GENERALIZED ANDR'E-QUILLEN COHOMOLOGY DAVID BLANC Abstract. We explain how the approach of Andr'e and Quillen to defining* * cohomol- ogy and homology as suitable derived functors extends to generalized (co* *)homology theories, and how this identification may be used to study the relations* *hip between them. Introduction After the cohomology of topological spaces was discovered in the 1930's, the* * concept was expanded to groups, and later to associative, commutative, and Lie algebras* *, in the 1940's and early 1950's. In the following decade the first generalized coho* *mology theories for spaces appeared (see [Mc2 , Mas ]). All these examples started ou* *t in the form of explicit constructions, and only later were their theoretical underpinn* *ings pro- vided: in particular, cohomology for general algebraic categories was described* * by Beck and others in terms of triples (see [Be ], and compare [D1 ]), and then by Andr* *'e and Quillen in terms of (non-abelian) derived functors (see [An , Q1 ]). In the lat* *ter version, cohomology groups are the derived functors of Hom into a fixed abelian group * *object (and homology groups are the derived functors of abelianization). However, for topological spaces the only abelian group objects are (products* * of) Eilenberg-Mac Lane spaces, which represent ordinary cohomology. Thus we need a different framework to describe generalized (co)homology: this is provided by s* *table homotopy theory (cf. [Br , Wh ]). Our goal here is to provide a uniform definition for homology and cohomology* * en- compassing the theories mentioned above, as well as some new ones. As a side be* *nefit, we clarify exactly what assumptions on an (algebraic) category C are needed in * *order for the approach of Andr'e and Quillen to work. (This is the reason for the som* *ewhat technical Section 3.) The approach given here applies, inter alia, to: (a) Homology and cohomology of groups and various types of algebras; (b) Versions of the above with local coefficients (x4.1-4.2); (c) Unstable generalized (co)homology of spaces (x5.7-5.10); (d) Generalized (co)homology of spectra and spaces (x2.18); (e) Cohomology of operads, and of algebras over an operad (x4.15); (f) Cohomology of diagrams of spaces or algebras (x4.7). The last two have applications to deformation theory (see [Mar1 , MS2 ] and * *[GS1 , GGS ], respectively). The cohomology of sheaves has a dual definition to the one presented here he* *re (see x4.17). Of course, there are other concepts of cohomology which do not fit int* *o our ____________ Date: August 10, 2007; revised: February 10, 2008 . 1 2 DAVID BLANC framework; most notably, a number of versions of the cohomology of categories (* *see x4.16). 0.1. Representing cohomology. In order to define a cohomology theory in a category C, we need a representingoobjectp G 2 C, as well as a suitable model * *category structure on the category sC = C of simplicial objects over C (see x2.7). * *However, in this generality Hom C(-, G) will take values in sets, and applying this f* *unctor to a simplicial resolution Vo ! X in sC just yields a cosimplicial set, for wh* *ich we have no appropriate model category. It turns out that in order to get an inter* *esting cohomology theory, two ingredients are generally needed: o The category C must be enriched over a symmetric monoidal category V; o The representing object G must have additional "algebraic" structure. We shall use the concept of a sketch - a straightforward generalization of * *Lawvere's concept of a theory - to describe this additional structure (see x1.1). In thi* *s language, we say that G is a -algebra in C, for a suitable FP-sketch . We also use sket* *ches to describe the kind of algebraic categories to which our approach applies: this w* *ill allow us to treat operads and their algebras, for example, uniformly with the usual u* *niversal algebras. o Note that the functor Hom C(-, G) now takes values in the category D* * of (cosimplicial) -algebras in V. Our final requirement is that the abov* *e two ingredients must combine to make D into a (semi-) triangulated model ca* *tegory (see x2.2). The question we consider here is in some sense dual to that of Brown Represe* *ntability in triangulated categories (cf. [CKN , F, K , N ]): rather than asking which c* *ohomology functors are representable, we seek conditions for a representable functor to b* *e a co- homology theory. 0.2. Examples. In the category of groups (where V = Set), with an abelian gro* *up G as the coefficients, the model category we consider is that of simplicial group* *s. The total left derived functor of Hom (-, G) then takes values in the semi-triang* *ulated category of cosimplicial abelian groups (equivalently, cochain complexes). On the other hand, for pointed simplicial sets or topological spaces (where * *V = S*), we may take = , and Hom (-, G) takes values in -spaces - again, a sem* *i- triangulated category. Note that the category of spectra is triangulated (and enriched over itself)* *, so we can take any spectrum G as coefficients. Our original motivation for creating a joint setting for algebraic and gener* *alized topological (co)homology theories was to try to gain a better understanding of * *the relationship between homology and cohomology. This is provided by a universal * *co- efficients spectral sequence (see Theorem 6.12 below). We obtain a similar res* *ult for homology (Proposition 6.14), as well as "reverse Adams spectral sequences" (The* *orems 6.17 and 6.18) relating homotopy to (co)homology. 0.3. Notation and conventions. The category of topological spaces is denoted by T , and that of pointed connected topological spaces by T*. The category of g* *roups is denoted by Gp , that of abelian groups by Abgp , and that of pointed set* *s by Set *. For any category C, grS C denotes the category of S-graded objects o* *ver C 'E-QUILLEN COHOMOLOGY 3 GENERALIZED ANDR (i.e., diagrams indexed by the discrete category S), sC that of simplicial ob* *jects over C, and cC that of cosimplicial objects over C. The category of simplical sets* * will be denoted by S, that of reduced simplicial sets by S*, and that of simplicial g* *roups by G. For any Z 2 C, we write c(Z)o for the constant simplicial object determi* *ned by Z, and c(Z)o for the constant cosimplicial object. If A is any abelian categ* *ory, we denote the category of chain complexes over A by Ch(A); however, we write ChR for Ch(R-Mod ), and similarly cCh R for cochain complexes of R-modules. 0.4. Organization. Section 1 provides background material on sketches, theorie* *s, and algebras over them. In Section 2 we give our abstract definition of homolo* *gy and cohomology, in the context of suitable model categories. Abelian group obje* *cts in sketchable categories are described in Section 3, and these are used in Section* * 4 to define the (co)homology of -algebras. Section 5 explains how generalized cohomologie* *s fit into our framework, using -spaces. Finally, the theory is applied in Section * *6 to construct universal coefficient and reverse Adams spectral sequences in this ge* *neral framework. 0.5. Acknowledgements. This paper is an outgrowth of joint work with George Pes* *chke, in [BP ], and I would like to thank him for many useful discussions and insight* *s. I also thank the referee for his or her helpful comments, and the Institut Mittag-Leff* *ler (Djur- sholm, Sweden) for its hospitality during the period when this paper was comple* *ted. 1. Algebras and theories As Lawvere observed (cf. [La ]), `varieties of universal algebras' in the s* *ense of Mac Lane (cf. [Mc1 , V,6]) can be corepresented by functors out of a fixed cate* *gory . This idea was later generalized by Ehresmann to sketches (see [BE ]), which* * turn out to be the most convenient language to describe both the algebraic categorie* *s we work in, and the representing objects for cohomology. 1.1. Definition. A sketch < , P, I> is a small category with distinguished * *sets P of (limit) cones and I of (colimit) cocones. In particular, a finite product (F* *P-)sketch is a sketch in which P consists only of finite products (and I = ;). A theory* * is an FP-sketch containing a zero object, for which P consists of all finite produc* *ts. We think of a map f : #1 x . .x.