STABLE HOMOTOPY OF ALGEBRAIC THEORIES Stefan Schwede 1 Abstract: The simplicial objects in an algebraic category admit an abstract ho* *motopy theory via a Quillen model category structure. We show that the associated stable homotopy t* *heory is completely determined by a ring spectrum functorially associated with the algebraic theory* *. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other * *theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ri* *ng spectrum which is related to Andre-Quillen homology via certain spectral sequences. We show that * *the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology o* *f the parameterizing ring spectrum. 0 Introduction The original motivation for this paper came from the attempt to generalize a ra* *tional result about the homotopy theory of commutative rings. For a map of commutative rings, D. Q* *uillen [31] defined the cotangent complex as the left derived functor of abelianization; th* *is construction is now referred to as Andre-Quillen homology. We wanted to obtain a topological v* *ariant of the cotangent complex by replacing "abelianization" by "stabilization", i.e., passa* *ge to spectra in the sense of stable homotopy theory. In [36] we made this precise by introducing a * *model category of spectra for simplicial commutative algebras. At the same time we showed that ov* *er the rational numbers, nothing really new is happening. More precisely, for a commutative Q-a* *lgebra B, the stable homotopy theory of commutative simplicial B-algebras is equivalent to th* *e homotopy theory of simplicial B-modules, see [36, Thm. 3.2.3]. Loosely speaking, stable homotop* *y and homology of commutative rings coincide rationally - just as they do for topological spac* *es. One aim of this paper is to understand some of the torsion phenomena in the sta* *ble homotopy theory of commutative rings. The main structural result is again concerned with* * the stable homo- topy category of commutative simplicial B-algebras, but where B is now an arbit* *rary commutative ring. We will see that this stable homotopy category is still a category of mod* *ules if one allows rings spectra rather than ordinary rings. For our purpose, the most convenient* * notion of ring spectrum is that of a Gamma-ring (see Definition 1.12). Gamma-rings are based o* *n a symmetric monoidal smash product for -spaces with good homotopical properties [38, 9, 25]* *. The homotopy theory of Gamma-rings and their modules is developed in [37]. The generalizatio* *n of the rational result [36, Thm. 3.2.3] then reads: Theorem: Let B be a commutative ring. Then the stable homotopy theory of augmen* *ted com- mutative simplicial B-algebras is equivalent to the homotopy theory of modules * *over a certain Gamma-ring DB. The graded ring of homotopy groups of DB is isomorphic to the ri* *ng of stable homotopy operations of commutative augmented simplicial B-algebras. If B is a Q* *-algebra, this DB is stably equivalent to the Eilenberg-MacLane Gamma-ring of B. This theorem is appropriately dealt with in a more general framework. We are le* *d to consider pointed simplicial algebraic theories, or just simplicial theories for short. A* * simplicial theory T has a category of algebras; definitions will be given later, but thinking of a T -a* *lgebra as a simplicial set _______________________________1 April 14, 1999; 1991 AMS Math. Subj. Class.: 55U35, 18C10 1 with certain algebraic operations and equational relations is a good guide line* *. Examples include simplicial sets, simplicial sets with an action of a group, (abelian) groups, m* *odules over a simplicial ring, augmented (commutative) algebras over a commutative ring, Lie algebras an* *d many more. The category of T -algebras is naturally a closed simplicial model category, th* *us allowing one to apply homotopy theoretic concepts. The category of spectra of T -algebras is al* *so a closed simplicial model category, and its homotopy theory will be referred to as the stable homot* *opy theory of T . The above theorem then becomes a special case of Theorem 4.4, which in particul* *ar states Theorem: To a simplicial theory T , there is functorially associated a Gamma-r* *ing T s. The graded ring of homotopy groups of T sis isomorphic to the ring of stable homoto* *py operations of T -algebras. The stable homotopy theory of T -algebras is equivalent to the hom* *otopy theory of modules over T s. For several examples of algebraic theories the parameterizing Gamma-ring can be* * identified with something familiar: for the theory of sets we obtain the standard model of the * *sphere spectrum; the theories of monoids and groups give different, but stably equivalent models* * for the sphere spectrum; for sets with an action of a fixed groups one gets the spherical grou* *p ring; the theory of modules over a fixed ring leads to the Eilenberg-MacLane Gamma-ring. More de* *tails on these examples can be found in Section 7; there we also list algebraic theories - suc* *h as the motivating example of commutative algebras - whose associated Gamma-rings give new homotop* *y types of ring spectra. With the help of Theorem 4.4 we can deduce several structural properties that t* *he homotopy theory of T -algebras shares with the ordinary homotopy theory of spaces. Among* * other things we provide Hurewicz and Whitehead theorems (Corollaries 5.3 and 5.4) as well as At* *iyah-Hirzebruch and universal coefficient spectral sequences (see 5.5) which relate the Quillen* *-homology of a T - algebra to its stable homotopy. In [20, Sec. 4] M. Jibladze and T. Pirashvili defined the cohomology of a theor* *y with coefficients in a functor that takes values in abelian group objects. We provide the link between* * the (co-)homology of an algebraic theory and its stable homotopy. Any coefficient functor G for t* *he (co-)homology of the theory T gives rise to a bimodule G!over the Gamma-ring T s. If T is the th* *eory of modules over some ring, a theorem of T. Pirashvili and F. Waldhausen [30, Thm. 3.2] ide* *ntifies theory homology with topological Hochschild homology (THH). In Theorem 6.7 we generali* *ze this from rings to arbitrary algebraic theories and provide a cohomological analogue: Theorem: Let T be a pointed discrete algebraic theory and G a coefficient funct* *or. Then there is a natural isomorphism H *(T ; G) ~= THH *(T s; G!) : If G is additive, then there is a natural isomorphism H *(T ; G) ~= THH *(T s; G!) : The paper is organized as follows. In Section 1 we review -spaces, Gamma-rings* * and their modules. Section 2 recalls algebraic theories. The unstable homotopy of algeb* *raic theories is discussed in Section 3. Section 4 deals with the stable homotopy of an algebra* *ic theory and proves the equivalence with modules over the associated Gamma-ring, Theorem 4.4* *. In Section 5 we describe the relationship between stable homotopy and Quillen homology for a* *lgebras over a theory. Section 6 establishes the equivalence of theory (co-)homology with topo* *logical Hochschild (co-)homology. Examples and applications are given in Section 7. The reader wit* *h little experience with the abstract notion of an algebraic theory might want to browse through th* *e examples first, 2 to get an idea of what the general theory is all about. We assume familiarity w* *ith the language of homotopical algebra [32, 16]. This paper is a substantially modified version of the main part of the author's* * thesis. It could not have been written without many discussions that I had with various colleagu* *es. I would like to thank Greg Arone, John Klein and Manos Lydakis for many conversations o* *n related and unrelated topics during my time as a graduate student at Bielefeld University. * *I am particularly indebted to Manos Lydakis for bringing to my attention the smash product of -sp* *aces (cf. [25]); this device was crucial in turning ideas into theorems. It is a pleasure to ac* *knowledge various helpful discussions I had with Bjorn Dundas, Tom Goodwillie, Teimuraz Pirashvil* *i, Charles Rezk, Brooke Shipley and Jeff Smith. Lastly, I would like to thank my adviser, Friedh* *elm Waldhausen, for his continuous support. 1 Review of -spaces and Gamma-rings The category of -spaces was introduced by G. Segal [38], who showed that it has* * a homotopy category equivalent to the stable homotopy category of connective spectra. A. K* *. Bousfield and E. M. Friedlander [9] considered a bigger category of -spaces in which the ones* * introduced by Segal appeared as the special -spaces (see 1.4). Their category admits a closed* * simplicial model category structure with a notion of stable weak equivalences giving rise again * *to the homotopy category of connective spectra. Then M. Lydakis [25] showed that -spaces admit * *internal function objects and a symmetric monoidal smash product with good homotopical properties. 1.1 -spaces. The category opis a skeletal category of the category of finite po* *inted sets. There is one object n+ = {0; 1; : :;:n} for every non-negative integer n, and morphis* *ms are the maps of sets which send 0 to 0. op is equivalent to the opposite of Segal's category [* *38]. A -space is a covariant functor from opto the category of simplicial sets taking 0+ to a on* *e point simplicial set. A morphism of -spaces is a natural transformation of functors. We denote t* *he category of -spaces by GS. We sometimes need to talk about -sets, by which we mean pointed * *functors from opto the category of pointed sets. Every -space can be viewed as a simplicial o* *bject of -sets. A symmetric monoidal smash product functor ^: opx op-! opis given by lexicograp* *hically ordering the elements of the set n+^ m+. This smash product extends to a smash * *product for all pointed sets. We denote by S the -space which takes n+ to n+, considered a* *s a constant simplicial set. The spectrum associated to the -space S (see 1.2) is the sphere* * spectrum. The representable -spaces n = op(n+; -) play a role analogous to that of the standa* *rd simplices in the category of simplicial sets. 1 is isomorphic to S. If X is a -space and K a* * pointed simplicial set, a new -space X^ K is defined by setting (X^ K)(n+) = X(n+)^ K. There are three kinds of hom objects for -spaces X and Y . There is the set of * *morphisms (natural transformations) GS(X; Y ). Then there is a simplicial hom set hom(X; Y ), defi* *ned by hom(X; Y )i = GS (X ^(i)+; Y ) ; where the `+' denotes a disjoint basepoint. In this way GSbecomes a simpliciall* *y enriched category. Finally there is an internal hom -space Hom(X; Y ) defined by Hom (X; Y ) (n+) = hom (X; Yn+^) ; where Yn+^(m+) = Y (n+^ m+). 3 A -space X can be prolonged, by direct limit, to a functor from the category of* * pointed sets to pointed simplicial sets. By degreewise evaluation and formation of the diagonal* * of the resulting bisimplicial sets, it can furthermore be promoted to a functor from the categor* *y of pointed simpli- cial sets to itself [9, x4]. This prolongation process has another description * *as the following coend [27, IX.6]. If X is a -space and K a pointed simplicial set, the value of the e* *xtended functor on K is given by Z n+2op Kn^ X(n+) : The extended functor preserves weak equivalences of simplicial sets [BF, Prop. * *4.9] and is auto- matically simplicial, i.e., it comes with coherent natural maps K ^X(L) -! X(K * *^L). We will not distinguish notationally between the prolonged functor and the original -sp* *ace. 1.2 Spectra. A spectrum X in the sense of [9, Def. 2.1] consists of a sequence * *of pointed simplicial sets Xn for n 0, together with maps S1^Xn -! Xn+1. A map of spectra X -! Y con* *sists of maps Xn -! Yn strictly commuting with the suspension maps. The homotopy gro* *ups of a spectrum X are defined as ssnX = colimissn+i|Xi| : A map of spectra is a stable equivalence if it induces isomorphisms on homotopy* * groups. A - space X extends to a simplicial functor from all pointed simplicial sets, so it* * defines a spectrum X(S) whose n-th term is the value of the prolonged -space at Sn = S1^: :^:S1 (n* * factors). For example, the -space S becomes isomorphic to the identity functor of the cat* *egory of pointed simplicial sets after prolongation. So the associated spectrum is given by S(S* *)n = Sn, i.e., S represents the sphere spectrum. The homotopy groups of a -space are those of th* *e associated spectrum, and they are always trivial in negative dimensions. A map of -spaces * *is called a stable equivalence if it induces isomorphisms of homotopy groups. 1.3 Smash products. In [25, Thm. 2.2], M. Lydakis defines a smash product for -* *spaces by the formula (X ^Y ) (n+) = colimk+^ l+-!n+X(k+ ) ^Y (l+) : The smash product is characterized by the universal property that -space maps X* * ^Y -! Z are in bijective correspondence with maps X(k+ ) ^Y (l+) -! Z(k+ ^l+) which are natural in both variables. By [25, Thm. 2.18], the smash product of * *-spaces is as- sociative and commutative with unit S, up to coherent natural isomorphism. Ther* *e is a natural isomorphism of -spaces Hom (X ^Y; Z) ~= Hom (X; Hom(Y; Z)) : In other words, the category of -spaces becomes a symmetric monoidal closed cat* *egory. 1.4 Special -spaces. A -space X is called special if the map X(k+ _l+) -! X(k+ * *) x X(l+) induced by the projections from k+ _l+ to k+ and l+ is a weak equivalence for a* *ll k and l. In this case, the weak map X(1+) x X(1+) --~-- X(2+) --X(r)---!X(1+) induces an abelian monoid structure on ss0(X(1+)). Here r : 2+ -! 1+ is the fol* *d map defined by r(1) = 1 = r(2). X is called very special if it is special and the monoid ss* *0(X(1+)) is a group. 4 By Segal's theorem ([38, Prop. 1.4], see also [9, Thm. 4.2]), the spectrum asso* *ciated to a very n)| -! |X(Sn+1)* *| adjoint special -space X is an -spectrum in the sense that the maps |X(S to the spectrum structure maps are homotopy equivalences. In particular, the ho* *motopy groups of a very special -space X are naturally isomorphic to the homotopy groups of t* *he simplicial set X(1+). 