RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY A. D. Elmendorf and M. A. Mandell Purdue University Calumet Indiana University May 28, 2005 Abstract. We give a new construction of the algebraic K-theory of small pe* *rmutative cat- egories that preserves multiplicative structure, and therefore allows us t* *o give a unified treat- ment of rings, modules, and algebras in both the input and output. This re* *quires us to define multiplicative structure on the category of small permutative categories. * *The framework we use is the concept of multicategory (elsewhere also called colored operad)* *, a generalization of symmetric monoidal category that precisely captures the multiplicative * *structure we have present at all stages of the construction. Our method ends up in the Hovey* *-Shipley-Smith category of symmetric spectra, with an intermediate stop at a category of * *functors out of a particular wreath product. 1. Introduction This paper offers a new treatment of multiplicative infinite loop space theo* *ry that expands and improves on the account in the literature. The motivation comes fr* *om the new tools provided by the modern categories of spectra such as those of [7] and* * [9], which provide cleaner versions of old questions as well as new ones that could not be* * asked before. We now know that any E1 ring spectrum is equivalent to a strictly commutative * *ring in these new categories of spectra. It has been known since the 1980's that the K-* *theory of a bipermutative category is an E1 ring spectrum, although there are gaps in th* *e proof in the literature which we describe below, and circumvent by our new methods. The* * next natural question, asked by Gunnar Carlsson, is: What structure on a permutative* * category makes its K-theory into a module over this commutative ring? We give a full ans* *wer to _____________ 1991 Mathematics Subject Classification. Primary 19D23; Secondary 55P43, 18D* *10. Key words and phrases. K-theory, permutative category, symmetric spectra, E1* * ring spectra. The second author was supported in part by NSF grant DMS-0203980 Typeset by AM S-* *TEX 1 2 A. D. ELMENDORF AND M. A. MANDELL this question, as well as corresponding ones about rings, modules, and algebras* * of all sorts in the context of permutative categories and their K-theory spectra. Our treatment of multiplicative structures relies on the concept of multicat* *egory, which is an old, familiar friend to category theorists and computer scientists, but m* *ay be foreign to topologists and K-theorists. It was introduced by Lambek in 1969 in [12], altho* *ugh without the symmetric group actions we require, and also by Boardman and Vogt in their * *1973 book [1] under the name "colored operad." A multicategory is a simultaneous generali* *zation of an operad and a symmetric monoidal category, and can be thought of as an "opera* *d with many objects" in precisely the same way that a category can be thought of as a * *"monoid with many objects." Indeed, an operad is precisely a multicategory with one ob* *ject. Any symmetric monoidal category has an underlying multicategory (more accuratel* *y, one for each choice of associating sums, all of which are canonically isomorphic), * *but there are many other multicategories besides these. In particular, restricting to a * *subclass of objects in a multicategory again results in a multicategory, in contrast to wha* *t happens with a symmetric monoidal category. The natural structure-preserving maps betw* *een multicategories are called multifunctors. Every multicategory has an underlying* * category, and a multifunctor gives a functor between underlying categories. Just as it is often fruitful to consider categories enriched over a symmetri* *c monoidal category other than sets, so too with multicategories. The multicategories we s* *tudy are all enriched over either small categories or simplicial sets, and these enrichments* * play a crucial role in our theory. If a multicategory is enriched over small categories, we al* *so consider it as enriched over simplicial sets via the nerve construction with no further com* *ment. Our use of multicategories in this paper is structural: we construct a multi* *category P enriched over small categories whose objects are the small permutative categori* *es - we could do so more generally for symmetric monoidal categories, but to no additio* *nal advan- tage. We give a new construction of the K-theory of a small permutative categor* *y which gives us an enriched multifunctor from P to the symmetric monoidal category of * *symmetric spectra constructed in [9]. The proof of the following theorem occupies Section* *s 3-7. Theorem 1.1. The category of small permutative categories forms a multicategory* * P which is enriched over the category of small categories. There is a multifunct* *or K from P to symmetric spectra, weakly equivalent to the usual K-theory functor, respec* *ting the enrichment over simplicial sets. As a consequence of this theorem, any structure on small permutative categor* *ies cap- tured by a map out of a "parameter" multicategory passes directly to K-theory s* *pectra. In the case of ring structures, the parameter multicategories have only one obj* *ect, i.e., they are operads. We define ring structures on permutative categories in Section 3 in terms of* * a second monoidal product and distributivity maps that satisfy certain coherence relatio* *ns. The noncommutative version we call "ring" categories, and the E1 version we call b* *ipermuta- tive categories. The second of these is the generalization for lax morphisms of* * the usual RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 3 definition (for example, in May [17]); see the discussion preceding Definition * *3.6, below. We prove the following theorem in Section 8, where we interpret these structure* *s in terms of operads. Theorem 1.2. There is an operad * for which a ring structure (Definition 3.3) * *on a small permutative category A determines and is determined by a multifunctor * ** ! P sending the single object of * to A. There is an E1 operad E * for which a bi* *permutative structure (Definition 3.6) on a small permutative category R determines and is * *determined by a multifunctor E * ! P sending the single object of E * to R. We will see that, as an immediate consequence of these two theorems, our K-t* *heory functor sends ring categories to ring symmetric spectra and bipermutative categ* *ories to E1 ring symmetric spectra. In Section 9, we prove analogous theorems that give par* *ameter multicategory interpretations of various types of module structures, defined in* * terms of a pairing of a ring or bipermutative category with a small permutative category, * *and also algebra structures, defined in terms of certain maps from a bipermutative categ* *ory to a ring category. Again, as immediate consequences of Theorem 1.1, all such ring, * *module, and algebra structures pass via K-theory to the corresponding structures in the* * category of symmetric spectra. Since we wish our output structures to be as rigid as possible, we prove a t* *heorem comparing E1 versions of rings, modules, and algebras with their strictly comm* *utative analogues. We do this by studying model category structures on categories of mu* *ltifunctors into the category S of symmetric spectra. We prove the following theorem in Sec* *tion 11. Theorem 1.3. Suppose M is a small multicategory enriched over simplicial sets, * *and let SM be the category of multifunctors from M to the category S of symmetric spec* *tra. There is a simplicial model structure on SM whose weak equivalences are the objectwi* *se stable equivalences and whose fibrations are the objectwise positive stable fibrations* * of symmetric spectra. The map of operads from the E1 operad E * describing bipermutative categorie* *s to the one point operad describing commutative monoids or commutative ring symmetric s* *pectra is an example of a "weak equivalence" of multicategories, as is the multifuncto* *r from the multicategory describing modules over E * algebras to the multicategory des* *cribing modules over a commutative monoid; see Definition 12.1 for the general definiti* *on of weak equivalence of multicategories. We prove the following theorem in Section 12. Theorem 1.4. Let M and M0 be small multicategories enriched over simplicial0se* *ts. If f: M ! M0 is a simplicial multifunctor, then the induced functor f* : SM ! S* *M is the right adjoint in a Quillen adjunction. If in addition f is a weak equivale* *nce, then the Quillen adjunction is a Quillen equivalence and therefore induces an equiva* *lence on homotopy categories. 4 A. D. ELMENDORF AND M. A. MANDELL As a corollary of this general rectification result, we conclude that any E1* * ring in symmetric spectra is equivalent to a strictly commutative ring spectrum (as was* * already well-known), but also that any E1 module over an E1 ring is equivalent to a s* *trict module over an equivalent commutative ring, as well as a wide range of similar results* * for many other structures. The need to use a multicategory structure on small permutative categories ra* *ther than a symmetric monoidal structure seems intrinsic: contrary to Thomason's claim in t* *he intro- duction to [22], small permutative categories appear not to support a symmetric* * monoidal structure consistent with a reasonable notion of multiplicative structure. We w* *ill explain in a later paper how this problem can be resolved by embedding into a larger sy* *mmet- ric monoidal category (whose objects are, ironically, multicategories), but the* * necessary complications are irrelevant to the present paper. On a technical note, our construction of the K-theory multifunctor is actual* *ly a two step process, with an intermediate stop at a new multicategory of functors out of a * *particular wreath product category. They are described in Section 5. Historically, the question of what additional structure to impose on a permu* *tative, or more generally a symmetric monoidal category in order to give its K-theory some* * sort of ring structure was first investigated by Peter May in [17]. He defined bipe* *rmutative categories, and offered a proof that their K-theory spectra are E1 ring spectr* *a. However, this argument contained a serious combinatorial error (found by Steinberger), a* *s explained in Appendix A of [20]. This led May to write [20], whose main results are entir* *ely correct, but its argument contains a further combinatorial error in [20], Section 7. Uwe* * Hommel developed a patch for this error (unpublished). Gerry Dunn also found an error* * in the category theory in Section 4 of [20], which he described and attempted to patch* * in [4], but there is a critical error in [4], Section 2 (the evaluation , of Lemma 2.2(* *ii) is not well- defined). The categorical error in [20] can apparently be fixed by making a cor* *rection to the left adjoints, although a detailed check has yet to be made. One benefit of* * the current paper is to give a new proof of this theorem. Since there were no reasonable co* *ncepts of module and algebra spectra available at the time [20] was written, the question* * of which permutative categories give rise to these sorts of K-theory spectra was not ful* *ly addressed, although [19] gives a start in this direction. In an appendix, May compared th* *e theory developed in [20] to the more combinatorial theory of structured ring spectra d* *eveloped by Woolfson ([23] and [24]). The paper is organized as follows: Section 2 contains a precise definition o* *f multicategory and a description of types of parameter multicategories giving ring, module, an* *d algebra structures. Section 3 constructs the multicategory structure on the category o* *f small permutative categories and describes our results on ring structure in greater d* *etail. In Section 4, we recall the construction of the K-theory of a permutative category* * in the literature, give our new construction as a functor (as opposed to a multifuncto* *r), and prove that our construction is equivalent to the old one. Section 5 is devoted to the* * description of a particular wreath product category we call G, and the multicategory struct* *ure on RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 5 the category of functors that is the intermediate stop in our construction. Se* *ction 6 constructs the multifunctor from permutative categories to this multicategory o* *f what we call G*-categories, and Section 7 constructs the multifunctor from G*-catego* *ries to symmetric spectra; the composite of these two is our K-theory multifunctor. Se* *ction 8 proves Theorem 1.2, describing ring categories and bipermutative categories in * *terms of actions of the operads * and E *. Section 9 describes the various sorts of mo* *dules and algebras in permutative categories in terms of parameter multicategories. S* *ection 10 describes various ways in which free permutative categories can have ring or bi* *permutative structure. Finally, Sections 11 and 12 contain the proofs of our model categor* *y results, Theorems 1.3 and 1.4. We would like to thank the referees for a very thorough report that resulted* * in a signif- icant improvement in content; the use of G*-categories is inspired by a suggest* *ion in their report. The first author would also like to thank Gunnar Carlsson for asking so* *me very interesting questions, and Peter May for both encouragement and criticism. 2. Multicategories Definition 2.1. A multicategory M consists of the following: (1) A collection of objects (which may form a proper class) (2) For each k 0, k-tuple of objects (a1, . .,.ak) (the "source") and sing* *le object b (the "target"), a set Mk(a1, . .,.ak; b) (the "k-morphisms") (3) A right action of k on the collection of all k-morphisms, where for oe * *2 k, oe* : Mk(a1, . .,.ak; b) ! Mk(aoe(1), . .,.aoe(k); b) (4) A distinguished "unit" element 1a 2 M1(a; a) for each object a, and (5) A composition "multiproduct" :Mn(b1, . .,.bn; c) x Mk1(a11, . .,.a1k1; b1) x . .x.Mkn(an1, . .,.ankn* *; bn) - ! Mk1+...+kn(a11, . .,.ankn; c). subject to the identities for an operad listed on pages 1-2 in [15], which stil* *l make perfect sense in this context. In greater detail, we require the diagrams (1)-(4) below* * to commute for all nonnegative integers k, js for 1 s k, and isq for 1 q js, and a* *ll objects d, cs for 1 s k, bsq for 1 s k and 1 q js,Pand asqpfor 1 P s k, 1 * *q js,Pand 1 p isq. In these diagrams, we write is for jsq=1isq, i for ks=1is, and* * j for ks=1js, and to compress the diagrams to fit on the page, we write lists like c1, . .,.c* *k as or as 6 A. D. ELMENDORF AND M. A. MANDELL ks=1when the index is ambiguous. (1) We require the following multiassociativity diagram to commute. Qk i j Mk(; d) x Mis(<psq=1>qs=1; c* *s) 55 s=1 2 idxjjjjj 2 jjjj 22 k _ js 22! M (; d) x Q M (js ; c ) x Q M (isq; b )222 k js sq q=1 s isq sqp p=1 sq 2 s=1 q=1 22 | 22 || ssss2 ~=| Mi(<<isq>js >k ; * *d). | p=1Eq=1Es=1 | ffff fflffl| ffff Qk j Qk Qjs i ffff Mk(; d) x Mjs(qs=1; cs) x Misq(psq=1;fbsq)fffff s=1 s=1q=1T ffff TTTTT ffffff x1TTT)) ff Qk Qjs i Mj(<jsq=1>ks=1; d) x Misq(ps* *q=1; bsq) s=1q=1 (2) We require the following unit diagrams to commute: ~= ~= Mk(; d) x {1}k____//_Mk(;8d),8{1} x Mk(; d)__//Mk(; d). ppp nnn77n idx1k|| ppppp 1xid|| nnnnn fflffl|ppp fflffl|nnn Qk M1(d; d) x Mk(; d) Mk(; d) x M1(cs; cs) s=1 (3) Given oe 2 k, we require the following equivariance diagram to commute: Qk j j Mk(; d) x Mjs(qs=1; cs)__________//_Mj(<qs=1>ks=1; d) s=1 | | oe*xoe-1|| |(oe)* |fflffl fflffl|| Qk j j Mk(ks=1; d) x Mjoe(s)(qoe(s)=1;/coe(s))/_Mj(<qo* *e(s)=1>ks=1; d), s=1 where oe permutes blocks as indicated. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 7 (4) Given os 2 js for 1 s k, we require the following equivariance diag* *ram to commute: Qk j j Mk(; d) x Mjs(qs=1;_cs)____//Mj(<qs=1>ks=1; d) s=1 | idxQ o*s|| |o*1|... o*k fflffl| fflffl|| Qk j j Mk(; d) x Mjs(qs=1;_cs)_//Mj(<qs=1>ks=1; d). s=1 This concludes the definition of a multicategory. However, we may also ask tha* *t the k- morphisms Mk(a1, . .,.ak; b) take values in a symmetric monoidal category other* * than sets; the examples we are interested in take values in either categories or sim* *plicial sets. This gives the concept of an enriched multicategory. Note that a multicategory * *enriched over small categories can be considered enriched over simplicial sets by applyi* *ng the nerve functor to the k-morphisms, since the nerve functor preserves Cartesian product* *s. Definition 2.2. For multicategories M and M0 , a multifunctor from M to M0 cons* *ists of a function f from the objects of M to the objects of M0 , and for all object* *s b and k-tuples of objects a1, . .,.ak, a function Mk(a1, . .,.ak; b) ! M0k(f(a1), . .