#n ! ` in as corepresenting a (possibly graded) n-ary operation. A theory is sorted by a set S Obj if every ob* *ject in is uniquely isomorphic to a finite product of objects from S (see [Bor , x* *5.6]). Lawvere originallyQconsidered only theories sorted by {1}, so that Obj ( ) =* * N, with n ~= ni=11 for n 0. If is an FP-sketch and C is any pointed category, a -algebra in C is a p* *ointed functor X : ! C which preserves all products in P. More generally, if is* * any sketch, a -algebra X : ! C is required to preserve all distinguished limit* *s (in P) and colimits (in I). The category of -algebras in C is denoted by -C; a -a* *lgebra in Set* will be called simply a -algebra, and we write -Alg for -Set*. * * We call a category D sketchable if it is equivalent to -Alg, and say that ske* *tches D. Such categories are accessible, in the sense of model theory, as well as be* *ing locally presentable (see [AR , Cor. 2.61 & 1.52]). A map of theories (or of sk* *etches) _ : ! 0 is a functor which preserves all products (respectively, all disting* *uished limits and colimits). Such a map _ induces a functor _* : 0-Alg ! -Alg. 4 DAVID BLANC More generally, if is a theory (or FP-sketch), a -algebra in any symmetr* *ic monoidal category (cf. [Bor , x6.1]) is a functor X : ! V takin* *g the (distinguished) products in to -products in V, with X(*) = I. 1.2. Remark. Since we can think of a -algebra X in C as a certain kind of diag* *ram in C (with specified products), we see that Hom C(-, X) takes values in -Alg.* * More generally, if C is enriched over a symmetric monoidal category via * * map C (cf. [Bor , x6.2]), and map C(A, -) takes products to , then map C(-, X) t* *ake values in -V. 1.3. Examples. (a) The category of groups is sketched by a theory G, with ~ : * *2 ! 1 representing the group operation, ae : 1 ! 1 the inverse, and e : 0 ! 1 the* * identity (satisfying the obvious relations). Similarly, the category of abelian groups i* *s sketched by a theory A (with the same maps, satisfying a further relation) and the inc* *lusion i : G A induces the inclusion of categories Abgp Gp. (b) An operad = ( (n))1n=0 is an O-algebra in a symmetric monoidal category , where O is a "universal" theory for operads. Similarly, an algebra* * over the operad (see [May2 , x14]) is just a -algebra in , where the the* *ory is obtained from in the obvious way (replacing with x). The same applies mo* *re generally to PROP's, colored operads, and other variants (see [MSS ] for a sur* *vey on operads, especially in the algebraic context). (c) Given a topological space X, let U denote the directed set of non-empty ope* *n sets in X, with inclusions` - so that Uop sketches presheaves of sets. By adding * *arbitrary formal coproducts ff2AUff for any collection {Uff}ff2A in U, we obtain a ca* *tegory U^, in which the diagram; ` _i__//_` ~ S (1.4) (ff,fi)2AxAUff\ Ufi___//_ff2AUff_____//ff2AUff j is a coequalizer (if the first term is empty, ~ is an isomorphism). If we now let U := ^Uop (sorted by U), with P consisting of the opposites * *of the formal coproducts and of all the coequalizers (1.4)(and I = ;), we obtain a * *sketch whose algebras F : U ! Set are sheaves of sets on X. Furthermore, for any V * *2 U, if: ( {*} ifU V CV (U) := ; ifU 6 V, there is a natural isomorphism Hom U-Alg(CV , F) = F(V ). 1.5. Definition. Given a theory X, an`X-theory (or sketch) is one equipped wi* *th a map of theories (or sketches) _ : S X ! which is bijective on objects, * *where the coproduct is taken in the category of theories (or sketches) over some inde* *x set S. If X is sorted by {1}, an X-structure at an object c in a category C is an X-al* *gebra ae : X ! C with ae(1) = c. A theory sorted by S is an X-theory if and only* * if it is equipped with an X-structure at each s 2 S. If all other maps of commute with those coming from _, we call a strong * *X-theory (or sketch). 'E-QUILLEN COHOMOLOGY 5 GENERALIZED ANDR ` 1.6. Example. If is a G-theory, then the map of theories _ : S G ! indu* *ces an "underlying`S-graded group" functor _*, which we denote by V : -Alg ! GpS = S G-Alg . is a strong G-theory if all the operations in are homomorphis* *ms of the underlying graded group. 1.7. Free -algebras. For any theory , let ffidenote the discrete theory w* *ith the same objects (and products) as . If is sorted by S, ffisketches the ca* *tegory of S-graded sets, and the inclusion I : ffi,! induces the forgetful functo* *r U = U : -Alg ! ffi-Alg. As usual, there is a free functor F = F : ffi-Alg! * *-Alg left adjoint to U . We denote by F the full subcategory of -Alg whose * *objects are free (that is, in the image of F ). Since all limit-sketchable categories are locally presentable, they are comp* *lete (see, e.g., [AR , Theorem 1.46]) and cocomplete. Thus for any theory , the category * * -Alg of -algebras has all limits and colimits. 1.8. Sketching -algebras in -Alg . If is a theory (sorted by S) and is another theory (singly sorted, for simplicity), the category - -Alg of -alg* *ebras in -Alg is sketched by a theory ( ) (sorted by S), defined as follows: ` (a) We first add an S-graded copy of to , setting := [S S , so that we now have each operation of acting on each ` 2 S. The inclus* *ion i : ,! induces a forgetful functor i* : -Alg ! -Alg. (b) Next, we force all operations of to commute with the new operations * *- that is, for each f : `1 ! `2 in and g : n ! k in , we require that g `n1_____//`k1 |fn| |fk| fflffl|gfflffl| `n2_____//`k2 commute, so we obtain a quotient of theories q : ! ! ( ). By construction ( ) -Alg ~= - -Alg . Note that q* and i* commute wi* *th the underlying S-graded set functors U , U , and U , which create all * *limits in their respective categories, so q* and i* commute with all (small) limits. * *Thus by [Bor , Theorem 5.5.7] each has a left adjoint. The adjoint of the composite i* ** O q* : - -Alg ! -Alg will be called the -localization of -Alg, and denoted by L : -Alg ! - -Alg . 1.9. Remark. Note that given G in - -Alg , by Remark 1.2 Hom -Alg(-, G) h* *as a natural structure of a -algebra. Furthermore, if i* O q* is a faithful embe* *dding of categories (which will happen if is a -theory, for example), then L is id* *empotent and any -algebra in -Alg is in the image of L , up to natural isomorphism. * *Thus Hom - -Alg(-, -) has a natural structure of a -algebra, in this case. By mi* *micking the construction of A x B ! A B for abelian groups, one can then make - -* *Alg into a closed symmetric monoidal category (see [Bor , x6.1.3]). 2. Generalized homology and cohomology We are now able to give a definition of homology and cohomology for model ca* *te- gories, somewhat more general than Quillen's original approach (cf. [Q1 , II, x* *5]): 6 DAVID BLANC 2.1. Triangulated categories. The target of a cohomology functor should be a model category whose homotopy category is triangulated. There are a number of variants of this concept, origi* *nally due to Grothendieck. For our purposes, a triangulated category is an additive categ* *ory C equipped with an automorphism T : C ! C (called the translation functor), and* * a f g h collection D of distinguished triangles of the form , sa* *tisfying the four axioms of [Ha , x1] (which codify the properties of cofibration sequen* *ces in pointed model categories - see [Q1 , I, x3]). 2.2. Definition. A semi-triangulated category is an additive category ^Cequippe* *d with a collection D of distinguished triangles of satisfying the above four axioms, * *as well as a translation functor T : ^C! ^C which is an isomorphism onto its image. In al* *l cases of interest, T can be formally inverted to yield a full triangulated category C* * = ^C[T -1] with ^Cas a full subcategory; however, this property is not needed in what foll* *ows. A set P of cogroup objects in ^Cwill be called a set of generators if the co* *llection of functors {Hom ^C(T iP, -)}P2P,i 0 detects all isomorphisms in ^C. 2.3. Example. Typically, (semi-)triangulated categories appear as the homotopy * *cate- gory of a suitable (semi-)stable model category, as defined axiomatically in [H* *o , Ch. 7] (see also [HPS ]). Thus, the motivating example of a triangulated category* * is the homotopy category of (unbounded) chain complexes over an abelian category A. An- other example is provided by Boardman's stable homotopy category ho Spec (cf. * *[V ]), where there are a number of different underlying stable model categories (see [* *HSS ], [Sc1], or [EKMM ]). The subcategory ^Cof non-negatively graded chain complexes is semi-triangula* *ted; if A has a projective generator P , then K(P, 0) (the chain complex with P concent* *rated in degree 0) is a generator for ^C. Similarly, the homotopy category of connective spectra, hoSpec (0), is sem* *i-triangu- lated (with generator S0). 