1.5 Eilenberg-MacLane -spaces. Simplicial abelian groups give rise to very spec* *ial -spaces via an Eilenberg-MacLane functor H. For a simplicial abelian group A, the -spac* *e HA is defined by (HA)(n+) = A eZ[n+] where eZ[n+] denotes the reduced free abelian group gen* *erated by the pointed set n+. HA is very special and the associated spectrum is an Eilen* *berg-MacLane spectrum for A. The homotopy groups of HA are naturally isomorphic to the homot* *opy groups of A. The functor H embeds simplicial abelian groups as a full subcategory of G* *S and it has a left adjoint, left inverse functor L. For a -space X, L(X) is the cokernel of t* *he map of simplicial abelian groups (p1)*+ (p2)*- r* : eZ[X(2+)] ----! eZ[X(1+)] : Here p1 and p2 are the two projections from 2+ to 1+ in op. The functor L is co* *mpatible with the smash product of -spaces (i.e., L is strong symmetric monoidal) and it preserve* *s finite products, see [37, Lemma 1.2]. For Q-cofibrant -spaces (see 1.6), L represents spectrum * *homology [37, Lemma 4.2]. 1.6 Model category structures. A. K. Bousfield and E. M. Friedlander introduce * *two model category structures for -spaces called the strict and the stable model categori* *es [9, 3.5, 5.2]. It will be more convenient for our purposes to work with slightly different model * *category structures, which we call the Quillen- or Q-model category structures (see [37, App. A]). T* *he strict and stable Q-structures have the same weak equivalences (hence the same homotopy ca* *tegories) as the corresponding Bousfield-Friedlander model category structures, but differen* *t fibrations and cofibrations. We call a map of -spaces a strict Q-fibration (resp. a strict Q-equivalence) if* * it is a Kan fibration (resp. weak equivalence) of simplicial sets when evaluated at every n+ 2 op. St* *rict Q-cofibrations are defined as the maps having the left lifting property with respect to all st* *rict acyclic Q-fibrations. The Q-cofibrations can be characterized in the spirit of [32, II.4 Remark 4] as* * the injective maps with projective cokernel, see [37, Lemma A3 (b)] for the precise statement. By* * [32, II.4 Thm. 4], the strict Q-notions of weak equivalences, fibrations and cofibrations make* * the category of -spaces into a closed simplicial model category. More important is the stable Q-model category structure. This one is obtained * *by localizing the strict Q-model category structure with respect to the stable equivalences. * * We call a map of -spaces a stable Q-equivalence if it induces isomorphisms on homotopy groups* *. The stable Q-cofibrations are the strict Q-cofibrations and the stable Q-fibrations are de* *fined by the right lifting property with respect to the stable acyclic Q-cofibrations. By [37, Thm* *. 1.5], these stable notions of Q-cofibrations, Q-fibrations and Q-equivalences make the category of* * -spaces into a closed simplicial model category. A -space X is stably Q-fibrant if and only if* * it is very special and X(n+) is fibrant as a simplicial set for all n+ 2 op. The Q-model category* * structure is compatible with Lydakis' smash product, see [37, Lemma 1.7]. For example, smas* *hing with a Q-cofibrant -space preserves stable equivalences. Bousfield and Friedlander also introduce strict and stable model category struc* *tures for spectra. A map of spectra X -! Y is a strict fibration (resp. strict weak equivalence) i* *f all the maps 5 Xn -! Yn are fibrations (resp. weak equivalences) of simplicial sets. It is a s* *trict cofibration if X0 -! Y0 and Xn [Xn-1 Yn-1 -! Yn (for n 1) are cofibrations of simplicial sets. The stable weak equivalences ar* *e the maps which induce isomorphisms on homotopy groups. The stable cofibrations coincide with t* *he strict cofibra- tions and the stable fibrations are the maps with the right lifting property wi* *th respect to stable acyclic cofibrations. There is a more explicit characterization of the stable f* *ibrations in [9, 2.2]. In [9, Thm. 2.3] it is shown that the stable notions of fibrations, cofibrations a* *nd weak equivalences make the category of spectra into a closed simplicial model category. We show i* *n Lemma A.3 that this model category structure is cofibrantly generated (see [15], [37, Def. A2]* *). 1.7 Quillen equivalences. An adjoint functor pair between model categories is c* *alled a Quillen pair if the left adjoint L preserves cofibrations and acyclic cofibrations. An * *equivalent condition is to demand that the right adjoint R should preserve fibrations and acyclic fi* *brations. Under these conditions, the functors pass to an adjoint functor pair on homotopy cate* *gories (see [32, I.4 Thm. 3], [16, Thm. 9.7 (i)]). A Quillen functor pair is called a Quillen equiva* *lence if the following condition holds: for every cofibrant object A of the source category of L and f* *or every fibrant object X of the source category of R, a map L(A) -! X is a weak equivalence if * *and only if its adjoint A -! R(X) is a weak equivalence. Sometimes the right adjoint functo* *r R preserves and detects all weak equivalences. Then the pair is a Quillen equivalence if fo* *r all cofibrant A the unit A -! R(L(A)) of the adjunction is a weak equivalence. A Quillen equivalenc* *e induces an equivalence of homotopy categories (see [32, I.4 Thm. 3], [16, Thm. 9.7 (ii)]),* * but it also preserves higher order structure like (co-)fibration sequences, Toda brackets and the hom* *otopy types of function complexes. If two model categories are related by a chain of Quillen e* *quivalences, they can be viewed as having the same homotopy theory. The functor that sends a -space X to the spectrum X(S) has a right adjoint [9, * *Lemma 4.6], and these two functors form a Quillen pair. One of the main theorems of [9] say* *s that this Quillen pair induces an equivalence between the homotopy category of -spaces, taken wit* *h respect to the stable structure of [9], and the stable homotopy category of connective spectra* * (see [9, Thm. 5.8]). Since every Q-cofibration is also a cofibration in the sense of Bousfield and F* *riedlander, and since the stable equivalences coincide in the two model category structures, the adjo* *int functor pair of [9, Lemma 4.6 and x5] is also a Quillen pair with respect to the stable Q-model* * category structure for -spaces. 1.8 The assembly map. Given two -spaces X and Y , there is a natural map X ^Y -* *! X O Y from the smash product to the composition product. The formal and homotopical * *properties of this assembly map are of fundamental importance to this paper. Since -space* *s prolong to functors defined on the category of pointed simplicial sets, they can be compos* *ed. Explicitly, for -spaces X and Y , we set (X O Y )(n+) = X(Y (n+)). This composition O is a mono* *idal (though not symmetric monoidal) product on the category of -spaces. The unit is the sam* *e as for the smash product, it is the -space S (alias 1) which as a functor is the inclusion* * of op into all pointed simplicial sets. The assembly map is obtained as follows. Prolonged -spaces are naturally simpli* *cial functors [9, x3], which means that there are natural coherent maps X(K) ^L -! X(K^ L). This * *simplicial structure gives maps X(n+) ^Y (m+) ----! X(n+^ Y (m+)) ----! X(Y (n+^ m+)) 6 natural in both variables. From this the assembly map X ^Y -! X O Y is obtain* *ed by the universal property of the smash product of -spaces. The assembly map is associa* *tive and unital, S being the unit for both ^and O. In technical terms: the identity functor on t* *he category of - spaces becomes a lax monoidal functor from (GS; ^) to (GS; O). The crucial homo* *topical property of the assembly map is that it is a stable equivalence whenever X or Y is cofib* *rant (see [25, Prop. 5.23]). 1.9 Stable excision. Every functor obtained from a -space by prolongation pres* *erves weak equivalences of simplicial sets and connectivity of maps [9, 4.9 and 4.10]. Bu* *t the homotopy functors arising as prolonged -space have further connectivity and excision pro* *perties, such as for example the one we prove now. Lemma 1.10 Let F be a -space and X -! Y an injective map between simply connec* *ted pointed simplicial sets. Then the map F (Y )=F (X) ----! F (Y=X) is at least as connected as the suspension of X ^(Y=X). Proof: Every -space admits a strict equivalence F c-! F from a Q-cofibrant -spa* *ce. Then the induced maps F c(X) -! F (X) are weak equivalences for all simplicial sets X, s* *o we can assume that F is Q-cofibrant. If F = n^ K for some simplicial set K, then F (X) ~=Xn^ * *K and the lemma can be verified for F by inspection. Now we consider an injection of -spaces F - ! G, we assume that the lemma holds* * for F and the quotient G=F , and we claim that the lemma follows for G. Since all sp* *aces in sight are simply-connected, it suffices to show that the homotopy cofibre (mapping co* *ne) of the map G(Y )=G(X) -! G(Y=X) is as connected as X ^(Y=X). We can calculate the total ho* *motopy cofibre of the square F (Y )=F (X)___wG(Y )=G(X) | | | | |u |u F (Y=X) ________G(Y=X)w in two ways. If we take horizontal homotopy cofibres first and use that the lem* *ma holds for the -space G=F , we conclude that the total homotopy cofibre is as connected as X ^* *(Y=X). By first taking vertical homotopy cofibres and using the lemma for F we see that t* *he lemma holds for G. We now know that if the lemma holds for a -space F and if G is obtained from F * *by cobase change along one of the generating Q-cofibrations (see proof of [37, Thm. 1.5]) n^(@i)* *+ -! n^(i)+, then the lemma holds for G. Also if the lemma holds for all -spaces in a (possi* *bly transfinite) sequence of cofibrations, then it holds for the colimit of the sequence. Finall* *y, the property we are interested in is preserved under retract. This finishes the proof since all -sp* *aces can be obtained from the trivial -space by these operations (by the small object argument [37, * *Lemma A1]). __|_| 7 1.11 Gamma-rings and their modules. Our notion of ring spectrum is that of a Ga* *mma-ring. Gamma-rings are the monoids in the symmetric monoidal category of -spaces with * *respect to the smash product and they are special cases of `Functors with Smash Product' (* *FSPs, cf. [5, 1.1], [30, 2.2]). One can describe Gamma-rings as `FSPs defined on finite sets'. It w* *as tempting to call these monoids `-rings' but since that term should refer to a functor from opto * *the category of rings, the name `Gamma-ring' was chosen instead. A more detailed discussion of * *the homotopy theory of Gamma-rings can be found in [37]. Definition 1.12A Gamma-ring is a monoid in the symmetric monoidal category of -* *spaces with respect to the smash product. Explicitly, a Gamma-ring is a -space R equip* *ped with maps S -! R and R ^R -! R ; called the unit and multiplication map, which satisfy certain associativity and* * unit conditions (see [27, VII.3]). A Gamma-ring R is commutative if the multiplication map is u* *nchanged when composed with the twist, or the symmetry isomorphism, of R ^R. A map of Gamma-r* *ings is a map of -spaces commuting with the multiplication and unit maps. If R is a Gamma* *-ring, a left R-module is a -space N together with an action map R ^N -! N satisfying associa* *tivity and unit conditions (see again [27, VII.4]). A map of left R-modules is a map of -s* *paces commuting with the action of R. We denote the category of left R-modules by R-mod. One similarly defines right modules. The unit S of the smash product is a Gamma* *-ring in a unique way. The category of S-modules is isomorphic to the category of -spaces. For a * *Gamma-ring R the opposite Gamma-ring Rop is defined by twisting the multiplication with th* *e symmetry isomorphism of R ^R. Then the category of right R-modules is isomorphic to the* * category of left Rop-modules. The smash product of two Gamma-rings is naturally a Gamma-rin* *g. An R-T - bimodule is defined to be a left (R ^T op)-module. Because of the universal pro* *perty of the smash product of -spaces (see 1.3), Gamma-rings are in bijective correspondence with * *lax monoidal functors from the category op to the category of pointed simplicial sets (both * *under smash product). Similarly, commutative Gamma-rings correspond to lax symmetric monoid* *al functors. Standard examples of Gamma-rings are monoid rings over the sphere Gamma-ring S * *and Eilenberg- MacLane models of classical rings. If M is a simplicial monoid, we define a -sp* *ace S[M] by S[M] (n+) = M+ ^n+ (so S[M] is isomorphic, as a -space, to S ^M+ ). There is a unit map S -! S[M] * *induced by the unit of M and a multiplication map S[M] ^S[M] -! S[M] induced by the multiplica* *tion of M which turn S[M] into a Gamma-ring. This construction of the monoid ring over S * *is left adjoint to the functor which takes a Gamma-ring R to the simplicial monoid R(1+). If B is a simplicial ring, then the Eilenberg-MacLane -space HB is naturally a * *Gamma-ring, simply because H is a lax monoidal functor [37, Lemma 1.2]. The functor H is fu* *ll and faithful when considered as a functor from the category of simplicial rings to the categ* *ory of Gamma-rings. The functor L is still left adjoint and left inverse to H. More examples of Gam* *ma-rings arise from simplicial algebraic theories and as endomorphism Gamma-rings, see 4.5 and 4.6 * *below. Modules over a Gamma-ring form a model category. A map of R-modules is called * *a weak equivalence (resp. fibration) if it is a stable Q-equivalence (resp. stable Q-f* *ibration) as a map of -spaces. A map of R-modules is called a cofibration if it has the left lift* *ing property with respect to all acyclic fibrations in R-mod. By [37, Thm. 2.2], the category of* * left R-modules becomes a closed simplicial model category this way. For a simplicial ring B, t* *he functors H and 8 L are a Quillen equivalence between the model category of HB-modules and the mo* *del category of simplicial B-modules [37, Thm. 4.4]. The category of R-modules inherits a smash product. More precisely, let M be a * *right R-module and N a left R-module. Then the smash product M^R N is defined as the coequali* *zer, in the category of -spaces, of the two maps M ^R ^N ----!