* *,.f(ak); f(b)) which preserves the oek action on the collection of all k-morphisms, preserves * *the units, and preserves the multiproduct. When M and M0 are enriched over simplicial se* *ts or small categories, the multifunctor is enriched when the maps on k-morphisms pre* *serve the enrichment; in this context, "multifunctor" always means enriched multifunctor. Example. In any symmetric monoidal category (M, , 0), we can define k-morphism* *s as Mk(a1, . .,.ak; b) := M(a1 . . .ak, b), with the sums associated in any fixed* * order. Example. An operad is simply a multicategory with one object. Remark. If we restrict our attention just to the objects and 1-morphisms of a m* *ulticat- egory, we get a category. A major theme of this paper is that rings, modules, and algebras can be desc* *ribed in any multicategory, and as we shall see in Section 8, the enrichments present in our* * examples of interest allow for E1 versions of these concepts as well. These are all de* *scribed by means of maps out of what we call parameter multicategories, which are simply s* *pecific, very small examples of multicategories. Since our construction of the K-theory * *of a small permutative category is a multifunctor, it follows automatically that ring, mod* *ule, and algebra structures on small permutative categories are preserved in their K-the* *ory spectra. We turn next to descriptions of our basic classes of parameter multicategories. Definition 2.3. Let O be an operad (a multicategory with only one object) and Q* * a multicategory. An O-ring in Q is a multifunctor from O to Q. Usually we speak o* *f the 8 A. D. ELMENDORF AND M. A. MANDELL target object in Q as being the ring. If the morphism spaces of O are all contr* *actible, then we say that the target object is an E1 ring. It is commonplace to mention that in any symmetric monoidal category, the ob* *jects have endomorphism operads: Given an object X in a symmetric monoidal category C* *, the endomorphism operad EX consists of the sets EX (n) := C(X n , X). However, this* * is just a special case of the observation that in any multicategory, restricting attent* *ion to one object gives a multicategory which, having only one object, is an operad. It i* *s natural to call this operad the endomorphism operad of the object, and the previous def* *inition amounts to specifying a map of operads from O to the endomorphism operad of the* * target object. As an example of an O-ring, if O is the final operad with Ok = * for all k, * *then an O- ring in a symmetric monoidal category is simply a commutative monoid in that ca* *tegory. In particular, if the target category is abelian groups under tensor product, a* *n O-ring is simply a commutative ring. As another example, if O = * is the "associative" * *operad with Ok = k (described in greater detail after the statement of Theorem 3.4), * *then an O-ring in a symmetric monoidal category is a monoid in the underlying monoidal * *category. In the case of abelian groups, we get a ring. We also define parameter multicategories for modules. Definition 2.4. Let M be a multicategory with two objects, R (the "ring") and M* * (the "module"). We say that M is a parameter multicategory for modules if the only nonempty morphism sets are Mk(Rk; R) and Mk(Rj-1, M, Rk-j; M) for 1 j k. If all the nonempty morphism spaces are contractible, then we say that M is a para* *meter multicategory for E1 modules. In the special case where the nonempty morphism sets consist of a single poi* *nt, we find that a multifunctor into a symmetric monoidal category consists of a commutativ* *e monoid (the image of R) and an action of that monoid on another object (the image of M* *). In the special case of abelian groups, we get a commutative ring and a module over* * it. As another example, if O is an operad, we can let Mk(B1, . .,.Bk; C) = Ok wh* *enever it is not required to be empty. This recovers the notion of O-module defined by* * Ginzburg and Kapranov in [8] and discussed by Kriz and May in Section I.4 of [11]. In pa* *rticular, if O = *, we get a monoid and a "bimodule" which has commuting left and right act* *ions. We describe algebra structures, further examples of module structures, and t* *heir appli- cations to permutative categories in Section 9. 3. The Multicategory of Permutative Categories In this section we describe the multicategory of permutative categories. We * *begin by recalling the definition of permutative category. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 9 Definition 3.1. A permutative category is a category C with a functor : C x C * *! C, an object 0 2 Ob (C), and a natural isomorphism fl: a b ~=b a satisfying: (1) (a b) c = a (b c) (strict associativity), (2) a 0 = a = 0 a (strict unit), and (3) The following three diagrams must commute: fl = a 0______~=_____//E0 a a b_________________//Ha;; b EEE yyyy HHH~=H ~=vvvv = EEE yy= flHHH vvflv E"" __yy ##H vv a, b a, fl a b Nc___________________//_Nc88pa b NNNN ppppp 1 flNNNN&&N ppppflp1 a c b. A permutative category is small if its underlying category is small. Any symmetric monoidal category is naturally equivalent to a permutative cat* *egory by a well-known theorem of Isbell [10]. We also have the following examples of* * small permutative categories from K-theory. Examples. Let A be a ring and let GL A be the category whose objects are the st* *an- dard free modules An and whose morphisms are the (left) A-module isomorphisms. * *Direct sum and the usual symmetry isomorphism makes GL A into a small permutative cate* *gory, whose K-theory is the "free module" algebraic K-theory of A. Let PrA be the fol* *lowing category. An object is a pair (An, i) where i: An ! An is an idempotent left A* *-module endomorphism. A map from (Am , i) to (An, j) is a left A-module isomorphism fro* *m Im(i) to Im(j). Again, direct sum (of modules and idempotents) and the usual symmetry* * iso- morphism makes PrA a small permutative category. The K-theory of PrA is the alg* *ebraic K-theory of the ring A. The functor GL A ! PrA that sends An to (An, id) induce* *s a map on K-theory that is an isomorphism on homotopy groups in all degrees except (po* *ssibly) degree zero. Before giving the full definition of the multicategory P of permutative cate* *gories, it is helpful to first describe the category formed by the 1-morphisms of P. We call * *these lax morphisms, although they are not as lax as they could be: we require them to s* *trictly preserve the 0-objects. Specifically, a lax map f : C ! D of permutative categ* *ories is a functor for which f(0) = 0, together with a natural transformation ~ : f(a) f(b) ! f(a b). 10 A. D. ELMENDORF AND M. A. MANDELL We require ~ = idwhen either a or b are 0, and for ~ to be associative and to r* *espect the commutativity isomorphisms, in the sense that the following diagrams must commu* *te: f(a) f(b) f(c)1_~//_f(a) f(b c) f(a) f(b)__~__//_f(a b) ~ 1 || ~|| fl|| |f(fl)| fflffl| fflffl| fflffl| fflffl| f(a b) f(c)__~____//f(a b c), f(b) f(a)_~__//_f(b a). Setting P(C, D) to be the set of lax maps from C to D, the obvious composition * *then makes P into a category. We then enrich P over small categories (thereby making it a 2-category) as f* *ollows. A transformation of lax functors is a natural transformation that also commutes w* *ith ~, in the sense that if we have the natural transformation OE : f ! g, then f(a) f(b)_~__//f(a b) OE OE|| |OE| fflffl| fflffl| g(a) g(b)_~__//_g(a b) must commute. We also require that OE(0) = id0. With these transformations as* * mor- phisms, each P(C, D) becomes a small category, and P becomes enriched over smal* *l cate- gories. The following definition generalizes the discussion above from 1-morphisms t* *o k-morph- isms for any k 0, making P a multicategory enriched over categories. Definition 3.2. Let C1, . .,.Ck and D be small permutative categories. We defin* *e cate- gories Pk(C1, . .,.Ck; D) that provide the categories of k-morphisms for the mu* *lticategory P of permutative categories as follows. The objects of Pk(C1, . .,.Ck; D) consi* *st of functors f: C1 x . .x.Ck ! D which we think of as k-linear maps, satisfying f(c1, . .,.ck) = 0 if any of the* * ci are 0, together with natural transformations for 1 i k, which we think of as distr* *ibutivity maps, ffii: f(c1, . .,.ci, . .,.ck) f(c1, . .,.c0i, . .,.ck) ! f(c1, . .,.c* *i c0i, . .,.ck). We conventionally suppress the variables that do not change, writing ffii: f(ci) f(c0i) ! f(ci c0i). RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 11 We require ffii = idif either ci or c0iis 0, or if any of the other cj's are 0.* * These natural transformations are subject to the commutativity of the following diagrams: f(ci) f(c0i) f(c00i)1/ffii/_f(ci) f(c0i fc00i)(ci) _f(c0i)ffii//_f(c* *i c0i) ffii|1| ffii|| fl~=|| ~=f(fl)|| fflffl| fflffl| fflffl| fflffl| f(ci c0i) f(c00i)ffii//_f(ci c0i c00i),f(c0i) f(ci)ffii//_f(c0i c* *i), and for i 6= j, f(ci c0i, cj) f(ci c0i, c0j) i44i ? ffii ffiiiiii ?? iiiii ?? iiiii ?? 0 0 0 0 ??ffij? f(ci, cj) f(ci, cj) f(ci, cj) f(ci, cj) ?? | ?? | ?? | ?? | ?OO 1 fl 1~=| f(ci c0i, cj c0j). | "?? | "" | "" fflffl| """ f(ci, cj) f(ci, c0j) f(c0i, cj) f(c0i, c0j) """ UUUU """ ffii UUUUU "" ffijUffijU**UUUUU """ f(ci, cj c0j) f(c0i, cj c0j) This completes the definition of the objects of Pk(C1, . .,.Ck; D). To specify * *its morphisms, given two objects f and g, a morphism OE: f ! g is a natural transformation com* *muting with all the ffii's, in the sense that all the diagrams ffifi f(ci) f(c0i)___//f(ci c0i) OE OE|| |OE| fflffl| fflffl| g(ci) g(c0i)ffig//_g(ci c0i) i commute. We also require that OE(c1, . .,.ck) = id0whenever any of the ci= 0. In order to make the Pk(C1, . .,.Ck; D)'s the k-morphisms of a multicategory* *, we must specify a k action and a multiproduct. The k action oe*f: Coe(1)x . .x.Coe(k)! D 12 A. D. ELMENDORF AND M. A. MANDELL is specified by oe*f(coe(1), . .,.coe(k)) = f(c1, . .,.ck), with the structure maps ffii inherited from f (with the appropriate permutation* * of the indices). We define the multiproduct as follows: Given fj: Cj1 x . .x.Cjkj ! * *Dj for 1 j n and g: D1 x . .x.Dn ! E, we define (g; f1, . .,.fn) := g O (f1 x . .x.fn). To specify the structure maps, suppose k1 + . .+.kj-1 < s k1 + . .+.kj, and l* *et i = s - (k1 + . .+.kj-1). Then ffis is given by the composite ffigj g(ffifji) g(fj(cji)) g(fj(c0ji))_//g(fj(cji) fj(c0ji))//_g(fj(cji c0ji)). The authors have checked that these definitions satisfy the required properties* * of the structure maps ffis, and the diligent reader will do so as well; the pentagonal* * diagram for the last structure map has two cases. These definitions extend easily to morphi* *sms, and we leave to the reader the straightforward task of checking that the necessary * *identities for a multicategory are satisfied. Variant. A strong map of permutative categories is a lax map for which the natu* *ral transformation ~ is a natural isomorphism. When we require the distributivity * *trans- formations ffii of the previous definition to be isomorphisms, we obtain a mult* *icategory structure whose underlying category is the category of strong maps of small per* *mutative categories. In the rest of this section, we describe the analogues of rings and commutat* *ive rings that appear to be most useful in the context of permutative categories, and giv* *e some ex- amples. We begin with the definition of ring category. This is the analogue in * *permutative categories of a ring with unit. Definition 3.3. A ring category is a permutative category A together with a str* *ictly associative 2-morphism : A x A ! A in the multicategory P, and a strict unit o* *bject 1. We think of the structure maps of the 2-morphism as natural distributivity maps dl: (a b) (a0 b) ! (a a0) b and dr: (a b) (a b0) ! a (b b0). Explicitly, we require that 1 a = a = a 1 for any object a of A and that th* *e following diagrams commute. Here (a)-(c) and (f) express the bilinearity of the distribut* *ivity maps (i.e., that they give the structure of a 2-morphism), while (d) and (e) expre* *ss the precise notion of associativity we require in this context: (a) a 0 = 0 a = 0 for all a. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 13 (b) The following diagram commutes, as does an analogous one for dr: (a b) (a0 b) (a00 _b)dl/1/_((a a0) b) (a00 b) 1 dl|| dl|| fflffl| fflffl| (a b) ((a0 a00) _b)_dl___//_(a a0 a00) b. (c) The following diagram commutes, as does an analogous one for dr: (a b) (a0 b)_dl_//_(a a0) b fl || |fl| 1 fflffl| fflffl| (a0 b) (a b)_dl_//(a0 a) b. (d) The following diagram commutes, as does an analogous one for dr: (a b c) (a0T b c) | TTTTTTdlT dl| TTT fflffl| TTT**T ((a b) (a0 b)) _cdl_1//_(a a0) b c (e) The following diagram commutes: (a b c) (a b0 c)dl_//((a b) (a b0)) c dr|| |dr|1 fflffl| fflffl| a ((b c) (b0 c))1_dl___//a (b b0) c. (f)The following diagram commutes: (a (b 44b0)) @ (a0 (b b0)) dr driiiiiii @@ iiiii @@ 0 i0 0 0 @@@dl (a b) (a b ) (a b) (a b ) @@ | @@ | @@ | @OO 1 fl 1| (a a0) (b b0). | "?? | """ fflffl| "" (a b) (a0 b) (a b0) (a0 b0) """ UUU "" dr UUUUU "" dl dlUUUU**U "" ((a a0) b) ((a a0) b0) 14 A. D. ELMENDORF AND M. A. MANDELL Example. The primary examples of ring categories are categories of endomorphism* *s of small permutative categories. Let C be a small permutative category. Then we ca* *n give the category of lax maps P1(C; C) the structure of a ring category as follows. * *Given objects f and g of P1(C; C) (i.e., lax maps from C to itself), define f g as the lax * *map for which (f g)(c) := fc gc, with lax structure map given by the composite fl (f g)(c) (f g)(c0)=_//_fc gc fc0 gc0~=_//fc fc0 gc gc0 __~f_~g_// f(c c0) g(c c0) = (f g)(c c0). (Notice that even if both lax structure maps ~f and ~g were the identity, the l* *ax structure map for f g would still involve the transposition isomorphism.) This gives us * *permutative structure on P1(C; C). The ring structure is given by composition of lax maps; * *we leave the necessary verifications to the reader. Example. If C is a small monoidal category with a strictly associative and unit* *al monoidal product, then the "free permutative category" on C is functorially a ring categ* *ory, in fact, in uncountably many ways. See Section 10 for details. As further motivation for the definition of the multicategory structure on p* *ermutative categories, we offer the following theorem, proved in Section 8. The operad * * *mentioned in the theorem is discussed immediately below. Theorem 3.4. A ring structure on a small permutative category A determines and * *is determined by a multifunctor * ! P sending the single object of * to A. Here, as above, * denotes the fundamental "associative" operad of sets whos* *e algebras in a symmetric monoidal category are the associative monoids. For convenience, * *we recall the definition. The component sets of * are the symmetric groups k and the mu* *ltiprod- uct is as follows: Let oe 2 k, OEi 2 jifor 1 i k. Then (oe; OE1, . .,.O* *Ek) 2 j (for j = j1 + . .+.jk) is the composite ` ` ` iOEi ` ` oe ` ` j1 . . .jk____//_j1 . . .jk_________//_joe-1(1) . . .joe-1(k), where oe permutes the blocks j1, . .,.jk as indicated. The right a* *ction of k is simply right multiplication. Since the algebras for the operad * in any symmetric monoidal category are * *simply the monoids in the underlying monoidal category, Theorem 1.1 now implies the fo* *llowing corollary. Corollary 3.5. If A is a ring category, then KA is a strict ring symmetric spec* *trum. We next consider commutativity in multiplication, which cannot be strict in * *our con- text; we must settle for E1 . To describe the relevant E1 operad, we need the * *following RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 15 construction. Consider the forgetful functor from small categories to sets that* * forgets the morphisms and remembers only the objects. This functor has a right adjoint E th* *at takes a set X and produces the category EX with X as its set of objects, and with exa* *ctly one morphism between each pair of objects; formally, the morphism set is X xX. We u* *se E for this construction because if the set is actually a group G, the classifying spa* *ce of the cat- egory EG is the usual construction of the universal principal G-bundle. Since E* * is a right adjoint, it preserves products, and therefore if O is any operad of sets, EO is* * an operad of categories. Applying E to the operad * defines the categorical Barratt-Eccl* *es operad E *. Since * is -free, so is E *, and EX is always contractible. The structur* *es in P induced by E * turn out to be bipermutative categories, as defined below. We no* *te that our bipermutative categories are more general than May's ([17], p. 154) both in* * requiring only distributivity morphisms rather than isomorphisms, and in deleting the req* *uirement that one of the distributivity morphisms be the identity. Laplaza's symmetric b* *imonoidal categories [13] are more general even than our bipermutative categories, and si* *nce they can be rectified to equivalent bipermutative categories in May's sense, so can * *ours. Our explicit definition is as follows: Definition 3.6. A bipermutative category is a permutative category (R, , 0) to* *gether with a second permutative structure (R, , 1) with symmetry isomorphism fl : a* * b ~= b a, and natural distributivity maps dl: (a b) (a0 b) ! (a a0) b and dr: (a b) (a b0) ! a (b b0). These are subject to the requirement that the diagrams for a ring category give* *n in Defi- nition 3.3 commute, except with diagram (e) replaced with the following diagram* * (e0): (a b) (a0 b)_dl_//_(a a0) b fl fl|| |fl| fflffl| fflffl| (b a) (b a0)dr_//b (a a0). As noted below, diagram (e) now follows from the remaining axioms, so any biper* *mutative category is automatically a ring category. Example. Let A be a commutative ring. The categories GL A and PrA described abo* *ve become bipermutative categories using the tensor product A , when we identify * *Am A An with Amn using lexicographical order on the standard basis. Of course, we want any bipermutative category to be a ring category, and the* * only issue is whether or not diagram (e), which we removed in the definition, is sti* *ll satisfied. However, it is a component of the proof of Theorem 3.8 below, given in Section * *8, that diagram (e) follows from the remaining diagrams. See Figure 1 on page 38. 