)- module M with the property that for any unital C(1)-module N, the canonical map Tor*(M; N) -! H*(M N) is an isomorphism. As a consequence, when a C-algebra E is -flat, the coproduct of E with any C-algebra represents the de- rived coproduct in the homotopy category of C-algebras. In Section 9, we prove * *the following proposition. Proposition 4.4.If A is a special CW C-algebra, then C*A is -flat. The next two theorems regard the geometric realization of simplicial G-algebr* *as. It follows from [3, X.1.3-4] that the geometric realization of a simplicial G-a* *lgebra is again a G-algebra. The purpose of the next theorem is to give a sufficient cond* *ition for the geometric realization of a simplicial special CW G-algebra to be a spec* *ial CW G-algebra. For this we introduce the concept of a "split" simplicial object. Recall that for a simplicial object Ao, the "degeneracy subobject" sAn of An-1 * *is defined`to be the amalgamation of n copies of An-1 over the n(n-1) degenerate (* *in An-1) copies of An-2; sAn maps to An via the amalgamation of the degeneracy maps s0, : :,:sn-1. The definition of the degeneracy subobject makes sense for simplicial objects in any cocomplete category, and the object sAn depends on wh* *at category we regard Ao as a simplicial object in. Definition 4.5.Let C be a cocomplete category, and let Ao be a simplicial object in C . We say that Ao is split if the map sAn -! An is the inclusion of a copro* *duct- factor for every n, i.e. if there exist objects AN0; AN1; : :a:nd maps ANn-! An* * such that the induced map sAn q ANn-! An; is an isomorphism. We prove the following theorem in Section 9. 10 MICHAEL A. MANDELL Theorem 4.6. Let Ao be a split simplicial object in the category of special CW G-algebras. The geometric realization |Ao| is naturally a special CW G-algebra. Furthermore, the map A0 -! |Ao|is a cellular map and is an isomorphism of special CW G-algebras when Ao is a constant simplicial object. Recall that a simplicial differential graded k-module Eo has a normalization N*(Eo) obtained by taking the total complex of the chain complex of differential graded k-modules obtained by normalizing degreewise. When Eo is a simplicial C-algebra, the normalization obtains the structure of a C-algebra via the graded shuffle map N*(Eo) N*(Eo) -! N*(Eo Eo) [5, p. 51]. The next theorem identifies the cellular chain complex of the realization of a split simplicial * *special CW G-algebra as the normalization of its cellular chain complex. Theorem 4.7. Let Ao be a split simplicial special CW G-algebra. There is a nat- ural isomorphism C*|Ao|~=N*(C*Ao) compatible with the inclusions of C*A0. The previous theorems apply in particular to the bar construction. Given spec* *ial CW G-algebras A, B, C, and maps of special CW G-algebras A -! B, A -! C, the bar construction fi(B; A; C) is the geometric realization of the split simp* *licial special CW G-algebra fio(B; A; C), given in degree n by fin(B; A; C) = B q A_q_._.q.A-z____"qC: n factors The face maps are induced by the codiagonal maps A q A -! A, B q A -! B q B -! B, and A q C -! C q C -! C. The degeneracies are induced by the inclusions of the form X q Y -! X q A q Y for each factor of A in the coproduct. We conclude from Theorem 4.6 that fi(B; A; C) is a special CW G- algebra, and from Theorem 4.7 that C*fi(B; A; C) is the C-algebra bar construct* *ion fiC(C*B; C*A; C*C). The bar construction fi(B; A; C) is the main tool we need to build our CW approximation functor. For an arbitrary special CW G-algebra B, we can take A to be GSnHkand C to be GCSnHk, and then the bar construction gives a special CW G-algebra model for attaching a cell. With this in mind, it is clear that we need to be able to form certain sequential colimits when the maps are similar to the inclusion of B in fi(B; GSnHk; GCSnHk). For the purposes of the next theore* *m, say that a map A -! B of special CW G-algebras is a special inclusion if there * *is a special CW G-algebra C and a split simplicial special CW G-algebra Bo for which there are isomorphisms of special CW G-algebras A q C ~=B0 and |Bo|~=B, such that the map A -! B is the the composite of the map A -! B0 and the map B0 -! B. The following propositions are proved in Section 9. Proposition 4.8.Let A0 -! A1 -! . .b.e a sequence of special inclusions of spe- cial CW G-algebras. Then the colimit in the category of CW G-algebras exists, i* *s a special CW G-algebra, and has as its underlying G-algebra the colimit of the un* *der- lying G-algebras. Furthermore, the canonical map ColimC*An -! C*(Colim An) is an isomorphism. Proposition 4.9.A special inclusion is a cofibration of the underlying Hk-modul* *es. In Section 6, we construct the CW approximation functors and of Theo- rem 3.1 as a colimit of cell attachments. The following proposition will be imm* *ediate from the definition of these functors and Theorems 4.6 and 4.8. ALGEBRAIZATION OF E1 RING SPECTRA 11 Proposition 4.10.The functors and take values in the category of special CW G-algebras. To explain the relationship between cell attachment by the bar construction a* *nd cell attachment in the sense of [3, VII.4.11], we need the following propositio* *n that gives the G-algebra analogue of [3, VII.3.7] and which can be proved by entirely similar arguments. Proposition 4.11.Let A -! B and A -! C be maps of G-algebras. There is an isomorphism of G-algebras under B q C and over B qA C fi(B; A; C) ~=B qA (A G I) qA C: Here A G I is the "tensor" of the G-algebra A with the space I; this is the G- algebra that corepresents the functor U (I; G (A; -)), where U denotes the cate* *gory of unbased (compactly generated weak Hausdorff) spaces. For us the relevance of A G I is contained in the following two facts. o For an Hk-module X, (GX) I = G(X ^ I+ ). o For a cofibrant G-algebra A, A I is a Quillen cylinder object [3, VII.4.1* *6]. It follows that on the underlying G-algebras, attaching cells using the bar con* *struc- tion is the same as attaching cells as in [3, VII.4.11]. In particular, since * * and are formed by inductively attaching cells by the bar construction, the followi* *ng proposition is an immediate consequence of the construction in Section 6. Proposition 4.12.For any G-algebra A, A is a cofibrant G-algebra. For any CW G-algebra A, A is a cofibrant G-algebra. 5. Proof of The Representability Theorem In this section, we prove the Representability Theorem (Theorem 3.3) by ver- ifying Brown's axioms [1, 2.1-4, 6-7]. Let A = hG , the homotopy category of G-algebras. Write Af for the full subcategory of A consisting of the finite c* *ell G-algebras, the cell G-algebras in the sense of [3, VII.4.11] with only finitel* *y many cells. Choose A0 to be a skeleton of the category Af, a full subcategory of Af containing exactly one object of each isomorphism class. We prove the following theorem. Theorem 5.1. The pair (A ; A0) is a "homotopy category" in the sense of [1, x2* *]. For a C-algebra E, the functor E is a "homotopy functor" in the sense of [1, x2* *]. The Representability Theorem is an immediate consequence of the previous the- orem and [1, 2.8]. Note that GSnHkis an object in Af and is therefore isomorphic to an object of A0. This implies that isomorphisms in A are detected by_maps out of the objects of A0, or in the the notation of [1, x2], that A = A 0. In t* *his situation, [1, 2.8] asserts that any "homotopy functor" on A is representable. The finite cell G-algebras are those G-algebras formed from Hk by iteratively attaching a finite number of "cells" GCSnHk along maps from SnHk. It follows by induction on the number of cells that A0 is a small category. We can form the "telescopes" in A required for [1, 2.4], by modeling a sequence in A by a sequence of cofibrant G-algebras and cofibrations in G and forming the colimit. This construction satisfies [1, 2.4.i-ii] by the usual arguments. Thus, to see * *that (A ; A0) is a homotopy category, it just remains to verify that A and A0 have 12 MICHAEL A. MANDELL "homotopy pushouts" that satisfy [1, 2.3]. These can also be formed by the usual construction, which we repeat here for use in verifying [1, 2.7]. Given maps A -! X1, A -! X2 in A , we can choose cofibrant G-algebras A0, X01, X02and cofibrations (in G ), A0- ! X01and A0- ! X02such that the resulting diagram in A is isomorphic to the diagram X1 - A -! X2. Define Z to be the pushout X01qA0 X02The isomorphisms in A give us maps in A , X1 -! Z and X2 -! Z that satisfy [1, 2.3]. Note that when A, X1, and X2 are in A0, we can choose A0, X01, and X02to be finite cell G-algebras, and then Z is in Af, a* *nd so replacing it by a homotopy equivalent finite cell G-algebra if necessary, we* * can choose Z to be in A0. In order to verify [1, 2.7], we compare the construction above with the analo* *gous construction in C-algebras by comparing each to the bar construction. Consider the bar construction fi(X01; A0; X02). Regarding Z as a constant simplicial object, the maps X01-! Z and X02-! Z agree on the restriction to A and therefore induce a simplicial map from fio(X01; A0; X02) to Z. The geometric realization of this map is a weak equivalence. Proposition 5.2.Let A0, X01, X02, and Z be as above. The map fi(X01; A0; X02) -! Z is a weak equivalence. Proof.Regarding Z and Z as constant simplicial G-algebras, naturality of im- plies that the following diagram of simplicial G-algebras commutes. fio(X01; A0; X02)_____//Z | | | | fflffl| fflffl| fio(X01; A0; X02)___//_Z By Proposition 4.12 and Proposition 4.11, the left vertical arrow is a degreewi* *se weak equivalence. By Proposition 4.11, the bottom horizontal map is a weak equi* *v- alence after geometric realization. The top horizontal map is therefore a weak_ equivalence after geometric realization. |__| Now choose cofibrant C-algebra approximations B -! C*A0, Y1 -! C*X01, Y2 -! C*X02and C-algebra cofibrations B - ! Y1, B - ! Y2 such that the following diagram commutes. Y1 oo______oBo_//_____//Y2 |~| ~|| |~| fflffl| fflffl| fflffl| C*X1 oo___C*A ____//_C*X2 Let W be the pushout Y1 B Y2, and consider the canonical map W -! C*Z. Proposition 5.3.The canonical map W -! C*Z is a quasi-isomorphism. Proof.By Theorems 4.3 and 4.7, we have a canonical isomorphism fiC(C*X01; C*A0; C*X02) ~=C*fi(X01; A0; X02) ALGEBRAIZATION OF E1 RING SPECTRA 13 that makes the following diagram commute. ~= 0 0 0 fiC(Y1; B; Y2)___//fiC(C*X01; C*A0; C*X02)____//C*fi(X1; A ; X2) | | | | fflffl| fflffl| W ____________________________________________//_C*Z Since C*A0, C*X01, and C*X02are all -flat, the top horizontal map is a quasi- isomorphism. The vertical maps are quasi-isomorphisms by [9, 1.6] and the previ* *ous_ proposition. It follows that the map W -! C*Z is a quasi-isomorphism. |__| Since any maps fi:Yi- ! E in hC that restrict to the same map on B can be extended to a map W -! E, it follows that E satisfies [1, 2.7]. Finally, to prove Theorem 5.1, it remains to show that E converts arbitrary coproducts in A to products. It suffices to check that C* converts arbitrary coproducts in A to coproducts in hC . Note that the inclusions A -! A q B and B -! A q B induce a map A q B -! (A q B) that is cellular by Theorem 4.2 and a weak equivalence by Proposition 4.12. The assertion then follows for fini* *te coproducts by Theorem 4.3 and induction and for arbitrary coproducts from an easy filtered colimit argument. 6. Proof of The CW Approximation Theorem The proof of the CW Approximation Theorem (Theorem 3.1) uses the properties of special CW G-algebras listed in Section 4 to construct the functors , , and the natural transformations , , O. We begin with the construction of , . Construction 6.1. For a G-algebra A, we construct a special CW G-algebra A and a G-algebra map :A -! A inductively as follows. Starting with 0A = Hk = G* and 0: 0A -! A the unique map, suppose inductively that we have constructed a special CW G-algebra nA and a G-algebra map n :nA -! A. Let Dn = Dn(A) be the set of all diagrams SmHk___i_//CSmHk g|| |h| fflffl| fflffl| nA __n___//_A whereWg is a cellular map and m varies through all integers. Write Xn or Xn(A) * *for Dn SmHk. The maps g and h in the diagrams in Dn assemble to maps g :Xn -! nA and h: CXn -! A that induce maps g:GXn -! nA and h:GCXn -! A. Define n+1A by n+1A = fi(nA; GXn; GCXn): It follows from Proposition 4.1 and Theorems 4.2 and 4.6 that n+1A is a special CW G-algebra. Let n+1 be the geometric realization of the simplicial map from fio(nA; GXn; GCXn) to the constant simplicial object on A. Define A to be ColimnnA and : -! A to be Colimn n. It follows from Theorem 4.8 that A is a special CW G-algebra. For a map of G-algebras f :A -! A0, we can use the naturality of the sets Dn(* *A) and the naturality of the bar construction to construct inductively a special CW 14 MICHAEL A. MANDELL G-algebra map nf :nA -! nA0. Passing to the colimit gives a special CW G-algebra map f :A -! A0. The following proposition is then clear from the construction. Proposition 6.2. is a functor from the category of G-algebras to the category of special CW G-algebras, and is a natural transformation -! Idin G . As part of the proof of Theorem 3.1, we need to identify as a weak equivale* *nce. Proposition 6.3.For any G-algebra A, :A -! A is a weak equivalence. Proof.By construction, the map 1: 1A -! A is surjective on homotopy groups, so it remains to show that the map is injective on homotopy groups. Let f :SnHk-! A be a map whose composite O f is null-homotopic. By Proposi- tion 4.9, the map f factors through nA for some n. We can choose a cellular map g :SnHk-! nA homotopic to f and a map h: CSnHk-! A that provides a null homotopy for O g. Then (g; h) is an element of Dn(A), and so g is null-homoto* *pic_ in n+1A. It follows that f is null homotopic in A. |__| Construction 6.4. We construct , , and O inductively. For a CW G-algebra A, let 0A be Hk, let 0: 0A -! A be the unique map Hk -! A, and let O0: 0A -! 0A be the identity map of Hk. Having constructed a functor n and natural transformations of special CW G-algebras n :n -! Idand On :n -! n such that n O On = n, we construct n+1, n+1, and On+1 as follows. Let En = En(A) be the set of all diagrams SmHk___i_//CSmHk g|| |h| fflffl| fflffl| nA _n____//_A W where g and h are cellular maps. Write Yn(A) for EnSmHk; the maps g and h in the diagrams in En assemble to maps g :Yn -! nA and h: CYn -! A. Define n+1A by n+1A = fi(nA; GYn; GCYn): We obtain the map n+1 as the geometric realization of the simplicial map from fio(nA; GYn; GCYn) to the constant simplicial object on A. We obtain the map On+1 from the observation that the map On gives us a function from the set of diagrams En(A) to the set of diagrams Dn(A) and therefore a map Yn(A) -! Xn(A) that allows the construction of a simplicial map of special CW G-algebras fio(nA; GYn; GCYn) -! fio(nA; GXn; GCXn): Let On+1 be its geometric realization. Define A to be ColimnA, to be Colimn and O to be ColimOn. It is clear from construction that is functorial in cellular maps of CW G- algebras, and that the transformations and O are natural. The following propo- sition can be proved the same way as Proposition 6.3. Proposition 6.5.For any CW G-algebra A, :A -! A is a weak equivalence. This completes the proof of Theorem 3.1. For the proof of Theorem A, we will need the following variation of the constructions in this section. ALGEBRAIZATION OF E1 RING SPECTRA 15 Variation 6.6.Let R be a special CW G-algebra, and consider the category of G- algebras under R. We can construct a functor R from the category of G-algebras under R to the category of special CW G-algebras under R and a natural transfor- mation of G-algebras under R, R -! Id. We begin with R0= R and proceed just as in the absolute case R = Hk. Similarly, we can build a functor R from t* *he category of CW G-algebras under R to the category of special CW G-algebras under R, and natural transformations of special CW G-algebras under R, OR :R -! R and R :R -! Idwith R = R O OR by starting with R0= R and proceeding as above. The same arguments as above show that R , R , and OR are always weak equivalences. 