2.4. Cohomology. In order to define cohomology functors on a model category E, we assume that E is equipped with: (a) An FP-sketch and a category V such that V and -V are symmetric monoidal, E is enriched over V via map E(-, -) : Eop x E ! V, and * * -E is enriched over -V via Hom__(-, -) : ( -E)opx -E ! -V. (b) An FP-sketch and a model category structure on -V for which ho -V* * is semi-triangulated. Then for any G 2 -E, we define the cohomology of X 2 E with coefficient* *s in G to be the total left derived functor L map E(-, G) of map E(-, G), applie* *d to X. Recall that total left derived functor of a "left exact" functor F : C ! D be* *tween model categories is defined by applying F to a cofibrant replacement of X (see * *[Q1 , I, x4] or [Hi, 8.4]). If ho -E has a set of generators P, then the P-graded group Hn(X; G) := [T nP, (L map E(-, G))X]P2P is called the n-th cohomology group of X with coe* *fficients in G. 2.5. Homology. To define homology, we need also a homotopy functor A : E ! -E ~= equipped with a natural isomorphism map E(E, X) -! Hom__(A E, X) in -V (cf. 'E-QUILLEN COHOMOLOGY 7 GENERALIZED ANDR x1.2) for E 2 E and X 2 -E. We then define the homology of X 2 E to be t* *he total left derived functor of A , applied to X (x2.4). Again the n-th homolog* *y group of X is: HnX := [T nA P, (LA )X]P2P . If -E is a symmetric monoidal model category (see [Ho , x4.2.6]), with Hom* *__(-, Y ) right adjoint (over -V) to - Y , then for any G 2 -E, homology with coefficie* *nts in G is the total left derived functor of A (-) G (assuming A E is always cofib* *rant). The homology groups Hn(X; G) are defined as above. Compare [BB , I]. 2.6. Example. If E = V = S* (or T*) and = A, then -C ~= -V ~= sAbgp * *and G is a (generalized) Eilenberg-Mac Lane space, so we have ordinary cohomology. * *The functor A : E ! -C is the usual `abelianization' X 7! ZX, which yields or* *dinary (singular) homology. 2.7. Resolution model categories. To provide a uniform treatment of the various kinds of (co)homology it will * *be convenient to use a framework originally conceived by Dwyer, Kan and Stover in * *[DKS ] under the name of "E2 model categories", and later generalized by Bousfield (see [Bou , J]. Recall that the concept of a model category was introduced by Quillen in [Q1* * ] to allow application of the methods and constructions of homotopy theory (of topol* *ogical spaces) in more general contexts. This is a category C, equipped with three di* *stin- guished classes of morphisms - weak equivalences, cofibrations, and fibration* *s - satisfying certain axioms (analogous to those which hold for the corresponding * *classes in T ). See [Hi] or [Ho ] for further details. Let C be a pointed cofibrantly generated right proper model category (cf. [H* *i, 7.1, 11.1]), equipped with a set M of cofibrant homotopy cogroup objects in C, calle* *d models (playing the role of the spheres in T*). Let M denote the smallest full su* *bcategory of C containing M and closed under coproducts, and suspensions (cf. [Q1 , I, x3* *]). For any X 2 C, M 2 M, and k 0, set ssM,kX := [ kM, X0], where X ! X0 is a fibrant replacement. We write ssM,kX for the M-graded group (ssM,kX)M2M . 2.8. Definition. A map f : V ! Y in sC is homotopically M-free if for each n * * 0, there is: a) a cofibrant object Wn 2 M , and b) a map 'n : Wn ! Yn in C inducing a trivial cofibration`(VnqLnV LnY )qWn* * ! Yn, where the n-th latching object for Y is LnY := 0 i n-1Yn-1= ~, * * with sj1sj2. .s.jkx 2 (Yn-1)i is equivalent to si1si2. .s.ikx 2 (Yn-1)j w* *henever sisj1sj2. .s.jk= sjsi1si2. .s.ik. The resolution model category structure on sC determined by M is now defin* *ed by declaring a map f : X ! Y to be: (i) a weak equivalence if ssM,kf is a weak equivalence of M-graded simpli* *cial groups for each k 0; (ii)a cofibration if it is a retract of a homotopically M-free map; (iii)a fibration if it is a Reedy fibration (cf. [Hi, 15.3]) and ssM,kf is* * a fibration of simplicial groups for each M 2 M and k 0. 8 DAVID BLANC 2.9. Remark. The resolution model category sC is simplicial (cf. [Q1 , II, x1* *], and is itself endowed with a set of models, of the form M^ := {Sn M | M 2 M, n 2 N}, where Sn 2 S is the simplicial sphere. 2.10. Examples. Typical resolution model categories include the following: (i) When C = Gp, let M := {Z}, so M is the subcategory of all free * *groups. The resulting resolution model category structure on the category G = * *sGp of simplicial groups is the usual one (see [Q1 , II, x3]). (ii) More generally, if is a G-theory (x1.5), let M := F0 denote the col* *lection of all monogenic free -algebras F (s) in F , with s a singleton * *in ffi-Alg (i.e., a graded set, indexed by the discrete sketch ffi, consisting * *of a single element in some degree). In this case M ~=F , and the model categ* *ory on s -Alg is that of [Q1 , II, x4]). (iii)For C = T*, let M := {S1}, so that M is the homotopy category of wedg* *es of spheres. In this case the model category of simplicial spaces is the* * original E2-model category of Dwyer, Kan and Stover (cf. [DKS ]). 2.11. Remark. The above discussion is also valid if we work in the comma catego* *ry -Alg=X (cf. [Mc1 , II,6]), for a G-theory and some fixed -algebra X. In fa* *ct, any p : F ! X in F =X is determined by its adjoint "p: T ! U X - in other * *words, by the U X-graded set {p-1(x)]}x2U X . Therefore, -Alg=X can be sketched* * by a theory =X, sorted by U X = {OEx | x 2 U X}. Note that =X is a G-ske* *tch over X in the sense that it has G-structures of the form: m(x1,x2): OEx1x OEx2 ! OEm`(x1,x2) for every ` 2 and x1, x2 2 U X` (and similarly for other morphisms in ). Equivalently, we can equate the discrete theory =Xffiwith ffi-Alg=U X, * * and use the adjointness of (F , U ) to define an adjoint pair: F ffi ffi =X -Alg = -Alg=X U -Alg=U X = =X . We can then take the monogenic free -algebras F0 =X (cf. x2.10(ii)) as our * *models, and obtain a resolution model category structure on s( -Alg=X). In particular* *, any free resolution Vo ! X in s -Alg is also a resolution (cofibrant replacemen* *t) in s( -Alg=X). 2.12. A simplicial version of (co)homology. In order to make the abstract description of (co)homology given in x2.4-2.5 * *more concrete, it is convenient to formalize the ingredients needed in the following: 2.13. Definition. A cohomological setting consists of: (1) A model category C, enriched via map C(-, -) over a symmetric monoidal category V. (2) A set of models M for C. (3) An FP-sketch , such that: (i) < -C, , I, Hom__> is a closed symmetric monoidal category (with Ho* *m__(G, -) right adjoint to - G). (ii)c -V has a model category structure for which ho c -V semi-triangul* *ated. 'E-QUILLEN COHOMOLOGY 9 GENERALIZED ANDR (4) A homotopy functor A : M ! -C, equipped with a natural isomorphis* *m: (2.14) : map C(F, G) ~=Hom__(A F, X) for F 2 M and G 2 -C. 2.15. Definition. Given a cohomological setting , take E := * *sC, with the resolution model category structure defined by M. Then for any object* * X and -algebra G in C, the cohomology of X with coefficients in G is the total l* *eft derived functor of map C(-, G), applied to X. The n-th cohomology group of X * * with coefficients in G is the M-graded group: Hn(X; G) := [T nc(A M )o, (L map C(-, G))X]M2M . 2.16. Definition. For as above, note that A M is a homotopy cogroup object in -C for each M 2 M, so we have a resolution model category structure on s -C determined by the set of models M := {A M}M2M . Define the homology of X to be the total left derived functor of A applied to X. T* *he n-th homology group of X 2 C is the M-graded group: HnX := ssM ,nLA X (cf. x2.9). (For this part of the definition we only require that -C be enr* *iched over itself via Hom__ - we do not need the symmetric monoidal structure.) If G 2 -C, we define the n-th homology group of X with coefficients in G * *to be: Hn(X; G) := ssM ,n(L(A (-) G)(X) 2.17. Example. The simplest example is when C = Gp (with M = {Z} as on x2.10(i)* *), = G (or A), and V = Set, so -C ~= -V ~=Abgp . In this case -C ~=Abgp , so the category c -C of cosimplicial -algebra* *s in C is equivalent to the category of cochain complexes. Thus K(Z, n) (a cochain comp* *lex concentrated in degree n) corepresents the n-th cohomology group of a cochain c* *omplex (n 2 N). This yields the usual cohomology groups of a group X with coefficient* *s in an abelian group G (as a trivial X-module). The functor A : M ! -C is the abelianization Ab : Gp ! Abgp , and t* *he closed symmetric monoidal structure yields the usual ho* *mology of groups. 