----!M ^N induced by the acti* *on of R on M and N respectively. If N is a cofibrant left R-module then the functor -^RN * *takes stable equivalences of right R-modules to stable equivalences of -spaces [37, Thm. 2.2* *]. We define the derived smash product M ^LRN of M and N in the usual way: we choose a cofibrant* * left R-module Nc and a weak equivalence Nc -~!N and set M ^LRN = M ^RNc. The derived smash pr* *oduct is well defined up to stable equivalence. There are certain standard Tor spectral * *sequences converging to the homotopy groups of M ^LRN, see [37, Lemma 3.1]. 2 Algebraic theories Algebraic theories, introduced by F. W. Lawvere [23], formalize the concept of * *an algebraic object as a set together with n-ary operations for various n 2 N and equational relati* *ons. A detailed exposition of algebraic theories can be found in [6, Sec. 3]. To do homotopy th* *eory, we use algebraic theories which are enriched over the category of simplicial sets; these simplic* *ial theories have been considered by C. L. Reedy [33]. The version of algebraic theories enriched over* * topological spaces can be found in [3, (6)], [4, Chpt. II] or [35, 34]. For many purposes, topolog* *ical and simplicial theories can be used interchangeably: the geometric realization and singular co* *mplex functors commute with finite products, so applying these to the simplicial hom set or th* *e hom spaces respectively gets one from simplicial to topological theories and vice versa. W* *e discuss examples of algebraic theories in 2.6 and Section 7. An algebraic theory is essentially a category with objects indexed by the natur* *al numbers in such a way that the n-th object is the n-fold product of the first object, in a spec* *ified way. The prototypical example (and in fact the initial algebraic theory) is the category* * opposite to the category opof finite pointed sets. Definition 2.1A simplicial theory is a pointed simplicial category T together w* *ith a functor -! T . It is required that T has the same discrete set of objects as , and tha* *t -! T is the identity on objects and preserves products. A morphism of simplicial theori* *es is a product preserving simplicial functor commuting with the functor from . One should think of the simplicial set homT(n+; 1+) as the space of n-ary opera* *tions in the theory T . If all the simplicial hom sets are discrete and if one omits the condition * *that T be pointed, one recovers the original definition of an algebraic theory. For emphasis we re* *fer to such theories as discrete theories. Since morphisms of theories are always the identity on ob* *jects, a simplicial theory is the same as a simplicial object of pointed discrete theories. There i* *s an initial simplicial theory, the theory of pointed sets. Formally it is the category together with * *the identity functor. Definition 2.2If T is a simplicial theory, a T -algebra is a product preserving* * simplicial functor X from T to the category of pointed simplicial sets. A morphism of T -algebras* * is a natural transformation of functors. X(1+) is called the underlying simplicial set of th* *e T -algebra X. 9 Note that even if T happens to be a discrete theory, T -algebras are still allo* *wed to have a simplicial * *+; 1+) gives direction, unless otherwise stated. For a T -algebra X, each morphism ' 2homT(n* * 0 rise to a map X(1+)n ~=X(n+) -X(')---!X(1+) : This justifies thinking of a T -algebra as the underlying simplicial set togeth* *er with n-ary operations parameterized by the simplicial set homT(n+; 1+). A morphism OE : R -! T of sim* *plicial theories induces a functor OE* : T -alg-! R-algby precomposition with OE. The functor OE* ** always has a left adjoint OE* [6, Thm. 3.7.7]. 2.3 Free T -algebras. The forgetful functor T -alg-! (pt. simpl. sets), X 7! X(* *1+) has a left adjoint, the free T -algebra functor F T. For a pointed simplicial set Y , the * *underlying simplicial set of F T(Y ) is given by the coend Z n+2 F T(Y ) (1+) = Y n^homT(n+; 1+) : (The forgetful functor is equal to the functor j* for the unique theory morphis* *m j : -! T . Hence the free functor F Tis isomorphic to j*.) For any n, the representable functor homT(n+; -) is a T -algebra isomorphic to * *the free T -algebra generated by n+. In fact, this gives an equivalence of simplicial categories be* *tween T opand the full subcategory of the finitely generated free T -algebras inside T -alg (cf. * *[6, Prop. 3.8.5]). The composite of the free T -algebra functor with the forgetful functor has the str* *ucture of a triple on the category of pointed simplicial sets. A triple also has a notion of algebras* * over it; however the category of algebras over the triple derived from T is equivalent to the catego* *ry of T -algebras, so there is no ambiguity as to what a T -algebra is. Note that the triple, cons* *idered as a functor from the category of pointed simplicial sets to itself, is degreewise evaluable* *, i.e., it comes from a -space. This leads to an alternative characterization of algebraic theories via the fre* *e T -algebra functor: the category of simplicial theories is equivalent to the category of those trip* *les on the category of pointed simplicial sets which are degreewise evaluable and commute with filt* *ered colimits. Yet another way of putting this is as follows: we denote by T sthe restriction of t* *he free T -algebra functor to the category op. So T sis a -space which comes with maps T sO T s-! T s and S -! T s which are associative and unital. This just says that T sbecomes a monoid in th* *e category of - spaces with respect to the composition product of 1.8. The functor that sends a* * simplicial theory T to the -space T swith the composition product structure is an equivalence of * *categories (simplicial theories)~=(monoids(inGS; O)) : This result can be found in [6, Prop. 4.6.2] for discrete theories, and in [4, * *Prop. 2.30] for topological theories. 2.4 Models. A T -algebra is also called a model of T in the category of pointed* * simplicial sets. We will also consider models of a theory in categories other than simplicial se* *ts. Let C be a pointed category which has finite products and which is enriched over simplicial sets. * *A model of T in C is a product preserving simplicial functor X : T -! C. A morphism of models in * *C is a natural transformation of functors. The object X(1+) 2 C is called the underlying objec* *t of X. We will denote by C(T ) the category of models of T in C. For example, a group object o* *f T -algebras is the 10 same thing as a model of T in the category of simplicial groups. Group objects * *of T -algebras can also be viewed as models of the theory G of groups (see 2.6) in the category of* * T -algebras. In this paper we will consider models of T in the categories of simplicial abelian grou* *ps Ab(T ), spectra Sp(T ) and -spaces GS(T ). We will later need the following lemma whose proof we omit. Lemma 2.5 Let C and D be two categories with finite products which are enriche* *d over simplicial sets. Let L : C -! D and R : D -! C be a simplicial adjoint functor pair such * *that the left adjoint L preserves finite products. Then for any simplicial theory T , composi* *tion with L and R is an adjoint functor pair between the categories of models of C(T ) and D(T ). A first instance of this lemma is the geometric realization and singular comple* *x functor pair between the categories of simplicial sets and compactly generated topological s* *paces. Another example to which we will apply the lemma is the adjoint functor pair H and L be* *tween -spaces and simplicial abelian groups. 2.6 Example: the theory of groups. To illustrate the above definitions, we rec* *all how the familiar example of the category of groups fits into the abstract framework. We* * denote the theory of groups by G. This is a discrete theory with homG(k+ ; n+) equal to the set * *of n-tuples of elements of the free group on k generators fl1; : :;:flk. The identity morphism* * of homG(k+ ; k+ ) is the tuple (fl1; : :;:flk) whose i-th component is the word consisting only o* *f the i-th generator. Composition is given by substitution: if (w1; : :;:wn) and (v1; : :;:vk) are tu* *ples of words in k resp. m generators, their composite is the tuple (w1(v1; : :;:vk); : :;:wn(v1; : :;:vk)) : Here wi(v1; : :;:vk) means that in the word wieach generator flj is substituted* * by the entire word vj. The functor -! G is given by hom (k+ ; n+) = {0; 1; : :;:k}n----!homG(k+ ; n+) (i1; : :;:in)7-! (fli1; : :;:flin) with the convention fl0 = 1. We claim that the category of G-algebras is equivalent to the category of simpl* *icial groups. So let X : G -!(pt. simpl. sets) be a G-algebra, i.e., a product preserving functo* *r. Then the underlying simplicial set X(1+) has a group structure as follows. The word fl1f* *l2 is an element of homG(2+; 1+), so it gives rise to a multiplication map = X(fl1fl2) : X(1+)2 ~=X(2+) -! X(1+) : The word fl-11is an element of homG(1+; 1+), so it gives rise to an inverse map = X(fl-11) : X(1+) -! X(1+) : The associativity and inverse conditions are codified in the category G. We ex* *plain this for associativity. We consider the two elements (fl1fl2; fl3) and (fl1; fl2fl3) of* * HomG(3+; 2+). Since multiplication in the free group on 3 generators is associative, the equality (fl1fl2) O (fl1fl2; fl3) = fl1fl2fl3 = (fl1fl2) O (fl1; fl2fl* *3) holds in homG(3+; 1+). The product preserving functor X takes this relation to * *the associativity condition O (xid) = O (idx). Hence the underlying simplicial set of a G-algeb* *ra is naturally a simplicial group. 11 For the converse let H be a simplicial group. Define a functor H : G -!(pt. sim* *pl. sets) on objects H(n+) = Hn. The behavior on morphisms is again given by substitution: for w * *= (w ; : :;:w ) by * * 1 n from homG(k+ ; n+), define H (w)(h1; : :;:hk) = (w1(h1; : :;:hk); : :;:wn(h1; : :;:hk)) : Here wi(h1; : :;:hk) means that the elements hj 2 H are substituted for the gen* *erators flj into the word wi, and then multiplication is carried out in the group H. We omit the ver* *ification that this H is a functor and that we in fact described an equivalence of categories G-alg~= (simplicial groups): This example illustrates the general pattern: an arbitrary algebraic theory can* * be recovered from its category of algebras as the opposite of the full subcategory of finitely ge* *nerated free objects. There is also a criterion for when a category with an adjoint functor pair to t* *he category of sets is (equivalent to) the category of algebras over some algebraic theory, see [6,* * Thm. 3.9.1]. The example of the theory of groups is somewhat special because here the operations* * are generated by unary and binary operations, and all relations involved at most three genera* *tors. This need not be true for general theories. In fact, what makes Lawvere's notion of algeb* *raic theories so elegant is that generating operations and relations are not mentioned at all. I* *nstead, the category underlying the theory encodes all possible operations and their relations at th* *e same time. 3 Unstable homotopy The model category structure for algebras over a discrete theory is due to Quil* *len [32, II.4 Thm. 4]. For simplicial theories it was established by Reedy [33, Thm. I] and for to* *pological theories by Schw"anzl and Vogt [34, Thm. B]. A map of T -algebras is a weak equivalence or * *fibration if it is a weak equivalence or fibration on underlying simplicial sets respectively. A map* * of T -algebras is a cofibration if it has the left lifting property with respect to all acyclic fib* *rations. A map A -! B of T -algebras is called a free map if there exists subsets Cn Bn which are st* *able under the simplicial degeneracy operators and such that in every simplicial dimension n, * *the induced map An q F Tn(Cn) -! Bn is an isomorphism of discrete Tn-algebras. Theorem 3.1 [33, Thm. I] Let T be a simplicial theory. Then the category of T* * -algebras is a closed simplicial model category. The cofibrations are precisely the retracts o* *f free maps. Proof: The model category structure follows from the lifting lemma A.2, applied* * to the forgetful and free T -algebra functors between the categories of T -algebras and the cate* *gory of pointed simplicial sets. The category of T -algebras is locally finitely presentable, s* *ee Lemma A.1. As the fibrant replacement functor Q we can take either Kan's functor Ex1 [21] or the * *composition of the singular complex and geometric realization functor (we then have to work in* * the category of compactly generated topological spaces). Each of these functors is simplici* *al and preserves finite products, so it passes to T -algebras. Quillen's argument [32, II.4 Rem.* * 4] shows that the cofibrations are the retracts of the free maps. * * __|_| 12 3.2 The bar resolution. It will be convenient to have at our disposal the stan* *dard cotriple resolution, also called bar resolution, for T -algebras. With the bar resoluti* *on, homotopical properties of free objects can be extended to all cofibrant objects. We denote* * by T the composite of the forgetful functor T -alg-!(pt. simpl. sets)with the free T -al* *gebra functor F T: (pt. simpl. sets)-! T.