16 A. D. ELMENDORF AND M. A. MANDELL Corollary 3.7. Any small bipermutative category is a ring category. We prove the following result in Section 8. Theorem 3.8. Bipermutative structure on a small permutative category R determin* *es and is determined by a multifunctor E * ! P sending the single object of E * to* * R. Since the map E * ! * of operads is a weak equivalence, and the algebras for* * the one-point operad in any symmetric monoidal category are the commutative monoids* * in that category, Theorems 1.1 and 1.4 now give the following corollary. Corollary 3.9. If R is a bipermutative category, then KR is equivalent to a str* *ictly commutative ring symmetric spectrum. 4. The K-Theory of Permutative Categories In this section, we construct the underlying functor of our K-theory multifu* *nctor from permutative categories to symmetric spectra, and show that it is equivalent to * *the K- theory functor in the literature. Since our functor is a modification of the u* *sual Segal construction of the K-theory spectrum of a small permutative category, we descr* *ibe that first, using the construction from [18]. Construction_4.1. For a small permutative category C and a finite based set A, * *let CSeg(A) denote the category whose objects are the systems {CS, aeS,T}, where (1) S runs through the subsets of A not containing the basepoint, (2) S, T runs through the pairs of such subsets with S \ T = ;, (3) the CS are objects of C and the aeS,T are isomorphisms CS CT ! CS[T , such that CS = 0 and aeS,T = idCT when S = ;, and the following diagrams commut* *e for all mutually disjoint S, T, U: aeS,T aeS,T idCU CS CT _____//CS[T CS CT CU ________//CS[T CU fl|| |||| idCS aeT,U|| |aeS[T,U| fflffl| || fflffl| fflffl| CT CS aeT,S//_CT[S CS CT[U __aeS,T[U__//CS[T[U . A morphism f: {CS, aeS,T} ! {C0S, ae0S,T} consists of morphisms fS: CS ! C0Sin * *C for all S, such that f; = id0, and the following diagram commutes for all S, T : aeS,T CS CT _____//_CS[T fS fT|| |fS[T| fflffl| fflffl| C0S C0T ae0__//C0S[T. S,T RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 17 Remark. The construction is described in [18] in terms of based subsets of a ba* *sed set as indices. This leads to some awkwardness in defining functoriality which the * *formalism above avoids. The description in [18] can be recovered simply by reattaching th* *e basepoint to all indexing subsets. ____ ____ Theorem 4.2. The assignment A 7! CSeg(A) defines a functor CSegfrom the categor* *y of finite based sets to the category of small categories. ____ Proof. A map of finite_based_sets ff: A ! A0 induces the functor_CSeg(ff)_that * *sends the object {CS, aeS,T} of CSeg(A) to the object {CffS, aeffS,T} of CSeg(A0), where * *CffS= Cff-1Sand aeffS,T= aeff-1S,ff-1T. Note_that since ff is basepoint-preserving, ff-1(S) do* *es not contain the basepoint. Likewise, CSeg(ff) sends_the map {fS} to the map {fffS} where ff* *fS=_fff-1S._ Clearly,_when_ff_is the identity, CSeg(ff) is the identity, and for ff0: A0! A0* *0, CSeg(ff0O ff) = CSeg(ff0) O CSeg(ff). In the conventions of [2], a " -space" is a functor from the category of fin* *ite based sets to the category of simplicial sets that takes the trivial based set (consisting* *_of only the base point) to a constant one point simplicial set. It follows that NCSeg is a * * -space, where N denotes the nerve functor. Standard notation is to use n to denote_the finite* * based set {0, 1, 2, . .,.n} with 0 serving as the basepoint. The category CSeg(1) is then* * canonically isomorphic to the original category C. For n > 0, the based maps n ! 1 that sen* *d all but one of the non-basepoint elements to the basepoint induce a functor ____ ____ ____ pn: CSeg(n) ! CSeg(1) x . .x.CSeg(1) ~=C x . .x.C that is easily identified as the functor_that_sends {CS, aeS,T} to (C{1}, . .,.* *C{n}) and is an equivalence of categories._The_ -space_NCSeg_is therefore_"special" in the term* *inology of [2] in that the map pn: NCSeg(n) ! NCSeg(1) x . .x.NCSeg(1) is a homotopy equiv* *alence for each n > 0. The spectrum associated to a -space X is constructed as follows. Let S1oden* *ote the following simplicial model of the circle: The set of n-simplices is S1n= n with* * face maps di the order-preserving maps that delete the element i and the degeneracy maps * *si the order-preserving maps that skip the element i. Then S1ohas one 0-simplex and on* *e non- degenerate 1-simplex; all n-simplices are degenerate for n > 1. Regarding S1oas* * a simplicial based set and applying the functor X degreewise, we obtain a bisimplicial set X* *(S1o), which we regard as a simplicial set by taking the diagonal. Writing Snofor the n-fold* * smash power of S1o(with S0othe constant simplicial set 1), we likewise get simplicial sets * *X(Sno). Since Sn-1q^ S1q= Snq, each q-simplex x of S1oinduces a map of based sets Sn-1q~=Sn-1q^ {0, x} ! Snq that assemble to a based map ` X(Sn-1q) ^ S1q~= (X(Sn-1q^ {0, x})) ! X(Snq) x2S1q\{0} 18 A. D. ELMENDORF AND M. A. MANDELL for each q. Taking these together for all q and n form the "structure maps" X(* *Sn-1o) ! X(Sno) that make {X(Sno)} into a spectrum. In fact, {X(Sno)} forms a symmetric * *spectrum, where the symmetric group action on X(Sno) is induced by permuting the smash fa* *ctors of Sno. The main theorem of [21] then can be phrased as saying that when X is a* * special -space, this spectrum is an "almost -spectrum" in that after geometric realiz* *ation, the maps |X(Sno)| ! |X(Sn+1o)| adjoint to the structure maps are homotopy equivalences for all n 1. ____ Although we have followed [18] in constructing CSegand [2] in constructing t* *he associated (symmetric) spectrum, we refer to this as Segal's construction. Definition 4.3. Segal's construction_of K-theory of the permutative category C * *is the symmetric spectrum KSegC = {NCSeg(Sno)}. Previously, the main difficulty with constructing ring and module structures* * on the spectra associated to permutative categories was the lack of a symmetric monoid* *al product on the target category of spectra. Even using the category of symmetric spectra* *, which does have a symmetric monoidal product, the previous definition does not carry ring * *structures (e.g., ring category structures) to ring structures. A suitable collection of m* *aps ____ ____ ____ NCSeg(m) ^ NCSeg(n) ! NCSeg(m ^ n) would give rise to a pairing KSegC^KSegC ! KSegC, but no reasonable definition * *of pairing on the permutative category C gives rise to such a collection of maps. We can i* *llustrate this by looking at just the zero simplices, or equivalently, the_objects in the* * categories. Given_some kind of pairing on C and objects {CS, aeS,T}_of_CSeg(m) and {C0S, * *aeS,T} of CSeg(n), we need to construct an object {C00S, aeS,T} of CSeg(m ^ n). It seems* * natural to take C00SxT= CS C0T on the subsets of the form S x T m ^ n, but how do we fill in the objects C00* *Ufor subsets U not of this form? ____ Our basic idea is to modify the construction of CSegso the objects correspon* *d only to those subsets of the_appropriate_form. The set of q-simplices Snqof Snois S1q^* * . .^.S1q; instead of using NCSeg(Snq) where we choose objects CT for all subsets T of S1q* *^ . .^.S1q not containing the basepoint, we can use a variant where we only choose them fo* *r the subsets of the form T1x . .x.Tn. We make one other alteration: Since we have de* *fined the multicategory of permutative categories using lax distributivity maps, we do no* *t require the morphisms ae to be isomorphisms. Before describing the construction, it is* * useful to introduce the following notation. Given finite basepoint-free (sub)sets S1, . .* *,.Sn, we write for the n-tuple (S1, . .,.Sn). Given a finite basepoint-free set T and i 2 * *{1, . .,.n}, we write for the n-tuple (S1, . .,.Si-1, T, Si+1, . .,.Sn) obtained by sub* *stituting T in the i-th position. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 19 Construction_4.4. For a small permutative category C and finite based sets A1, * *. .,.An, let C(A1, . .,.An)denote the category whose objects are the systems {C, ae;i,T,U}, where (1) = (S1, . .,.Sn) runs through all n-tuples of basepoint-free subsets * *Si Ai, (2) For ae;i,T,U, i runs through 1, . .,.n, and T, U run through the base* *point-free subsets of Si with T \ U = ; and T [ U = Si, (3) The Care objects of C, and (4) The ae;i,T,Uare morphisms C C! Cin C, such that (1) C= 0 if Sk = ; for any k, (2) ae;i,T,U= idif any of the Sk (for any k), T , or U are empty, (3) For all ae;i,T,Uthe following diagram commutes: ae;i,T,U C C_____________//_C | || fl|| |||| | || fflffl| || C Cae;i,U,T__//_C, (4) For all , i, and T, U, V Ai with T [ U [ V = Si and T , U, and V mu* *tually disjoint, the following diagram commutes: ae;i,T,U id C C C______________//_C C | | id ae;i,U,V|| |ae;i,T[U,V| | | fflffl| fflffl| C C___ae;i,T,U[V_______//C, 20 A. D. ELMENDORF AND M. A. MANDELL (5) For all ae;i,T,Uand ae;j,V,Wwith i 6= j, the following diagram com* *mutes: C32C ggg (ae;i,T,U) (ae;i,T,U)gg22ggggg ggggggg 22 22ae;j,V,W2 CCC C 2 | 22 | 22 | 2ssss id fl id|| C. | fEEifi | fifi fflffl| fifi CC CC fae;i,T,Uifi WWWW fifi WWWWWW ffii (ae;j,V,W) (ae;j,V,W)++WWWWWWWfifi C C A morphism f: {C, ae;i,T,U} ! {C0, ae0;i,T,U} consists of morphisms* * fS: C ! C0in C for all such that f is the identity id0 when Si = ; for any i,* * and the following diagram commutes for all ae;i,T,U: ae;i,T,U C C________//_C f f|| |f| fflffl| fflffl| C0 C0______//C0. ae0;i,T,U If_any_of the Ai are trivial (consist of just the basepoint) in the definiti* *on above, then C(A1, . .,.An) is a trivial category with one object and one morphism. To* * avoid a_(unique) isomorphism later, we choose and fix a particular trivial category ** *, and set C(A1, . .,.An) = * in this case. __ We make C into a functor from_n-tuples of finite based sets to categories ju* *st as in Theorem 4.2. The categories C have further functoriality as well: Permutation Functors. A permutation oe in n induces a functor __ __ oe!: C(A1, . .,.An)! C(Aoe-1(1), . .,.Aoe-1(n)), which is an isomorphism of categories, as follows: The object {C, ae;i,T,* *U} is sent to the object {Coe, aeoe;i,T} where Coe= Coe, aeoe;i,T,U= aeoe;oe(i),T,U, oe = (S* *0oe(1), . .,.S0oe(n)), so if S0i= Soe-1(i) Aoe-1(i), then oe = . The morphism {f} is sent * *to the morphism {foe} where foe= foe. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 21 Extension Functors. We have an isomorphism of categories __ __ e: C(A1, . .,.An)! C(A1, . .,.An, 1) defined as follows: The object {C, ae;i,T,U} is sent to the object {Ce, aee;i,T,U} where Ce(S1,...,Sn,{1})= C,aee(S1,...,Sn,{1});i,T,U= ae;i,T,Ufor i < n + 1, Ce(S1,...,Sn,;)= 0, aee(S1,...,Sn,;);i,T,U= id, aee(S1,...,Sn,{1});n+1,T,* *U= id. The morphism {f} is sent to the morphism {fe} where fe(S1,...,Sn,{1})= f, fe(S1,...,Sn,;)= id. This description of the components of the objects and morphisms is complete sin* *ce the only two basepoint-free subsets of 1 are {1} and ;. The inverse of this isomor* *phism is induced by dropping the {1} from (n + 1)-tuples of the form (S1, . .,.Sn, {1}).* * Of course, for any_other set {*,_x} with precisely one non-basepoint, we have an extension* * functor ex: C(A1,...,An)! C(A1,...,An,{*,x})given by the composite of e and the functor* * induced by the unique based bijection 1 ! {*, x}. The various functors above satisfy formal properties that we describe implic* *itly in the next section, by abstracting them into the definition of a G*-category. We can * *also extend such functors naturally to functors from finite simplicial based sets to simpli* *cial categories by applying the functor degreewise. The nerve of a simplicial category is forme* *d by taking the nerve degreewise and then taking the diagonal. The underlying functor of th* *e K-theory multifunctor we describe in Sections 6 and 7 is naturally isomorphic to the K-t* *heory functor in the following definition. Definition 4.5. For a small_permutative category_C, the symmetric spectrum_Knew* *C is defined by (KnewC)(0) = NC (S0), (KnewC)(1) = NC (S1o), (KnewC)(2) = NC (S1o, S* *1o), and in general, __ (KnewC)(n) = NC (S1o,_._.,.S1o_-z___"), n with symmetric group actions induced by the permutation functors above and stru* *cture maps __ 1 1 1 ` __ 1 1 __ 1 1 1 NC (Sq, . .,.Sq)^ Sq ~= NC (Sq, . .,.Sq, {0,!x})NC (Sq, . .,.Sq, S* *q) x2S1q\{0} induced by the extension functors above. We close this section by showing that the symmetric spectra KSegC_and_KnewC * *are weakly_equivalent. First we note that we have a canonical functor CSeg(A1 ^ . .* *^.An) ! C(A1, . .,.An)that takes the object {CS, aeS,T} to the object C= CS1x...xSn, ae;i,T,U= aeS1x...xTx...xSn,S1x...xUx...xSn. This functor is natural in C and A1, . .,.An and commutes with the permutation * *and extension functors. We therefore get a natural map of symmetric spectra KSeg ! * *Knew. 22 A. D. ELMENDORF AND M. A. MANDELL Theorem 4.6. The natural map of symmetric spectra KSegC ! KnewC is a level equi* *va- lence for every C. ____ __ Proof. It suffices to show that the map NCSeg(m1^ . .^.mn) ! NC is a weak e* *quiva- lence for_all_n, . Write ^ as an abbreviation for m1^. .^.mn and let m = * *m1 . .m.n. Let pm : CSeg(^) ! Cm denote the functor that takes {CS, aeS,T} to the m-tu* *ple whose (i1, ._.,.in)'th_coordinate is C{(i1,...,in)}. Then pm is an equivalence of c* *ategories. Let q: C ! Cm denote the functor that takes {C, ae;i,T,U} to the m-tup* *le whose (i1, . .,.in)'th coordinate is C({i1},...,{in}). Then q is not an equivalen* *ce of categories but does have a left adjoint, namely the functor that sends an object with coor* *dinates (Xi1,...,in) to the object with M M C= . . . Xi1,...,in i12S1 in2Sn (ordered using the natural order on Si m), with the convention that the empty * *sum is the unit 0 of C; the ae's are defined by the appropriate rearrangement using the co* *mmutativity isomorphism fl. The functor q therefore induces a homotopy_equivalence_on_n* *erves. Since the functor pm factors as the composite of the functor CSeg(^)_! C* *_we are interested in and the functor q, we conclude that the map NCSeg(^) ! NC <* *m> is a homotopy equivalence and therefore a weak equivalence. This completes the pro* *of. 5. The Multicategory of G*-categories Extending the K-theory functor to a multifunctor from the multicategory of p* *ermuta- tive categories to the multicategory of symmetric spectra requires a detailed s* *tudy of the properties of the constructions of the previous section. Instead of carrying al* *ong the de- tails, it is useful to abstract the essential properties, and this leads us to * *the G*-categories that form the objects of the multicategory we define in this section. These G*-* *categories will be certain functors out of a wreath product category G built from the cate* *gories of n-tuples of finite based sets together with maps corresponding to the permutati* *on functors and extension functors studied briefly in the previous section. In the two next* * sections we define the K-theory multifunctor in terms of a_multifunctor from permutative ca* *tegories to G*-categories (extending the construction C of the previous section) and a m* *ultifunctor from the G*-categories to symmetric spectra. We begin with the definition of G. Let Injbe the category with objects the u* *nbased sets r_= {1, . .,.r} for r = 0, 1, 2, 3, . .,.and morphisms the injections. Let F be* * the skeleton of the category of finite based sets consisting of the objects n = {0, 1, . .,.n} * *with basepoint 0. Then there is a functor F* : Inj! Cat described by F*(r_) := Fr on objects. On morphisms, F* rearranges the coordina* *tes according to the given injection and, most crucially, inserts the object 1 in t* *he slots that RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 23 are missed. Formally, if we are given an injection q : r_! s_, then F*(q) is th* *e functor from Fr to Fs that takes an object = (m1, . .,.mr) to the s-tuple q* = (m01, * *. .,.m0s) in which ae mi if q-1 (j) = {i} m0j= 1 if q-1 (j) = ;, and takes a morphism (ff1, . .,.ffr) to the s-tuple (fi1, . .,.fis) where aeff if q-1 (j) = {i} fij = i id1 if q-1 (j) = ;. As withRany functor to Cat , there is associated to this functor F* a wreath pr* *oduct functor Inj F*. R Definition 5.1. G = Inj F*. The category G can be described explicitly as follows. The objects of our c* *ategory G are the tuples of objects of F, say (n1, . .,.ns). Each tuple has a specific l* *ength; the empty tuple () has length 0. A morphism between tuples, say from = (m1, . .* *,.mr) to = (n1, . .,.ns), consists of a pair (ff, q), where q : r_! s_is a morphi* *sm in Inj, and ff : q* ! is a morphism in Fs. For a morphism (fi, t) : !