2.18. Example. Another simple example is provided by a symmetric monoidal categ* *ory of spectra, such as the symmetric spectra of [HSS ], or the S-modules of [EKMM * * ]. In the latter version, for example, we take E = MS, with the symmetric mon* *oidal smash product ^S, and the internal function complexes FS(-, -) 2 V = E (cf. [EKMM , II, 1.6]). Since ho MS is the usual stable homotopy category, it i* *s trian- gulated, with generator S. Thus we can take = * to be the trivial FP theor* *y, any S-module M yields a cohomology theory FS(-, M), and A : E ! -E is the identity. Similarly if E = MR for some S-algebra R. 2.19. Remark. These definitions may appear somewhat convoluted; they have been * *set up to describe both the algebraic and (generalized) topological theories in a u* *niform way, as appropriate derived functors. Note that in general the total homology * *and cohomology functors, as well as the homology and cohomology groups, take values* * in different categories. 10 DAVID BLANC 3.Theories and Abelianization In this section we describe the necessary background for defining (co)homolo* *gy in a category C = -Alg of -algebras. Most of it should be familiar from the case C* * = Gp, and the generalizations of Beck and Quillen for algebras (see [Be , Q3 ]); howe* *ver, it seems that the literature lacks a full description in this generality. We start* * with the concept of (abelian) group objects, which are to play the role of -algebras in* * C. 3.1. Group objects. In general, for a sketchable category C = -Alg we do n* *ot expect any enrichment beyond V = Set; so the natural choice for a cohomologic* *al setting is = A. Recall that an (abelian) group object structure on an object G in a category* * C is a natural (abelian) group structure on Hom C(X, G) for all X 2 C - in other* * words, a lifting of the functor Hom C(-, G) from Set to Gp (or Abgp); this is e* *quivalent to a G- (resp., A-) structure at G. 3.2. Remark. Note that if C = -Alg for some G-theory , any group object struc* *ture on G commutes with the underlying (graded) G-structure, so that the two necessa* *rily agree and are commutative. In particular, in this case a -algebra can have at * *most one (necessarily abelian) group object structure. This is of course not true fo* *r general C (as is shown by the example of sets). 3.3. Abelianization of -algebras. If is any theory (sorted by S), the categ* *ory of abelian group objects in -Alg is sketched by the theory ab := A of x1* *.8. We call the A-localization LA : -Alg ! ab-Alg the abelianization functor for * *, and denote it by A . Note that A (F T ) = F abT . 3.4. Examples. (a) When is a G-theory, ab := G( ), by Remark 3.2, and we can take G := in x1.8, so q : ! ! ab is a quotient of theories, * *and q* is simply the inclusion of the full subcategory of abelian -algebras i* *n -Alg (cf. [BP , x2.8]). Note that by Remark 1.9 we can then make ab into a* * closed symmetric monoidal category. (b) On the other hand, if = ffi, then ab = A sketches S-graded abe* *lian groups, q* : ab-Alg ! -Alg is the forgetful functor U : grSAbgp ! g* *rSSet, and its left adjoint A is the free graded abelian group functor. 3.5. -algebras over X. We now show how the above discussion extends to the category -Alg=X of - algebras over a fixed object X (see x2.11). First, we need a: 3.6. Definition. If is any theory and X 2 -Alg, then: (a) An X-algebra is an object K in -Alg equipped with maps f^: K(#)xX(#) ! K(#0) for each f : # ! #0 in , satisfying: ^g(f^(k, x), X(f)(x)) = [g O(fk, x) for every (k, x) 2 K(#) x X(#), and g : #0! #00, with ^f(k, 0) = K* *(f)(x). (b) The semi-direct product of a -algebra X by an X-algebra K is the -alg* *ebra K o X over X given by: (i) (K o X)(#) := K(#) x X(#) (as sets); 'E-QUILLEN COHOMOLOGY 11 GENERALIZED ANDR (ii)For each f : # ! #0 in , (K o X)(f)(k, x) := (f^(k, x)), X(f)(x)* *). If we want K o X to be a group object in -Alg=X, we must require more. * *From now on, let be a G-theory (sorted by S), and X a (fixed) -algebra. 3.7. Definition. An X-module is an X-algebra K which is an abelian group object* * in -Alg, such that for each fixed x 2 X(#), each ^f(-, x) : K(#) ! K(#0) is * *additive (in the sense that it commutes with the given abelian group structure). The cat* *egory of X-modules will be denoted by X-Mod (see [Be , x3]). 3.8. Remark. In this case the underlying S-graded group V K is an S-graded V* * X- module in the traditional sense (a module over the graded group ring Z[V X]), * * and the group operation at each ` 2 is given by m`((k, x), (`, y) = (k + x .* * `, xy), as usual. 3.9. Definition. Assume that p : Y ! X is a map of -algebras, and K is an X-module. A function , : Y ! K (preserving the products of ) will be calle* *d a derivation with respect to p if ,(Y (f)(y)) = ^f(,(y), p(y)) for any f : # !* * #0 in . The set of all such will be denoted by Der p(Y, K). In particular, a derivati* *on with respect to IdX will be called simply a derivation, and Der(X, K) := Der Id(X* *, K). 3.10. Remark. Note that this holds in particular for f = m` : ` x ` ! `, so t* *hat by Remark 3.8: ,(m`(y1, y2))) = ^m`((,(y1), p(y1)), (,(y2), p(y2)) = ,(y1) + p(y1) .* * ,(y2) . Thus , is a derivation (crossed homomomorphism) with respect to the G-structure. Furthermore, Derp(Y, K) is an abelian group (under the addition of K), and* * any map of X-modules ff : K ! L induces a homomorphism ff* : Der p(Y, K) ! Der p(Y, L). The following results do not appear in this form in the literature, but thei* *r proofs are straightforward generalizations of the corresponding (classical) results fo* *r groups (see, e.g., [Be , x3-4] and [R , x11.1]). 3.11. Proposition. Any group object structure on p : Y ! X in -Alg=X is necessarily abelian. Moreover, K := Ker(p) is an X-module, with Y ~= K o X, * * and for some derivation , : X ! K, the group operation map ~ : Y xX Y ! Y is gi* *ven (under the identification UY = UK x UX) by ~(k, k0, x) = (k + k0+ ,(x), x), * * the zero map by (k, x) 7! (-,(x), x), and the inverse by (k, x) 7! (-k - 2,(x), * *x). Conversely, for any X-module K and derivation , : X ! K, the above formulas make K o X into an abelian group object over X. 3.12. Corollary. There is an equivalence of categories `*G-( -Alg=X) ! A-( -Al* *g=X) , induced by the quotient map ` : G ,! A. 3.13. Lemma. Any homomorphism OE : K o X ! L o X between group objects over X (with group operations determined by oe 2 Der (X, K) and o 2 Der (X, * *L), respectively) is of the form OE(k, x) = (ff(k) + ,(x), x), where ff : K ! L * * is a homomorphism of X-modules and , := ff O oe - o. In particular, any two group object structures over X on the semi-direct pro* *duct K o X are canonically isomorphic, so we deduce: 12 DAVID BLANC 3.14. Proposition. The functor ~ : X-Mod ! A-( -Alg=X) , defined ~(K) := K* * o X (with the group operation map determined by the zero derivation), is an equi* *valence of categories, with inverse ~ : A-( -Alg=X) ! X-Mod which assigns to an abe* *lian group object p : Y ! X the X-module Ker(p). 3.15. Remark. Since the forgetful functor U = U : -Alg ! ffi-Algis faithful* *, for any -algebra Y and semi-direct product K o X 2 -Alg we have: (3.16) U Hom -Alg(Y, K o X) ,! Hom ffi-Alg(UY, U(K o X)) = Hom ffi-Alg(UY, UK x UX) = Hom ffi-Alg(UY, UK) x Hom ffi-Alg(UY, U* *X) . Thus given p : Y ! X, we can write any map OE : Y ! K o X over X in the f* *orm OE(y) = (ff(y), p(y)), and the requirement that OE be a map of -algebras mean* *s that ff : F T ! K is a derivation with respect to p (x3.9), so in fact: (3.17) Hom -Alg=X(Y, K o X) ~= Derp(Y, K) as abelian group (once we choose a fixed group structure on K o X). Three special cases should be noted: (a) For p = Id : X ! X, we see that Der(X, K) is the space of sections * *for K o X, as usual. (b) If Y = L o X for some L 2 X-Mod , then by Proposition 3.14: Hom X-Mod(L, K) ~= Hom -Alg=X(L o X, K o X) = Derp(L o X, K) . On the other hand, by Lemma 3.13 any map of X-modules ff : L ! K induc* *es a homomorphism of group objects OE = ~(ff) : L o X ! K o X (where we * *use the zero derivation to define the group structures on the semi-direct p* *roducts). Thus in fact: (3.18) Hom A-( -Alg=X)(L o X, K o X) = Derss2(L o X, K) as abelian groups. (c) If Y = F T is free, then by adjointness we actually have equalities * *of sets: Hom -Alg(F T, K o X) = Hom ffi-Alg(T, UK) x Hom ffi-Alg(T, UX) in (3.16), so for p : F T ! X in F =X, we have: (3.19) Hom -Alg=X(F T, K o X) ~= Hom ffi-Alg(T, UK) ~= Hom -Alg(F T, W* * K) , where W : X-Mod ! -Alg is the forgetful functor. In particular: Der p(F T, K) ~= Hom -Alg(F T, W K) as sets (though this identification is not natural in the full subcateg* *ory F in -Alg). 