-algThis T is a cotriple on the category of T -algeb* *ras, so for any T -algebra X a simplicial object of T -algebras B(X) is defined by B(X)n = (T)n* *+1(X) and with the usual simplicial face and degeneracy maps (see [28, 9.6]). By construction,* * B(X)n is a free T -algebra and all structure maps except one of the face maps are free maps wit* *h respect to the defining generators. If Y* is any simplicial object of T -algebras, then the di* *agonal of the under- lying bisimplicial set is naturally another T -algebra. We denote this diagonal* * T -algebra by |Y*| and refer to it as the geometric realization. The bar resolution is augmented o* *ver the constant simplicial T -algebra X, so there is a map of T -algebras |B(X)| -! X. Lemma 3.3 The augmentation map |B(X)|-! X is a weak equivalence of T -algebras* *. If X -! Y is a map of T -algebras which is injective on underlying simplicial sets, then * *|B(X)| -! |B(Y )| is a free map of T -algebras. In particular, |B(X)| is cofibrant as a T -algebra. Proof: The fact that the augmentation map |B(X)|-! X is a weak equivalence on u* *nderlying simplicial sets is a well known property of the bar construction, see e.g. [28* *, Prop. 9.8]. It remains to show that |B(-)| takes injective maps to free maps. In simplicial di* *mension n, the map B(X)n -! B(Y )n is freely generated, in the category of discrete Tn-algebras, b* *y the injective map underlying (T)n(X)n -! (T)n(Y )n. So if we let Cn denote the complement of the * *image of (T)n(X)n in (T)n(Y )n, then the sets Cn satisfy the conditions in the definitio* *n of a free map. __|_| As an application of the bar resolution we obtain a homotopy invariance propert* *y, which can be found in [33, Cor. p. 37]. In the context of topological theories, results of a* * similar kind can be found in [3, Thm. (8)] and [4, Thm. 4.58]. Lemma 3.4 Let OE : T -! R be a morphism of simplicial theories which is a weak* * equivalence on all simplicial hom sets. Then the adjoint functors OE* and OE* are a Quillen* * equivalence between the model categories of T -algebras and R-algebras. Proof: OE* preserves underlying simplicial sets, hence weak equivalences and f* *ibrations, so OE* and OE* form a Quillen pair. Since OE* detects and preserves all weak equivalen* *ces, it remains to check that for every cofibrant T -algebra X, the unit of the adjunction X -! OE* **OE*X is a weak equivalence. If X is freely generated by a finite set, this unit map is a weak* * equivalence by assumption. If X is freely generated by an arbitrary set, it is the filtered co* *limit, over cofibrations, of finitely generated free T -algebras. If X is freely generated by a simplicia* *l set, the realization lemma (degreewise weak equivalences of bisimplicial sets induce weak equivalenc* *es on the diagonal simplicial sets, see [19, Prop. 2.4]) reduces to the discrete case. The bar re* *solution and the realization lemma reduce the general case to the free case. * * __|_| 3.5 Homotopy fibre sequences. Since T -algebras have underlying simplicial sets* *, we can in- troduce the usual notions of connectivity of maps and objects. By the homotopy* * groups of a T -algebra we always mean the homotopy groups of the geometric realization of t* *he underlying simplicial set. A T algebra will be called n-connected if all homotopy groups b* *elow and including dimension n are trivial. A map of T -algebras is called n-connected if it induc* *es isomorphisms on homotopy groups below dimension n and an epimorphism in dimension n (for all ch* *oices of base- point in the underlying simplicial set of the source). The homotopy fibre of a * *map of T -algebras X -! Y is defined by choosing a factorization in the category of T -algebras X _____wW~ _____Yww 13 of the map into a weak equivalence and a fibration and then taking the categori* *cal fibre of the fibration W -! Y . The homotopy fibre is independent up to weak equivalence of* * the choice of factorization and it is also a homotopy fibre in the underlying category of * *simplicial sets. If X -! Y -! Z are two maps of T -algebras whose composite is trivial, there is an* * induced map from X to the homotopy fibre of Y -! Z. We call the sequence X -! Y -! Z an n-h* *omotopy fibre sequence if the map X -! hofibre(Y -! Z) is n-connected. Theorem 3.6 Let X -! Y be an (n + k)-connected cofibration between n-connected* * cofibrant T -algebras (with n 1, k 0). Then the cofibration sequence of T -algebras X ----! Y ----! Y==X is a (2n+k)-homotopy fibre sequence. Here Y==X denotes the quotient in the cate* *gory of T -algebras which has to be distinguished from the quotient of the underlying simplicial se* *ts. Proof: We first prove the theorem in the special case where the map X -! Y is o* *btained from an (n + k)-connected cofibration A -! B between n-connected simplicial sets by * *application of the free T -algebra functor. In this case the quotient of T -algebras Y==X is * *isomorphic to the free T -algebra generated by the quotient of simplicial sets B=A. The free T -a* *lgebras functor is a prolonged -space, so by Lemma 1.10 the map Y=X = T (B)=T (A) ----! T (B=A) = Y==X is (2n + k + 2)-connected. By the Blakers-Massey homotopy excision theorem, th* *e sequence X -! Y -! Y=X is a (2n + k)-homotopy fibre sequence of simplicial sets. The the* *orem follows in the special case. In the general case we use the bar resolution. The cofibration sequence of T -* *algebras X -! Y -! Y==X admits a map from the cofibration sequence |B(X)| -! |B(Y )| -! |B(X)* *|==|B(Y )|. Since all objects in sight are cofibrant and the maps |B(X)| -! X and |B(Y )| -* *! Y are weak equivalences, the map induced on cofibres |B(X)|==|B(Y )| -! Y==X is also a wea* *k equivalence. So it suffices to show that the sequence |B(X)| ----! |B(Y )| ----! |B(X)|==|B(Y )| is a (2n + k)-homotopy fibre sequence. Geometric realization of simplicial T -a* *lgebras commutes with taking quotients, so all three T -algebras are realizations of simplicial * *objects. In a fixed simplicial dimension, all objects are freely generated by certain simplicial se* *ts, and all maps are free maps. So by the previous paragraph, the sequence of bar resolutions is a (* *2n + k)-homotopy fibre sequence in every simplicial degree. But geometric realization preserves* * connectivity and homotopy fibre sequences of connected objects [9, Thm. B.4], which finishes the* * proof. __|_| 4 Stable homotopy The goal of this section is to show that the stable homotopy theory of T -algeb* *ras is equivalent to the homotopy theory of modules over a certain Gamma-ring T s. The theorem we ar* *e heading for has a well known algebraic analog which we first recall. The notion of an abeli* *an group object in the category of T -algebras coincides with that of a model of T in the category* * Ab of simplicial abelian groups. The forgetful functor from abelian group objects to T -alg has * *a left adjoint and the category Ab(T ) of abelian group objects is in fact an abelian category. Mo* *reover, the category 14 Ab(T ) is equivalent to the category of modules over a certain simplicial ring * *T ab, namely the endomorphism ring of the free abelian group object on one generator: Ab(T ) ~= T ab-mod: In Theorem 4.4 we are proving the homotopy theoretic analog of this fact. The i* *dea is to replace abelian group objects by its homotopy analogue, namely infinite loop objects or* * spectra. Spectra of T -algebras form a model category Sp(T ). Then there is another `ring' T swh* *ose modules are the spectra of T -algebras. We only have to allow Gamma-rings instead of simpli* *cial rings, and instead of an equivalence of categories we obtain a Quillen equivalence of mode* *l categories Sp(T )conn ' T s-mod: T sis the endomorphism Gamma-ring of the free T -algebra on one generator (see * *4.6 for the precise meaning). The homotopy groups of T sare isomorphic to the ring of stable homoto* *py operations of T -algebras. The two rings arising from the theory T are closely related. Th* *ere is a 1-connected map of Gamma-rings T s-! HT abwhich governs the relationship between stable hom* *otopy and homology of T -algebras. The simplicial ring T abcan be obtained from the Gamma* *-ring T sby applying the functor L left adjoint to the Eilenberg-MacLane functor (see Theor* *em 5.2). 4.1 Spectra of T -algebras. To define spectra, we need to recall the definition* * of the suspension of a T -algebra X. Since T -algebras form a simplicial model category, the prod* *uct X T S1 with the simplicial circle is defined. The suspension of X is then obtained as the c* *ofibre, in the category of T -algebras, of the map X -! X T S1 induced by the inclusion of the unique v* *ertex into S1. X is a bar construction with respect to the coproduct of T -algebras. This mea* *ns that it is the geometric realization of the simplicial T -algebra which in simplicial degr* *ee k consists of the coproduct of k copies of X. The suspension functor has an adjoint which is def* *ined dually. The loop functor commutes with the forgetful functor, i.e., the underlying simplici* *al set of X is the simplicial set of pointed maps of S1 into the underlying simplicial set of X. * *and are a Quillen adjoint functor pair. The total derived functors of and are the suspension an* *d loop functors on the homotopy category of T -algebras [32, I.2]. The simplicial structure thu* *s provides liftings of these functors from the homotopy category to functors on the actual model ca* *tegory. One has to remember that X can have the `wrong' homotopy type if X is not cofibrant jus* *t as Y can have the `wrong' homotopy type if Y is not fibrant. For our purposes, the naive definition of a spectrum suffices. The following is* * an elaboration on the construction of [9, x2]. Definition 4.2A spectrum X of T -algebras is a collection of T -algebras Xn; n * * 0; and T - algebra homomorphisms Xn -! Xn+1. A morphism of spectra f : X -! Y is a collect* *ion of maps fn: Xn -! Yn such that all the diagrams fn Xn ______Ynw | | | | | | | | |u |u Xn+1 _____Yn+1wfn+1 commute. We denote the category of spectra by Sp(T ). 15 If T is the theory of sets, a T -algebra is just a simplicial set and the above* * definition reduces to the notion of a spectrum as in [9, x2]. In general, a spectrum of T -algebras i* *s the same thing as a model of T in the category of spectra of [9]. We call a map of spectra of T -* *algebras a weak equivalence (resp. fibration) if it is a stable weak equivalence (resp. stable * *fibration) as a map of spectra of simplicial sets. We call a map a cofibration if it has the left lift* *ing property with respect to all acyclic fibrations of spectra of T -algebras. Theorem 4.3 If T is a simplicial theory, then the category Sp(T ) of spectra o* *f T -algebras is a closed simplicial model category. Proof: We apply Lemma A.2 to lift the stable model category structure of spectr* *a of simplicial sets to the category Sp(T ). The adjoint functor pair to use consists of the fo* *rgetful and the free T -algebra functor, applied dimensionwise to spectra. All limits, colimits as w* *ell as tensors and cotensor with simplicial sets are inherited from the category of T -algebras an* *d they are defined dimensionwise for spectra of T -algebras. The category Sp(T ) is locally finite* *ly presentable (Lemma A.1) and the model category of spectra of simplicial sets is cofibrantly genera* *ted (Lemma A.3) .So it remains to describe a functor Q that provides fibrant replacements. In [* *9, x2] Bousfield and Friedlander use a functor Q given by (QX)n = colimiiSing|Xn+i| : (Kan's functor Ex1 can be substituted for the geometric realization of the sing* *ular complex). A priori, the functor Q is defined for spectra of simplicial sets; but this choic* *e of Q is simplicial and preserves finite products, so it passes to the category of spectra of T -algebr* *as. For every spectrum X, QX is an -spectrum and degreewise a fibrant simplicial set. It is thus fibra* *nt in the stable model category of spectra by [9, A.7]. So the lifting lemma A.2 applies. * * __|_| Now we can state the main theorem of this section, saying that the (connective)* * stable homotopy theory of T -algebras is equivalent to the homotopy theory of modules over a ce* *rtain Gamma-ring T s. The fact that we only get connective spectra of T -algebras stems from the* * fact that -spaces only represent connective spectra. Theorem 4.4 To a simplicial theory T there is functorially associated a Gamma-* *ring T s. The ring ss*T sis isomorphic to the ring of stable homotopy operations of T -algebr* *as. There is a Quillen adjoint functor pair T s-mod _____wu_____Sp(T ) whose total derived functors are inverse equivalences between the homotopy cate* *gory of T s-modules and the homotopy category of connective spectra of T -algebras, Ho (T s-mod) ~= Ho (Sp(T ))conn: 4.5 The Gamma-ring T s. The -space underlying the Gamma-ring T sis defined as t* *he com- posite of the free T -algebra functor, restricted to the category op, with the * *forgetful functor from T -algebras to pointed simplicial sets. The composite of the free T -algeb* *ra functor with the forgetful functor is a triple on the category of simplicial sets. As we pointed* * out in 2.3, this means that T scomes with associative and unital maps T sO T s-! T s and S -! T s making it a monoid in the category of -spaces with respect to the composition p* *roduct O. Com- position with the assembly map (see 1.8) T s^T s-! T sO T s-! T s gives T sa multiplication with respect to the smash product. Since the assembly* * map is associative and unital, T sbecomes a Gamma-ring. 16 4.6 A generalization: endomorphism Gamma-rings. The construction of the Gamma-r* *ing sis a special case of a more general construction of endomorphism Gamma-rings* *. As input we T can use any pointed category C which has finite coproducts. Then C is tensored * *over the category op, i.e., the assignment X^ n+ = X_q_:_:q:X-z____" n