7.Proof of Theorem A In this section we show how to adapt the arguments in the proof of the Main Theorem to prove Theorem A. For this, fix a cofibrant commutative Hk-algebra A. Then by Proposition 4.12, the map :A -! A is a weak equivalence from a cofi- brant G-algebra. Since the equivalence of Theorem 1.4 is by a Quillen equivalen* *ce, we have in addition the following theorem. Theorem 7.1. The forgetful functor from the category of commutative A-algebras to the category of G-algebras under A induces an equivalence of homotopy cate- gories. Corollary 7.2.The restriction of to a functor from the category of commuta- tive A-algebras to the category of G-algebras under A induces an equivalence of homotopy categories. Proof.The map can be regarded as a natural transformation from to the forgetful functor in the category of G-algebras under A. Since is always a we* *ak equivalence, is a natural isomorphism in the homotopy category of G-algebras_ under A. |__| For a special CW G-algebra R we introduced in the previous section the functor R from the category of G-algebras under R to the category of special CW G- algebras under R and the functor R from the category of special CW G-algebras u* *n- der R to itself. In addition, we introduced the natural weak equivalence R :R * *-! Idof G-algebras under R, and the natural weak equivalences OR :R -! R and R :R -! Idof special CW G-algebras under R, with = O O. Taking R = A, we have a natural zigzag of weak equivalences of special CW G-algebras under A. (-) - A (-) -! A (-) Applying C* gives a natural zigzag of quasi-isomorphisms of C-algebras under C*A relating C* and C*A . Thus, Theorem A follows from Corollary 7.2 and the special case R = A of the following theorem. Theorem 7.3. If R is a special CW G-algebra, then the functor C*R induces an equivalence of the homotopy category of G-algebras under R and the homotopy category of C-algebras under C*R. Write h(G \R) for the homotopy category of G-algebras under R and h(C \C*R) for the homotopy category of C-algebras under C*R. For an object E in h(C \C*R), define a functor REfrom h(G \R) to sets by RE(-) = h(C \C*R)(C*R (-); E): 16 MICHAEL A. MANDELL Just as in the absolute case, we have a representability theorem. Theorem 7.4. The functor REis representable. Theorem 7.3 follows from Theorem 7.4 and Variation 6.6 by arguments entirely similar to the arguments used in Section 3 to prove the Main Theorem from the CW Approximation Theorem and the Representability Theorem. Theorem 7.4 is proved by appealing to Brown's representability theorem, and most of the arguments given in Section 5 for the proof of the Representability Theorem carry over for the proof of Theorem 7.4. Essentially, the only thing to check is that A carries coproducts in h(G \R) to coproducts in h(C \C*R). Let X1; : :;:Xn be a finite collection of G-algebras under R. For each i, we * *can regard fio(R; R; R Xi) as a simplicial special CW G-algebra under R, where the map from R is via the inclusion of the coproduct-factor in fi0. Since the inclu* *sion is via a coproduct factor, clearly the coproduct of simplicial special CW G-alg* *ebras under R, ` So = fio(R; R; R Xi) exists, and we have a natural isomorphism ` C R C*So ~= fio (C*R; C*R; C* Xi); i2I where the coproduct is taken in the category of C-algebras under C*R. The usual arguments then compare |So|with the coproduct of the Xiin h(G \R) and N*(C*So) with the coproduct of the C*R Xi in h(C \C*R). For an infinite collection of Xff, the natural map from the coproduct of C*R * *Xff to C*R of the coproduct is a quasi-isomorphism since the restriction to the co- product of any finite subcollection is a quasi-isomorphism and the induced map * *on homology is the colimit of these restrictions. 8. The Category of G(1)-Modules In order to define special CW G-algebras in the next section, we need to in- troduce the category of G(1)-modules. This category is the topological analogue of the category of C(1)-modules, the E1 k-modules introduced in [5, V]. It has* * a symmetric weak monoidal structure for which the G-algebras are the commutative monoids. We begin with the definition of the category of G(1)-modules. For this, the only observation that needs to be made is that G(1)+ ^ (-) is a monad on the category of Hk-modules. Definition 8.1.The category of G(1)-modules, MG(1), is the category of algebras for the monad G(1)+ ^ (-). We define a weak symmetric monoidal product ^G in the category of G(1)- modules analogous to the weak symmetric monoidal product in the category of C(1)-modules. The construction begins with the observation that the operad G inherits the "split coequalizer property" of the operad L: the following diagra* *m is a split coequalizer for i,j 1 G(2) x G(1) x G(1) x G(i) x G(j)//_//_G(2) x G(i)_x_G(j)//_G(i + j); where the maps are induced by the operadic multiplication. The splitting of this diagram is obtained by applying geometric realization to the singular simplicia* *l set functor on the splitting of the analogous diagram for L in the proof of [3, I.5* *.4]. ALGEBRAIZATION OF E1 RING SPECTRA 17 The space G(2) has a left G(1)-action G(1)xG(2) -! G(2) and a right G(1)xG(1)- action G(2) x G(1) x G(1) -! G(2) induced by the operadic multiplication and which commute. Therefore, for an Hk-module Z, we may regard G(2)+ ^ Z as a G(1)-module via the left action and the right action induces a natural map of G(1)-modules (G(2) x G(1) x G(1))+ ^ Z -! G(2)+ ^ Z: In addition, for G(1)-modules M and N, there is a natural map of G(1)-modules (G(2) x G(1) x G(1))+ ^ (M ^Hk N)~=G(2)+ ^ (G(1)+ ^ M) ^Hk (G(1)+ ^ N) ! G(2)+ ^ (M ^Hk N): We use these natural maps define the functor ^G as follows. Definition 8.2.Define the bifunctor ^G :MG(1)x MG(1)-! MG(1)by the co- equalizer diagram (G(2) x G(1) x G(1))+ ^ M ^Hk_N_////_G(2)+ ^ (M ^Hk_N)//_M ^G N: Proposition 8.3.The bifunctor ^G is coherently associative and commutative. The proof is identical to the proof of the analogous assertions in [3, I.5]. Regarding Hk as a G(1)-module with trivial action, Hk provides a "weak unit" [3, II.7.1] for the product ^G. In other words, for a G(1)-module M, there is a natural map : Hk ^G M -! M such that the maps ; O o :Hk ^G Hk -! Hk coincide, where o is the natural commutativity isomorphism above. We produce by noting that the following diagram of coequalizers commutes (G(2) x G(1) x G(1))+ ^ Hk ^Hk M____////_G(2)+ ^ Hk ^Hk_M_//Hk ^G M __ | | _____ | | _____ fflffl| fflffl| fflffl___ (G(1) x G(0) x G(1))+ ^_M__________////_G(1)+ ^_M________//_M where the map G(2) -! G(1) is the operadic multiplication fl(-; *; 1). We define a functor FG :MGop(1)x MG(1)-! MG(1)as the equalizer FG(M; N) _____//FHk (G(2)+ ^ M; N)__//_//_FHk ((G(2) x G(1) x G(1))+ ^ M; N) where one map is induced by the right action of G(1) x G(1) on G(2) and the oth* *er by the composite of the isomorphism FHk ((G(2) x G(1) x G(1))+ ^ M; N) ~=FHk ((G(2) x G(1))+ ^ M; F (G(1)+ ; N)) with the map induced by the action G(1)+ ^ M -! M and the "co-action" N -! F (G(1)+ ; N), adjoint to the action map G(1)+ ^ N -! N. Chasing through the diagrams and comparing universal properties reveals that the functor FG(M; -) is right adjoint to the functor - ^G M. Recall from [3, II.7.1] the definition of a closed weak symmetric monoidal ca* *te- gory: a category with "product" and "function" functors that satisfy the condit* *ions of a closed category [7, p. 180] except that the unit map is not required to be* * an isomorphism. In this language, we summarize the previous results on ^G in the following proposition. Proposition 8.4.The product ^G is a closed weak symmetric monoidal product [3, II.7.1] on the category MG(1). 