3.20. Abelianization over a -algebra. Recall from x2.11 that for a fixed - algebra X, -Alg=X can be sketched by =X (sorted by U X). Similarly, A-( -Alg=X) can be sketched by A =X, obtained from =X as in x1.8 by addi* *ng: (a) a section - i.e., constants in each OEx (in the notation of x2.11); (b) group structure maps ~ : OEx x OEx ! OEx and ae : OEx ! OEx, 'E-QUILLEN COHOMOLOGY 13 GENERALIZED ANDR satisfying the obvious identitites. Again the map of theories i : =X ,! A =X induces the forgetful functor i* : A-( -Alg=X) ! -Alg=X, with an adjoint A* * =X : -Alg=X ! A-( -Alg=X) called the abelianization of -Alg=X. This is needed * *in order to define homology for -algebras (see x4.2 below). Note that the category X-Mod can also be sketched by an A-theory X , obta* *ined from ab (x3.3) by adding operations x . (-) : ` ! ` for each x 2 U X, sa* *tisfying the obvious identitites. The inclusion j : ab ,! X induces the forgetful f* *unctor j* : X -Alg ! ab-Alg. If we define ~ : -Alg=X ! -Alg as in Proposition 3* *.14, we obtain the commutative outer diagram: * ( -Alg=X)ab ________i___________//_ -Alg=X OO A^ =X g g gg g ~ ||~~=|| g gg g |~| fflffl|| ssggg fflffl| X-Mod = X -Alg __j*_// ab-Alg_q*__//_ -Alg in which the horizontal arrows are forgetful functors (and q*, i* have adjoi* *nts A , A =X , respectively, with ^A =X:= ~ O A =X : -Alg=X ! X-Mod ). Note that by (3.19), the abelianization functor A^ =X takes any free -a* *lgebra p : F T ! X over X to the corresponding free X-module F X T 2 X -Alg = X-Mod . Moreover, for any ' 2 Derp(F T, K) (determined by '(ti) = ki 2 K for ti 2* * T ), the corresponding ^' 2 Hom X-Mod(F X T, K) is also determined by requiring t* *hat '^(ti) = ki. Now assume given a map _ : F T 0! F T in F =X, determined * *by the condition that, for each t02 T 0, _(t0) = f0*(ti1, . .,.tin) for some f0 * *in . Then: (_*')(t0) = ^f0((ti1, . .,.tin), (p(ti1), . .,.p(tin))) 2 K . 3.21. Remark. Evidently, the discussion of abelian group objects and abelianiza* *tion over a -algebra X extends the absolute case of x3.1ff., taking X = 0. More generally, K will be called a trivial X-module if f^(k, x) = f(k) for* * every f 2 (x3.6) - so that K is simply an abelian -algebra, K o X is the produ* *ct in -Alg, and a derivation into K is just a map of -algebras. 4. (Co)homology of -algebras Andr'e (in [An ]) and Quillen (in [Q1 , II, x5] and [Q3 , x2]) defined homol* *ogy and cohomology groups in categories of universal algebras. Quillen also showed how* * this generalized the earlier definition of triple cohomology (see [Be , x2]). We now* * indicate briefly how this definition fits into the setup of x2.4. 4.1. Cohomology of -algebras. Let be a G-theory, and C := -Alg (or -Alg=X for a fixed -algebra X), with the resolution model category structur* *e on sC described in x2.10(ii) (or x2.11). As in Example 2.17, here V = Set, so we must take = A (or equivalently* *, by Corollary 3.12: = G), since cosimplicial sets do not have any useful model c* *ategory structure (see however [Bou ]). Thus if G is an abelian group object in C, and * * Vo ! Y is a free simplicial resolution (cofibrant replacement in sC), then the cosimp* *licial abelian group W o:= Hom C(Vo, G) corresponds under the Dold-Kan equivalence * *(cf. [DP , x3] and [We , 9.4]) to a cochain complex W *, and the category cCh Z o* *f non- negatively graded cochain complexes of abelian groups embeds in the category C* *h Z of unbounded (co)chain complexes, which is a stable model category (cf. [Ho , C* *h. 7]. 14 DAVID BLANC Suspensions of g := K(Z, 0) detect homology in cCh Z (or ChZ), so cAbgp * *~=cCh Z is semi-triangulated in the sense of x2.2, and in fact [T ig, W *] = Hi(Y ; G* *) is the i-th Andr'e-Quillen cohomology group of Y . Remark 3.15 shows that these can be thought of as usual as the derived funct* *ors of Der (-, G), in the case C = -Alg=X, and as Exti(Y, G) in the case C = -* *Alg (x3.21). This identification has been the basis for a number of definitions of * *cohomology in various topological settings - see, e.g., [MS2 ], and the survey in [BR ]. 4.2. Homology of -algebras. In this situation one can define the homology of* * a -algebra Y as the total left derived functor of abelianization A : -Alg ! * *ab-Alg (x3.3), which takes values in the category s ab -Alg of simplicial ab-algeb* *ras (as usual, we only need to evaluate A on F , so LA actually takes values in s* *F ab). Since ab -Alg is an abelian category (with enough projectives, namely: F a* *b), s ab -Alg is equivalent to the stable model category Ch ( ab) of chain compl* *exes over ab, and the homology groups [T iK(F abs, 0), A Vo] = HiY (for s an S-gr* *aded singleton) are themselves ab-algebras. The same holds for Y 2 -Alg=X: using x3.20, we may define Hi(Y=X) as the i* *-th derived functor of A X : F =X ! ( -Alg=X)ab, taking values in ( -Alg=X)ab - or equivalently (Proposition 3.14) in X-modules. For groups, H*(G=G) is the homo* *logy of G with coefficients in Z[G]. For a pointed connected space X with G = ss1* *(X, x), H*(X=BG) is the homology of X with coefficients in the local system Z[G]. 4.3. Definition. To define homology of Y ! X with coefficients in an arbitra* *ry X-module G, we need a monoidal structure on X-Mod ~=( -Alg=X)ab, induced via the adjoint pair F X ffi X -Alg U -Alg X from the usual monoidal structure ( ffi-Alg, x) of Cartesian products of grad* *ed sets. More precisely, define : F X x F X ! F X by F X T F X S := F X (T x * *S). The 0-th derived functor in the second variable defines F X T G for any X -a* *lgebra (X-module) G; and the n-th left derived functor of A X (-) G (in the first va* *riable) is by definition Hn(Y=X; G). 4.4. Example. When = G, a free simplicial resolution Vo of a group G in * *sGp is actually a cofibrant model for the classifying space BG (in S*). Applyi* *ng the functor A^ =X of x3.20 to Vo dimensionwise yields a model for the chains on* * the universal contractible G-space EG (since conversely, taking the free Z-module* * on the bar construction model for EG and dividing out by the free G-action yields Z* *BG, so ZEG ' Z[G]Vo). Taking homotopy groups of Z[G]Vo is the same as taking the homology of the chain complex corresponding to ZEG, which is just H*(G; Z[G]* *). 4.5. Remark. Note that the previous discussion actually defines homology and co* *ho- mology for any simplicial -algebra Yo, not only for the constant ones. Moreo* *ver, if = G, the adjoint pairs of functors: |-| G (4.6) T* S* G = sGp S W~ induce equivalences of the homotopy categories of pointed connected topologic* *al spaces, reduced simplicial sets, and simplicial groups. Here | - | is the ge* *ometric 'E-QUILLEN COHOMOLOGY 15 GENERALIZED ANDR realization functor, S is the singular set functor, W~ is the Eilenberg-Mac Lan* *e classify- ing space functor, and G is Kan's loop functor (cf. [May1 , x26.3] and [Q1 , I,* *4 & II,3]). Thus Quillen's approach provides an algebraic description of ordinary homology * *and cohomology of spaces (with local coefficients). Note, however, the shift in ind* *exing: in particular, we lose H0, since we can deal only with connected spaces from thi* *s point of view. There is also an algebraic model for not-necessarily-connected spaces due to* * Dwyer and Kan, using simplicial groupoids (see [GJ , V, x5]), and Quillen's approach,* * as well as much of the discussion here, carries over to that setting (compare [D2 ]). H* *owever, in order to avoid further complicating the description, we restrict attention h* *ere to simplicial groups. 