is the object function of a functor ^ : C x op -! C. As a consequence, the cate* *gory C is also enriched over the category of -sets. This means that for any two objects X and * *Y of C there is a homomorphism -set HOM(X; Y ) defined by HOM (X; Y )(n+) = C(X; Y ^n+) : Furthermore there is a unit morphism S -! HOM (X; X) induced by the identity o* *f X and associative and unital composition pairings HOM (Y; Z) ^HOM (X; Y ) ----! HOM (X; Z): The composition pairing is induced by the universal property of the smash produ* *ct of -sets from the maps C(Y; Z ^n+) ^C(X; Y ^m+)----! C(X; Z ^n+^ m+) f ^ g 7! (f ^idm+) O g : In particular, for every object X, the endomorphism -set HOM(X; X) becomes a Ga* *mma-ring. If the category C is also enriched over the category of simplicial sets (as is * *the case for algebras over a simplicial theory), then the enrichment over -sets extends to one over -* *spaces. This basic observation gives a rich supply of (endomorphism) Gamma-rings. For a simplicial* * theory T the category of T -algebras is pointed, simplicially enriched and has coproducts. T* *he value at n+ of the endomorphism Gamma-ring of the free T -algebra on one generator is given by HOM (F T(1+); F T(1+))(n+) ~= Hom T-alg(F T(1+); F T(n+)) which is naturally isomorphic to the value of the Gamma-ring T sat n+. Since th* *e isomorphism also preserves the multiplications we obtain the following Lemma 4.7 Let T be a simplicial theory. Then the Gamma-ring T sis isomorphic t* *o the endo- morphism Gamma-ring of the free T -algebra on one generator. Warning: The homomorphism -spaces HOM(X; Y ) just defined are usually not adjoi* *nt to any kind of smash product pairing between C and the category of -spaces. For exampl* *e HOM(X; Y ) usually does not preserve limits in the second variable (although it does take * *colimits in the first variable to limits). If C = GS is the category of -spaces, then the homom* *orphism -space HOM(X; Y ) is different from the internal hom -space Hom(X; Y ) which is adjoin* *t to the smash product. Indeed the former is made up from homomorphisms into -spaces of the fo* *rm Y ^n+, whereas the latter uses maps into -spaces of the form Yn+^-. The natural maps * *of -spaces Y ^n+ -! Yn+^- induce a natural map HOM(X; Y ) -!Hom(X; Y ). 4.8 Comparison of the stable categories. We now proceed to compare the two cat* *egories Sp(T ) and T s-mod. This will be done through an intermediate category GS(T ), * *the category of pointed functors op-! T -alg, alias the models of T in the category of -spaces.* * We obtain three model categories with Quillen adjoint functor pairs _*___ (-)(S)_ T s-mod u_____w GS(T ) u_____ wSp(T ) * (S;-) The left pair is a Quillen equivalence, the right is a Quillen pair which passe* *s to an equivalence of the homotopy category of GS(T ) with that of connective spectra of T -algebras. 17 Step 1. We first establish the stable model category structure for GS(T ). We* * call a map in GS (T ) a weak equivalence (resp. fibration) if and only if it is a stable equi* *valence (resp. stable Q-fibration) of underlying -spaces. A map is called a cofibration if it has the* * left lifting property with respect to all acyclic fibrations. Theorem 4.9 With these notions of fibrations, cofibrations and weak equivalenc* *es, the category GS (T ) becomes a closed simplicial model category. If X -! Y is a cofibration * *in GS(T ), then for every pointed simplicial set K, the map X(K) -! Y (K) is a cofibration of T -al* *gebras. Proof: We want to apply Lemma A.2 to lift the stable Q-model category structure* * from -spaces to GS(T ). The adjoint functor pair to use consists of the forgetful and the fr* *ee T -algebra functor, applied objectwise to -objects. As a functor category into a complete and cocom* *plete simplicially enriched category, GS(T ) has all limits and colimits as well as tensors and co* *tensor with simplicial sets. The category GS(T ) is locally finitely presentable, see Lemma A.1. The c* *rucial ingredient is the stably fibrant replacement functor Q. One possible choice of such Q is give* *n by (QX)(n+) = colimiiSing|X(Si^n+)| : Again we can use Ex1 [21] instead of geometric realization and singular complex* *. A priori, the functor Q is only defined on the category of -spaces. However, Q is a simplicia* *l functor and it preserves finite products, so it passes to an endofunctor on the category GS(T * *). There is a natural stable equivalence X -! QX, and QX is pointwise fibrant and very special, so it* * is fibrant in the stable Q-model category structure. Thus the lifting lemma A.2 applies. To get the statement about cofibrations we first consider generating cofibratio* *ns. These are of the form X = T sO A -! T sO B = Y for A -! B a cofibration of -spaces. Then th* *e map X(K) -! Y (K) is obtained from a cofibration of simplicial sets by application * *of the free T - algebra functor, so it is a cofibration of T -algebras. The general case follow* *s by the small object argument [37, Lemma A1] since the property in question is preserved under cobas* *e change, trans- finite composition and retract. * * __|_| Step 2. The comparison of the category Sp(T ) of spectra of T -algebras with th* *e category GS(T ) of -objects of T -algebras follows easily from the work of Bousfield and Friedl* *ander. In [9, x5], they show that the functor X 7! X(S) from -spaces to spectra has a right adjoin* *t (S; -). Both functors are simplicial and the left adjoint preserves finite products, so they* * pass to adjoint functors between the categories GS(T ) and Sp(T ) (see Lemma 2.5). Since (stable) weak e* *quivalences and fibrations in GS(T ) and Sp(T ) are defined on underlying -spaces or spectra re* *spectively, the functors (-)(S) and (S; -) still form a Quillen pair. We have to note here that* * the stable Q- model category structure for -spaces has more fibrations than the stable Bousfi* *eld-Friedlander model category structure. If X is a connective fibrant spectrum of T -algebras,* * then the adjunction map (S; X)(S) -~!X is a stable equivalence. So if A is a cofibrant object in GS* *(T ), then a map A(S) -! X is a stable equivalence if and only if the adjoint map A -! (S; X) is* *. Thus the Quillen adjoint functor pair passes to an equivalence between the homotopy cate* *gory of GS(T ) and the homotopy category of connective spectra of T -algebras. Step 3. In this last step we want to construct a Quillen equivalence between th* *e category GS(T ) and the category of T s-modules. The associative and unital assembly map X ^Y -* *! X O Y of 1.8 gives a morphism : T s^- -! T sO - of triples on the category of -spaces. An algebra over the triple T s^- is not* *hing but a T s- module, an algebra over the triple (T sO -) is a -object of T -algebras. Pullin* *g back along the 18 triple morphism gives a functor * : GS(T ) -! T s-mod. This functor has a left* * adjoint * * preserves fibrations and weak equivalen* *ces since these are (see [24, Cor. 1]). The right adjoint defined everywhere on underlying -spaces; so the functors form a Quillen pair. * *Since the right adjoint in fact detects and preserves all weak equivalences, it suffices to show Lemma 4.10 For a cofibrant T s-module A, the unit map A -! **A of the adjuncti* *on is a stable equivalence. Proof: We assume first that the cofibrant T s-module A is induced, i.e., it is * *of the form A = T s^Y for some cofibrant -space Y . Pushforward along a map of triples takes free obj* *ects to free objects, so in this case the map in question is the assembly map A = T s^Y -! T sO Y = **A ; which is a weak equivalence by [25, Prop. 5.23]. Now we assume that the cofibra* *nt T s-module A can be written as the pushout of a diagram of T s-modules A0 u_____K v_____Lw: in which K -! L is a cofibration and such that Lemma 4.10 holds for the cofibra* *nt T s-module A0 and the quotient module L=K ~=A=A0. We claim that then the lemma also holds for* * A. Indeed, cofibrations of T s-modules are injective and cofibres of T s-modules are calcu* *lated on underlying -spaces [37, Thm. 2.2]. So the cofibre sequence of T s-modules A0 -! A -! A=A0* * gives rise to a long exact sequence of homotopy groups by [37, Lemma 1.3]. As a left adjoi* *nt in a Quillen functor pair * preserves pushout, cofibres and cofibrations. Hence the five le* *mma gives the desired conclusion once we know that the cofibre sequence in GS(T ) *A0----! *A ----! *(A=A0) also gives rise to a long exact sequence of homotopy groups. If we evaluate at * *the simplicial n- sphere Sn, we obtain a cofibre sequence of (n - 1)-connected cofibrant T -algeb* *ras (by Theorem 4.9). By Theorem 3.6, we obtain a long exact sequence of homotopy groups in a s* *table range, and we let n go to infinity. The previous two paragraphs together show that the conclusion of Lemma 4.10 hol* *ds for all T s- modules which can be obtained from the trivial module by finitely many cobase c* *hanges along cofibrations between modules that are induced from -spaces. Then it also holds* * for modules which are filtered direct limits of such modules. But an arbitrary cofibrant T * *s-module is a retract of one of this sort by the small object argument [37, Lemma A1]. * * __|_| 4.11 Stable homotopy operations. Interpreting ss*T sin terms of stable homotopy* * operations is a standard representability argument. We will be brief since this result wil* *l not be used in the rest of this paper. A homotopy operation of T -algebras is a natural transforma* *tion ssn -! ssm of functors T -alg-!(sets)for some n; m. Homotopy operations can be composed if th* *e source of one is the target of the other, and they form a category with objects the natural n* *umbers. The functor ssn is represented by F T(Sn) in the homotopy category of T -algebras, i.e., ss* *nX ~=[F T(Sn); X] (the right hand side denotes maps in the homotopy category). Consequently, the* * category of homotopy operations is isomorphic the opposite of the full subcategory of Ho (T* * -alg)generated by the F T(Sn). In particular, homotopy operations ssn -! ssm are in bijective cor* *respondence with elements of [F T(Sm ); F T(Sn)]. Homotopy operations can be suspended, i.e., if* * o : ssn -! ssm is one, one defines o : ssn+1 -! ssm+1 on a T -algebra X as the composite o|X| ssn+1|X| ~= ssn|X| ----! ssm |X| ~= ssm+1|X| : 19 The suspension of operations corresponds to the suspension : [F T(Sm ); F T(Sn)] -! [F T(Sm+1 ); F T(Sn+1)] in the homotopy category of T -algebras. A stable homotopy operation of degree n is represented by a sequence (oi)ii0 of* * homotopy oper- ations oi: ssi- ! ssi+n with the property that oi+1= oi. Two such sequences def* *ine the same stable operation if the components eventually agree, i.e., if almost all compon* *ents are equal. Sta- ble homotopy operations can always be composed, the degrees add under compositi* *on, and they form a graded ring. The natural isomorphisms of ssn+i|F T(Si)| with the sets of* * homotopy classes [F T(Sn+i); F T(Si)] assemble into an isomorphism of ssn T s= colimissn+i|F T(S* *i)| with the colimit of the sets [F T(Sn+i); F T(Si)] over suspension; but the elements of colimi[F * *T(Sn+i); F T(Si)] are nothing but the stable homotopy operations of degree n. The fact that the isomo* *rphism between ss*T sand stable homotopy operations is multiplicative follows from the fact bo* *th products are (suitable kinds of) composition products, see Lemma 4.7. 5 Stable homotopy versus homology We have seen that a simplicial theory T gives rise to two "rings" and a multipl* *icative map between them. There is the simplicial ring T abwhose modules are the abelian group obje* *cts in the cate- gory of T -algebras. And there is the Gamma-ring T swhose modules are (Quillen * *equivalent to) connective spectra of T -algebras. Abelianization induces a Gamma-ring map T s-* *! HT ab. This map encodes the relationship between stable homotopy and homology in the homoto* *py theory of T -algebras. In this section we will show that the map T s-! HT abis 1-connecte* *d and we will establish Hurewicz and Whitehead Theorems for T -algebras as well as universal * *coefficient and Atiyah-Hirzebruch spectral sequences. 5.1 Quillen homology. We recall Quillen's definition of homology as the left de* *rived functor of abelianization [32, II.5]. By definition, the abelianization functor -ab: T -alg-! Ab(T ) is the left adjoint to the forgetful functor. If X is a T -algebra, one chooses* * a cofibrant replace- ment Xc -~!X and defines the homology of X to be the homotopy of the abelianiza* *tion of the replacement: H *X = ss*(Xcab) : The unit of the adjunction is a map of T -algebras X -! Xab which we refer to a* *s the Hurewicz map. More generally there is (co-)homology of a T -algebra with coefficients. * * If M is a right simplicial T ab-module, then the homology of X with coefficients in M is define* *d as the homotopy of the tensor product H*(X; M) = ss*(M TabXcab) : If N is a left simplicial T ab-module, then the cohomology groups of X with coe* *fficients in N are defined as the homotopy classes of T ab-module maps H *(X; N) = [Xcab; *N]Tab-mod: Note that unless the simplicial ring T abis commutative, homology and cohomolog* *y need different kinds of coefficients. In the case of the theory of sets this notion of homolog* *y specializes to singular homology of simplicial sets. For commutative rings it specializes to Andre-Quil* *len homology [31, Sec. 4]. 20 Given a T -algebra X, we need a model for its suspension spectrum as a T s-modu* *le. There is sas the endomorphism Gamma-rin* *g of the free a slick definition using the interpretation of T T -algebra on one generator as in Lemma 4.7. In the notation of 4.6 (with C = T* * -alg), we can set 1 X = HOM (F T(1+); X) as a -space, and with T s-module structure given by the * *composition action of T s~=HOM (F T(1+); F T(1+)). An equivalent description of 1 X is as * *follows. First define a -object of T -algebras g1 X by (g1 X) (n+) = X ^n+ = X_q_._.q.X-z____"; n the coproduct being taken in the category of T -algebras. As a functor T -alg-!