18 MICHAEL A. MANDELL In a weak symmetric monoidal category, monoids and commutative monoids [3, II.7.1] are defined by the same diagrams defining monoids and commutative monoids in symmetric monoidal categories [7, p. 166]. The following proposition* * is immediate from the definition of the product ^G. Proposition 8.5.The category of G-algebras coincides with the category of com- mutative monoids in MG(1). In a weak symmetric monoidal category, it is convenient to introduce the mixed and pointed products of [3, XIII]. The mixed product is a functor C on a G(1)- module under Hk and a G(1)-module and the pointed product ? is a bifunctor on the category of G(1)-modules under Hk. These are defined as follows. Definition 8.6.For Hk -! L, a G(1)-module under Hk and M, a G(1)-module, define L C M to be the G(1)-module that is defined by the pushout diagram to the left below. Define M B L by the symmetric diagram. For Hk -! K and Hk -! L, G(1)-modules under Hk, define K ? L to be the G(1)-module defined by the pushout diagram to the right below. Hk ^G M _____//L ^G M Hk ^G L [Hk^GHk K ^G Hk ____//_K ^G L || || [ || || fflffl| fflffl| fflffl| fflffl| M ________//L C M K [Hk L _____________//_K ? L We regard K?L as a G(1)-module under Hk via the map Hk -! K[Hk L -! K?L. The following proposition summarizes the associativity and commutativity prop- erties of C and ?. This proposition is an immediate consequence of the definiti* *ons and the commutativity, associativity, and weak unity of ^G. Proposition 8.7.The product ? is a symmetric monoidal product on the cate- gory of G(1)-modules under Hk. There is a natural commutativity isomorphism L C M ~= M B L. The products C and ? have the following additional natural associativity isomorphisms for G(1)-modules M and N, and G(1)-modules under Hk, K and L. (K ? L) C M ~= K C (L C M) (K C M) B L ~= K C (M B L) (L C M) ^G N ~= L C (M ^G N) The following proposition is now a categorical consequence of the first part * *of the previous proposition and Proposition 8.5. Proposition 8.8.The category of commutative monoids for the product ? coin- cides with the category of G-algebras, and ? gives the coproduct in the categor* *y of G-algebras. 9. Pointed Extended CW G(1)-Modules In this section we describe a combinatorial structure on G(1)-modules under H* *k. We define a "pointed extended CW G(1)-module" in terms of gluing certain "cells" along attaching maps. The importance is that the underlying G(1)-module of a special CW G-algebra has a pointed extended CW G(1)-module structure, and the behavior of this structure with respect to the ? product allows us to deduce the ALGEBRAIZATION OF E1 RING SPECTRA 19 results of Section 4. We begin with the definition of the basic building block * *of the pointed extended CW G(1)-modules, the extended G(1)-module cell. Definition 9.1.We define an extended G(1)-module cell to be a pair of G(1)- modules (H ^ Bm+; H ^ Sm-1+) where (Bm ; Sm-1 ) is the standard m-ball (space) and its boundary, and where H = G(n)+ ^n X for X some finite CW Hk-module with a cellular n-action, n 1. We regard H, H ^ Bm+, and H ^ Sm-1+as G(1)- modules via the left action of G(1) on G(n). Since G(n) is a free n CW space, t* *he underlying Hk-module of H ^ Bm+ is a CW Hk-module that contains H ^ Sm-1+ as a subcomplex. We refer to H ^ Sm-1+ as the boundary (@) of the extended G(1)-module cell. Starting with Hk and attaching cells by cellular attaching maps, we obtain pointed extended CW G(1)-modules. Definition 9.2.A pointed extended CW G(1)-module M is a G(1)-module under Hk together with the structure of an almost CW Hk-module on its underlying Hk-module and a sequential filtration M = ColimMn that satisfies the following conditions. (i)M0 = Hk (ii)Mn+1 is formed from Mn as the pushout of a wedge of extended G(1)-module cells under attaching maps from their boundaries that are G(1)-module maps and cellular maps of the underlying almost CW Hk-modules. We say that a map of G(1)-modules under Hk between pointed extended CW G(1)- modules under Hk is cellular if it is a cellular map of the underlying almost CW Hk-modules. A cellular map of pointed extended CW G(1)-modules L -! M is an inclusion of a subcomplex if it preserves the sequential filtration and if for each n, the r* *estriction Ln -! Mn sends each attaching map of each extended G(1)-module cell of Ln+1 to the attaching map of some extended CW G(1)-module cell of Mn+1 such that the restriction Ln+1 -! Mn+1 is the induced map of pushouts. We denote the category of extended CW G(1)-modules and cellular maps by pE C W . Since the underlying Hk-modules of pointed extended CW G(1)-modules come with the structure of an almost CW Hk-module, we can in particular apply the cellular chain functor. The G(1)-module structure is so tightly tied in with t* *he cellular structure that the action map G(1)+ ^ M -! M is forced to be a cellular map. The cellular chain complex then inherits the following C(1)-module structu* *re. Proposition 9.3.The functor C* lifts to a functor from the category of pointed extended CW G(1)-modules to the category of C(1)-modules under k, with C(1)- module action given by the following composite. C(1) C*M ~=C*(G(1)+ ^ M) -! C*M The C(1) structure behaves well with respect to the -product. Proposition 9.4.For any pointed extended CW G(1)-module M, C*M is -flat. Proof.Clearly C*Hk = k is -flat. The arguments of [9, x9] show that C*(H ^ Bn+_ Hk) and C*(H ^ Sn-1+_ Hk) are -flat. The proposition follows by induction up the sequential filtration. * * |___| 20 MICHAEL A. MANDELL We have introduced the pointed extended CW G(1)-modules because we can relate their ? product to the product of their cellular chain complexes. The precise relationship is stated in the following theorem, proved in Section 11. Theorem 9.5. The product ? extends to a symmetric monoidal product on the category of pointed extended CW G(1)-modules that has the following properties * *for pointed extended CW G(1)-modules M and N: (i)The natural maps M ~=M ? Hk -! M ? N and N ~=Hk ? N -! M ? N are inclusions of pointed extended CW subcomplexes. (ii)For any almost CW Hk-module Z under Hk, a map of Hk-modules M?N -! Z under Hk is cellular if and only if the composite map M _ (G(2)+ ^ M ^Hk N) _ N -! M ? N -! Z is cellular. Moreover, there is a symmetric monoidal isomorphism C*(M ? N) ~=C*M C*N that makes the following diagram commute. ~= C* M _ (G(2)+ ^ M ^Hk N _ N) _____//C*M (C(2) C*M C*N) C*N | | | | fflffl| fflffl| C*(M ? N) ____________~=___________//C*M C*N We can now define special CW G-algebras. Definition 9.6.A special CW G-algebra consists of a G-algebra A and a pointed extended CW G(1)-module structure on its underlying G(1)-module such that the multiplication A ? A -! A is a cellular map. The rest of the section is devoted to reducing the properties of special CW G- algebras described in Section 4 to properties of pointed extended CW G(1)-modul* *es. Proposition 4.1 is clear from the