4.7. Diagrams of -algebras. If D is a small category and is a G-theory, the* *re is a model category structure on the functor category s -AlgD , and the object* *wise descriptions of abelian group objects and abelianization (for each d 2 D) pro* *vide definitions of (co)homology for diagrams of -algebras, too (see [BJT , x4] for* * the details). Moreover, even for C = -Alg or -Alg=X, we can allow our coefficients t* *o be diagrams G : D ! A- -Alg of abelian group objects (or X-modules). This enables us to treat a map such as Z!!Z=p (reduction mod p), say, as the coefficients * *for a cohomology theory (rather than a natural transformation). In particular, we can* * apply any general machinery, such as universal coefficient theorems, to H*(-; G), t* *oo. 4.8. Spherical model categories. When C = -Alg for some G-theory , the resolution model category sC (and the models M = F0 - cf. x2.10(ii)) will have additional useful structure wh* *ich is familar to us from topological spaces: 1. For any n 1, ssM,n (-) is naturally an abelian group object over * *ssM,0(-). 2. Each Vo 2 sC has a functorial Postnikov tower of fibrations: p(n) p(n-1) . .!.PnVo --! Pn-1Vo ---! . .!.P0Vo , as well as a weak equivalence r : Vo ! P1 Vo := limn PnVo and fibrati* *ons (n) P1 Vo r--! PnVo such that r(n-1)= p(n)O r(n) for all n, and (r(n)O * *r)# : ssM,kVo ! ssM,kPnVo is an isomorphism for k n, and zero for k > n. 3. For every -algebra X, there is a classifying object BX with BX ' P0* *BX and ssM,0BX ~=X, unique up to homotopy. 4. Given a -algebra X and an X-module G, there is an extended G-Eilenberg- Mac Lane object E = EX (G, n) in sC=X for each n 1, unique up to homotopy, equipped with a section s for p(0): E ! P0E ' BX, such that ~ssM,n E ~=G as X-modules; and ssM,kE = 0 for k 6= 0, n. If G is * *a trivial X-module (x3.21), we write simply E(G, n). Any resolution model category with this additional structure (as well as fun* *ctorial k-invariants) is called a spherical model category. See [B3 , x1-2] for the det* *ails. 4.9. Remark. The homotopy groups ssM,n in the resolution model category s -A* *lg are corepresented by Sn F (s) for M = F (s) 2 F0 , s 2 S (cf. x2.9* *). Thus 16 DAVID BLANC by adjointness for any Vo 2 s -Alg we have: ssM,nVo = [Sn F (s), Vo]* = [Sn, (U Vo)s] = ssn(U Vo)s , so that the group ssM,nX (induced by the homotopy cogroup structure of Sn) * *is the usual n-th simplicial homotopy group of the graded simplicial group U Vo * *in the appropriate degree. This works also in s -Alg=X: more precisely, ssM,n Vo as defined above * *is an abelian group object in -Alg=ssM,0Vo, and applying ~ of Proposition 3.14 yie* *lds a ssM,0Vo-module, whose underlying S-graded set is ssnU Vo (see [BP , x4.14]). 4.10. Cohomology in s -Alg . It may appear more natural to take as a represent* *ing object an abelian group object in the model category s -Alg itself. In most * *cases this will yield no new cohomology groups, but it will enable us to define, and * *in some cases compute, the primary cohomology operations - as we do for topological s* *paces (see, e.g., [P ]). The obvious examples are those of the form E(G, n) as above (or EX (G, n)* * in s -Alg=BX, if we want local coefficients). In most cases of interest - inclu* *ding T*, S*, G = sGp - the only objects in A-s -Alg are products of the above. Further* *more, since E(-, n) : A- -Alg ! s -Alg is a functor, we can define an Eilenberg-Mac * *Lane diagram E(G, n) for any diagram G : D ! A- -Alg as in x4.7. Thus for any cofibrant Wo in s -Alg and coefficients M 2 A- -Alg D, for each n 1 we define the n-th cohomology group of Wo with coefficients in G, denoted by Hn(Wo; G), to be the set of components of map (Wo, E(G, n)) (whi* *ch is a D-diagram of simplicial abelian groups, so the components constitute a D-diag* *ram of abelian groups). Again, there is also a local version, for G in -Alg=X or M : D ! A- -Alg* * =X, yielding: Hn(Wo=X; G) := ss0map s -Alg=X(Wo, EX (G, n)) for each n 1. 4.11. Proposition. If a G-theory, X is a -algebra, and G is in A-( -Alg=X) * * , then cohomology with coefficients in G as defined in x4.1 is naturally isomorph* *ic to that defined in x4.10. Compare [D1 , x3]. Proof. Let K be the X-module corresponding to G = K o X, so Eo := EX (G, n) is of the form E(K, n) o X, where E(K, n) is obtained from the analogous ch* *ain complex (over X-Mod ) by the Dold-Kan equivalence (cf. [May1 , p. 95]). Thus: 8 >>X for0 i < n >< K o X i = n (4.12) Ei = L n >>( sjK) o X i = n + 1 >: j=0 MiEo i n + 2 , (where MiEo is the i-th matching object - see [BK , X, x4.5] or [BJT , x2.* *1]), with the differential: n+1X (4.13) @n+1(x, ~)) := ( dix, ~) for every(x, ~) 2 En+1 . i=0 'E-QUILLEN COHOMOLOGY 17 GENERALIZED ANDR Let Wo be a free simplicial object in sC, with " : W0 ! X inducing ss* *0Wo ~=X (for example, Wo could be a resolution of X). From (4.12)and (4.13)we see th* *at Hom sC=X(Wo, Eo) is naturally isomorphic to the subgroup of Hom C=X(Wn, K * *o X) consisting of maps f : Wn ! K o X (over X) for which f O di is the projectio* *n to X (the zero of Hom (C=X)(Wn+1, K o X) for each 0 i n + 1. Here Wn maps to X by " O d0 O . .O.d0. Again by the Dold-Kan equivalence, there is a path object EIofor Eo in s -Al* *g=X with 8 >>X for 0 i < n - 1 >< K o X i = n - 1 (4.14) EIi = L n-1 >>(K K sjK) o X i = n >: j=0 MiEo i n + 1 , with d0 the identity on the first copy of K o X in EIn, and minus the id* *entity on the second copy. There are two obvious projections p0, p1 : EIo! Eo, and a homo* *topy between two maps f0, f1 : Wo ! Eo over X is a map F : Wo ! EIo with piO F = * *fi (i = 0, 1), which in turn corresponds to a map F 0: Wn-1 ! K o X over X for * *which F 0O d0 represents f0, f1 respectively on the two copies of K o X. Thus we see that Hn(Wo=X, M) := [Wo, Eo]sC=X is canonically isomorphic to * *the n-th cohomotopy group of the cosimplicial abelian group Hom (C=X)(Wo, K o X), * * as claimed. 4.15. Cohomology of operads and their algebras. As noted in x1.3(b), our definition of sketchable categories covers both the* * category of operads, O-Alg , and that of algebras over a given operad P. Of course, O is not a G-theory; however, essentially all known applications * *are to operads of (connected) topological spaces or of chain complexes (see [MSS ]). * * In the first case, we can use (4.6)to replace T* by G, so that in both cases we may * *assume, without loss of generality, that our operad takes value in s -Alg for some G-* *theory . Note that the category of O-algebras in s -Alg is equivalent to s "-Alg, * * where " = Ox (product of FP-sketches) is now an G-theory (see x1.8). Thus the defin* *ition of x4.10 (applied to " ) is valid for operads of spaces or chain complexes. The same applies to algebras over a fixed operad P taking values in T* or * *Chk for some field k (see [May2 , x2]), as well as to the cohomology of a k-linear cate* *gory (that is, algebras over a k-linear PROP) considered in [Mar2 ]. We should observe, however, that the various cohomology theories constructed* * - in the context of deformation theory - in [Mar2 ], in [MS1 ] (for Drinfel'd a* *lgebras), in [GS2 ] (for bialgebras), and so on, are defined in terms of a specific diffe* *rential graded resolution. To show that these agree with our general definition requir* *es a generalization of Quillen's equivalence between simplicial and differential gra* *ded Lie algebras over Q (see [Q2 , I, x4], and compare [DP , x3]). One can expect such* * an equivalence only for suitable k-linear categories over a field k of characteris* *tic 0. 4.16. Remark. We should point out that a different definition of (co)homology f* *or -algebras, based on the Baues-Wirsching and Hochschild-Mitchell cohomologies of categories (cf. [BW , Mit ]), is given by Jibladze and Pirashvili in [JP ]. * *See [Sc2, Theorem 6.7] for an equivalent formulation in terms of the topological Hochschi* *ld (co)homology of suitable ring spectra. 18 DAVID BLANC 4.17. Cohomology of sheaves. We have assumed so far that was a G-theory. This is necessary for the approach described here at two points: in order to id* *entify the (abelian) group objects in -Alg (see Section 3), and to define the model cat* *egory structure on s -Alg (see x2.10(ii)). This is a resolution model category (in* *duced by the adjoint pair (F , U ) of x1.7) only with some such additional assump* *tion (cf. [B2 ]): otherwise the free -algebras are not necessarily cogroup objects. One obvious example where this fails is the category of sets, where we appar* *ently have no meaningful concept of cohomology. A more interesting case is the catego* *ry of sheaves on a topological space X, sketched by U (see x1.3). Note that there * *is no free/forgetful adjoint pair between ffiU-Algand U -Alg or ab = A- U ~= U -* *Abgp, since sheaves of abelian groups rarely have any projectives (e.g., ZCU in x1.