, we de* *fine the composite (fi, t) O (ff, q) to be (fi O t*ff, t O q). We define the category of G*-categories as a certain category built out of t* *he category of functors from G into the category of small categories. In order to avoid possib* *le confusion as to the meaning of "functor" and "natural transformation" where they occur be* *low, we define G*-objects in an arbitrary bicomplete category C. We write * for a chos* *en final object in this category, and C* for the category of objects of C equipped with * *a structure map from *. We think of C* as the category of based objects in C. In our applic* *ations C is always either Cat , the category of small categories, or SS , the category* * of simplicial sets. Definition 5.2. A G*-object in C consists of a functor F : G ! C together with * *a map * ! F () such that F (m1, . .,.mr) = * whenever any mi = 0 and such that the fo* *llowing diagram commutes, * __=___//_F (0) | | | | fflffl| fflffl| F ()_____//F (1), where the left hand map is the given map, the top map is the unique map, the ri* *ght hand map is induced by the unique map (0) ! (1) in G, and the bottom map is induced * *by the map () ! (1) in G from the unique map 0_! 1_in Inj and the identity map on 1 in* * F. 24 A. D. ELMENDORF AND M. A. MANDELL A map of G*-objects F ! G is a natural transformation f: F ! G making the follo* *wing diagram commute: * B "" BBB """ BBB """"" B!! F ()_____f()____//_G(). We denote the category of G*-objects as G*-C. We remark that for a G*-object F , the objects F of C are based: the map* * from * is the explicitly given one for = (), and the map * = F (0, . .,.0) ! F (m1* *, . .,.mr) is induced from the unique map (0, . .,.0) ! (m1, . .,.mr) in Fr for r > 0. It * *is easy to see from the universal property of the terminal object and the diagram in the d* *efinition, that any map ! in G induces a based map F ! F . Likewise, for a * *map f: F ! G in G*-C, the maps F ! G are based for all in G. The follow* *ing proposition is now clear. Proposition 5.3. The category G*-C is the full subcategory of the category of f* *unctors G ! C* consisting of those functors F with F (m1, . .,.mr) = * whenever any mi=* * 0. In order to define the multicategory structure on G*-objects, we take advant* *age of additional structure on the category G: it is actually a permutative category. * *The product operation is given on objects by concatenation of tuples, with the obvious exte* *nsion to morphisms. We denote this operation as . This allows us to regard a G*-object * *G as a functor from Gk = Gx. .x.G to C* by the formula G(, . .,.) = G( . .* * .). We will also exploit the smash product (written ^) in C*. For X and Y object* *s of C*, the smash product X ^ Y is the pushout (X x *) q (* x Y )____//X x Y | | | | fflffl| fflffl| *_____________//X ^ Y. It is well-known that ^ is a closed symmetric monoidal product on SS *and Cat*,* * and more generally, when C is bicomplete and Cartesian closed, ^ is a closed symmetric m* *onoidal product on C*. Given G*-objects F1, . .,.Fk, and G, the set of k-morphisms in G*-C from (F1* *, . .,.Fk) to G is the set of natural transformations f: F1 x . .x.Fk ! G of functors Gk !* * C which take the map F1()x. .x.Fi-1()x*xFi+1()x. .x.Fk() ! F1()x. .x.Fk() induced by the given map * ! Fi() to the given map * ! G(). Equivalently and more concisely in* * the case when C is Cartesian closed, this is the set of natural transformations f: F1 ^ * *. .^.Fk ! G RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 25 of functors Gk ! C*: F1x...xFk k Gk ________//_C* || f |^| fflffl| fflffl| G ___G____//_C*. We obtain an action of k from the symmetry isomorphism of , and we obtain a m* *ulti- product from composition in C*. Proposition 5.4. G* -C forms a multicategory with the definitions above. When C is enriched over small categories or simplicial sets, the conditions * *defining the objects and k-morphisms of G*-C translate into limits on the categories or simp* *licial sets of maps, and the multicategory G*-C therefore inherits an enrichment. In the ca* *se when C is the category of simplicial sets, the description of simplicial sets of k-mor* *phisms is clear. In the case when C is the category of small categories, we can describe the enr* *ichment of G*-Cat over Cat explicitly as follows. First, since * is the trivial category with one object and one morphism, the* * map * ! G() in the definition of G*-category is equivalent to specifying a distinguished "b* *asepoint" object of G(). For G*-categories F1, . .,.Fk and G, the category of k-morphism* *s from (F1, . .,.Fk) to G has as its objects the natural transformations f: F1 ^ . .^.* *Fk ! G of functors from Gk to Cat *, i.e. collections of based functors f : F1 ^ .* * .^.Fk ! G( . . .) natural in Gk. A map of such k-morphisms OE: f ! g assigns * *to each object = (, . .,.) of Gk a natural transformation OE: f ! g such that the value of OE() at the basepoint object of F1() ^ . .^.Fk() is the identity m* *ap on the basepoint object of G() and such that for any morphism (h1, . .,.hk): (, . * *.,.) ! (, . .,.) in Gk, the transformations given by the following two pasting* * diagrams 26 A. D. ELMENDORF AND M. A. MANDELL coincide: ___________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *___________________________________________________f ________________________________________________* *_____________________________________________________________________________* *____**OO F1 ^ . .^.Fk____________OOff'OEG( . . .) ________________________________________________* *_____________________________________________________________________________* *_________________________________44 | _________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____________| | g | F1(h1)^...^Fk(hk)|| G(h1|...|hk) | | fflffl| fflffl| F1 ^ . .^.Fk_____g______//_G( . . .) F1 ^ . .^.Fk _____f_____//_G( . . .) | | | | = F1(h1)^...^Fk(hk)|| G(h1|...|hk) | __________________________|___________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *___________f fflffl|__________________________fflffl|_____________* *___________________________________________________**OO F1 ^ . .^.Fk____________OOff'OEG( . . .). ________________________________________________* *_____________________________________________________________________________* *_________________________________44 _________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *____________ g Note, however, that since the left vertical arrows in both diagrams are induced* * by maps using Cartesian products rather than smash products, the coincidence of the tra* *nsforma- tions given in the diagrams could equally well be specified by replacing the sm* *ash products with Cartesian products. This will be of use to us in the next section. A colle* *ction of nat- ural transformations satisfying the coherence condition in the display above is* * called a modification. It turns out that in the case when C is the category of simplicial sets or t* *he category of small categories, or more generally, a bicomplete Cartesian closed category, th* *en G*-C is a bicomplete closed symmetric monoidal category, and the multicategory structur* *e asso- ciated to the symmetric monoidal structure is the one considered above. Since t* *his is not needed in the remainder of the paper, we give only a brief sketch of the argume* *nt. Let F(0)be the category with objects * and () where * is a null object (both* * initial and final) and the set of maps from () to itself consists of just the zero map and * *the identity. For r > 0, let F(r)be the r-th smash power of the based category F. (Note that* * F(0) is not the usual zeroth smash power of based categories, although it is the zer* *oth smash power in the full subcategory of Cat *of categories whose base object is null.)* * As above, we write = (m1, . .,.mr) but now (for r > 0), = *, the basepoint object* *, if any mi= 0. The categories F(r)have a based action of Injinduced from the action des* *cribed above for the Cartesian powers Fr; in particular, the object () of F(0)gets sen* *t to the RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 27 R constant string (1, . .,.1). We define G* to be the wreath product Inj F(-) f* *ormed in based categories. Specifically, the set of objects of G* is ` i j Ob F(r) r_2Ob(Inj) and the set of maps from to in G* form the based set ` `^r ' ` ^ ' F(mi, nq(i)) ^ nj q:r_!s_i=1 q-1(j)=; where the wedge is over the maps q in Inj. The empty wedge is of course the one* * point set, and the empty smash is 1. Note that the basepoint object * of G* is a null* * object, and the basepoint in each mapping set is the unique map that factors through *. We have a canonical functor G ! G*. In fact, we can identify the category G** *as the cate- gory obtained from G*by attaching a new null object * and identifying with * ** whenever any mi= 0. In particular, every map in G*(, ) is either the trivial morph* *ism (factor- ing through *) or in the image of G(, ). The function G(, ) ! G*(, ) is in fact one-to-one on the subset of G(, ) that does not map to the trivia* *l morphism. A based functor G* ! C* is a functor that takes the null object * of G* to the * *null object * of C*. The following proposition is now clear from the discussion and Proposi* *tion 5.3. Proposition 5.5. The category G*-C is isomorphic to the category of based funct* *ors G* ! C*. Concatenation again makes G* into a permutative category (where concatenatio* *n with * on either side yields *). It follows from theorems of Day ([3], Theorems 3.3 * *and 3.6) that the category of based functors from G* to C* has a closed symmetric monoidal st* *ructure, enriched over C*, in which the product of functors F1 and F2 is given by the le* *ft Kan extension F1 ^ F2 in the diagram on the left below. The universal property of * *the Kan extension is that maps from F1 ^ F2 to G are in one-to-one correspondence with * *natural transformations f as in the diagram on the right below. G*x G* F1xF2//_C* x C*_^__//C*44ii G*x G* __F1xF2_//_C* x C* i i || i i i || iiiiffi ^|| fflffl|iiiiF1^F2 fflffl| fflffl| G* G* ______G______//_C* Because G is the final object * whenever any mi = 0, a natural transformatio* *n f in the diagram above is precisely the same as a 2-morphism in G*-C under the ident* *ification of the previous proposition. The analogous observation for iterated smash produ* *cts and consistency with the multiproduct and symmetric group actions then imply the fo* *llowing theorem. 28 A. D. ELMENDORF AND M. A. MANDELL Theorem 5.6. Let C be a bicomplete Cartesian closed category. Then G*-C is a cl* *osed symmetric monoidal category enriched over C. The multicategory structure of Pr* *oposi- tion 5.4 coincides with the multicategory structure inherited from the symmetri* *c monoidal structure. Although we have no need for it in this paper, the discussion of this sectio* *n may be generalized to the context of a bicomplete symmetric monoidal closed category C* *, where x in C is replaced with the symmetric monoidal product in C; however, * remains* * the final object in C and not the unit object. 6. From Permutative Categories to G*-categories We turn next to the description of our enriched multifunctor from permutativ* *e categories to G*-categories. This section is devoted to the proof of the following theorem. Theorem 6.1. Construction 4.4 extends to an enriched multifunctor J from permut* *ative categories to G*-categories. To each permutative category C we need to associate a G*-category JC : G ! C* *at *. Since this construction is_to_extend Construction 4.4, we must define it on obj* *ects = (n1, . .,.ns) by JC := C. We then have JC = * if any ni= 0. We take JC* *() = C_ and we use the 0-object of C as the basepoint object. The canonical isomorphism* * C ~=C(1) takes the unit of C to the image of the single object of C(0). Next we specify JC on morphisms of G. Given (ff, q) : ! , where = (m1 , . .,.mr) and = (n1, . .,.ns), the functor __ __ JC(ff, q) : C ! C is obtained by composing the isomorphism __ __ C ~=Cq* induced by the permutation and extension functors described in Section 4 with t* *he functor __ __ ff* : Cq* ! C described in the proof of Theorem 4.2. This constructs a G*-category JC. Next we describe the functor J on k-morphisms. For this we need to describe * *functors J: Pk(C1, . .,.Ck; D) ! G*-Cat(JC1, . .,.JCk; JD) between categories of k-morphisms that preserve the symmetric group actions and* * the multiproduct. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 29 We begin by giving J on objects of Pk(C1, . .,.Ck; D); for this fix a k-line* *ar map f: C1x . .x.Ck ! D. The k-morphism Jf: (JC1, . .,.JCk) ! JD in G*-Cat then consists * *of a natural transformation of functors Jf as in the following diagram: JC1x...xJCk k Gk _________//_Cat* || fflflfllJf^|| fflffl| fflffl| G ____JD____//Cat*. This means that for each k-tuple of objects of G, (, . .,.), where we h* *ave = (ni1, . .,.nisi), we need to specify a functor Jf(, . .,.) : JC1 x . .x.JCk ! JD( . . .) which returns 0 or id0 whenever any of the input objects or morphisms are 0 or * *id0, respectively. An object of JCi is a system of objects Cof Ci, where = (Si1, .* * .,.Sisi) and Sijruns over all subsets of {1, . .,.nij}, together with structure maps as * *specified in Construction 4.4. Given a k-tuple of such systems (C, . .,.C), we need * *to construct an object of JD( . . .), which is a system of objects Dof D, wher* *e runs over all s1 + . .+.sk-tuples of subsets of the sets {1, . .,.nij} for 1 i k* * and 1 j si. But such a is simply the concatenation of a collection of lists for 1* * i k, each of which determines an object Cof Ci. We therefore define the object D a* *s simply f(C, . .,.C) for the component sublists of ; note that this ob* *ject is 0 if any of the inputs are 0, by the k-linearity of f. It is now a lengthy but strai* *ghtforward exercise to check that the evident structure maps satisfy the requirements for * *an object of JD( . . .). The definition easily extends to morphisms of JC1 x . .x* *.JCk. We need to check that this construction is natural in morphisms of Gk. But t* *he mor- phisms of Gk are generated by the morphisms in each factor of G, which in turn * *are generated by maps in the component Fs's and induced maps ! q* for inject* *ions q : r_! s_. For maps ffi: ! in Fsi, we need the following diagram to* * commute: JC1 x . .x.JCk _Jf__//_JD( . . .) JC|| |JD| fflffl| fflffl| JC1 x . .x.JCkJf__//JD( . . .). However, going around the square either way sends a k-tuple of systems (C, * *. .,.C) to the system D, where runs over s1+. .+.sk-tuples of subsets of the se* *ts {1, . .,.n0ij} for 1 i k and 1 j si, and D is defined by breaking up into comp* *onent 30 A. D. ELMENDORF AND M. A. MANDELL lists = where has length si. The subsets in the list <* *Ti> are then pulled back along the ffi's to give a list of subsets ff-1i of {1, . .,.nij* *}, and D is then f(Cff-11, . .,.Cff-1k). A similar formula gives the composite in e* *ither direction on morphisms. The induced morphisms from the maps ! q* for an injection q : r_!_s_a* *re the isomorphisms induced by the permutation and extension isomorphisms in the C's. * *Again, given injections qi: ri_! si_, we need the following diagram to commute: JC1 x . .x.JCk ______Jf______//JD( . . .) JC|| |JD| fflffl| fflffl| JC1(q1)* x . .x.JCk(qk)* _Jf__//JD((q1)* . . .(qk)*). Again, an object in the upper left category is a k-tuple of systems (C, . .* *,.C), and we produce a Dfrom going around the square in either direction. But is a c* *oncatenation of lists of subsets of {1, . .,.miq-1i(j)}, where if q-1i(j) is empty, we * *set miq-1i(j)= 1 for consistency with our construction of JCi as a G*-category. If any of the c* *omponent subsets Sijare empty, then Dmust be 0, while if all of the Sij's for q-1i(j)* * = ; are {1}, then the 1's can be dropped and we get a permutation of lists indexing the give* *n object (C, . .,.C), say (C, . .,.C). The corresponding obj* *ect under either composition is then f(C, . .,.C). A similar (but easier) ch* *eck shows that the diagram commutes on morphisms as well. We have therefore specified a k-morp* *hism Jf in G*-Cat from (JC1, . .,.JCk) to JD. We also need to specify a modification JOE from Jf to Jg whenever we have a * *morphism OE : f ! g in Pk(C1, . .,.Ck; D). This means that for each object (, . .,.<* *nk>) of Gk, we need to specify a natural transformation ____________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________________________________________Jf __________________________________________________* *_____________________________________________________________________________* *__**OO JC1 x . .x.JCk____________OOff'JOEJD( . . .). _________________________________________________* *_____________________________________________________________________________* *________________________________44 ___________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________ Jg This means in turn that for each object > = (C, . .,.C) of JC1 x . .x. JCk, we need a morphism in JD( . . .) from Jf> to Jg* *>. But Jf> is a system of objects of the form f>, and Jg> is a syste* *m of objects of the form g>, and OE provides a natural transformation from one system * *of objects to the other. We leave to the reader the tedious but straightforward verificati* *ons necessary to show that we have, in fact, specified a map J of multicategories enriched ov* *er Cat . RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 31 7. From G*-categories to Symmetric Spectra We turn next to the description of our multifunctor from G*-categories to sy* *mmetric spectra. As before, to avoid the confusion of the different levels of functors* * and natural transformations, it is convenient to work as long as possible with G*-objects i* *n a general Cartesian closed bicomplete category C; we are only really interested in the ca* *se when C is the category of small categories Cat and the case when C is the category of * *simplicial sets SS . As before, let C* be the category of based objects in C. The constr* *uctionoinp this generality is a multifunctor into the multicategory of symmetric spectra i* *n C* . We begin with a review of this multicategory. The standard definition of the category of symmetric spectra in C*op in the * *case when C is the category of sets is usually phrased in terms of the smash product of b* *ased sim- plicial sets, which is a special case of the smash product in C* introduced in * *Section 5. The formulation of the category of symmetric spectra that follows is therefore * *a simple generalization of the category of symmetric spectra of [9]. Definition 7.1. Let C be bicomplete and Cartesian closed. Let C*op denote the c* *ategory of simplicial objects in C*, i.e., contravariant functors from the simplicial c* *ategory to theocategorypC*. We use * to denote both the null objectoofpC* and also the nul* *l object in C* , the constant simplicial object on *. For X in C* and K a finite based s* *implicial set, write X ^ K for the tensor of X with K; concretely, X ^ K has n-simplices ` (X ^ K)n = Xn, Kn\{*} W op where denotes theocoproductpin C*. A symmetric spectrum in C* consists of objects X(p) in C* for all non-negative integers p, an action of the symmetri* *c group p on X(p), and "suspension" maps oep : X(p) ^ S1o! X(p + 1), such that for each q 1 the composite X(p) ^ (S1o)q ! X(p + q) preserves the (* * p x q)- action. A k-morphism in symmetric spectra in C*op from (X1, . .,.Xk) to Y consists o* *f maps X1(p1) ^ . .