definition. Theorems 4.2 and 4.3 are immediate consequences of Theorem 9.5. Proposition 4.4 follows from Proposition 9.4. Theo- rem 4.6 is a corollary of the following general theorem about the geometric rea* *liza- tion of simplicial pointed extended CW G(1)-modules proved in the next section. Theorem 9.7. Let Mo be a simplicial pointed extended CW G(1)-module, each of whose degeneracy maps is the inclusion of a pointed extended CW subcomplex. Then the geometric realization of Mo is a pointed extended CW G(1)-module such that the map M0 -! |Mo|is the inclusion of a pointed extended CW subcomplex and a map from |Mo|to an an almost CW Hk-module Z is a cellular map of almost CW Hk-modules if and only if each composite Mn ^ [n]+ -! |Mo|-! Z is a cellular map of almost CW Hk-modules. Proof of Theorem 4.6.Let Ao be a split simplicial special CW G-algebra. Then by part (i) of Theorem 9.5, the degeneracy maps are inclusions of pointed extended CW subcomplexes. Theorem 9.7 then gives |Ao|the structure of a pointed extended CW G(1)-module. To see that the multiplication is cellular, it suffices to chec* *k that the isomorphism of [3, X.1.4] G(2)+ ^ |Ao|^Hk |Ao|~=|G(2)+ ^ Ao ^Hk Ao| ALGEBRAIZATION OF E1 RING SPECTRA 21 is a cellular map. This isomorphism is induced by the Milnor homeomorphism [11] ~= [m] x [n] -! |o[m] x o[n]| ; which is cellular. Finally, when Ao is the constant simplicial object on A0, th* *e map A0 -! |Ao|is an isomorphism of Hk-modules; since it is also the inclusion of a pointed extended CW subcomplex, it must be an isomorphism of pointed extended_ CW G(1)-modules, and therefore an isomorphism of special CW G-algebras. |__| Theorem 4.7 is a consequence of the following corollary of Theorem 9.7. Corollary 9.8.If Mo and No are simplicial pointed extended CW G(1)-modules, each of whose degeneracy maps is the inclusion of a pointed extended CW subcom- plex, then the natural map |Mo|? |No|-! |Mo ? No|is cellular and the composite |C*Mo| |C*Mo|~=C*(|Mo| ? |No|) -! C*|Mo ? No|~=|C*(Mo ? No)| is the shuffle map. Proof.It suffices to check that the isomorphism G(2)+ ^ |Mo|^Hk |No|~=|G(2)+ ^ Mo ^Hk No| induces the shuffle map on cellular chain complexes. This follows from the obse* *r- vation that the Milnor homeomorphism [11] ~= [m] x [n] -! |o[m] x o[n]| induces the shuffle map on cellular chain complexes. |__* *_| A special inclusion, as defined in Section 4, is in particular the inclusion * *of a pointed extended CW subcomplex; Proposition 4.8 follows from the easy observa- tion that the colimit of a sequence of inclusions of subcomplexes of pointed ex* *tended CW G(1)-modules is naturally a pointed extended CW G(1)-module. Proposi- tion 4.9 follows from the easy observation that the inclusion of a pointed exte* *nded CW subcomplex L -! M is a cofibration of the underlying Hk-modules. 10.Proof of Theorem 9.7 Let Mo be a simplicial pointed extended CW G(1)-module, each of whose degen- eracy maps is the inclusion of a subcomplex. In constructing a pointed extended CW G(1)-module structure on |Mo|, it is convenient to keep each step of the con- struction as a pointed extended CW G(1)-module. Since Mn ^ [n]+ is not a pointed extended CW G(1)-module, we define the product Mn [n]: If N is a G(1)-module under Hk and X is an unbased space, define N X by the following pushout diagram. Hk ^ X+ ____//_N ^ X+ | | | | fflffl| fflffl| Hk ________//N X When N is a pointed extended CW G(1)-module and X is a CW space, it is easy to see how to give N X the structure of a pointed extended CW G(1)-module that has one extended CW G(1)-module cell for each extended CW G(1)-module 22 MICHAEL A. MANDELL cell of N and each cell of X and for which the map N ^ X+ -! N X is cellular. Furthermore, |Mo|is isomorphic to the coend Z Mn [n]: Inductively |Mo|is formed from M0 by attaching Mn [n] along (sMn [n]) [(sMn@[n]) (Mn @[n]); where sMn is the degeneracy subobject. Since each degeneracy is the inclusion o* *f a subcomplex, the map sMn -! Mn is the inclusion of a subcomplex and therefore so is sMn [n] -! Mn [n]. The map Mn @[n] -! Mn [n], and therefore the map (sMn [n]) [(sMn@[n]) (Mn @[n]) -! Mn [n] is the inclusion of a subcomplex. We obtain a pointed extended CW G(1)-module structure on |Mo|by application of the following theorem and passage to the col* *imit. Theorem 10.1. If L -! M is the inclusion of a subcomplex of pointed extended CW G(1)-modules, and L -! N is a cellular map of pointed extended CW G(1)- modules, then M [L N can be given the structure of a pointed extended CW G(1)- module such that it contains N as a subcomplex and has the property that a map out of M [L N is cellular if and only if its restrictions to M and N are cellul* *ar. Although the theorem looks simple, the proof is slightly complicated by the f* *act that we have required the sequential filtration of a pointed extended CW G(1)- module to be sequential instead of just filtered. If we knew that the map L -! N in the above theorem preserved the sequential filtrations, the pointed extended CW G(1)-module structure on M [L N would be straightforward. To generalize to the case when this map does not preserve the sequential filtrations, we require* * the following two lemmas that are the analogues of [3, III.1.7,III.2.2]. Lemma 10.2. Let C = H ^Bm+or C = H ^Sm+, where H = G(n)+ ^n X for some finite CW Hk-module X with cellular n-action, n 1. If N1 -! N2 -! . .i.s a sequence of maps in MG(1)that are cofibrations of Hk-modules, then the natural map ColimMG(1)(C; Ni) -! MG(1)(C; ColimNi) is a bijection. Proof.Let Y = X ^ Bm+ or Y = X ^ Sm+. Since passage to orbits is a finite colimit, for any G(1)-module N, the mapping space MG(1)(C; N) is a finite limit* * of a diagram of self-maps on the set MG(1)(G(n)+ ^ Y; N). Since finite limits of s* *ets commute with filtered colimits, it suffices to prove the statement for G(n)+ ^ * *Y . Choose an isometric isomorphism f in L(n). Then right composition with f-1 induces an isomorphism L(n) -! L(1) that commutes with the left action of L(1). Applying the geometric realization to the singular simplicial sets, we obtain an isomorphism G(n) -! G(1) that commutes with the left action of G(1). In other words, G(n)+ ^ Y is isomorphic in MG(1)to G(1)+ ^ Y . Thus, it suffices to show that the natural map ColimMHk (Y; Ni) -! MHk (Y; ColimNi) is an isomorphism. The result now follows by [3, III.1.7]. |* *___| ALGEBRAIZATION OF E1 RING SPECTRA 23 Lemma 10.3. Let M and N be pointed extended CW G(1)-modules and L a sub- complex of M. Let f :L -! N be a map in MG(1). Then there exists a pointed extended CW G(1)-module M0, a subcomplex L0of M0, and an isomorphism M0 -! M in pE C W that restricts to an isomorphism L0- ! L in pE C W , such that the composite L0- ! N preserves sequential filtrations. Proof.The proof is essentially the same as that of [3, III.2.2]. We form M0 as a sequential colimit of inclusions of subcomplexes M0(n) -! M0(n+1), starting with M0(0) = Hk. By induction, suppose that we have constructed L0(n) -! M0(n) isomorphic in pE C W to Ln -! Mn satisfying the following properties: (i)The composite L0(n) -! N preserves sequential filtrations. (ii)Every extended G(1)-module cell of M0(n) is a cell of Mn (possibly attached in a different order) and vice versa. (iii)L0(n) is the subcomplex of M0(n) consisting of those extended G(1)-module cells which are in Ln. To form M0(n + 1), we add each extended G(1)-module cell in Mn in the sequential filtration level in which it occurs in M0(n), so that M0(n) is a subcomplex of M0(n + 1). In addition, for each extended G(1)-module cell (C; @C) added in Ln+1 the restriction of C - ! N factors through Nk for some k and the restriction @C - ! M0(n) factors through M0(n)l for some l; we add C in M0(n + 1) in sequential filtration level max(k; l+1). For each extended G(1)-module cell (C;* * @C) added in Mn+1 not in L, again the restriction of @C -! M0(n) factors through M0(n)lfor some l and we add C in M0(n + 1) in sequential filtration level l + 1* *. We let L0(n+1) be the subcomplex of M0(n+1) consisting of the cells from Ln+1, so * *in particular L0(n) is a subcomplex of L(n+1). The obvious map M0(n+1) -! Mn+1 is cellular and has an inverse that is also cellular, hence is an isomorphism in pE C W . Clearly this isomorphism restricts to an isomorphism L0(n + 1) -! Ln+1 in pE C W , whose composite L0(n + 1) -! N preserves the sequential filtrations. Let M0 = ColimM0(n) and let L0= ColimL0(n). It is easy to see that L0and M0_ have the required properties. |__| 11. Products of Pointed Extended CW G(1)-Modules In this section, we discuss the behavior of pointed extended CW G(1)-module structures under the ?