* * GS(T ), g1 is left adjoint to evaluation at 1+. Note that when g1 X is extended (by direct li* *mit and degreewise application) to a functor from simplicial sets to T -algebras, then we get (g1 * *X) (K) = X ^K (this smash product refers to the enrichment of the category of T -algebras ove* *r pointed simplicial sets). Hence the spectrum associated to g1 X is isomorphic to the suspension sp* *ectrum of X as a T -algebra, which justifies the name. Recall from 4.8 that -objects in T -alg c* *an be pulled back to T s-modules via a functor *. This means that the underlying -space of the -T -a* *lgebra g1 X is endowed with a left T s-action via the assembly map (1.8) T s^g1 X -! T sO g1X -! 1 X : The T s-module *^1 X is isomorphic to 1 X. So the two suspension spectrum objec* *ts 1 X and g1 X have the same underlying -spaces, but one is viewed as a T s-module, t* *he other one as an object of GS(T ). Recall from [37, Lemma 1.2] that the left adjoint functor L to the Eilenberg-Ma* *cLane functor H is strong symmetric monoidal and preserves finite products. In particular, it t* *akes Gamma-rings to simplicial rings. Furthermore, the functors L and H pass to an adjoint funct* *or pair between the category GS(T ) of -objects of T -algebras and the category Ab(T ) of abeli* *an group objects of T -algebras by Lemma 2.5. In the following theorem we combine these formal p* *roperties with some homotopical input to obtain information on the relationship between stable* * homotopy and homology for T -algebras. Theorem 5.2 For a T -algebra X, the object L(g1 X) 2 Ab(T ) is naturally isomo* *rphic to Xab. The map of Gamma-rings T s-! HT abinduces an isomorphism on ss0 and an epimorph* *ism on ss1 and its adjoint is an isomorphism of simplicial rings L(T s) ~=T ab. In particu* *lar, if T is a discrete theory, then T ab~=ss0T s. For a right T ab-module M and a T -algebra X, there * *is a natural map of -spaces HM ^LTs1 X ----! H(M TabXab) which is a stable equivalence whenever X is cofibrant. Proof: The forgetful functor from abelian group objects factors as a composite + Ab(T ) ---H--! GS(T ) -eval.-at-1---!T -alg: Since L : GS(T ) -! Ab(T ) is left adjoint to H and g1 is left adjoint to evalu* *ation at 1+, their composite is a left adjoint to the forgetful functor. The adjoint of the map T* * s-! HT abis a homomorphism of simplicial rings so it suffices to show that it is an isomorphi* *sm in Ab(T ). But the -space T sunderlies the suspension spectrum of the free T -algebra on one g* *enerator, so L(T s) is isomorphic to the free abelian group object on one generator T abby what we * *already proved. The fact that T s-! HT abinduces an isomorphism on ss0 and an epimorphism on ss* *1 then follows from [37, Lemma 1.2]. 21 There is a functorial cofibrant replacement (1 X)c -! 1 X in the category of T * *s-modules. sprovides a natural stable equivalence of -spac* *es from Then [37, Lemma 4.2] with R = T HM^Ts(1 X)c to H(M TabL((1 X)c)). Since Xab ~=L(g1 X), it remains to show that the map of cofibrant T ab-modules L((1 X)c) -! L(g1 X) is a weak equivalence. T* *his follows if we can show that for any T ab-module W , the induced map on homomorphism spa* *ces hom Tab-mod(L(g1 X); W ) ----! homTab-mod(L((1 X)c); W ) is a weak equivalence. By the various adjunctions, this map is isomorphic to th* *e map homGS(T)(g1 X; HW ) ----! homGS(T)(*(1 X)c; HW ) ~= homTs-mod((1 X)c; HW ) induced by the stable equivalence of cofibrant objects *(1 X)c -! g1 X (this us* *es Lemma 4.10). Hence the latter map of homomorphism spaces is a weak equivalence, which* * finishes the proof. __* *|_| Corollary 5.3 (Hurewicz theorem)Let T be a simplicial theory and X a cofibrant * *(n- 1)- connected T -algebra (n 2). Then the Quillen homology of X vanishes below dime* *nsion n and the Hurewicz map induces an isomorphism ssnX ~=HnX and an epimorphism ssn+1X -!* * Hn+1X. Proof: An application of Theorem 3.6 to the cofibre sequence X -! Cone(X) -! X * *shows that the map |X| -! |X| is (2n - 1)-connected. Since n 2, the n-th homotopy gr* *oup of X is thus isomorphic to the n-th stable homotopy group and ssn+1X surjects onto the * *(n + 1)-st stable homotopy group. By Theorem 5.2, the -space HXabis stably equivalent to HT ab^LT* *s1 X. So the Tor spectral sequence for the derived smash product [37, Lemma 3.1] takes t* *he form s ab 1 Torss*Tp(ss*T ; ss* X)q =) Hp+q(X) : Since the map T s-! HT abis 1-connected, this spectral sequence gives the desir* *ed answer for the first non-trivial homology groups of X. * * __|_| Corollary 5.4 (Whitehead theorem) Let T be a simplicial theory and X -! Y a ma* *p of simply connected T -algebras which induces an isomorphism in Quillen homology. * *Then the map is a weak equivalence. Proof: We can assume that X and Y are 1-reduced and cofibrant and that the map * *is a cofibra- tion. Then the cofibre Y==X is 1-connected and has vanishing Quillen homology, * *hence is weakly contractible by the Hurewicz Theorem 5.3. Again by the Hurewicz Theorem, the ma* *p X -! Y is 2-connected. An application of Theorem 3.6 to the cofibration X -! Y and induct* *ion gives that the map X -! Y is m-connected for all m * * __|_| 5.5 Spectral sequences relating Quillen homology and stable homotopy. Let X be* * a cofibrant T -algebra and M a coefficient module for Quillen homology. By Theor* *em 5.2, the homology groups of X with coefficients in M are isomorphic to the stable homoto* *py groups of the -space HM ^LTs1 X. So the spectral sequence for the derived smash product [37, * *Lemma 3.1] gives a universal coefficient spectral sequence s s E2p;q= Torss*Tp(ss*M; ss*X)q =) Hp+q(X; M) : 22 This spectral sequence lies in the first quadrant. If X is (n - 1)-connected, i* *f we take M = T ab and if for simplicity we take T to be a discrete theory we can read off the fol* *lowing six term exact sequence (ss1T s sssn+1X) (ss2T s sssnX)----!sssn+2X ----! H n+2X ----! ----! ss1T s sssnX----! sssn+1X ----! H n+1X ----! 0 which refines the part of the Hurewicz theorem that claims the surjectivity of * *the Hurewicz map in dimension n + 1. If W is a right T s-module, then the homotopy groups of the derived* * smash product W*X = ss*(W ^LTs1 X) are a generalized homology theory in the T -algebra X. Th* *e spectral sequence [37, Lemma 3.1] together with Theorem 5.2 thus gives an Atiyah-Hirzebr* *uch spectral sequence E2p;q= H p(X; ssqW ) =) Wp+q(X) : 6 Relation to theory cohomology In this section we provide the link between the cohomology and the stable homot* *opy of an algebraic theory. In [20, Def. 4.2], M. Jibladze and T. Pirashvili introduce the cohomolo* *gy of an algebraic theory as Ext groups in an abelian functor category _ see Remark 6.4 for some b* *ackground and explanation about their cohomology theory. The main result of this section, The* *orem 6.7, says that the Jibladze-Pirashvili homology groups of a theory T with coefficients in* * a functor G are isomorphic to the topological Hochschild homology groups of the Gamma-ring T sw* *ith coefficients in a bimodule G!associated to G. This generalizes a theorem of T. Pirashvili an* *d F. Waldhausen [30, Thm. 3.2]. We also show that the analogous statement in cohomology holds * *provided the coefficient functor G is additive. 6.1 Homological algebra in functor categories. Let C be a small category with z* *ero object and R any ring. We denote by F(C; R) the category of covariant pointed functors* * from C to the category of left R-modules. This is an abelian category in which exactness is d* *efined objectwise. For every object c of C there are functors Pc and Ic defined by Pc(d) = Re[C(c; d)] and Ic(d) = map *(C(d; c); Rinj) : Here eR[-] denotes the reduced free R-module on the pointed set of morphisms fr* *om c to d, Rinj= Hom Z(R; Q=Z) is the injective cogenerator in the category of left R-modu* *les and `map*' denotes the set of pointed maps into Rinjwith the pointwise left R-module struc* *ture. Because of the Yoneda-type isomorphisms Hom F(C;R)(Pc; G) ~= G(c) and Hom F(C;R)(G; Ic) ~= Hom R-mod(G(c); Ri* *nj) ; Pcis a projective object and Icis an injective object in the abelian category F* *(C; R). Furthermore, the functors Pc form a set of projectives generators and the functors Ic form a* * set of injectives cogenerators for F(C; R) when c runs over the objects of C. 23 6.2 (Co-)homology of algebraic theories. We consider a discrete, pointed algebr* *aic theory op; T ab) of pointed funct* *ors from T opto T . We abbreviate to F(T ) the abelian category F(T the category of left T ab-modules. We recall from [6, Prop. 3.8.5] that T opis* * equivalent to the full subcategory of T -alg given by the finitely generated free T -algebras. A* *lso the category of left modules over the ring T abis equivalent to the category Ab(T ) of abelian * *group objects of T -algebras. T abis the endomorphism ring of the free abelian group object on * *one generator, and by Theorem 5.2 it is isomorphic to the ring ss0T s. The category F(T ) has * *a special object Iab, the abelianization functor for T -algebras, restricted to T op. Every (R * *(T ab)op)-module M defines a functor M TabIab in F(T op; R). These are precisely the additive func* *tors, i.e., those functors which commute with coproducts. The functor R-mod-T ab-! F(T op; R) whi* *ch sends M to M TabIabis right adjoint to the functor which sends G 2 F(T op; R) to the R-* *T ab-bimodule Gadd = coker(G(2+) (p1)*+(p2)*-r*---------!G(1+)) : Here the right T ab-action is induced, through the functor G, from the action o* *f the monoid homT(1+; 1+) on the free T -algebra on one generator. In the case where R = T aband G 2 F(T ), the abelian group Gaddthus has a two-s* *ided action of the ring T ab. In this case we can equalize the actions and define QG = Gadd=(tx - xt) ; i.e., we divide out the subgroup generated by elements of the form tx - xt for * *x 2 Gaddand t 2 T ab. Then Q is an additive, right exact functor from F(T ) to the category* * of abelian groups and so it has left derived functors LiQ. Definition 6.3[20, Def. 4.2] Let T be a pointed discrete algebraic theory and G* * 2 F(T ). The homology and cohomology of T with coefficients in G are then defined as H *(T ; G) = (L*Q)(G) and H *(T ; G) = Ext*F(T)(Iab; G) : Remark 6.4 The notion of (co-)homology of a theory T with coefficients in a fu* *nctor in the sense of Jibladze and Pirashvili has to be distinguished from the Quillen (co-)homolo* *gy of a T -algebra X with coefficients in an abelian group object which we reviewed in 5.1. If the* * theory T is fixed, then Quillen homology provides a homology theory for varying T -algebras. For e* *xample, Quillen homology satisfies excision and is homotopy invariant. The Jibladze-Pirashvili cohomology plays the same role for algebraic theories t* *hat is played by Hochschild cohomology for algebras over a field, and it generalizes MacLane coh* *omology [26] for arbitrary rings. For example in [20, Sec. 4], Jibladze and Pirashvili give int* *erpretations of the theory cohomology groups in dimensions 0, 1 and 2 as suitable `center', `outer * *derivation' and `singular extension' groups respectively. For the theory of modules over a ring* * and for an additive coefficient functor these reduce to the corresponding classical interpretations* * of the MacLane cohomology groups. Indeed, by [20, Theorem A] the cohomology groups of the the* *ory of R- modules with coefficients in a bimodule are isomorphic to the MacLane cohomolog* *y groups. 6.5 Topological Hochschild (co-)homology. Let S be a Gamma-ring and M an S-bimo* *dule. We choose a cofibrant approximation cS -! S in the model category of Gamma-ring* *s of [37, Thm. 24 2.5]. For us the topological Hochschild homology groups of S with coefficients * *in M are defined as the homotopy groups of the derived smash product of S and M as cS-S-bimodules, THH n(S; M) = ssn (S ^LcS^SopM) : This is not the original definition of topological Hochschild homology given by* * B"okstedt [5]. How- ever Shipley [39, Sec. 4] shows in the context of symmetric spectra that the tw* *o definitions are equivalent; a proof of the analogous statements in the context of Gamma-rings i* *s similar, but easier. The topological Hochschild cohomology groups of S with coefficients in * *M are defined as the homotopy classes of cS-S-bimodule maps from S to M, ae n THH n(S; M) = [S;[-M]cS-SnS; ifnM]0 if n cS-S< 0. Here refers to the suspension functor in the homotopy category of cS-S-bimodul* *es. 6.6 The bimodule construction. To a functor G 2 F(T op; R) there is a functoria* *lly associated cHR-T s-bimodule G!. This generalizes the construction of [30, Ex. 2.6]. Here c* *HR is a cofibrant approximation of HR in the model category of Gamma-rings of [37, Thm. 2.5]. As * *a -space, G! is equal to the composite functor T G forget op -F---!T op----! R-mod ----! (pt. simpl. sets): In other words, the value of the -space G!on n+ is the underlying set of the va* *lue of G on the free T -algebra on n generators. There is a map HR O G!O T s-! G!given at n+ 2 * *opby (HR O G!O T s)(n+) = eR[G(F T(F T(n+)))] ----! G(F T(n+)) = G!(n+) ; this map is evaluation both inside and outside of G and it uses that G takes va* *lues in R-modules and that F Tis a triple. Composition with the assembly map (1.8) and the stabl* *e equivalence of Gamma-rings cHR -~! HR gives the bimodule structure cHR ^G!