^.Xk(pk) ! Y (p1 + . .+.pk) for all p1, . .,.pk that preserve the p1 x . .x. pk action and that make the f* *ollowing diagram commute for 1 i k: (X1(p1) ^ . .^.Xk(pk)) ^ S1o__________//Y (p1 + . .+.pk) ^ S1o |~=| || fflffl| fflffl| (7.1) X1(p1) x . .^.(Xi(pi) ^ S1o) ^ . .^.Xk(pk) Y (p1 + . .+.pk + 1) | | | |ci fflffl| fflffl| X1(p1) ^ . .^.Xi(pi+ 1) ^ . .^.Xk(pk)______//Y (p1 + . .+.pk + 1), 32 A. D. ELMENDORF AND M. A. MANDELL where ci denotes the permutation in p1+...+pk+1that moves the last element to * *the (p1+ . .+.pi+ 1)-st position but otherwise preserves the order, i.e., the cycle (q +* * 1, . .,.p, p + 1) where q = p1 + . .+.pi and p = p1 + . .+.pk. The k action on the k-morphisms * *is induced by permuting the product factors and the symmetric group action on the * *target, permuting blocks. The multiproduct is induced by smash products and compositio* *ns in C*. By the simplicial nature of the construction, the multicategory is enriched * *over simplicial sets. When C* is enriched over small categories or simplicial sets, the condit* *ions in the previous definition translate into limitsoonpthe categories or simplicial sets * *of maps, and the multicategory of symmetric spectra in C* becomes enriched over simplicial cat* *egories or bisimplicial sets. Proposition 7.2. The multicategory of symmetric spectra in based simplicial set* *s as de- fined above is isomorphic to the multicategory associated to the symmetric mono* *idal cate- gory of symmetric spectra of [9]. Proof. This is an easy consequence of the external formulation of the smash pro* *duct of symmetric spectra. Technically, the paper [9] considers the category of "left S* *-modules" whereas the (external) formulation above specifies the category of right S-modu* *les, but the identity isomorphism S ~=Sop induces a strong symmetric monoidal isomorphism be* *tween these categories. Now we describe our multifunctor I from G*-objects in C to symmetric spectra* * in C*op. Recall from Section 4 that we have defined our model of the circle S1oso that i* *ts based set of n-simplices is n, giving S1oas a functor from op to F. Construction 7.3. For F a G*-object in C and for p 0 let IF (p) be given by t* *he composite in the following diagram: (S1o)p F op __D__//( op)p____//Fp_____//G____//C*, where D is the diagonal, and the unlabelled arrow is the canonical inclusion of* * Fp into G. In particular, IF (0) is the constant simplicial object F (). We give IF (p) t* *he p action arising from the action of p on Fp. We have maps IF (p) ^ S1o! IF (p + 1) induced by the maps in G (n1, . .,.np) ! (n1, . .,.np, np+1) indexed by the nonzero elements of np+1, with the map indexed by x being the ma* *p given by the injection including {1, . .,.p} in {1, . .,.p + 1} and the map in the p * *+ 1'st copy of F sending 1 to np+1 by the unique based map sending 1 to x. The composite map IF (p) ^ (S1o)q ! IF (p + q) RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 33 has a similar description and so is easily seen to be p x q equivariant. It f* *ollows that these objects and maps assemble to a symmetric spectrum which we write as IF . Theorem 7.4. I extends to a multifunctor fromothepmulticategory of G*-objects i* *n C to the multicategory of symmetric spectra in C* . Proof. Let F1, . .,.Fk and G be G*-objects in C, and consider a k-morphism f f* *rom (F1, . .,.Fk) to G, which consists of a natural transformation as indicated in * *the following diagram: F1x...xFk k Gk ________//_C* || f |^| fflffl| fflffl| G ___G____//_C*. It is straightforward to verify that the following diagram commutes: Fp1 x . .x.Fpk_____//Gk ~=|| || fflffl| fflffl| Fp1+...+pk________//G, so by pasting we obtain the following composite diagram, which gives the induce* *d map of symmetric spectra. (For reasons of space, we have written F(p1,...,pk)for Fp1 x* * . .x.Fpk, and similarly for other superscripted k-tuples.) k (S1o)(p1,...,pk) F1x...xFk ( op)kO____D_____//_(Oop)(p1,...,pk)_____//_F(p1,...,pk)_____//_Gk_________//_* *Ck* D | ~=|| ~=|| || ffififiif* *^|| || D fflffl|(S1o)p1+...+pk fflffl| fflffl|G f* *flffl| op ___________//_( op)p1+...+pk_______//_Fp1+...+pk________//G__________//C* The suspension diagram 7.1 commutes by naturality of f and the definition of th* *e sus- pension maps and symmetric group action because the following diagram of maps i* *n G commutes for all i, all objects , . .,., and all based maps 1 ! n. For * *reasons of space, let = . . ., and let = . . . Then we * *have (1)___// (n) id o|| |id|o fflffl| fflffl| (1) __// (n) , where the horizontal maps are induced by the given map 1 ! n. We leave to the r* *eader the exercise of correlating definitions to check that this association preserve* *s the symmetric group action on the k-morphisms, the units, and the multiproduct. 34 A. D. ELMENDORF AND M. A. MANDELL When we regard the k-morphisms of G*-objects as discrete simplicial sets, th* *e multicat- egory G*-C is enriched over simplicial sets and the multifunctor described abov* *e is enriched (for trivial reasons). When C is enriched over small categories or simplicial s* *ets, we can regard the multicategory of G*-objects as enriched over simplicial categories o* *r bisimpli- cial sets by taking the (other) simplicial direction to be discrete. A straight* *forward check then shows that the multifunctor described above is enriched over simplicial ca* *tegories or bisimplicial sets. Composing the multifunctor J from the previous section, the multifunctor I, * *the nerve functor, and the diagonal functor (from bisimplicial sets to simplicial sets), * *we obtain a multifunctor K from the multicategory of small permutative categories to the mu* *lticate- gory of symmetric spectra. By inspection, the underlying functor is naturally i* *somorphic to the functor Knew described in Definition 4.5. This completes the proof of Th* *eorem 1.1. 8. Ring Categories, Bipermutative Categories, and the Operads * and E * This section is devoted to the proofs of Theorems 3.4 and 3.8. Proof of Theorem 3.4. First, suppose we are given a small ring category A; we m* *ust produce a multifunctor * ! P sending the single object of * to A. In this ca* *se, a multifunctor as specified in the theorem is precisely a map of operads (in Cat * *) from * to the endomorphism operad of A in P, whose component categories are the k-line* *ar maps Pk(A, . .,.A; A). In other words, we must define a sequence of functors Tk* *: k ! Pk(A, . .,.A; A), and show that they specify a map of operads. Since k is a d* *iscrete category, specifying the functor Tk is equivalent to specifying a k-morphism Tk* *oe for every element oe in the group k. As per Definition 3.2, the k-morphism T oe consists* * of a functor foe: Ak ! A and natural distributivity maps ffioeifor 1 i k. We define foeby foe(a1, . .,.ak) = aoe-1(1) . . .aoe-1(k). For notational convenience in defining ffioei, let P = aoe-1(1) . . .aoe-1(oe(* *i)-1), and Q = aoe-1(oe(i)+1) . . .aoe-1(k). We then define ffioeias the common diagonal of t* *he following square, which commutes by Definition 3.3, condition (e): (P ai Q) (P a0i Q)__dl_//((P ai) (P a0i)) Q dr || |dr|1 fflffl| fflffl| P ((ai Q) (a0i Q))___1_dl__//_P (ai a0i) Q. The reader may now verify that the requirements for distributivity maps are sat* *isfied. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 35 We must verify that the Tk's give a map of operads. Equivariance is element* *ary; we check preservation of the multiproduct. This follows as a consequence of the f* *ollowing commutative diagram, where oe 2 k and OEi2 jifor 1 i k: Aj1x . .x.AjkQQ OE x...xOE| QQQQfOE1x...xfOEkQQ 1 | k QQQ fflffl| QQQ(( Aj1x . .x.Ajk _____k___//AkQQQ QQQQ oe oe|| |oe| QfQQQQ fflffl| fflffl| QQQQ(( Ajoe-1(1)x . .x.Ajoe-1(k)_k//_Ak______________//A. We must also check that the distributivity maps of (T oe; T OE1, . .,.T OEk) c* *oincide with those of T (oe; OE1, . .,.OEk). However, both distribute to the same ending po* *int, which may be written P1 P2 (ai a0i) Q2 Q1, where P1 is the tensor product of blocks preceding the one in which ai a0iappe* *ars, and P2 is the tensor product of the terms in the same block which precede ai a0i. * * Q1 and Q2 are described analogously. Now (T oe; T OE1, . .,.T OEk) distributes first * *P1 and Q1, and then P2 and Q2, while T (oe; OE1, . .,.OEk) does it all at once. The resulting* * maps coincide by property (d) of the distributivity maps in Definition 3.3. Therefore T pres* *erves the multiproduct, and we get a map of operads, i.e., a multifunctor T : * ! P. Now suppose given a map of operads T : * ! {Pk(Ak; A)}; we must produce a r* *ing structure on A. First, the tensor product functor : A2 ! A is the functor par* *t of the image of 1 2 2, and the unit object is the image of the unique element of 0. * *Write 1n for the identity element of n. Then the strict associativity of follows from* * the fact that (12; 12, 11) = 13 = (12; 11, 12), and the unit condition follows from (12; 1* *1, 10) = 11 = (12; 10, 11). The distributivity maps dland dr arise as part of the structure of the targe* *t of 12 2 2. Properties (a), (b), (c), and (f) follow immediately from requirements for k-mo* *rphisms in P. Properties (d) and (e) follow from the facts that T is a map of operads, and* * also that (12; 11, 12) = (12; 12, 11). The distributivity maps for the images of these* * composites must therefore coincide, and both (d) and (e) follow. We therefore have a ring * *structure whenever we have a map of operads * ! {Pk(Ak; A)}. Finally, we must verify that these correspondences are inverse to each other* *. First suppose given a ring structure on A, and let T : * ! {Pk(Ak; A)} be the induce* *d map of operads. By definition, T (12) is the tensor product on A, together with both d* *istributivity maps, and the multiplicative unit is given by T (10). We therefore recover the* * original structure from its induced map of operads. 36 A. D. ELMENDORF AND M. A. MANDELL Now suppose we start with a map of operads T : * ! {Pk(Ak; A)}, and give A * *the induced ring structure. By induction using the fact that (12; 1k-1, 11) = 1k, * *we find that f1k(a1, . .,.ak) = a1 . . .ak, and from equivariance it follows that, for oe 2 k, foe(a1, . .,.ak) = aoe-1(1) . . .aoe-1(k). We therefore recover the map of operads T on underlying functors f, and we are * *left with the recovery of the distributivity maps. By equivariance, it suffices to * *recover the distributivity maps ffi1ki, which we do by induction on k. This is trivial if k* * 2. Since T is a map of operads, we have (T (12); T (1i), T (1k-i)) = T (1k). If i < k, assume by induction that ffi1iiis given by (P ai) (P a0i)dr_//P (ai a0i). Then by the definition of distributivity maps in the multiproduct (T (12); T (* *1i), T (1k-i)), we have ffi1kigiven by the composite (P ai Q) (P a0i Q)__dl_//((P ai) (P a0i))_drQ_1//P (ai a0i) * * Q, as required. In the remaining case, where i = k, we use the fact that the (sing* *le) distribu- tivity map of T (11) is the identity, together with (T (12); T (1k-1), T (11)) = T (1k), to exhibit ffi1kkas simply (P ak) (P a0k)dr_//P (ak a0k), as required. This completes the proof. Proof of Theorem 3.8. First suppose given a map of operads E * ! {Pk(Rk; R)}. T* *hen we have the composite multifunctor * _____//E *__R__//P, so by Theorem 3.4, R is associative. We therefore get all of the bipermutative * *structure except for: (1) fl , (2) The coherence diagram for fl from the requirement that (R, , 1) form a* * permu- tative category, and (3) Diagram (e0). RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 37 The symmetry isomorphism fl is the image of the isomorphism between the two ob* *jects of E 2. The coherence diagram fl a b Mc___________________//c88 a b MMM qqqq MMM qqq 1 fl MMM&& qqq fl 1 a c b now follows as a consequence of there being exactly one isomorphism in E 3 betw* *een 13 2 3 and the permutation sending (abc) to (cab). Diagram (e0) is simply the * *requirement that fl , being the image of a morphism in E 2, must be a morphism in P2(R2; R* *). A map of operads E * ! {Pk(Rk; R)} therefore determines a bipermutative structure* * on R. Suppose now that we are given that R is a small bipermutative category; we n* *eed to construct the multifunctor T : E * ! P. From Theorem 3.4, we get the map of ope* *rads on the objects * once we know that R is a ring category, and the only issue here * *is diagram (e) in Definition 3.3, which we have replaced with (e0). However, diagram (e) f* *ollows as a consequence of the commutativity of the diagram in Figure 1 (see page 38), all * *of whose subdiagrams are instances of the coherence requirements for a bipermutative cat* *egory. We therefore get a map of operads T : * ! {Pk(Rk; R)}, and it remains to ex* *tend this to the morphisms in the E k's. These consist of one isomorphism between each pa* *ir of objects. Given any pair of elements oe and OE in k, the permutative structure * *on (R, , 1) gives a canonical isomorphism aoe-1(1) . . .aoe-1(k)~=//_aOE-1(1) . . .aOE-1(k), as a composite of the maps fl ; we take this as the image of the unique morphi* *sm from oe to OE. The coherence condition for fl implies that any ways of composing * *various instances of fl that lead to the same permutation of the tensor factors give * *the same isomorphism; we use this fact multiple times below, and refer to it as "uniquen* *ess of the permutation isomorphisms". Compatibility of these permutation isomorphisms wit* *h the given distributivity maps follows from coherence of the bipermutative structure* *, specifically property (e0) using the fact that k is generated by transpositions. The unique* *ness of the permutation isomorphisms implies that Tk is a functor E k ! Pk(Rk; R). In orde* *r to see that T defines a map of operads on the morphisms, we apply a little more co* *herence theory. Given objects (oe; OE1, . .,.OEk) and (oe0; OE01, . .,.OE0k) of E k x * *E j1x . .x.E jk, there is a unique isomorphism from one to the other in E k x E j1x . .x.E jk. T* *he target of this morphism under T first permutes within blocks, and then permute* *s the blocks, while the target under T does this all at once; these are the same is* *omorphism by the uniqueness of the permutation isomorphisms. This concludes the proof tha* *t T is a map of operads, and consequently the given data determine a multifunctor E * * *! P. The proof that the passages back and forth are inverse to each other is exactly* * as in the proof of Theorem 3.4. 38 A. D. ELMENDORF AND M. A. MANDELL | | (a b c)|_E(a__ b0 _c)______________dl_________________//((a b)| (a b0* *)) c |____________ mm66 | |__EEE______________________ flmmmm | ||___EEE_________________________ mmmmm || | ____EEEflEfl_______________________ m 0 | | _______EE____________________c 6((a6 b) (a b )) | | ________EEE____________________drmm|mmm | | _________EEE____________________mm||mmm | | __________EE""__________________________|mmm | || __________________(c a b) |1(drc a b0) || | (1 fl)_2________________________________________|QQQQ| | | _______________________drQQQ|| | | _______________________QQQQ|| | | _______________________((QQfflffl|| | | ________________________ 2 | 0 | ________________________(fl|1)c1 a (b |b ) ________________________||1 dr|| _______________________||111| |dr|1 || __AEAE_____________________fflffl|111|| || 0) | (a c b) (aQQ c b fl 1|| 11 | | | QQQdrQ | 11 | | | QQQQ | 11 | | | QQ((Q fflffl| 11 | | | 01fl1 | | dr| a c (b b )1 | | | mm66m BBB 11 | | | mmmmm BB 11 | | | mmmm1 dr BBB 11 | | fflffl|m BB 11 | | a ((c b) (c b0)) BB 1 | | ll 1 fl BBB 11 | | 1 (fllfl)lll BB 11| | llll BB 1|1 fflffl|vvlll B!fflffl|,,! a ((b c) (b0 c))________________1_dl_________________//a (b b0) c Figure 1 9. Modules and Algebras in Permutative Categories In this section, we describe some of the module and algebra structures in P,* * the multi- category of permutative categories. We first define each structure in terms of * *functors and natural transformations; we then reinterpret the structure in terms of paramete* *r multi- categories. All of the parameter multicategories we describe below have contrac* *tible com- ponents in their k-morphism categories, so collapsing each component to a singl* *e point gives a map of multicategories that is the identity on objects and a weak equiv* *alence on k-morphisms. From Theorems 1.3 and 1.4, it follows that the structures we descr* *ibe pass to structures on K-theory spectra equivalent to the associated strict structure* *s. 9.1. Modules. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 39 Definition 9.1.1. Let A be a ring category and D a permutative category. A lef* *t A- module structure on D consists of a functor : A x D ! D that is strictly assoc* *iative in the sense that the diagram A x A x D _1x__//A x D x1 || || fflffl| fflffl| A x D _________//_D commutes, strictly unital in the sense that the composite D ~={1} x D _____//A x D_____//D coincides with the identity, together with natural distributivity maps dl: (a d) (a0 d) ! (a a0) d and dr: (a d) (a d0) ! a (d d0) subject to the commutativity of all the diagrams in Definition 3.3. Left module structure over a ring category can be described in terms of a pa* *rameter multicategory; recall that a ring category structure is given by a map out of t* *he operad * (Theorem 3.4). Definition 9.1.2. The multicategory `M * is the following parameter multicatego* *ry for modules: It has two objects, A (the "ring") and M (the "module"). In the case i* *n which all inputs and the output are A, we have `Mk*(Ak; A) = k, and if exactly one i* *nput is M and the output is also M, we set `Mk*(Aj-1, M, Ak-j; M) = {oe 2 k : oe(j) = * *k}. All other k-morphism sets are required to be empty. The multiproduct and *-ac* *tion are defined in exactly the same way as in the operad *; see the discussion fol* *lowing Theorem 3.4. Note that restricting our attention to the single object A gives a multifunc* *tor * ! `M *, so if we have a multifunctor `M * ! P, the image of A is a ring category. The f* *undamental theorem about left module structures on permutative categories is the following: Theorem 9.1.3. Left A-module structures on D determine and are determined by mu* *lti- functors `M * ! P sending A to A and M to D such that the restriction * ! `M * ! P 40 A. D. ELMENDORF AND M. A. MANDELL gives the structure map for A as a ring category. Proof. First suppose given a left A-module structure on D; we must produce a mu* *ltifunctor T : `M * ! P. The ring structure on A gives us the multifunctor on the k-morph* *isms of `M * involving only A, so consider oe 2 `Mk*(Aj-1, M, Ak-j; M), i.e., oe 2 * *k and oe(j) = k. We define T oe: Aj-1 x D x Ak-j ! D by the formula T oe(a1, . .,.aj-1, d, aj+1, . .,.ak) = aoe-1(1) . . .aoe-1(k-1) * *d. Since oe(j) = k, all of the objects aoe-1(1), . .,.aoe-1(k-1)are indeed objects* * of A, and this formula is simply a special instance of the usual formula T oe(b1, . .,.bk) = boe-1(1) . . .boe-1(k). The proof that this formula determines a multifunctor now proceeds exactly as i* *n the proof of Theorem 3.4. On the other hand, given a multifunctor `M * ! P sending A to A and M to D, and which restricts on A to the ring category structure map for A, we must prod* *uce a left A-module structure on D. The tensor pairing : A x D ! D is the image of * *the single element of `M *(A, M; M), and the distributivity maps are part of the st* *ructure of the target of this element. The rest of the proof now follows exactly as in * *the proof of Theorem 3.4. We have the following immediate consequence. Corollary 9.1.4. If D is a left A-module, then KD is a left KA module. When A is not just ring but actually bipermutative, we can describe a parame* *ter multi- category that captures this further structure using the translation category co* *nstruction E applied to `M *: for a multicategory of sets M, let EM denote the multicategory* * enriched over small categories for which EMk(B1, . .,.Bk; C) is the category obtained by* * applying E to Mk(B1, . .,.Bk; C). There is an obvious inclusion of multicategories M ! * *EM, where we consider M enriched over small categories with all the categories disc* *rete. Lemma 9.1.5. Let * ! `M * be the inclusion of the k-morphisms of `M * involving only A. Then the diagram of multicategories * ______//_`M * | | | | fflffl| fflffl| E * _____//E`M * RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 41 is a pushout. In other words, making the k-morphisms in * all canonically iso* *morphic forces all the other k-morphisms in `M * to be canonically isomorphic as well. Proof. Let Q be another multicategory, and suppose we have a commutative diagram * ______//`M * | | | | fflffl| fflffl| E * _______//Q of multicategories. We must show that there is a unique dashed arrow making the* * diagram of multicategories * ______//_`M4* | | 444 | | 4 fflffl| fflffl|444 E * _____//TTTTE`MG44* TTTTT GG 44 TTTTT G44 TTTT4aeae))T##G Q commute. Certainly there is no choice about the values on the objects of the k-* *morphism category E`Mk*(B1, . .,.Bk; C), since the objects are the same as the objects o* *f `M *. The values on morphisms of E * are also determined. We show that whenever oe1 * *and oe2 are objects in E`Mk*(Aj-1, M, Ak-j; M), the image of the map from oe1 to oe* *2 is also determined. Since oe2 O oe-11fixes k, we can think of it as an element of k-1,* * and let OE be the unique map in E k-1 from the identity permutation to oe2O oe-11. Then we ca* *n express the unique map from oe1 to oe2 in E`Mk*(Aj-1, M, Ak-j; M) by the formula (id,; OE, 1M ) . oe1 where , is the single object of `M2*(A, M; M). This establishes uniqueness of * *such a multifunctor, and it remains to show existence. Using the formula above to def* *ine the functors, it is straightforward to show that they preserve the symmetric group * *action and the multiproduct and therefore define a multifunctor E`M * ! Q. Corollary 9.1.6. Let R be a small bipermutative category, D a small permutative* * cate- gory. Then left R-module structures on D determine and are determined by multif* *unctors E`M * ! P sending M to D and restricting on A to the bipermutative structure map E * ! P for R. Proof. This follows immediately from Lemma 9.1.5 with Q replaced by P. Applying Theorem 1.4 we obtain the following corollary. 42 A. D. ELMENDORF AND M. A. MANDELL Corollary 9.1.7. If D is a left module over a bipermutative category R, then KD* * is weakly equivalent to a strict module over a strictly commutative ring spectrum weakly * *equivalent to KR. For right modules, the relevant definitions are as follows. Definition 9.1.8. Let A be a ring category, D a permutative category. Then the * *structure of a right A-module on D consists of a functor : D x A ! D that is strictly as* *sociative and unital in the analogous sense as in Definition 9.1.1, together with distrib* *utivity maps again defined analogously and satisfying the corresponding diagrams. Definition 9.1.9. The multicategory rM * is the following parameter multicatego* *ry for modules: It has two objects, A and M, with k-morphism sets being empty unless a* *ll inputs are A and the output is A or exactly one input is M and the output is M. In th* *e first case, the k-morphisms are k, so the endomorphism operad of A is * (as in `M ** *), but we set rMk*(Aj-1, M, Ak-j; M) = {oe 2 k : oe(j) = 1}. The *-action and multiproduct are defined exactly as in *. Theorem 9.1.10. Let A be a small ring category and D a small permutative catego* *ry. Then right A-module structures on D determine and are determined by multifuncto* *rs rM * ! P sending M to D and restricting on A to the structure map for A as a ri* *ng category. The proof is safely left to the reader, given the proof of Theorem 9.1.3. T* *he obvious analogs to Corollaries 9.1.4, 9.1.6, and 9.1.7 also hold. Just as in ordinary algebra, a right module over A is the same thing as a le* *ft module over the opposite structure "Aop", which we now define. Definition 9.1.11. The opposite map is the particular map of operads op: * ! * defined as follows. For k 0, define rk 2 k by rk(j) = k + 1 - j, so rk rever* *ses order. We then define op: k ! k by op(oe) = rk O oe. We leave to the reader the check that op defines a map of operads. Definition 9.1.12. Let A be a ring category. The opposite of A, written Aop, i* *s the ring category given by the composite op A * _____// *____//P. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 43 Corollary 9.1.13. Right A-module structures on a small permutative category D d* *eter- mine and are determined by left Aop-module structures on D. Proof. The automorphism * -op! * extends to an isomorphism `M * -op! rM * for which the diagram op * ________// * | | | | |fflffl fflffl| `M * _op__//rM * commutes. The extension is given by exactly the same formula: using the element* *s rk 2 k defined by rk(j) = k + 1 - j, we define op(oe) = rk O oe, and clearly if oe(j) * *= k, then op(oe)(j) = 1. The result now follows immediately. Corollary 9.1.14. If R is bipermutative, so is Rop. Proof. The map "op" of operads extends to the map of operads E(op): E * ! E *. 9.2. Bimodules. The following is the explicit definition of a bimodule in the context of per* *mutative categories. Definition 9.2.1. Let A and B be ring categories, and D a permutative category.* * We say that D is an A-B bimodule if D is a left A-module and a right B-module, the ass* *ociativity diagram A x D x B __x1_//D x B 1x || || fflffl| fflffl| A x D _________//D commutes, and diagrams (e) and (f) from Definition 3.3 commute in all situation* *s in which the maps are defined. For bimodule structures, the fundamental parameter multicategory is as follo* *ws. Definition 9.2.2. The bimodule parameter multicategory B * has objects A, B (the "rings", with A acting on the left and B on the right) and M (the "module"). A* *ll sets of k-maps are empty with the exception of those in which M appears exactly once* * in the input and is the output, those where all inputs and the output are A, and those* * where all inputs and the output are B. In the latter two cases the set of k-maps is * *k. In the 44 A. D. ELMENDORF AND M. A. MANDELL case of Bk*(C1, . .,.Ck; D) with Cj = D = M and all other entries either A or B* *, we set Bk* = {oe 2 k : oe(i) < oe(j) , Ci = A}. These are precisely the oe's for whic* *h the list Coe-1(1), . .,.Coe-1(k)is the list Aoe(j)-1, M, Bk-oe(j). In particular, oe(j) * *must always be one plus the number of A's occurring in the input. The k action and the multiprodu* *ct are defined exactly as for the operad *. Note in particular that restriction to either of the single objects A or B d* *etermines a multifunctor * ! B *. Theorem 9.2.3. Let A and B be small ring categories. Then an A-B bimodule struc* *ture on a small permutative category D determines and is determined by a multifuncto* *r B * ! P sending M to D, restricting on the single object A to the structure multifunc* *tor * ! P for A and on the single object B to the structure multifunctor for B. Proof. Given a bimodule structure on D and an element oe 2 Bk*(C1, . .,.Ck; D),* * we need to define a functor T oe, and we use the usual formula T oe(c1, . .,.ck) = coe-1(1) . . .coe-1(k). The proof that this gives a multifunctor B * ! P now proceeds in exactly the sa* *me way as in the proof of Theorem 3.4. Conversely, suppose we are given a multifu* *nctor T : B * ! P satisfying the conditions in the theorem. Restricting to pairs of * *objects (A, M) or (B, M) gives us restriction multifunctors `M * ! B * and rM * ! B *, * *and we immediately obtain a left A-module structure on D and a right B-module struc* *ture on D. The associativity diagram commutes because B3*(A, M, B; M) has only one elem* *ent, and diagrams (e) and (f) commute exactly as in the proof of Theorem 3.4. This c* *oncludes the proof. Corollary 9.2.4. If D is an A-B bimodule for ring categories A and B, then KD i* *s a KA-KB bimodule in symmetric spectra. In the case where A = B, we can collapse the parameter multicategory further* * using a special case of the parameter multicategory in the second example after Definit* *ion 2.4: Definition 9.2.5. The parameter multicategory bM * has two objects, A and M, and is a parameter multicategory for modules, so there are no k-morphisms unless M * *is the output and appears exactly once in the input, or else A is the output and only * *A appears in the input. In these cases the k-morphisms are k, with the multiproduct defi* *ned as in *. To compare this multicategory with the previous one, we use the following le* *mma: Lemma 9.2.6. Consider the diagram of multicategories * _____////_B_*__//bM * RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 45 where the two arrows on the left are the inclusions of the endomorphism operads* * of the ob- jects A and B, and the arrow on the right sends both A and B to A, and sends pe* *rmutations in B * to corresponding ones in bM *. This is a coequalizer diagram of multicat* *egories. Proof. The key point here is that each permutation in bMk*(Aj-1, M, Ak-j; M) ha* *s ex- actly one preimage in B *. Once we realize this, extending an equalizing multif* *unctor to bM * is simply a matter of sending all permutations to their images under the m* *ultifunc- tor. The characterization of A-A bimodules in terms of a parameter multicategory * *now follows immediately. Corollary 9.2.7. If A is a small ring category and D is a small permutative cat* *egory, then an A-A bimodule structure on D determines and is determined by a multifunc* *tor bM * ! P sending M to D and restricting on A to the ring category structure mul* *tifunctor * ! P for A. The analog of Corollary 9.2.4 now follows as well. If one or both of A and B are bipermutative, one can also describe A-B bimod* *ules with this extra structure in terms of parameter multicategories. We leave this to th* *e interested reader. We can also ask for an analogous characterization of A-A bimodules as in Cor* *ollary 9.2.7 in the case where A is bipermutative. The answer is NOT to apply E to all the m* *ulticat- egories in the diagram in Lemma 9.2.6. (This illustrates the fact that E does n* *ot preserve coequalizers). Instead, we get a multicategory described as follows. Definition 9.2.8. The multicategory bE M * is a parameter multicategory for mod* *ules, so has objects A and M, with the k-morphisms empty except in the cases where M * *appears exactly once in the input and is the output, or else all inputs and the output * *are A. We set bE Mk*(Ak; A) = E k. The objects of bE Mk*(Aj-1, M, Ak-j; M) are the elemen* *ts of k, but the objects are not all isomorphic. Instead, we look at the equivalence* * relation on k in which oe ~ oe0 if and only if oe(j) = oe0(j) and oe and oe0 are in the sa* *me coset of the left action of oe(j)-1x k-oe(j)on k. Equivalently, we could say that oe ~ oe* *0 means that oe(i) < oe(j) , oe0(i) < oe0(j) whenever 1 i k. There is exactly one morphi* *sm from oe to oe0 when oe and oe0 are equivalent and no morphisms when they are not equiva* *lent. We leave it to the reader to check that the same formula for the multiproduct in * ** extends to give multicategory structure on bE M *. Lemma 9.2.9. Consider the diagram of multicategories E * _____////_EB_*__//bE M * where the two arrows on the left are the inclusions of the endomorphism operads* * of the ob- jects A and B, and the arrow on the right sends both A and B to A, and sends pe* *rmutations to themselves. This is a coequalizer diagram of multicategories. 46 A. D. ELMENDORF AND M. A. MANDELL Proof. Given Lemma 9.2.6, the only issue is the morphisms. However, the definit* *ion of the morphisms in bE M * is precisely the requirement that two k-morphisms are isomo* *rphic in bE M * if and only if they come from isomorphic k-morphisms in EB *. The re* *sult follows. Corollary 9.2.10. Let R be a small bipermutative category. Then R-R bimodule st* *ruc- tures on a small permutative category D determine and are determined by multifu* *nctors bE M * ! P sending A to R and M to D, and which restrict on A to the bipermutat* *ive structure map E * ! P for R. Consequently, the K-theory spectrum KD is equivale* *nt to a bimodule over a strictly commutative ring spectrum equivalent to KR. This still leaves the question of what sort of bimodule structure is paramet* *erized by EbM *. The relevant definition is as follows. Definition 9.2.11. Let R be a bipermutative category. The structure of a symmet* *ric bimodule over R on a permutative category D consists of an R-R bimodule structu* *re together with a natural isomorphism fl: r d ~=d r for r an object of R and d an object of D. The isomorphism fl must be compatibl* *e with the multiplicative symmetry isomorphism fl for R, in the sense that all possib* *le diagrams of the form given in part 3 of Definition 3.1 must commute (with the 's replac* *ed with 's). We also require diagram (e0) given in Definition 3.6 to commute. Theorem 9.2.12. Let R be a small bipermutative category and D a small permutati* *ve category. Then symmetric bimodule structures for D over R determine and are det* *ermined by multifunctors EbM * ! P sending M to D and restricting on A to the structure* * map E * ! P for R as a bipermutative category. Consequently, the K-theory spectrum * *KD is equivalent to a module over a strictly commutative ring spectrum equivalent * *to KR. The proof is the same as the proof of Theorem 3.8 with bM * in place of *. 9.3. Algebras. We turn our attention next to algebras. The parameter multicategories we wi* *ll be interested in here are of the following form. Definition 9.3.1. A parameter multicategory for algebras is a multicategory A w* *ith two objects, R (the "ring") and A (the "algebra"), subject to the following conditi* *on. Suppose given inputs B1, . .,.Bk with at least one of the Bj's being equal to A. Then w* *e require that Ak(B1, . .,.Bk; R) = ;. If all the other k-morphism spaces are contractibl* *e, then we say that A is a parameter multicategory for E1 algebras. Again, we can look at the example in which all the nonempty k-morphism space* *s are a single point, and we map to a symmetric monoidal category. Then the images of b* *oth R RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 47 and A are commutative monoids, and the rest of the structure is induced by a st* *rict map of monoids from R to A given by the single element of A1(R; A). A more interesting example is given by letting S = {j : Bj = A} in the expre* *ssion Ak(B1, . .,.Bk; C) and, if not required to be empty, setting this k-morphism sp* *ace equal to k= ~, where ~ is the equivalence relation on k given by requiring oe ~ oe0* * if and only if, for all elements i and j of S, oe(i) < oe(j) , oe0(i) < oe0(j). Then a mul* *tifunctor to a symmetric monoidal category makes the image of R again a commutative monoid, * *the image of A is now a noncommutative monoid, and the map induced by the single el* *ement of A1(R; A) is central in the obvious sense. For a third example, let O be an operad. Then we can let Ak(B1, . .,.Bk; C)* * = Ok whenever it is not required to be empty. Then the images of both R and A are O-* *rings, and there is a map of O-rings given by the identity element of O1 = A1(R; A) wh* *ich determines the entire algebra structure. The explicit characterization of a central algebra over a bipermutative cate* *gory depends on the following notion of a central map from a bipermutative category to a rin* *g category. Definition 9.3.2. Let R be a bipermutative category and A a ring category. A ce* *ntral map from R to A is a lax map OE: R ! A (i.e., (OE, ~) 2 Ob (P1(R; A))) and a na* *tural isomorphism fl: OE(r) a ~=a OE(r) for r an object of R and a an object of A* *, satisfying the following conditions: (1) OE preserves the tensor product in the sense that the diagram OExOE R x R _____//A x A || || fflffl| fflffl| R ____OE___//_A commutes strictly and OE(1) = 1. (2) The lax structure map ~ preserves the distributivity maps in the sense t* *hat the diagram (OEr1 OEr2) (OEr1 _OEr3)dr//_OEr1 (OEr2 OEr3) = || 1|~| fflffl| fflffl| OE(r1 r2) OE(r1 r3) OEr1 OE(r2 r3) ~|| =|| fflffl| fflffl| OE[(r1 r2) (r1 r3)]OE(dr)//_OE(r1 (r2 r3)) and a similar diagram involving dl commute. 