-product. In particular we prove Theorem 9.5. We begin with some observations on the extended G(1)-module cells. For this, let H = G(nH )+ ^nH XH and I = G(nI)+ ^nI XI and let J = G(nH + nI)+ ^nH+nI (nH+nI ^nHxnI (XH ^Hk XI)) for some finite CW Hk-modules XH and XI with cellular nH and nI actions respectively, for nH ; nI > 0. Clearly, we have an isomorphism of G(1)-modules J ~=H ^G I. The following lemma shows the sense in which this is an isomorphism of CW Hk-modules. Lemma 11.1. Let H, I, and J be as above. If B is a CW space and Z is an almost CW Hk-module, an Hk-module map f :J ^B+ -! Z is cellular if and only if the composite map G(2)+ ^ H ^Hk I ^ B+ -! J ^ B+ -! Z is cellular. Proof.The "only if" part follows since the map G(2)+ ^ H ^Hk I ^ B+ -! J ^ B+ is cellular. 24 MICHAEL A. MANDELL For the "if" part, it suffices to show that the map G(nH + nI)+ ^ XH ^Hk XI ^ B+ -! Z is cellular whenever the map (G(2) x G(nH ) x G(nI))+ ^ XH ^Hk XI ^ B+ -! Z is cellular. The diagram G(2) x G(1) x G(1) x G(nH ) x G(nI)//_//_G(2) x G(nH )_x_G(nI)//G(nH + nI) is actually a split coequalizer in the category of CW spaces and cellular maps.* * It follows that the diagram (G(2) x G(1) x G(1) x G(nH ) x G(nI))+ ^ XH ^Hk XI ^ B+ || ffflffl|flffl| (G(2) x G(nH ) x G(nI))+ ^ XH ^Hk XI ^ B+ | fflffl| G(nH + nI)+ ^ XH ^Hk XI ^ B+ is a split coequalizer in the category of almost CW Hk-modules. The conclusion_ now follows. |__| The inductive arguments in the construction of the pointed extended CW struc- ture on the ? product of pointed extended CW G(1)-modules require the following proposition, describing the behavior of these products on certain colimits. Proposition 11.2.The products C and ? have the following colimit properties. (i)The product C preserves all colimits in the right variable. (ii)The products C and ? preserve sequential colimits in each variable. (iii)Let P = L [M N be a pushout. If Hk -! L is a G(1)-module under Hk and we regard P as a G(1)-module under Hk via Hk -! L -! P , then for any G(1)-module Q and any G(1)-module under Hk, Hk -! K, the following diagrams are pushouts. M ^G Q ____//_N ^G Q M B K ____//_N B K | | | | | | | | fflffl| fflffl| fflffl| fflffl| L C Q ______//P C Q L ? K ______//P ? K Proof.The first statement follows easily by observing that the pushout defining* * C commutes with all colimits in the right variable. Actually, it is not hard to a* *rgue that the functor L C (-) has a right adjoint. Similarly, the second statement follows from an easy commutation of colimits. For the third statement, define X and Y by the following pushout diagrams. M ^G Q ____//_N ^G Q M B K _____//N B K | | | | | | | | fflffl| fflffl| fflffl| fflffl| L C Q ________//X L ? K _______//_Y ALGEBRAIZATION OF E1 RING SPECTRA 25 We want to show that X ~= P C Q and Y ~= P ? K, naturally in their implicit variables. Analysis of the universal property defining X and the universal prop* *erty defining P C Q reveals that each is the colimit of the following diagram. Hk ^G QM M ^G QM ttt MMMMM qqqqq MMMMM ttt MMM qqq MMM zzttt && xxq && Q L ^G Q N ^G Q The argument for Y is slightly more complicated. Examination of the universal property of Y reveals that Y is the colimit of the following diagram. M ^G Hk? oooo | ??? oooo | ?? wwooo fflffl|?? Hk ^ Hk M M ^ K ?? r G OO " >> G ? ?? rrr | OOO "" >> ??? ?? rr | OOOO"" >> ?? ?? yyrrr fflffl| "" ''O "" >> ?? OO? Hk Hk ^G K "" L ^G Hk >> ???N ^G Hk LLL r OOO"" oo >> ??o | LLLrrr ""OOOooo | >> ooo??o | | rrLLLr ""oooOOO | >> ooo ?? | fflffl|yyL&&L""""rrrww''Ooo""oofflffl>OO|wwooo OOfflffl| K L L ^G K N N ^G K Grouping together the L, M, and N pieces, we see that the above diagram defines the same colimit as the diagram Hk ^G HkO rrr | OOOOO rrr | OOO yyrrr fflffl| O'' Hk LL Hk ^G KO P ^G Hk | LLLLrrrr OOOOOooooo| | rrrLLL oooOOO | |fflfflyyLL&&rrrwwooooO'' fflffl| K P P ^G K whose colimit is evidently P ? K. |___| We now have all the ingredients necessary for the proof of Theorem 9.5. We begin with the construction of the pointed extended CW G(1)-module structure on the ? product of two pointed extended CW G(1)-modules. Construction 11.3. Let K and L be pointed extended CW G(1)-modules. We construct a pointed extended CW Hk-module P ~=K ? L as follows. We build the sequential filtration {Pm } inductively starting with P0 = Hk. Assume by induction that we have constructed Pm , with compatible maps Ki? Lj -! Pm for all i + j m, such that Km - ! Pm and Lm - ! Pm are cellular maps of extended CW G(1)-modules and such that all composite maps G(2)+ ^ Ki^Hk Lj -! Pm (i + j m) are cellular maps. We form Pm+1 by adding the following extended G(1)-module cells. (i)For each extended G(1)-module cell (H ^ Bn+; H ^ Sn-1+) added in forming Km+1 from Km , we add a cell (H^Bn+; H^Sn-1+) along the map H^Sn-1+-! Km -! Pm , which is cellular by the inductive hypothesis. 26 MICHAEL A. MANDELL (ii)For each extended G(1)-module cell (H^Bn+; H^Sn-1+) added in forming Lm+1 from Lm , we add a cell (H ^ Bn+; H ^ Sn-1+) along the map H ^ Sn-1+-! Lm -! Pm , which is cellular by the inductive hypothesis. (iii)For each extended G(1)-module cell (H ^ Bq+; H ^ Sq-1+) added in forming * *Ki from Ki-1, and each extended G(1)-module cell (I ^ Br+; I ^ Sr-1+) added in forming Lj from Lj-1 with i + j = m + 1, i; j 6= 0, let J be as in Lemma 1* *1.1; we add a cell (J ^ Bq+r+; J ^ Sq+r-1+) along the maps J ^ Bq+^ Sr-1+~= (H ^ Bq+) ^G (I ^ Sr-1+) -! Ki? Lj-1 -! Pm J ^ Sq-1+^ Br+~= (H ^ Sq-1+) ^G (I ^ Br+) -! Ki-1? Lj -! Pm : These maps agree on the restriction to J ^ Sq-1+^ Sr-1+since by the induct* *ive hypothesis the maps Ki? Lj-1 -! Pm and Ki-1? Lj -! Pm are compatible with the map Ki-1? Lj-1 -! Pm . The standard homeomorphism of Bq+r+ with Bq+^ Br+is a cellular map Bq+r+-! Bq+^ Br+. Thus, the two maps above taken together specify a map J ^ Sq+r-1 -! Pm that is cellular by Lemma 11.1 since by the induction hypothesis the following maps are cellul* *ar: G(2)+ ^ (H ^ Bq+) ^Hk (I ^ Sr-1+)-! G(2)+ ^ Ki^Hk Lj-1 -! Pm G(2)+ ^ (H ^ Sq-1+) ^Hk (I ^ Br+)-! G(2)+ ^ Ki-1^Hk Lj -! Pm : The maps Km+1 -! Pm+1 and Lm+1 -! Pm+1 are induced by the maps Km -! Pm -! Pm+1 and Lm -! Pm -! Pm+1 and the cells added as (i) and (ii). It follows immediately that the maps Km+1 - ! Pm+1 and Lm+1 - ! Pm+1 are inclusions of subcomplexes, and the maps G(2)+ ^ Km+1 ^Hk Hk -! Pm+1 and G(2)+ ^ Hk ^Hk Lm+1 -! Pm+1 are cellular. We produce the maps Ki? Lj -! Pm+1 for i + j = m + 1 and i; j 6= 0, compatible with the maps Kk ? Ll -! Pm -! Pm+1 for k + l m as follows. Let {(Cff; @Cff)} be the collection of extended G(1)-module cells u* *sed in forming Ki from Ki-1, and {(Dff; @Dfi)} those for Lj. By Proposition 11.2, a mapWKi? Lj -! Pm+1 is uniquely specified by