^T s-! G!. For ex* *ample, the cHR-T s-bimodule associated to the additive functor M TabIab is the Eilenbe* *rg-MacLane module HM. Theorem 6.7 Let T be a pointed discrete algebraic theory and G 2 F(T ). There* * is a natural isomorphism H *(T ; G) ~= THH *(T s; G!) : For T ab-bimodules M, the groups THH*(T s; HM) are trivial in negative dimensio* *ns and for * 0 there is a natural isomorphism H*(T ; M TabIab) ~= THH *(T s; HM) : Remark 6.8 A special case of interest is the case when T is the theory of left* * R-modules for a ring R. We are then looking at functors G 2 F(R) from the category of finitely * *generated free R-modules to all R-modules. In this case the homotopy groups ss*G!are (essentia* *lly by definition) the stable derived functors of G in the sense of A. Dold and D. Puppe [13, 8.3]* *. The homological case of Theorem 6.7 then specializes to [30, Thm. 3.2]. By a theorem of Jibladz* *e and Pirashvili [20, Thm. A] the groups Ext*F(R)(I; M R -) are naturally isomorphic to the MacLane c* *ohomology groups H*ML(R; M) introduced in [26]. So the cohomological part of Theorem 6.7* * implies that MacLane cohomology coincides with topological Hochschild cohomology. 25 The cohomology of T with coefficients in a non-additive functor can differ from* * the topological Hochschild cohomology, see Remark 6.15. To prove the homological part of Theore* *m 6.7 we use the same strategy as [30]: we show that topological Hochschild homology has the uni* *versal properties of the derived functors of Q. The cohomological part follows from a comparison * *of the derived category of the abelian category F(T op; R) with the homotopy category of cHR-T* * s-bimodules. We start with three short lemmas. Lemma 6.9 Let Pn be the projective object of F(T op; R) represented by n+ 2 T * *op(see 6.1). Then Pn!is stably equivalent to cHR ^n+^ T sas a cHR-T s-bimodule. Proof: The -space underlying Pn!is the composite of three other -spaces, Pn!= H* *R O n O T s. The cHR-T s-bimodule structure comes through the left and right composition fac* *tors. Since cHR is cofibrant as a Gamma-ring, it is also cofibrant as a -space [37, Thm. 2.5]. * *By [25, Prop. 5.23] the assembly map cHR ^n^ T s--~--! HR O n O T s from the smash to the composition product is thus a stable equivalence. The lem* *ma follows since the cofibrant -spaces n and S ^n+ are stably equivalent. * * __|_| Recall from [37, Sec. 4] that the functor L which is adjoint to the Eilenberg-M* *acLane functor H passes to a functor L : cHR-mod-T s-! R-mod-T ab(using the isomorphism T ab~=L(* *T s)). Lemma 6.10 There is a natural isomorphism Gadd~= LG! of functors F(T op; R) -!* * R-mod- T ab. Proof: The evaluation map eZ[G(1+)] -! G(1+) passes to a natural map of R-T ab-* *bimodules from LG!to Gadd. By Lemma 6.9 and [37, Lemma 1.2], LPn!is isomorphic to the fre* *e bimodule (R T ab)n, so the map is an isomorphism for the projective generators. Since b* *oth expressions are right exact in G, the map is an isomorphism in general. * * __|_| We denote by sF(T op; R) the category of simplicial objects in F(T op; R) (whic* *h is the same as the category of pointed functors from T opto the category of simplicial left R-modu* *les). The bimodule construction 6.6 which takes G to G!can be applied dimensionwise to simplicial * *functors. Lemma 6.11 The bimodule construction G 7! G!takes short exact sequences of sim* *plicial func- tors in sF(T op; R) to homotopy cofibre sequences of cHR-T s-bimodules. Proof: When the underlying -spaces of the bimodules associated to a short exact* * sequence are evaluated at a simplicial sphere Sn, one obtains a short exact sequence of simp* *licial R-modules which give rise to a long exact sequence in homotopy. When n tends to infinity,* * these assemble into a long exact sequence for the homotopy groups of the cHR-T s-bimodules. * * __|_| Proof of the homological part of Theorem 6.7: We show that the functors THH*(T * *s; (-)!) have the universal properties of the derived functors of Q. By [37, Lemmas 1.2 * *and 4.1] we can identify THH0(T s; G!) as THH 0(T s; G!) ~= L(T s^cTs^(Ts)opG!) ~= T abT ab(Tab)opLG!: So by Lemma 6.10, the group THH0(T s; G!) is naturally isomorphic to QG. Short * *exact sequences of objects in F(T ) go to homotopy cofibre sequences of bimodules (Lemma 6.11),* * which become homotopy cofibre sequences of -spaces after derived smash product with T sover * *cT s^(T s)op. So the functors THH*(T s; (-)!) have a connecting homomorphism with respect to * *which short exact sequences of objects in F(T ) go to long exact sequences in homology. So * *it remains to show that topological Hochschild homology vanishes in positive dimensions for each o* *f the projective generators Pn of F(T ). Using Lemma 6.9 we calculate THH *(T s; Pn!) ~= ss*(T s^LcTs^(Ts)op(HT ab^n+^ T s)) ~= ss*(HT ab^n+) ~=* * (ss*T ab)n which is indeed trivial in positive dimensions since T abis a discrete ring. * * __|_| 26 6.12 Model structures for simplicial functors. The cohomological part of Theore* *m 6.7 is a special case of a more general statement about the relationship between the cat* *egory of simplicial functor from T opto R-modules and the category of cHR-T s-bimodules. Quillen [3* *2, II.4 Thm. 4] provides a standard model category structure on the category sF(T op; R) of sim* *plicial functors. The weak equivalences (resp. fibrations) are the maps which are objectwise weak* * equivalences (resp. fibrations) of simplicial R-modules. We refer to this model category str* *ucture as the strict structure for simplicial functors. By the Dold-Kan theorem the normalized chain* * complex functor induces an equivalence of the strict homotopy category of sF(T op; R) with the * *derived category D+ (F(T op; R)) of non-negative dimensional chain complexes over the abelian ca* *tegory F(T op; R). The bimodule construction takes objectwise weak equivalences of simplicial func* *tors to stable equivalences of cHR-T s-bimodules. Lemma 6.11 implies that the induced functor * *on the level of homotopy categories (-)! : D+ (F(T op; R)) ----! Ho(cHR-mod-T s) is a triangulated functor. We call a map of simplicial functors F -! G a stable equivalence (resp. stable * *fibration) if the associated map of cHR-T s-bimodules F !-! G!is a stable equivalence (resp. stab* *le fibration). The stable cofibrations coincide with the strict cofibrations. A simplicial functor* * G is stably fibrant if and only if it is homotopy-additive, i.e., if for all X; Y 2 T opthe map F (X) * * F (Y ) -! F (X q Y ) is a weak equivalence of simplicial R-modules. Theorem 6.13 The stable notions of fibrations, cofibrations and weak equivalen* *ces make the cat- egory sF(T op; R) of simplicial functors into a closed simplicial model categor* *y. The functor (-)! is the right adjoint of a Quillen equivalence between the stable model category* * of simplicial functors sF(T op; R) and the model category of cHR-T s-bimodules. Proof: The functor G 7! G!preserves all limits and we first want to see that it* * actually has a left adjoint. It suffices to show this for discrete functors in F(T op; R). The* * category F(T op; R) is complete, it has a set of cogenerators (see 6.1) and it is well-powered (i.e* *., every object has only a set of subobjects). So Freyd's Special Adjoint Functor Theorem (see e.g.* * [27, V.8, Cor.]) provides a left adjoint (-)!. To obtain the model category structure we apply t* *he lifting lemma A.2. The category sF(T op; R) of simplicial functors is complete, cocomplete, s* *implicially enriched and locally finitely presentable (Lemma A.1). The model category structure of c* *HR-T s-bimodules is cofibrantly generated. It remains to find a stably fibrant replacement funct* *or Q for the category sF(T op; R). We first note that a simplicial functor G 2 sF(T op; R) can be ext* *ended to a functor from the category of T -algebras to simplicial R-modules by the coend construct* *ion Z k+2Top G(X) = Xk^ G(k+ ) for X 2 T -alg. Then the functor Q is given by (QG)(k+ ) = colimnnG(nF T(k+ )) : The underlying -space of (QG)!is a stably fibrant replacement on the -space und* *erlying G!, so Q in fact has the properties needed to apply the lifting lemma A.2. We conc* *lude that the stable notions of cofibrations, fibrations and weak equivalences make the categ* *ory sF(T op; R) of simplicial functors into a closed simplicial model category. By definition the right adjoint (-)!preserves and detects weak equivalences and* * fibrations. So it remains to show that for every cofibrant cHR-T s-bimodule A the unit map A -* *! (A!)!is a 27 stable equivalence. This is very similar to Lemma 4.10. We first consider the c* *ase when A is one n^ K) ^T sf* *or some pointed of the generating bimodules, i.e., when it is of the form A = cHR ^( simplicial set K. Then A!~=Pn eZ[K] and the unit map A = cHR ^(n^ K) ^T s----! HR O (n^ K) O T s= (A!)! is the composite of the assembly map and the map induced by the stable equivale* *nce cHR -! HR. The assembly map is a stable equivalence when all except possibly one of the fa* *ctors are cofibrant [25, Prop. 5.23]. Since cHR is cofibrant as a Gamma-ring, it is also cofibrant * *as a -space [37, Thm. 2.5]. Hence the map A -! (A!)!is a stable equivalence if A is a generating* * bimodule. An arbitrary cofibrant cHR-T s-bimodule is obtained from the trivial bimodule b* *y iterated pushouts along cofibrations between bimodule of the above form, transfinite composition * *and retract. So the rest of the argument is exactly as in Lemma 4.10. We only have to observe t* *hat the functor (-)!takes cofibre sequences of cHR-T s-bimodules to cofibre sequences of simpli* *cial functors. But cofibre sequences in sF(T op; R) are in particular short exact sequences which * *posses long exact sequences in homotopy by Lemma 6.11. * * __|_| If we take R = T aband F = Iab in the following corollary, the left hands side* * becomes H*(T ; M TabIab) and the right hand side becomes [HT ab; HM]*cHTab-Ts. Change o* *f rings gives an isomorphism [HT ab; HM]*cHTab-Ts~=[T s; HM]*cTs-Ts= THH *(T s; HM) ; so the cohomological part of Theorem 6.7 is a special case of Corollary 6.14Let F be any functor in F(T op; R) and M an R-T ab-bimodule. Then* * the groups [nF !; HM]cHR-Tsare trivial for n > 0 and the functor (-)!induces natural isomo* *rphisms ExtnF(Top;R)(F; M TabIab) ~= [F !; nHM]cHR-Ts: Proof: For an arbitrary cHR-T s-bimodule W , the group [W; HM]cHR-Tsis isomorph* *ic to the group of R-T ab-bimodule homomorphism from ss0W to M. Since ss0(nF !) is trivia* *l for n > 0, the first claim follows. When we regard functors F and G in F(T op; R) as cons* *tant simplicial objects, ExtnF(Top;R)(F; G) is isomorphic to the maps from F to nG in the stric* *t homotopy category of simplicial functors sF(T op; R). The functor M TabIabis additive an* *d the associated bimodule HM is stably fibrant, so M TabIabis fibrant in the stable model catego* *ry structure of simplicial functors. So the maps from F to M TabIabcoincide in the strict and s* *table homotopy categories. Since (-)!is the right adjoint of a Quillen equivalence of model ca* *tegories (Theorem 6.13), the groups of maps from F to n(M TabIab) in the stable homotopy category* * of simplicial functors is mapped isomorphically to the group of maps from F !to HM in Ho(cHR-* *mod-T s). __|_| Remark 6.15 The map ExtnF(Top;R)(F; G) -! [F !; nG!]cHR-Tsis not bijective for* * arbitrary functors in F(T op; R). For example, if we take F to be one of the projective g* *enerators Pn then HomF(Top;R)(Pn; G) ~=G(n+), but [Pn!; G!]cHR-Ts~=(ss0G!)n by Lemma 6.9. These t* *wo expressions are different unless G is additive. In particular we can take T to be the theor* *y of pointed sets and R = T ab= Z. In this case the abelianization functor Iab is isomorphic to the p* *rojective object P1, so H0(T ; G) ~=G(1+) and H*(T ; G) is trivial for * 1. On the other hand T* * sis the sphere spectrum, so THH*(T s; G!) ~=ss*G!, which can have higher homotopy groups. 28 7 Examples 7.1 Sets. In the theory of pointed sets, the algebras are the pointed simplicia* *l sets and the stable category is (a model for) the usual stable homotopy category. The associated Ga* *mma-ring is the sphere spectrum S, so Theorem 4.4 reduces to [9, Thm. 5.8] saying that the homo* *topy theory of -spaces is equivalent to that of connective spectra. 7.2 Simplicial sets with G-action. Let G be a simplicial monoid and consider th* *e theory of pointed simplicial sets with pointed G-action. The stable category is the categ* *ory of spectra with G-action (i.e., G-objects in the category of spectra in the sense of [9]). The * *stable equivalences are equivariant maps which induce isomorphisms of the homotopy groups of underlying* * spectra. The associated Gamma-ring is S[G], the monoid ring of G over the sphere spectrum (s* *ee 1.11). The map from stable homotopy to homology is represented by the map of monoid rings S[G]* * -! H(Z[G]). If G is a simplicial group (not just a simplicial monoid), then the homotopy th* *eory of pointed G-simplicial sets is the same as the homotopy theory of retractive spaces over * *the classifying space BG. This is well known and can be seen as follows: we let EG denote a universal* * principal G- space, i.e., any weakly contractible simplicial set with a free G-action, and w* *e take the orbit space of EG by the G-action as our model for the classifying space. Then pullback alo* *ng the orbit map EG -! BG is an equivalence of categories between the category of simplicial set* *s containing BG as a retract, and the category of (unpointed) G-simplicial sets containing EG a* *s an equivariant retract (in both cases the section and retraction are part of the data). The appropriate model category structure for retractive G-spaces over EG is the* * one in which fibrations and weak equivalences are those morphisms that are fibrations and we* *ak equivalences of simplicial sets after forgetting the G-action, the retraction and the sectio* *n. The functor that collapses the retract EG to a point is then the left adjoint of a Quillen equiv* *alence between the category of retractive G-simplicial sets over EG, and the category of pointed G* *-simplicial sets. So altogether Theorem 4.4 can be interpreted as saying that the stable homotopy th* *eory of spaces retractive over BG is equivalent to the homotopy theory of S[G] modules, or spe* *ctra with an action of G. This is exploited by J. Klein and J. Rognes to prove a chain rule * *for the Calculus of Functors [22]. 