48 A. D. ELMENDORF AND M. A. MANDELL (3) fl must be consistent with the symmetry isomorphism fl in R in the sens* *e for all objects r1, r2 of R, the diagram OE(r1) OE(r2)fl_//OE(r2) OE(r1) = || |=| fflffl| fflffl| OE(r1 r2)OE(fl_/)/OE(r2 r1) commutes. (4) fl satisfies all instances of the diagrams in part (3) of Definition 3.1* *, and diagram (e0) of Definition 3.6. An R-algebra structure on A consists of a central map from R to A. Definition 9.3.3. Let A * be the multicategory with two objects, R (the ground * *ring) and A (the algebra). The category Ak*(B1, . .,.Bk; C) is empty if C = R and one* * or more of the Bj's are A. Otherwise, Ak*(B1, . .,.Bk; C) has k as its set of objects* *, and has morphisms as follows. Let S = {j : Bj = A} and consider the equivalence relatio* *n on the elements of k where oe ~ oe0 means that for all i and j in S, oe(i) < oe(j) , * *oe0(i) < oe0(j). We have precisely one morphism from oe to oe0 when oe ~ oe0, and no morphisms b* *etween inequivalent elements. In the previous definition, if we restrict our attention to the object R, we* * get E *, while if we restrict our attention to the object A, we get *. We wish to show* * that R- algebra structures on a small ring category A correspond to multifunctors from * *A * to P extending the structure multifunctors for both R and A. To do this, we need the* * following combinatorial lemma about permutations. Lemma 9.3.4. Suppose T k_= {1, . .,.k} and that ae 2 k is order-preserving o* *n T in the sense that if i and j are elements of T with i < j, then ae(i) < ae(j). * *Then ae can be written as a product of transpositions of consecutive integers in k_, say ae* * = t1 . .t.m, in such a way that for 1 n m, tn does not transpose two elements of tn+1 . .t.* *mT . Proof. Let the elements of T be written in order as {a1, . .,.aq}. First, we us* *e transposi- tions of the required form to map T to {1, . .,.q}; we do this by first transpo* *sing a1 with its predecessors, in order, and then repeating the process with a2 through aq. * *Then use transpositions of adjacent elements of {q + 1, . .,.k} to rearrange this set in* * the same order that ae rearranges k_\ T . Finally, start with q and transpose it with its succ* *essors, in order, until it reaches ae(aq), and repeat the process with q - 1 back through 1. The * *result is ae, with the transpositions involved having the required property. Theorem 9.3.5. Let R be a small bipermutative category and A a small ring categ* *ory. Then R-algebra structures on A determine and are determined by multifunctors fr* *om A * to P restricting on the object R to the structure multifunctor for R as a biper* *mutative RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 49 category and on the object A to the structure multifunctor for A as a ring cate* *gory. Con- sequently, KA is equivalent to a central algebra over a strictly commutative ri* *ng spectrum equivalent to KR. Proof. Suppose we are given a multifunctor from A * restricting as required. T* *hen we obtain a functor OE: R ! A as the image of the unique element 11 of A1*(R; A); * *we claim that this functor is a central map. First, we have the formula (11; 12) = (12* *; 11, 11) = 12 in A *, which we can express by saying that the diagram in A * (11,11) (R, R)_____//(A, A) 12|| |12| fflffl| fflffl| R ___11____//A commutes, and consequently its image in P OExOE R x R _____//A x A || || fflffl| fflffl| R ____OE___//_A commutes as well. A similar argument shows that OE(1) = 1. Since the commutativ* *ity of this diagram in P also requires that the distributivity maps coincide, we get t* *he diagrams showing that ~ preserves the distributivity maps. The natural isomorphism fl: O* *E(r) a ~= a OE(r) is the image of the isomorphism between the two elements of A2*(R, A;* * A) = 2. Because the diagram (11,11) (R, R)_____//(R, A) | | | | fflffl| fflffl| R ___11____//A in A * commutes when the downward arrows are both one of the two elements of 2, the isomorphism between the two possible elements on the left gets taken by OE * *to the isomorphism between the two possible elements on the right, i.e., fl = OE(fl )* *, as required. Further, diagram (e0) of Definition 3.6 is satisfied because fl is a morphism i* *n P2(R, A; A). We therefore get a central map OE: R ! A given a multifunctor A * ! P restricti* *ng to the structure multifunctors of R and A on the objects R and A, respectively. Now suppose we are given a central map OE: R ! A; we must show that this ext* *ends uniquely to a multifunctor A * ! P by requiring the multifunctor to restrict to* * the structure multifunctors for R and A and also by requiring the single element of* * A1*(R; A) 50 A. D. ELMENDORF AND M. A. MANDELL to map to OE. The functor on Ak*(B1, . .,.Bk; C) is already determined when C =* * R or when C = A and all the Bj's are A. In the other cases, set S = {i : Bi = A} as* * in the definition. It remains to determine the images of the categories Ak*(B1, .* * .,.Bk; A) with S 6= ; and S 6= {1, . .,.k}. By equivariance, it suffices to consider the* * special case S = {1, . .,.q} for q < k. The objects are the elements of k, and it is clear * *that the image of 1k is the composite 1xOEk-q Aq x Rk-q ______//_Ak_____//_A, and the images of the rest of the objects are determined by equivariance. We mu* *st also determine the images of the isomorphisms in Ak*(B1, . .,.Bk; A). For this, not* *e that when oe ~ oe0 as in the definition, oe0oe-1 is order-preserving on oeS, so by L* *emma 9.3.4, can be written as a product of transpositions of adjacent integers which are no* *t both elements of oeS. Now the image of a typical k-tuple (b1, . .,.bk) under the el* *ement oe is boe-1(1) . . .boe-1(k), and we need to produce an isomorphism between this and * *the image under oe0. Write oe0oe-1 as t1 . .t.m, where tj is a transposition of adjacent* * integers not both in tj+1 . .t.moeS, and say tm transposes i and i + 1. Then the term boe-1(* *i) boe-1(i+1) appears as part of the image under oe, and since oe-1 (i) and oe-1 (i+1) are no* *t both elements of S, the two b's are not both objects of A, so they can be transposed using fl* *. We get an isomorphism between a tensor product of elements of the form boe-1(i)= boe0-1oe0oe-1(i)= boe0-1t1...tm (i) and elements of the form boe0-1t1...tm-1(i). By iterating the process m times, we get an isomorphism between the image under* * oe and the image under oe0. The isomorphism is uniquely determined by oe0oe-1 an* *d not its presentation, because the fl's satisfy the relations among transpositions i* *n k. This completes the proof. In the special case where A is also a bipermutative category and the symmetr* *y isomor- phism is given by the isomorphism already present in A, we can give a somewhat * *simpler description. Definition 9.3.6. Let R and A be bipermutative categories. A map of bipermutat* *ive categories OE: R ! A is a lax map that preserves the tensor product, distributi* *vity maps, and multiplicative unit in the same sense that a central map does, and for whic* *h also OE(flR ) = flA . The corresponding definition in terms of a parameter multicategory is as fol* *lows. Definition 9.3.7. The multicategory AE * is a parameter multicategory for algeb* *ras, so by Definition 9.3.1 has two objects, A and R, and with AEk* (B1, . .,.Bk; C) = * *; if S 6= ; RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 51 and C = R, where S = {i : Bi = A}. Otherwise, we set AEk* (B1, . .,.Bk; C) = E * *k, so this is an example of the sort discussed as the third example following Definit* *ion 9.3.1. The proof of the following theorem can now be safely left to the reader. Theorem 9.3.8. Let R and A be small bipermutative categories. Then a map of bip* *er- mutative categories OE: R ! A determines and is determined by a multifunctor AE* * * ! P which restricts on the object R to the structure multifunctor for R and on the * *object A to the structure multifunctor for A. Consequently, KOE is equivalent to a map o* *f strictly commutative ring spectra. 10. Free Permutative Categories This section is devoted to the construction of additional examples of both r* *ing and bipermutative categories via the "free permutative category" construction. This* * associates to any small category C a small permutative category PC as follows. Let E k be* * the translation category of k. Then we define a PC = E k x k Ck. k 0 The objects of PC are the elements of the free monoid on the objects of C, with* * 0 given by the empty string and the direct sum given by concatenation, which is the mon* *oid operation. The symmetry isomorphism arises from the isomorphism in E 2 between * *the two elements of 2. Although implicit in [16], Dunn [5] apparently first obser* *ved that P defines a monad in Cat whose algebras are precisely the small permutative cat* *egories. The resulting morphisms are called the strict morphisms and are even more restr* *ictive than the strong morphisms. In fact, they are too restrictive to form a multicat* *egory. The following theorem shows how additional structure on C gives rise to addi* *tional structure on PC. Theorem 10.1. Let C be a small strict monoidal category (i.e., one equipped wit* *h a strictly associative and unital "tensor product" operation). Then PC supports the struc* *ture of a ring category. If C is permutative, then PC becomes a bipermutative category. Proof. There are actually uncountably many different ways of constructing such * *struc- ture, depending on one's choice of what we call a priority order. Let m_ denot* *e the set {1, . .,.m} for positive integers m. Then a priority order is a choice of * *bijection !m,n: mn__! m_ x n_for each m and n that is coherent in the sense that all diag* *rams of the form mnp___!mn,p_//mn_x p_ !m,np || !m,nx1|| fflffl| fflffl| m_ x np_1x!_//_m_x n_x p_ n,p 52 A. D. ELMENDORF AND M. A. MANDELL commute. By ordering m_ x n_using lexicographic order and taking the inverse o* *f the resulting bijection, we get a priority order, as we do using reverse lexicograp* *hic order, but there are uncountably many other choices as well. For example, we can use lexic* *ographic order to define a bijection m_ ! 2_(m)_x ^m_, where ^m is odd, and then for any* * m and n, use the inverse of the bijection m_ x n_____//2_(m)_x ^m_x 2_(n)_x ^n_ _1xox1_//_ 2_(m)_x 2_(n)_x ^m_x_^n__//2_(m)2_(n)^m^n_= mn__, where the unlabelled arrows are given by lexicographic order or its inverse. W* *e can use the same sort of trick for any set of primes, not just 2, to get uncountably ma* *ny additional priority orders. In any case, pick one, and call it !. Let !1 and !2 denote ! f* *ollowed by projection onto the first or second factor, respectively. Then we define a rin* *g structure on PC as follows. Write a typical object (a1, . .,.am ) of PC as mi=1(ai), an* *d write the monoidal operation in C as . Then we define the tensor product on PC by the fo* *rmula Mm Mn mnM (ai) (bj) := (a!1(k) b!2(k)). i=1 j=1 k=1 In the case where C is permutative, we can then use the symmetry isomorphism in* * C to map this to Mmn (b!2(k) a!1(k)), k=1 and then shuffle inside of PC to map this to Mmn (b!1(k) a!2(k)), k=1 defining the multiplicative symmetry isomorphism necessary for a bipermutative * *category. The reader can check that one needs only the associativity condition on a prior* *ity order to show that these definitions satisfy the requirements for a ring or a bipermutat* *ive category, respectively. An example of particular importance of this form is the free permutative cat* *egory P(*) on a one point category, which becomes a bipermutative category via this constr* *uction. The reader should be aware, however, that modules over P(*) depend strongly on * *the priority order chosen. We leave as an exercise to the reader that if we use lex* *icographic order, then any permutative category is a left module over P(*), while if we us* *e reverse lexicographic order, every permutative category is a right module over P(*). Of* * course, the two orders give opposite bipermutative structures on P(*), so the duality is to* * be expected. Other choices of priority order seem to give far fewer modules over P(*). RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 53 11. Model Categories of Rings, Modules, and Algebras in Symmetric Spectra In this section we prove Theorem 1.3. Fix a small multicategory M enriched * *over simplicial sets, and let O denote its set of objects. Let SO denote the catego* *ry obtained as the product of copies of the category S of symmetric spectra indexed on the * *set O. As a product category, SO inherits a simplicial closed model structure for each* * simplicial closed model structure on S, precisely, one with its fibrations, cofibrations, * *and weak equivalences formed objectwise (i.e., coordinatewise). Our goal is to prove tha* *t the category SM of simplicial multifunctors from M to S has a simplicial closed model struc* *ture with the fibrations and weak equivalences the maps that are fibrations and weak equi* *valences respectively in SO for the positive stable model structure on S. Throughout thi* *s section, we use the terminology stable equivalence, positive stable fibration, and acycl* *ic positive stable fibration in SM to indicate those maps in SM whose underlying* * maps in SO are weak equivalences, fibrations, and acyclic fibrations in the positive* * stable model structure. The first step is to show that the category SM has limits and colimits. For* * this, it is convenient to observe that SM is the category of algebras over a monad M on SO* * . Definition 11.1. For b 2 O, and T in SO , let 0 1 ` ` (MT )b = @ M(a1, . .,.an; b)+ ^ (Ta1^ . .^.Tan)A= n, n 0 a1,...,an2O let j: T ! MT be the map Tb ~={idb}+ ^ Tb ! M(b; b)+ ^ Tb ! (MT )b, and ~: MMT ! MT the map induced by the multiproduct of M. The proof of the following theorem in the special case of operads [15] easil* *y generalizes to multicategories. Theorem 11.2. M is a simplicial monad on the category SO . An M-algebra structu* *re on an object of SO is equivalent to an M-multifunctor structure, and the simplici* *al category of M-algebras is isomorphic to SM . Corollary 11.3. M, viewed as a functor SO ! SM , is left adjoint to the forgetf* *ul functor SM ! SO . Corollary 11.4. The category SM is complete and cocomplete (has all small limi* *ts and colimits), and is tensored and cotensored over simplicial sets. 54 A. D. ELMENDORF AND M. A. MANDELL Proof. As a category of algebras over a monad on a complete category, SM is co* *mplete, with limits and cotensors formed in SO . Since M preserves reflexive coequalize* *rs (by the argument of [7] Proposition II.7.2), SM is cocomplete with reflexive coequaliz* *ers created in SO by [7] Proposition II.7.4. General colimits are formed by rewriting the c* *olimit as a reflexive coequalizer, and the tensor of an object A of SM and a simplicial se* *t X is formed as a (reflexive) coequalizer of the form M((MA) ^ X+ ) _____////_M(A ^ X+_)__//A X. In order to prove the required factorization and lifting properties, we need* * to review briefly the positive stable model structure on S. Recall that in any category C* * with small colimits, for any set I of maps, a relative I-complex ([14] Definition 5.4) is * *a map X ! Y in C where Y = Colim Xk, with X0 = X, and Xk+1 is formed from Xk as a pushout o* *f a coproduct of maps in I. In this terminology, a map of symmetric spectra is a co* *fibration in the positive stable model structure if and only if it is a retract of a rela* *tive I+ -complex, where I+ = {Fm @ [n]+ ! Fm [n]+ | m > 0, n 0}, and Fm is the functor from simplicial sets to symmetric spectra left adjoint t* *o the m-th space functor. A map is an acyclic cofibration if and only if it is a retract * *of a relative J+ -complex for a certain set of maps J+ (q.v. [9] Definition 3.4.9 and [14] Se* *ction 14). A complete description of the maps in J+ is not difficult but would require an u* *nnecessary digression; all we need to know about the maps is that the domain and codomain * *are small, meaning that the sets of maps out of them commute with sequential colimits. For a 2 O, let 'a denote the functor S ! SO that is left adjoint to the proj* *ection functor ssa: SO ! S. For a symmetric spectrum T , the object 'aT of SO satisfies aeT b = a ('aT )b = * b 6= a. The positive stable model structure on SO then has a similar description of its* * cofibrations and acyclic cofibrations: Let '*I+ = {'af | f 2 I+ , a 2 O} '*J+ = {'af | f 2 J+ , a 2 O}. A map in SO is cofibration if and only if it is the retract of a relative '*I+ * *-complex and is an acyclic cofibration if and only if it is a retract of a relative '*J+ -compl* *ex. Let I+= M'*I+ = {M'af | f 2 I+ , a 2 O} = {Mf | f 2 '*I+ } J+= M'*J+ = {M'af | f 2 J+ , a 2 O} = {Mf | f 2 '*J+ }. The adjunction of Corollary 11.3 and the lifting properties in SO then imply th* *e following. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 55 Proposition 11.5. A map in SM is an acyclic positive stable fibration if and o* *nly if it has the right lifting property with respect to I+, if and only if it has the ri* *ght lifting property with respect to retracts of relative I+-complexes. It is a positive stable fibr* *ation if and only if it has the right lifting property with respect to J+, if and only if it has * *the right lifting property with respect to retracts of relative J+-complexes. Because the domains and codomains of the maps in I+ and J+ are small in symm* *etric spectra, the domains and codomains of the maps in I+ and J+ are small in SM . * *The Quillen small object argument then gives the following. Proposition 11.6. A map in SM can be factored as a relative I+-complex followe* *d by an acyclic positive stable fibration or as a relative J+-complex followed by a * *positive stable fibration. The proof of the following lemma is complicated but similar to the analogous* * lemma in the case of commutative ring symmetric spectra. Since we need some specific* *s of the argument in the next section, we provide the proof at the end of that section. Lemma 11.7. A relative J+-complex is a stable equivalence. The usual lifting and retract argument then gives the following. Proposition 11.8. A map in SM has the left lifting property with respect to th* *e acyclic positive stable fibrations if and only if it is a retract of a relative I+-comp* *lex. A map in SM has the left lifting property with respect to the positive stable fibration* *s if and only if it is a retract of a relative J+-complex. We have now collected all the facts we need to prove Theorem 1.3. Proof of Theorem 1.3. We have shown (in Corollary 11.4) that SM has all finite* * limits and colimits. It is clear by their definition that weak equivalences (the stable e* *quivalences) are closed under retracts and have the two-out-of-three property. Also clear f* *rom the definition is that the fibrations (the positive stable fibrations) are closed u* *nder retracts, and if we define the cofibrations in terms of the left lifting property, then i* *t is clear that these are closed under retracts. The lifting properties follow from Propositio* *n 11.5 and Proposition 11.8, and the factorization properties follow from Proposition 11.6* *. Thus, all that remain is SM7. We need to show that when i: T ! U is a cofibration and p: X ! Y is a fibrat* *ion, the map of simplicial sets SM (U, X) -! SM (U, Y ) xSM (T,Y )SM (T, X) is a fibration, and a weak equivalence if either i or p is. Using the characte* *rization in Proposition 11.8 of cofibrations and acyclic cofibrations as the maps that are * *retracts of 56 A. D. ELMENDORF AND M. A. MANDELL relative I+- and J+-complexes respectively, this easily reduces to the case whe* *n i is a map in I+ or a map in J+. Using the adjunction of Corollary 11.3, this reduces to S* *M7 in SO , which reduces to SM7 in S, proved in [9]. 12. Multifunctors and Quillen Adjunctions In this section we prove Theorem 1.4. Before we can begin the proof, we need* * to complete the statement, by giving the full definition of weak equivalence of multicatego* *ries. The definition of weak equivalence of multicategories is a generalization of* * the definition of a weak equivalence of categories enriched over simplicial sets from [6], and* * for this, we need to recall the category of components. When C is a category enriched over s* *implicial sets, the sets of components ss0C(x, y) for objects x, y have the composition ss0C(y, z) x ss0C(x, y) ! ss0C(x, z) induced by the composition in C. This composition and the identity components * *make ss0C into a category, called the category of components. A simplicial functor f* *: C ! C0 is a weak equivalence when the induced functor ss0f is an equivalence of catego* *ries of components and for all objects x, y in C, the map of simplicial sets C(x, y) ! * *C0(fx, fy) is a weak equivalence. In the following definition, we understand the category of * *components of a enriched multicategory to be the category of components of its underlying * *enriched category. Definition 12.1. A simplicial multifunctor f: M ! M0 is a weak equivalence whe* *n the induced functor ss0f is an equivalence of categories of components and for all * *a1, . .,.an, b in O, the map of simplicial sets M(a1, . .,.an; b) ! M0 (fa1, . .,.fan; fb) is * *a weak equiv- alence. We now begin the proof of Theorem 1.4 by constructing the Quillen adjunction* * asso- ciated to a simplicial multifunctor. Let f: M ! M0 be a simplicial multifunctor* * between small multicategories enriched over simplicial sets. Let O denote the set of ob* *jects of M and O0 the set of0objects of M0 . The multifunctor0f in particular induces a p* *rojection functor ssf: SO ! SO . Let 'f: SO ! SO be the left adjoint: For T an object i* *n SO and b in O0, ` ('fT )b = Ta. a2f-1(b) The multifunctor f induces a natural transformation 'fM ! M0'f, where M0 is the monad on SO0 from Definition 11.1. For an object A of SM ,0we u* *se this natural transformation and the structure map MA ! A to construct f*A in SM by* * the (reflexive) coequalizer diagram M0'fMA _____////_M0'fA__//f*A. Unwinding the universal property and the adjunctions, we obtain the following r* *esult. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 57 Proposition 12.2. f*: SM ! SM0 is left adjoint to the pullback functor f*: SM* *0 ! SM . Since the functor f* clearly preserves weak equivalences and fibrations, the* * first state- ment of Theorem 1.4 is an immediate consequence of the previous proposition. Corollary 12.3. Given small multicategories M and M0 , enriched over0simplicial* * sets and f: M ! M0 a simplicial multifunctor, the induced functor f* : SM ! SM is * *the right adjoint in a Quillen adjunction. For the rest of the section, we assume that f is a weak equivalence. We need* * to show that (f*, f*) is a Quillen equivalence. The following lemma is the first step. Lemma 12.4. A map OE: T ! U is a stable equivalence in SM0 if and only if f*OE* * is a stable equivalence in SM . Proof. By definition, f*OE is a stable equivalence in SM if and only if it is * *a stable equiva- lence0in SO , i.e., if and only if ssfOE is a stable equivalence.0Since OE is a* * stable equivalence in SM if and0only if it is a stable equivalence in SO , it follows that f* tak* *es stable equiv- alences in SM to stable equivalences in SM . Thus, it remains to show that OE* * is a stable equivalence when f*OE is. Assume that f*OE is a stable equivalence. Then for any a in O0 in the image* * of f, OEa: Ta ! Ua is a stable equivalence. If b is an arbitrary element of O0, then * *the hypothesis that f is a weak equivalence implies that we can find an a in the image of f an* *d an isomorphism from a to b in the category of components of M0 . Choosing maps in * *M0 (a, b) and M0 (b, a) in the components giving such an isomorphism and its inverse, the* *re are generalized simplicial intervals connecting the composites with the appropriate* * identity map (on a and on b). Using the naturality of OE, it follows that OEb is (level* *wise) weakly equivalent to OEa, and is therefore a positive stable equivalence. We spend much of the rest of the section proving the following theorem. Theorem 12.5. If A is a cofibrant object of SM , then the unit A ! f*f*A of the* * (f*, f*) adjunction is a stable equivalence. Assuming the previous theorem for the moment, we have all we need to prove T* *heo- rem 1.4. Proof of Theorem 1.4. It remains to show that when f is a weak equivalence, the* * Quillen adjunction (f*, f*) is0a Quillen equivalence. Let A be a cofibrant object of SM* * and B a fibrant object of SM ; we need to show that a map OE: f*A ! B is a stable equi* *valence if and only if the adjoint map _: A ! f*B is a stable equivalence. By Lemma 12.* *4, we know that OE is a stable equivalence if and only if f*OE is a stable equivalenc* *e. Since _ is the composite * A -! f*f*A f-OE!f*B, 58 A. D. ELMENDORF AND M. A. MANDELL Theorem 12.5 implies that _ is a stable equivalence if and only if f*OE is. Thi* *s concludes the proof. We now move on to the proof of Theorem 12.5. The proof requires an analysis * *of the pushouts in SM of the form B qM'xX M'xY for a map of symmetric spectra X ! Y a* *nd a map 'xX ! B in SO . For this we need to set up two constructions. For the first* *, for each x1, . .,.xk in O, construct Ux1,...,xkB as the coequalizer in SO _ ! ` ` M(a1, . .,.an, x1, . .,.xk; -)+ ^ (MB)a1,...,an= n n 0 _a1,...,an ! ____//_//_` ` M(a1, . .,.an, x1, . .,.xk; -)+ ^ Ba 1,...,an= n n 0 a1,...,an ____//_Ux1,...,xkB. where Ba1,...,anis shorthand for Ba1 ^ . .^.Ban and similarly for MB. (One map* * is induced by the action map MB ! B and the other by the multiproduct.) The purpos* *e of introducing U*B is that for any T in SO , the underlying object in SO of the c* *oproduct B q MT in SM is _ ! ` ` Ux1,...,xkB ^ Tx1 ^ . .^.Txk= k. k x1,...,xk When x1 = . .=.xk = x and x is understood, we write UkB for Ux1,...,xkB. The second construction is defined for maps of symmetric spectra g: X ! Y .* * We construct symmetric spectra Qki(g) (or Qkiwhen g is understood) for k 0, 0 * *i k inductively as follows: Qk0= X(k), Qkk= Y (k)(the k-th smash power of X and Y )* *, and for 0 < i < k, we define Qkiby the pushout square: k+ ^ k-ix i X(k-i)^ Qii-1_____// k+ ^ k-ix i X(k-i)^ Y (i) | | | | fflffl| fflffl| Qki-1__________________________//_Qki Essentially, Qkiis the k-sub-spectrum of Y (k)of with i factors of Y and k - i* * factors of X: The quotient Y (k)=Qkk-1is naturally isomorphic to (Y=X)(k). When g is Fm * * of an injection of simplicial sets X ! Y , Qkiis precisely Fmk of the subspace of Y * *k where at most i factors are in Y \ X. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 59 Combining these constructions, we get a filtration on B qM'xX M'xY as follo* *ws. Let B0 = B, and let Bk be the pushout in SO UkB ^ k Qkk-1 _____//UkB ^ k 'xY (k) | | | | fflffl| fflffl| Bk-1 _______________//Bk, where the map UkB ^ k Qkk-1! Bk-1 is induced by the map 'xX ! B. Let B1 = Colim Bk. Proposition 12.6. With notation above, B1 is isomorphic to the underlying obje* *ct of B qM'xX M'xY in SO . In order to use this below, we need to know that the map Bk-1 ! Bk is object* *wise a level cofibration of symmetric spectra. Lemma 12.7. Let T be any right k object in symmetric spectra. If g: X ! Y i* *s a cofibration, then T ^ k Qkk-1(g) ! T ^ k Y (k)is a level cofibration, i.e., lev* *el injection. Proof. It suffices to consider the case when X ! Y is a relative I+ -complex, a* *nd a filtered colimit argument reduces to the case when X ! Y is formed by attaching a singl* *e cell, i.e., is the pushout over a map Fm i: Fm @ [n]+ ! Fm [n]+ in I+ . Then the map in the statement is the pushout over the map T ^ k Qkk-1(Fm i) ! T ^ k (Fm [n]+ )(k). We can identify this as T ^ k (-) applied to the map Fmk @( [n]k)+ ! Fmk [n]k+. It is easy to check explicitly that this is a level cofibration. Proof of Theorem 12.5. It suffices to consider the case when A is an I+-complex* *, i.e., the map from the initial object M(; -)+ ^S to A is a relative I+-complex. Then A = * *ColimAn where A0 = M(; -)+ ^ S, and An+1 is formed from An as a pushout over a coproduct of maps in I+ . Since f*f*A = Colim f*f*An, it suffices to show that An ! f*f*A* *n is a weak equivalence for all n. We prove this by induction on n for all An. Specifically, we say that an I+-* *complex B can be built in n stages if, starting with B0 = M(; -)+ ^ S, we can construct B* * as a 60 A. D. ELMENDORF AND M. A. MANDELL sequence of n pushouts over coproducts of maps in I+, B0 ! B1 ! . .!.Bn = B. Our inductive hypothesis is that for any I+-complex B that can be built in n stages* *, B ! f*f*B is a stable equivalence. Since f is a weak equivalence, M(; -)+ ^ S ! M0 (; -)+* * ^ S is a stable equivalence, and this gives the base case n = 0. Our argument also needs* * the base case n = 1, where we are looking at a map of the form MT ! f*M0'fT for some T i* *n SO that is objectwise cofibrant. Using the explicit formula for M and M0 in Defini* *tion 11.1, we see that this is a stable equivalence. For the inductive step from n to n + 1, a filtered colimit argument reduces * *to the case of C = B qM'xX M'xY for X ! Y in I+ , where B can be built in n stages. We have* * the filtration preceding Proposition 12.6, B = B0 ! B1 ! . .,. C = B1 = ColimBk, whose associated graded is ` UkB ^ k (Y=X)(k), k which is isomorphic in SO to B q M'x(Y=X), with the coproduct in SM . Let B0 =* * f*B and C0 = f*C. Since C0 = B0qM0'fxXM0'fxY , we have the analogous filtration B0= B00! B01! . .,. C0 = B01 = ColimB0k, whose associated graded is isomorphic in SO0 to B0q M0'fx(Y=X). The map C ! f*C* *0 = ssfC0 preserves the filtrations, and the map of associated gradeds B q M'x(Y=X) ! ssf(B0q M0'fx(Y=X) ~=f*f*(B q M'x(Y=X)) is a stable equivalence, because B q M'x(Y=X) can be built in n stages (since n* * 1). By Lemma 12.7, the maps in the filtration are objectwise level cofibrations, and i* *t follows that each map Bk ! ssfBk is a stable equivalence. The map C ! ssfC0 = f*f*C is there* *fore a stable equivalence. The constructions in this section also provide what is needed for the proof * *of Lem- ma 11.7. Proof of Lemma 11.7. A filtered colimit argument reduces to showing that the ma* *p B ! B qM'xX M'xY is a stable equivalence for X ! Y in J+ . Let B = B0 ! B1 ! . .b.e* * as above Proposition 12.6; it suffices to show that each Bk-1 ! Bk is a stable equ* *ivalence. The quotient Bk=Bk-1 is naturally isomorphic to UkB ^ k (Y=X)(k). Moreover, Y=X is positive cofibrant and stably equivalent to the trivial symmetric spectrum ** *, and so Bk=Bk-1 is stably equivalent to the trivial object * in SO . Since the map Bk-1* * ! Bk is objectwise a level cofibration, it follows that it is a stable equivalence. RINGS, MODULES, AND ALGEBRAS IN INFINITE LOOP SPACE THEORY 61 References 1.J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on to* *pological spaces, Lecture notes in Mathematics vol. 347, Springer-Verlag, Berlin, Heidelberg, * *New York, 1973. 2.A. K. Bousfield and E. M. Friedlander, Homotopy theory of -spaces, spectra,* * and bisimplicial sets, Geometric applications of homotopy theory II, Lecture Notes in Mathematics v* *olume 658, Springer, 1978, pp. 80-130. 3.Brian Day, On closed categories of functors, Reports of the Midwest Category* * Theory Seminar IV, Lecture Notes in Mathematics vol.137, Springer, Berlin, 1970, pp. 1-38. 4.G. Dunn, En-ring categories, J. Pure Appl. Algebra 119 (1997), 27-45. 5.G. Dunn, En-monoidal categories and their group completions, J. Pure Appl. A* *lgebra 95 (1994), 27-39. 6.W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topolo* *gy 19 (1980), 427- 440. 7.A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and * *algebras in stable homotopy theory, with an appendix by M. Cole, Mathematical Surveys and Monog* *raphs, vol. 47, American Mathematical Society, Providence, RI, 1997. 8.V. Ginzburg and M. Kapranov, Koszul Duality for Operads, Duke Math. J. 76 (1* *994), 203-272. 9.M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 1* *3 (2000), 149-208. 10.J. R. Isbell, On coherent algebras and strict algebras, J. Pure Appl. Algebr* *a 13 (1969), 299-307. 11.I. Kriz and J. P. May, Operads, algebras, modules and motives, Ast'erisque 2* *33 (1995), 1-145. 12.J. Lambek, Deductive systems and categories. II. Standard constructions and * *closed categories, in Category Theory, Homology Theory and their Applications, I (Battelle Institu* *te Conference, Seattle, Wash., 1968, Vol. One), Lecture Notes in Mathematics v. 86, Springer-Verlag,* * Berlin, 1969, pp. 76-122. 13.M. L. Laplaza, Coherence for distributivity, in Coherence in Categories, Lec* *ture Notes in Mathematics v. 281, Springer-Verlag, Berlin, 1972, pp. 29-65. 14.M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of di* *agram spectra, Proc. London Math. Soc. 82 (2001), 441-512. 15.J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematic* *s vol. 271, Springer- Verlag, Berlin-Heidelberg-New York, 1972. 16.J. P. May, E1 spaces, group completions, and permutative categories, New dev* *elopments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), London Math. Soc. Lecture * *Note Ser., No. 11, Cambridge Univ. Press, 1974, pp. 61-93. 17.J. P. May, E1 ring spaces and E1 ring spectra, with contributions by Frank Q* *uinn, Nigel Ray, and Jorgen Tornehave, Lecture Notes in Mathematics vol. 577, Springer-Verlag, Be* *rlin-New York, 1977. 18.J. P. May, The spectra associated to permutative categories, Topology 17 (19* *78), 225-228. 19.J. P. May, Pairings of categories and spectra, J. Pure Appl. Algebra 19 (198* *0), 299-346. 20.J. P. May, Multiplicative infinite loop space theory, J. Pure Appl. Algebra * *26 (1982), 1-69. 21.G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312. 22.R. W. Thomason, Symmetric monoidal categories model all connective spectra, * *Theory Appl. Categ. 1 (1995), 78-118 (electronic). 23.R. Woolfson, Hyper- -spaces and hyperspectra, Quart. J. Math. Oxford Ser. (2* *) 30 (1979), 229-255. 62 A. D. ELMENDORF AND M. A. MANDELL 24.R. Woolfson, -spaces, orientations and cohomology operations, Quart. J. Mat* *h. Oxford Ser. (2) 31 (1980), 363-383. Department of Mathematics, Purdue University Calumet, Hammond, IN 46323 E-mail address: aelmendo@calumet.purdue.edu Department of Mathematics, Indiana University, Bloomington, IN 47405 E-mail address: mmandell@indiana.edu