maps Ki-1?WLj -! Pm+1 and CffB Lj -! Pm+1 that agree on their restrictions to @CffB Lj. Again by Proposition 11.2, aWmap CffB Lj -! Pm+1 is uniquely specified by maps CffBWLj-1 -! Pm+1 and Cff^G Dfi-! Pm+1 that agree on their restrictions Cff^G @Dfi-! Pm+1 . The cells added in (iii) therefore induce a map Ki?Lj -! Pm+1 , which by construction is compatible with the map Ki? Lj-1 -! Pm+1 ; we could similarly construct a map Ki? Lj -! Pm+1 that is compatible with the map Ki-1? Lj -! Pm+1 , but by uniqueness, these two maps Ki? Lj -! Pm+1 agree. Finally, since G(2)+ ^ Ki-1^Hk Lj -! Pm and G(2)+ ^ Ki^Hk Lj-1 -! Pm and G(2)+ ^ Cff^Hk Dfi-! Pm+1 are cellular maps, it follows that the map G(2)+ ^ Ki^Hk Lj -! Pm+1 is cellular. We now compare P with K ? L. For each i + j = m, let P(i;j)denote the subcomplex of Pi+jconsisting of those cells from Ki and Lj. These subcomplexes can be built by an inductive construction like that above, but using Ki and Lj * *in place of K and L. In particular, the map Ki?Lj -! Pi+jfactors through P(i;j). I* *t is an easy inductive argument using Proposition 11.2 that this map Ki? Lj -! P(i;j) is an isomorphism. But by construction P is the colimit of the P(i;j), and by Proposition 11.2, K ? L is the colimit of Ki? Lj, so the induced map K ? L -! P is an isomorphism. ALGEBRAIZATION OF E1 RING SPECTRA 27 We have the following immediate consequence of the construction. Proposition 11.4.The maps K -! K ? L and L -! K ? L are inclusions of subcomplexes for the pointed extended CW G(1)-module structure on K ? L given by Construction 11.3. The pointed extended CW G(1)-module structure on K ? L above has the fol- lowing universal property. Proposition 11.5.With the pointed extended CW G(1)-module structure on K ?L given by Construction 11.3, a map from K ? L to an almost CW Hk-module Z is cellular if and only if the composite K _ (G(2)+ ^ K ^Hk L) _ L -! K ? L -! Z is cellular. Proof.By construction, the map K_(G(2)+ ^K^Hk L)_L -! K?L is cellular. The "if" part follows by induction for each P(i;j)using Lemma 11.1 and Proposition_* *11.2. |__| We can use this to prove the first part of Theorem 9.5. Proposition 11.6.The ? product extends to a symmetric monoidal product on the category of pointed extended CW G(1)-modules. Proof.To see that the pointed extended CW G(1)-module structure on K ? L is natural in K and L, let K - ! K0 and L -! L0 be cellular maps of pointed extended CW G(1)-modules. Then the following diagram commutes. K _ (G(2)+ ^ K ^Hk L) _ L____//K0_ (G(2)+ ^ K0^Hk L0) _ L0 | | | | fflffl| fflffl| K ? L ________________________//K0? L0 The top and right composite is cellular, and it follows from the previous propo* *sition that the map K ? L -! K0? L0 is cellular. A similar argument shows that the symmetry map K ? L ~=L ? K is cellular. The cellularity of the associativity map (K ?L)?M ~=K ?(L?M) can be deduced by showing that a map from (K ?L)?M to an almost CW Hk-module Z is cellular if and only if the composite (G(3)+ ^ K ^Hk L ^Hk M) _ K _ L _ M -! (K ? L) ? M -! Z is a cellular map. |___| The following proposition describes how this ? product behaves with respect to the cellular chain functor. Proposition 11.7.For pointed extended CW G(1)-modules K and L, there is a natural symmetric monoidal isomorphism C*K C*L ~=C*(K ? L) such that the following diagram commutes ~= C*K (G(2) C*K C*L) C*L ______//_C*(K _ (G(2)+ ^ K ^Hk L) _ L) | | | | fflffl| fflffl| C*K C*L ____________~=___________//_C*(K ? L) 28 MICHAEL A. MANDELL Proof.Rewriting slightly the definition of the product in [5, V.2], we see that maps from C*K C*L to a C(1)-module Z are in one to one correspondence with C(1)-module maps C*K (C(2) C*K C*L) C*L -! Z such that the composite maps G(2) (G(1) G(1)) (C*K C*L) _____////_G(2) C*K C*L___//Z coincide, the composite maps k_____////_C*K C*L___//Z coincide, and the diagram G(2) (C*K k k C*L) _____//G(2) C*K C*L | | | | fflffl| fflffl| C*K C*L ___________________//_Z commutes. This universal property gives a natural map C*K C*L -! C*(K ? L) that makes the required diagram commute. To check that it is an isomorphism, it suffices to check that the maps C*Ki C*Lj -! C*P(i;j) are isomorphisms where Ki, Lj, and P(i;j)are as in Construction 11.3. This is obvious for i = j = 0, and follows by inductive use of Proposition 11.2 and_its_ analogue in the category of C(1)-modules. |__| Proof of Theorem 9.5.Theorem 9.5 follows from Propositions 11.4-11.7. |__* *_| 12.Operadic Algebras and Closed Model Categories In this section, we verify that the category of G-algebras satisfies the cond* *itions of [3, VII.4.9] that construct a closed model structure, and we prove Theorem 1* *.4. We write these arguments in the more general context of categories of algebras * *(of S-modules or R-modules) over an operad (of spaces), since this is no more diffi* *cult and since we already have applications for the more general results in other pa* *pers in progress. In this section, R denotes a cofibrant commutative S-algebra, e.g. R = S or R = Hk, and we work in the category of R-modules. Let U be an operad of spaces. We denote the associated monad of R-modules U: W (n) UX = U(n)+ ^n X n0 for an R-module X, with unit induced by the inclusion of 1 in U(1) and multipli- cation induced by the multiplication of U. We call the U-algebras U-algebras and denote the category of U-algebras by U . If V is another operad of spaces, with associated monad of R-modules V, then a map of operads U -! V induces a map of monads U -! V and therefore a "forgetful" functor V -! U from the category of V-algebras to the category of U-algebras. The context of Section 1 is the sp* *ecial case U = G and V the trivial operad V(n) = *. The main results we prove in this section are the following two theorems. ALGEBRAIZATION OF E1 RING SPECTRA 29 Theorem 12.1. The monad U satisfies the conditions of [3, VII.4.9]: it is cont* *in- uous, preserves reflexive coequalizers, and satisfies the cofibration hypothesi* *s of [3, VII.4]. Theorem 12.2. Let OE: U - ! V be a map of operads of spaces. The forgetful functor V -! U is the right adjoint in an adjoint pair that satisfies the hypot* *hesis of [2, 9.7.(i)]. If in addition each space U(n) has the (non-equivariant) homot* *opy type of a CW complex and each map U(n) -! V(n) is a (non-equivariant) homotopy equivalence, then the adjoint pair also satisfies the hypothesis of [2, 9.7.(ii* *)], and therefore induces an equivalence of homotopy categories. For Theorem 12.1, it is clear that U is a continuous monad that preserves ref* *lexive coequalizers since it is the composite of such functors. For the proof that U s* *atisfies the cofibration hypothesis of [3, VII.4], we follow the outline of the proof of* * the analogous result for the monad P given in [3, VII.3]. According to [3, VII.2-3* *], since U preserves reflexive coequalizers, we can make sense of the "tensor" of a U-algebra with a space, and thereby define an internal geometric realization fo* *r the category of U-algebras. In more detail, for a simplicialRU-algebra Ao, the inte* *rnal U-algebra geometric realization |Ao|U is the coend A U [n] formed in the category of U-algebras. In other words, |Ao|U is the quotient U-algebra of the coproduct of the U-algebras An U [n] by the relations implied by the face and degeneracy operators. It is clear from this description that the internal U-alg* *ebra geometric realization has a universal property in the category of U-algebras. F* *or example, the following proposition is an immediate consequence of this universal property. Proposition 12.3.Let A -! B and A -! C be maps of U-algebras. There is a natural isomorphism of U-algebras |fiUo(B; A; C)|U ~=B qA (A