* *3 (c) is not generally a sheaf). However, they do have enough injectives, so if we repla* *ce left derived functors by right derived functors in x2.4, with E = U -Alg, V = Set* *, and = A, we may define Hn(X; F), for any F 2 -E, to be the right derived fu* *nctors of Hom E(CX , -), applied to F. This also explains why our definition of ho* *mology does not make sense for sheaves. 5. Generalized cohomology For simplicial -algebras over a G-theory - and thus for simplicial sets* * or topological spaces - the only strict abelian group objects are generalized Ei* *lenberg- Mac Lane objects (cf. [Moo , 19.6]). Of course, in any model category D, any ab* *elian group object G in hoD defines a functor [-, G] : hoD ! Abgp ; but such funct* *ors do not usually satisfy the axioms of a cohomology theory. From our point of view, * *this is because the structure maps on the higher products Gk (k 3) which are neede* *d to make G an G- or A-algebra in D are not uniquely defined. One way to deal with this problem would be to require that G have an E1 -op* *erad acting on it (cf. [May2 , x14]). If D = T* (or S*), by a result of Boardman * *and Vogt, under mild topological restrictions any E1 H-space is homotopy equivalent to a* * strict abelian monoid in D (cf. [BV , Theorem 4.58]. 5.1. -spaces. Homotopy-coherent abelian monoids may be conveniently described in terms of a lax version of A, representing -spaces (cf. [Se2]): Let denote the category of finite pointed sets, and choose a set n+ = {0,* * . .,.n} (with basepoint 0) for each n 2 N. A -object in a pointed category C is a po* *inted functor G : ! C; the category of all such will be denoted by -C. Note th* *at if C is cocomplete, we can extend G to all of Set* by assuming it commutes with arbitr* *ary colimits. A -space G - that is, an object in -S* (or -T*) - is called* * special if for A, A02 , the natural map G(A _ A0) ! G(A) x G(A0) is a weak equivalenc* *e. This implies that for each n 2 N, the obvious map (5.2) G(n+ ) ! G(1+ ) x . .x.G(1+ ) _________-z________" n is a weak equivalence. Such a G is called very special if in addition ss0G(1* *+ ) is an abelian group under the induced monoid structure. 5.3. Definition. A special -space G has a classifying -space BG, which is i* *tself special, defined by setting (BG)(n+ ) := G(n+ xn+ ), with the diagonal structur* *e maps 'E-QUILLEN COHOMOLOGY 19 GENERALIZED ANDR (see [Se2, 1.3] and compare [Mil]). By iterating the functor B we obtain a -sp* *ectrum BG := <(BiG)(1+ )>1i=0. Thus G(1+ ) itself is an infinite loop space (with a specified H-space str* *ucture) if and only if G is very special. 5.4. The + -construction. For any pointed`simplicial set K 2 S*, Barratt def* *ines the free simplicial monoid + K to be n 1 Kn x n W n= ~, where ~ is genera* *ted by the obvious inclusions Kn ,! Kn+1 and n ,! n+1 (cf. [Ba , x4]). Then + * *K is actually a -space (see [A1 , x8]). To avoid confusion in the notation we shall* * denote this functor by fl+ : S* ! -S*. The (dimensionwise) group completion flK := Bfl* *+ K is a very special -space, which models the infinite loop space 1 1 K. The functor fl : S* ! -S* is left adjoint to G 7! G(1+ ). If K is conne* *cted, then fl+ K ' flK (cf. [Ba , Theorem 6.1]). Note that we can think of S := flS0 a* *s the inclusion functor ! S* (cf. [Ly , 2.7]). 5.5. The model category of -spaces. In [BF , x3], Bousfield and Friedlander d* *efine a proper simplicial model category structure on -S* as a diagram category with* * n- action on each G(n+ ), which they call the strict model category: a map f : G !* * G0 is a weak equivalence if f(n+ ) : G(n+ ) ! G0(n+ ) is a n-equivariant weak equ* *ivalence for each n 1, and it is a (co)fibration if it is a n-Reedy (co)fibration (* *see [Hi, x15.3]). They show that the homotopy category of very special -spaces is equivalent * *to that of connective spectra (see [BF , Theorem 5.1]), with Quillen equivalences provi* *ded by iterations of the functor B and its adjoint. They then define a stable weak equ* *ivalence of -spaces to be a map inducing a weak equivalence of the corresponding spectr* *a, and so obtain a new simplicial model category structure on -S* (with the same cofibrations, but fewer fibrations), whose homotopy category is again equivalen* *t to the usual stable category of connective spectra (see [BF , Theorem 5.8]). Variants on these two model category structures (with the same weak equivale* *nces) are provided in [Sc1, App. A]. 5.6. -simplicial groups. In view of (4.6), it is natural to think of the cat* *egory -G of -simplicial groups as representing connected infinite loop spaces; not* *e that every special -object here is trivially very special, because of the shift in * *indexing for homotopy groups. A -simplicial group G also known as a chain functor (cf. [A2 , x1]), since * *one can associate to it a generalized homology theory by setting Hn(X; G) := ssn(GoX) * * for each X 2 S*, where the simplicial group GoX is defined by GnX := G(Xn)n. * *Here each G(Xn) 2 G is defined as above by extending G from to Set*, so that * *GoX is actually the diagonal of a bisimplicial group. Equivalently, given a -space G 2 -S*, extend it via colimits from to S* *et* and thus via the diagonal to a functor "G: S* ! S*, which in fact takes a (pre)sp* *ectrum (Xn)n2N to a (pre)spectrum (G"Xn)n2N using: S1 ^ "G(Xn) ! "G(S1 ^ Xn) ! "G(Xn+1) . Thus for each X 2 S*, one may evaluate the homology theory associated to G on* * X by: Hn(X; G) ~= ssSn"G(S ^ X) = colim ssn+kG"(Sk ^ X) , k ! 1 20 DAVID BLANC where S := 1n=0 is the sphere spectrum. Note that if G is very special, then "G(S ^ X) is the -spectrum correspon* *ding to Anderson's GoX (see [BF , x4]. 5.7. Generalized cohomology. We now explain how the definitions of x2.4 apply in this context: first, note that the usual model category structure on E = S* ** is symmetric monoidal and enriched over V = S* (cf. [Q1 , II, x3]). Now for * *= , Lydakis (in [Ly ]) defined a smash product of -spaces making -V = -S*, too, * *into a symmetric monoidal category, with unit S. He also defines internal function com* *plexes Hom____ -S*(G, H) 2 -S* for G, H 2 -S* by setting: (5.8) Hom__ -S*(G, H)(n+ ) := map -S*(G, H(n+ ^ -)) , where H(n+ ^-))(k+ ) := H(n+ ^k+ ) and map -S*(-, -) 2 S* is the usual simpli* *cial function complex. Thus -E = -S* is indeed enriched over -V (cf. [Ly , 2.1]). Moreover,* * -V is semi-triangulated, with the delooping B : -S* ! -S* (x5.3) as the "suspe* *nsion automorphism" T of x2.3. The deloopings of the 0-sphere {BnS}1n=0 corepresent homotopy groups in ho -S*, since its homotopy category is equivalent to that* * of connective spectra, with generator S (corresponding to S0). Now for any -space G 2 -E and any pointed simplicial set K 2 E, Hom E(K,* * G) is a fibrant -set (x1.2), so the S*-function complex M := map *(K, G) is a -s* *pace. If G is (very) special, so is M, since map *(K, -) has homotopy meaning and pres* *erves products. Moreover, applying Barratt's functor yields a special -space flK, and the* * adjunc- tion isomorphism: ~= (5.9) M = map *(K, G) -! Hom__ -E(flK, G) induces an isomorphism between the homotopy groups of M and those of Hom__ -E(* *flK, G) (corepresented by S and its suspensions). Therefore, for special G the homotopy groups of M are determined by those of M(1+ ) = map *(K, G(1+ )), which are by definition H*(K; G), the generalized* * coho- mology groups associated to the -spectrum for G. 5.10. Generalized homology. Barratt's functor fl : E ! -E is the required functor A , by (5.9), so its left derived functors are ss*flK (since eve* *ry K is cofibrant). These turn out to be the stable homotopy groups of K, and are by definition the homology groups of K in this context. Finally, since the smash product of (cofibrant) -spaces is taken to the sma* *sh product of spectra under the equivalence of homotopy categories (see [Ly , Lemma 5.16])* *, we see that the groups H*(K; G) of x2.5 are just the generalized homology groups assoc* *iated to the -spectrum for G. 5.11. The (co)simplicial version. We next show how these definitions can be made to fit the description in x2.