7.3 Monoids and groups. The theories of sets, monoids and groups have equivale* *nt stable homotopy theories. This follows from the fact (see [29, Thm. 1]) that the free * *monoid and the free group generated by a connected simplicial set are weakly equivalent to the* * loop space on the suspension of the simplicial set. Since the map from a simplicial set to th* *e loop space of its suspension is twice as highly connected as the space itself, the maps of Gamma-* *rings S ----! (monoids)s ----! (groups)s are stable equivalences. 7.4 Nilpotent groups. The lower central series of a group G is a filtration by * *normal subgroups rG. These subgroups are defined inductively by 1G = G and rG = [r-1G; G], the s* *ubgroup generated by commutators. A group is called nilpotent of class r if r+1G is tri* *vial. We denote by Nilr the theory of class r nilpotent groups. We obtain a tower of theories (groups) -! . . .-! Nilr -! Nilr-1- ! . . .-! Nil1= (abelian groups) : It follows from a theorem of E. Curtis [12, Thm. 1.4] that the unit map S -!(Ni* *lr)s is (log2r -1)- connected. So the associated sequence of Gamma-rings interpolates between S and* * (Nil1)s = HZ. 29 7.5 p-local groups. Fix a prime number p. By considering p-local nilpotent grou* *ps we obtain a Gamma-ring model for the p-local sphere spectrum, together with a `multiplicati* *ve filtration'. This provides a different view at the mod p-lower central series spectral sequence o* *f [8]. A nilpotent group G is called p-local if for all primes q 6= p the set map x 7! xq is a bij* *ection of G onto itself. On the category of nilpotent groups there exists a p-localization funct* *or G 7! G(p)which is left adjoint to the inclusion of nilpotent p-local groups [41, Sec. 8]. p-lo* *calization is exact and commutes with the terms in the lower central series, i.e., (G=rG)(p)~= G(p)=r(G(p)) : p-local groups of fixed nilpotence class r form a theory which we denote by Nil* *r(p). By [10, Ch. IV] the group-theoretic localization map G -! G(p)induces p-localization on homotop* *y groups for every simplicial nilpotent group G. This implies that the map of Gamma-rings (N* *ilr)s -! (Nilr(p))s is the p-localization map on the associated spectra. The category of simplicial theories has inverse limits and these are calculated* * pointwise [6, Prop. 3.11.1]. We denote by Nil^(p)the inverse limit theory of the Nilr(p). If X is a* * reduced simplicial set, GX its Kan loop group, then the inverse limit of the simplicial groups (GX=r(GX* *))(p)is weakly equivalent to the loop group of the Z(p)-completion of X by [10, Ch. IV Prop. 4* *.1]. In particular, the free Nil^(p)-algebra generated by a reduced simplicial set X is a model for* * the p-localization of |X|. So the Gamma-ring (Nil^(p))s is a model for the p-local sphere spectrum. * * i We define Ji as the pointwise fibre of the map of Gamma-rings associated to Nil* *^(p)-! Nilp(p). Then i Ji -! (Nil^(p))s -! (Nilp(p))s is a homotopy fibre sequence of -spaces: when evaluated at any simplicial set i* *t gives a short exact sequence of simplicial groups. Since Jiis the fibre of a multiplicative m* *ap between Gamma- rings, it inherits a multiplication (but no unit), and it behaves like an ideal* * of (Nil^(p))s. One can show that the Ji's even form a multiplicative filtration of (Nil^(p))s, i.e., t* *he image of Ji^Jj in (Nil^(p))s under the Gamma-ring multiplication is contained in Ji+j. Altogether* * we have obtained a convergent multiplicative filtration on a Gamma-ring model of the p-local sph* *ere spectrum. This filtration in turn gives rise to a multiplicative spectral sequence. There is a* * variant which starts with the p-lower central series, and gives a multiplicative filtration on the p* *-completed sphere spectrum. In that case, the spectral sequence obtained from the filtration is * *the mod-p lower central series spectral sequence of [8]. From the E2-term on this spectral sequ* *ence is the Adams spectral sequence. 7.6 Infinite loop spaces. In our simplicial set-up, the Barratt-Eccles model [* *1, 2] gives an algebraic theory modeling infinite loop spaces. M. G. Barratt and P. J. Eccles * *define a functor + from the category of pointed simplicial sets to itself [1, Def. 3.1]. To avoid * *notational confusion with the category opof finite pointed sets, we use the notation fl+ for the fun* *ctor of Barratt and Eccles. The functor fl+ is degreewise defined and commutes with filtered colimi* *ts, i.e., it comes from a -space, and fl+ has the structure of a triple [1, Prop. 3.6]. So fl+ is* * the free algebra functor of a simplicial theory. This is in fact the only example of a simplicia* *l theory which we consider explicitly and which is not a discrete theory. The algebras over this * *theory are called `simplicial set with +-structure' in [1]. For connected pointed simplicial sets* * X, fl+ X is a model for 1 1 |X| (this is proved for Kan complexes in [1, Thm. 4.10, 5.4], but since* * the functor fl+ is a prolonged -space, it preserves weak equivalences of simplicial sets [9, 4.* *9] so that property holds for arbitrary X). Every fl+ -algebra X is naturally a simplicial monoid. * *Barratt and Eccles show furthermore [2, Thm. A] that if ss0X is a group, then the fl+ -structure p* *rovides natural 30 infinite deloopings of X. In this sense, the algebraic theory fl+ models infini* *te loop spaces. The + )s arising from the theory fl+ is yet another model for the sph* *ere spectrum; it Gamma-ring (fl has the property that its underlying -space is special. 7.7 Modules. Let B be a simplicial ring and consider the theory of simplicial l* *eft B-modules. The Gamma-ring obtained from this theory is the Eilenberg-MacLane Gamma-ring HB* * as defined in 1.5. The homotopy theory of B-modules remains unchanged under stabilization * *(cf. [36, Thm. 2.2.2]). Theorem 4.4 thus says that the homotopy theory of simplicial B-modules* * is equivalent to the homotopy theory of HB-modules; we recover [37, Thm. 4.4]. 7.8 Associative algebras. Let B be a commutative simplicial ring and consider t* *he theory of augmented associative B-algebras (alias associative B-algebras without unit). W* *e claim that the map from the theory of B-modules to the theory of augmented associative B-algeb* *ras induces a weak equivalence on associated Gamma-rings HB --~--! (Ass. B-alg)s : The connective stable homotopy theory of augmented associative B-algebras is th* *us equivalent to the homotopy theory of simplicial B-modules. This fact could have been prove* *n without ever introducing Gamma-rings by the methods of [36, Sec. 3]. To prove the claim we n* *ote that the free associative non-unital B-algebra generated by a pointed simplicial set K decomp* *oses as the direct sum 1 M eB[ K^_._.^.K_-z___"] n=1 n where eB[-] denotes the reduced free B-module. If K is taken to be a k-dimensio* *nal sphere, all homogeneous components of degree 2 are at least (2k - 1)-connected, so the map* * from the free B-module on Sk to the free non-unital associative B-algebra on Sk is (2k - 2)-c* *onnected. 7.9 Commutative algebras. Let B be a commutative simplicial ring and consider t* *he theory of augmented commutative B-algebras (alias commutative B-algebras without unit) Co* *mmutative simplicial algebras have been the object of much study [31, 14, 17, 18, 36]. Th* *e homology theory arising as the derived functor of abelianization in this case is known as Andre* *-Quillen homology for commutative rings. We denote by DB the Gamma-ring arising from the theory of augmented commutative* * B-algebras. If B is a Q-algebra, the map HB -! DB induced from the symmetric algebra functo* *r is a stable equivalence (cf. [36, Thm. 3.2.3]). In general, the Eilenberg-MacLane spectrum * *splits off DB, but the category of commutative augmented B-algebras can have higher stable homotop* *y operations, in which case DB is not equivalent to HB. We claim that as a -space, DB is stably equivalent to HB ^LHZ. In particular, t* *he homotopy groups of DB are additively isomorphic to the integral spectrum homology of the* * Eilenberg- MacLane spectrum HB. To prove our claim we use that the free commutative B-alge* *bra without unit on a pointed set X is additively isomorphic to the reduced free B-module o* *n the infinite symmetric product of X. Hence if we let SP denote the -space that sends a poin* *ted set to the infinite symmetric product, then DB is isomorphic to the composite -space H* *B O SP. For every connected simplicial abelian monoid the map to its group completion is a * *weak equivalence [40, Cor. 5.7], so the group completion map SP-! HZ is a stable weak equivalenc* *e of -spaces. 31 Hence DB is weakly equivalent to the derived smash product of HB and HZ by [25,* * Prop. 5.23]. LHZ can be constructed as an E -ring spectrum, but the weak equivalence to* * DB can not be HB ^ 1 multiplicative in any sense since the ring of homotopy groups of HB ^LHZ is gra* *ded commutative, that of DB is generally not. By 4.11 the ring ss*DB is isomorphic to the ring of stable homotopy operations * *of commutative simplicial B-algebras. These operations are also referred to as the stable Cart* *an-Bousfield-Dwyer algebra (since these authors calculated the unstable operations for B = Fp, see* * [11, 7, 14]). Additively, ss*DB is the direct sum of the stable derived functors, in the sens* *e of Dold and Puppe [13, 8.3], of the symmetric power functors on the category of B-modules. In [7,* * x12], Bousfield calculates the ring ss*DFp under the name of `stable algebra of the functor alg* *ebra' of symmetric powers. For p = 2 ([7, Thm. 12.3]; see also [14]) it is the associative unital* * F2-algebra with generators ffifor i 2 subject to the relations X n X n ff2m+iff1+m+j = 0 and ff1+2m+iff1+m+j = 0 i+j=n i i+j=n i for m; n 0. [7, Thm. 12.6] gives a similar but more complicated description fo* *r odd primes. If X is an augmented commutative simplicial B-algebra, its stable homotopy is d* *efined as the homotopy groups of the suspension spectrum of any cofibrant replacement: sss*X * *= ss*1 Xc. Then the stable homotopy and Andre-Quillen homology of X are related by the universa* *l coefficient and Atiyah-Hirzebruch spectral sequences of 5.5 Torss*DBp(ss*B; sss*X)q=)HAQp+qX HAQp(X; ssqDB) =) sssp+qX : 7.10 Divided power and Lie-algebras. In the spirit of the previous two examples* *, one can consider other types of algebras over a commutative ring B and study the Gamma-* *rings they give rise to. For divided power algebras over Fp, Bousfield [7, Thm. 12.3 and 1* *2.6] calculates the graded ring of homotopy groups of the associated Gamma-ring, again under the na* *me of `stable derived functors' of the divided power functors. For the case of restricted Lie* *-algebras over Fp, this calculation is carried out in [8, 2.4 and 2.4']. The result is known as the -al* *gebra, and it shows up as a E1-term of the Adams spectral sequence for the stable homotopy groups o* *f spheres. The case of restricted Lie-algebras is closely related to Example 7.5 since the ass* *ociated graded to the p-lower central series filtration of a free group is the free restricted Lie-al* *gebra on the abelianized group. A Cofibrantly generated model categories In [32, p. II 3.4], Quillen formulates his small object argument, which is now * *a standard device for producing model category structures. An example of this is the lifting lemm* *a A.2 below which we use several times in this paper. After Quillen, various other authors have * *axiomatized and generalized the small object argument. We work with the `cofibrantly generated * *model categories' of [15]. We have given a review of cofibrantly generated model categories in [* *37, App. A] and we will continue to use that terminology. In all the cases we treat in this pap* *er, category theory automatically takes care of the smallness conditions. The basic reason is that* * we are dealing with suitable functor categories with values in simplicial sets. The relevant c* *ategory theoretical 32 notion is that of a locally presentable category. The categories we consider ar* *e even locally finitely presentable. In general, categories involving actual topological spaces tend n* *ot to be locally presentable. An object K of a category C is called finitely presentable if the hom functor h* *omC(K; -) preserves filtered colimits. A set G of objects of a category C is called a set of strong* * generators if for every object K and every proper subobject there exists G 2 G and a morphism G -! K wh* *ich does not factor through the subobject. A category is called locally finitely presentable* * if it is cocomplete and has a set of finitely presentable strong generators. Lemma A.1 Let T be a simplicial theory. Then the categories T -alg, GS (T ) * *and Sp(T ) are locally finitely presentable. If T is a discrete theory and R a ring, then the * *category sF(T op; R) of simplicial functors is locally finitely presentable. If S is a Gamma-ring, t* *hen the category of S-modules is locally finitely presentable. Proof: All the above categories are cocomplete. Finitely presentable strong g* *enerators exist because objectwise evaluation is representable in all these categories. More p* *recisely, possible choices of generators are as follows. In T -alg, we can choose the T -algebras * *freely generated by the simplicial standard simplices (i)+. In GS(T ) we can take the objects T sO * *(n^ (i)+). In Sp(T ) we take the spectra of T -algebras Fnidefined by ae (Fni)j = j-nF T*((i)+) ifijf