U I) qA C under B q C and over B qA C. Here and elsewhere, fiUodenotes the bar construction of U-algebras fiUn(B; A; C) = B q A_q_._.q.A-z____"qC n factors with the usual face and degeneracy operators. For a simplicial U-algebra Ao, we denote the geometric realization of its und* *er- lying simplicial R-module as |Ao|. The universal property of |Ao|gives a natural map |Ao|-! |Ao|U. Proposition 12.4.The natural map |Ao|-! |Ao|U is an isomorphism. Proof.When Ao is UXo for some simplicial R-module Xo, then we have isomor- phisms |UXo| ~=U |Xo| and U |Xo|~=|UXo|U; the former by [3, X.1.3] and the latter by a comparison of universal properties* *. It is straightforward to check that the composite isomorphism is the canonical map. In general, we can write Ao as a coequalizer UUAo ____//_//_UAo_//Ao; 30 MICHAEL A. MANDELL where one map is induced by the action UAo -! Ao, and the other is the multipli- cation UU -! U. Since this is a reflexive coequalizer, it is a coequalizer both* * in the category of simplicial R-modules and in the category of simplicial U-algebras. * *By commutation of colimits, we see that each of the geometric realizations preserv* *es_ this reflexive coequalizer. |__| Proof of Theorem 12.1.It remains to check that U satisfies the cofibration hypo* *th- esis of [3, VII.4]. Since the functor U commutes with sequential colimits, the * *colimit of a sequence of U-algebras is the colimit of the underlying R-modules. Now, by the previous two propositions, it sufficesftoicheckfthatifor any U-algebra A an* *d any R-module X, the inclusion of A in fifiUo(A; UX; U*)fiis a cofibration of R-modu* *les. Since A = fiU0(A; UX; U*), it suffices to check that fiUo(A; UX; U*) is proper * *in the sense of [3, X.2.2], and for this, it is enough to check that each degeneracy m* *ap is the inclusion of a wedge summand of the underlying R-modules. Each degeneracy map can be identified as an inclusion B -! B q UX for some U-algebra B (the co- product of A with copies of UX). We check that such a map is always the inclusi* *on of a wedge summand of the underlying R-modules. By a check of universal properties, the coproduct B q X makes the following diagram a coequalizer in the category of R-modules. U(UB _ X) _____////_U(B __X)_//B q X One map is induced by the action UB -! B, and the other is the composite of the multiplication UU -! U and the map U(UB _ X) -! UU(B _ X) induced by the inclusion UB _ X -! U(B _ X). Expanding out the definition of U, we have W W (n) (m) U(UB _ X) = U(m + n)+ ^(nxm ) (UB) ^Hk X m0 n0 W W (n) (m) U(B _ X) = U(m + n)+ ^(nxm ) B ^Hk X : m0 n0 Both maps preserve the number of factors of X, sending the summands containing X(m) only to summands containing X(m). In other words, in the category of R- modules, this diagram breaks up as the direct sum of the diagrams corresponding to the different values of m. For the value m = 0, we obtain the diagram, UUB ____//_//_UB; whose coequalizer is B. |___| The remainder of this section is devoted to the proof of Theorem 12.2. Since * *the model structure on the category of U-algebras has its fibrations and weak equiv- alences created in the category of R-modules, the first part of Theorem 12.2 is clear once we see that the forgetful functor from V-algebras to U-algebras has * *a left adjoint. We use the following construction. Construction 12.5. For a U-algebra A, define A by the following coequalizer diagram in the category of V-algebras OOE_//_ VUA _V__//_VA____//A where denotes the action UA -! A and denotes the multiplication VV -! V. A straightforward check of the universal property proves the following. ALGEBRAIZATION OF E1 RING SPECTRA 31 Proposition 12.6. is left adjoint to the forgetful functor from V-algebras to U-algebras. Now assume that OE: U - ! V satisfies the hypotheses of the second part of Theorem 12.2. To check that this adjunction satisfies [2, 9.7.(ii)], we need to* * see that for every cofibrant U-algebra A and for every V-algebra A0, a map of U-alg* *ebras A -! A0is a weak equivalence if and only if the corresponding map of V-algebras A -! A0 is a weak equivalence. Since any map of U-algebras A -! A0 factors through the unit A -j!A -! A0; it suffices to show that the unit map A -! A is a weak equivalence for any cofibrant U-algebra A. Recall that the cofibrant U-algebras are exactly the U-algebras that are retr* *acts of cell U-algebras [3, 4.11]. To show that the unit map A -! A is a weak equivalence for all cofibrant U-algebras, it suffices to prove it is an equival* *ence for all cell U-algebras. The simplest cell U-algebras are the algebras UX where X i* *s a wedge of (cell) sphere R-modules, SnR. The following proposition therefore prov* *ides the simplest case of this equivalence, and can be proved by the same argument u* *sed to prove [3, III.5.1]; in the context of Section 1, it is a consequence of [3, * *III.5.1]. Proposition 12.7.If X is a CW R-module, in particular if X is a wedge of (cell) sphere R-modules, the canonical map UX -! VX is a weak equivalence. The proof of Theorem 12.2 is then completed by the following lemma. Lemma 12.8. If A is a cell U-algebra then j :A -! A is a weak equivalence. Proof.We begin with the case when A can be built in a finite number of stages. If only one stage is required then A = UX, for some X, a wedge of (cell) spheres R-modules, and the statement follows by Proposition 12.7. Now assume by induction that the lemma has been shown for all cell U-algebras that can be built in n or fewer stages, and let A be a cell U-algebra that can * *be built in n + 1 stages. Then A can be written as a pushout A = B qUX UCX where X is a wedge of (cell) sphere R-modules and B is a cell U-algebra that can be b* *uilt in n or fewer stages. By Proposition 12.3, we have an isomorphism A ~=fiU (B; UX; U*) = |fiUo(B; UX; U*)|U: Since geometric realization is given by a topological colimit, it commutes with* * the continuous left adjoint , and it suffices to show that the map fiU (B; UX; U*) -! |fiUo(B; UX; U*)| ~=fiV (B; VX; V*) is a weak equivalence. Each of the simplicial R-modules fiUo(B; UX; U*) and fiVo(B; VX; V*) is proper, so it suffices to show that each map fiUm(B; UX; U*) -! fiVm(B; VX; V*) is a weak equivalence. Writing Y for the wedge of m copies of X, fiUm(B; UX; U*) ~=B q UY: Thus, we need to see that the unit map BqUY -! (BqUY ) is a weak equivalence. But since n > 0, B q UY is a cell U-algebra that can be built in n or fewer sta* *ges, 32 MICHAEL A. MANDELL and hence by the inductive hypothesis, the map B q UY -! (B q UY ) is a weak equivalence. In the general case, A is a sequential colimit of cell U-algebras An built in finitely many stages. Likewise A is the sequential colimit of An. The maps in these sequences are cofibrations of the underlying R-modules, and so the induce* *d_ map on colimits A -! A is a weak equivalence. |__| References [1]E. H. Brown, "Abstract Homotopy Theory," Trans. Amer. Math. Soc. 119 (1965)* *, pp. 79-85. [2]W. G. Dwyer, J. Spalinski, "Homotopy Theories and Model Categories," in I. * *M. James, ed, Handbook of Algebraic Topology, Elsevier Science B.V., 1995. [3]A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, Rings, Modules, and Alg* *ebras in Stable Homotopy Theory, Amer. Math. Soc. Surveys & Monographs, vol. 47, 1996* *. Errata http://www.math.uchicago.edu/"mandell/ekmmerr.dvi. [4]I. 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Schwede, "Stable Homotopical Algebra and -Spaces," to appear in Proc. Ca* *mb. Math. Soc. [14]D. Stanley, "Closed Model Categories and Monoidal Categories," thesis, Univ* *ersity of Toronto, 1997. Department of Mathematics, M. I. T., Cambridge, MA E-mail address: mandell@math.mit.edu