* *12: First, note that sS, as well as sT* and sG (cf. x4.5), have resolutio* *n model category structures with M = {S1} - this is the original E2-model category of [* *DKS , x5.10], which was constructed precisely so that if Vo is a resolution of X * *2 S, then the diagional diagVo (or equivalently, the realization of the corresponding s* *implicial 'E-QUILLEN COHOMOLOGY 21 GENERALIZED ANDR space) is weakly equivalent to X. Moreover, S, as well as T* and G, are enrich* *ed over V := S with its usual closed symmetric monoidal structure. We also need a suitable model category structure on the category c -S* of co* *simpli- cial -spaces - namely, the dual of Moerdijk's model category of bisimplicial* * sets (cf. [Moe , x1]), in which a map f : Xo ! Y o of cosimplicial -spaces is a weak equ* *ivalence (resp., cofibration) if Tot f is a weak equivalence (resp., cofibration) of -* *spaces. This implies that Tot : c -S* ! -S* induces an equivalence of homotopy categories* *, so for all practical purposes we can avoid working with cosimplicial objects altog* *ether (but see Theorem 6.18 below). The inverse equivalence c -S* ! c -S* is define* *d by 7! c( )o (the constant cosimplicial object). Thus ho(c -S*) (with this str* *ucture) is equivalent to the stable category of connective spectra, which is semi-trian* *gulated, with c(B)o O Tot : c -S* ! c -S* (x5.3) as the suspension automorphism T , and c(S)o as generator. Now, given a special -space G 2 -S* and a free simplicial resolution Vo * *! X in the original resolution model category sS, for any simplicial set Y - in pa* *rticular, for Y = G(1+ ) - we have: (5.12) map *(diag Vo, Y ) ~=Tot map *(Vo, Y ) (see [BK , XII, x4.3]). Thus in our case the cosimplicial -space map *(Vo, G)* * is weakly equivalent to the (constant cosimplicial) space c(map *(X, G(1+ )))o, whose h* *omotopy groups are H*(K, G) (x5.7). Finally, note that Barratt's functors fl+ and fl are defined dimensionwis* *e on a simplicial set K, so that diagflVo = fl diagVo for any bisimplicial set Vo. Thus we may define A : E ! -E to be fl, and* * its total left derived functor is naturally equivalent to fl (in Moerdijk's model c* *ategory sS*), since diagVo '-!K for any free simplicial resolution Vo ! K. Thus ag* *ain the (unadorned) homology groups are the stable homotopy groups of K, and H*(K; G) are the generalized homology groups associated to the -spectrum for G. 6. The spectral sequences We now want to use this machinery to try to understand relationships among t* *he various homology and cohomology theories. First, we shall need a preliminary no* *tion: 6.1. Definition. If M is a set of models in a model category C (with M C * * as in x2.7), then C- := (ho M )op is a G-theory, which sketches the category C* *- -Alg of C- -algebras (cf. [BS , x3]). 6.2. Remark. If we think of M and its suspensions as corepresenting homotopy gr* *oups in C (cf. x4.9), then C- -algebras are graded groups equipped with an action* * of the corresponding primary homotopy operations - the motivating example being ssM,*X for any X 2 sC. This notion may be extended to any concrete category C by the conventions of [BS , x3.2.2], and may also be dualized as in [Bou ] by* * taking C- := ho M , rather than the opposite category (cf. [BP , x1.13]). Note that the derived functors of any functor into C actually take values in* * C- -Alg . 6.3. Examples. (a) If C has a trivial model category structure, and M consis* *ts of (enough) projective generators - e.g., if C = -Alg and M = F0 - * *then C- -Alg ~=C. 22 DAVID BLANC (b) If C = sD or cD for some abelian category D, and M again consists* * of (enough) projective generators - e.g., for C = s X and M as above * * - then C- -Alg ~=grN D (where we use lower or upper indices for the gr* *ading according to the usual convention). (c) For C = T* or S*, with M = {S1}, then C- -Alg ~= -Alg is the category * *of ordinary -algebras, modeling the usual homotopy groups of topological * *spaces. (d) If C = -S* and M = {S}, then C- -Alg is equivalent to the catego* *ry of graded connected ss-modules for ss = ssS*S0 (homotopy groups of the s* *phere spectrum), since ssM,*G are just the stable homotopy groups of the -sp* *ectrum corresponding to G 2 -S*. Using the Quillen equivalence of (4.6), we see that when C = s -Alg we o* *ften have interesting categories of C- -algebras (see, e.g., [BS , x3.2.1]). We shall also need the following version of [BS , Prop. 3.2.3]: 6.4. Proposition. Any contravariant functor T : C ! cB from a model category C (equipped with a set of models M) to a concrete category B induces a graded fun* *ctor T~* : sC- -Alg ! sB- -Alg by setting ~T(kssM,*Vo) := ssk(T Vo) for cofibran* *t Vo 2 sC, and extending by taking 0-th derived functor. Proof. Since ssM,* : ho M ! FC- is an equivalence of categories (onto the fr* *ee C- - algebras), in particular ssM,*Vo ~=ssM,*Wo , Vo ' Wo for cofibrant Vo, Wo * *2 sC, so ~T *is well-defined on free C- -algebras. 6.5. A general setting. In Sections 3-5 the algebraic and topological versions of homology and cohom* *ology have been treated separately. We now show how the Procrustean framework of x2.12 may be used in order to obtain a uniform description of various relations betwe* *en them. 6.6. Examples. We wish to concentrate on the following list of cohomological se* *ttings (Definition 2.13), discussed above: (a) for some G-theory ; (b) More generally, * * for some G-theory and fixed X 2 -Alg. (c) where i* *s some strong A-sketch. (d) (with the symmetric monoidal* * struc- ture on -S* of x5.7). In all these examples we have additional properties which we shall require i* *n our applications, which we may formalize as follows: 6.7. Definition. A cohomological setting is complete if if * *it is equipped with: (1) A left adjoint diag: sC ! C to the inclusion c(- )o : C ! sC, which* * induces diag: s -C ! -C, as well as a convergent first-quadrant spectral sequ* *ence with: (6.8) E2s,t~=sssssM,tVo =) ssM,s+t(diag Vo) , for each Vo 2 sC and M 2 M; 'E-QUILLEN COHOMOLOGY 23 GENERALIZED ANDR (2) A right adjoint Tot : cV ! V to the inclusion c(- )o : V ! cV, which * *induces Tot : c -V ! -V, as well as a second-quadrant spectral sequence with: (6.9) Es,t2~= sssssM0,tXo =) ssM0,t-s(Tot Xo) , for each Xo 2 cV and M0 2 M (we do not address questions of conver- gence); (3) A natural " -C-adjointness" isomorphism: ~= (6.10) Tot(Hom__(Vo, G) -! Hom__(diag Vo, G) for any Vo 2 s -C and G 2 -C. 6.11. Proposition. Each of the examples of x6.6 is a complete cohomological set* *ting. Proof. Since (a) and (b) are instances of (c), we have only two cases to consid* *er: (1) Assume C = s -Alg=X for some G-theory sorted by S. Then Vo 2 sC is a bisimplicial -algebra (over X), and let diag Vo be the usual diagonal (* *with (diag Vo)n := (Vn)n). Note that U Vo is just an an S-graded bisimplicial se* *t, with U diagVo = diagU Vo (even though colimits are not generally preserved by U * * ). By Remark 4.9 we see that the Bousfield-Friedlander spectral sequence for U V* *o in each degree (cf. [BF , Theorem B.5]) has the form (6.8). Similarly, given a cosimplicial object Xo 2 c(s - -Alg=X ), the usual To* *t for the (S-graded) cosimplicial simplicial set U Xo is defined to be the simplic* *ial set To with Tn := Hom cSet( o [n], Xo), and this has a natural structure of * *a - algebra in -Alg=X by Remarks 1.2 and 2.11 and x1.8. Thus Tot U Xo lifts * *to Tot Xo 2 s -Alg. The homotopy spectral sequence for the cosimplicial space U * * Xo, with: Es,t2= ssssstU Xo =) sst-s(Tot U Xo) , (see [BK , X, 6.1 & 7.2]) gives (6.9) (though it does not necessarily converg* *e!). Finally, (6.10) follows from (5.12). (2) For C = S* we can use the usual diagonal and Tot and the original spect* *ral sequences for (co)simplicial spaces. For (6.10), consider the cosimplicial * *-space Eo := Hom__ -S*(Vo, G): Definition (5.8) of Hom__ -S* in terms of the simp* *licial function complex map -S* shows that Tot Eo ~= Hom__ -S*(diag Vo, G) again* *, by (5.12). With this at hand, we can describe several spectral sequences connecting the* * various functors we have defined so far. First, a universal coefficients theorem for co* *homology: 6.12. Theorem. Let be a complete cohomological setting, and* * let G be a -algebra in C. Then for any Y 2 C there is a natural cohomological sp* *ectral sequence with ____ Es,t2~= Exts,t(H*Y, G) =) Ht-s(Y ; G) , ____ where Ext s,t(C, G) := (Ls~T(C))t for any C 2 ( -C)- -Alg , and T := Hom__(* *-, G). Proof. Let Z ! Y be a cofibrant replacement in C, and assume G is fibrant. W* *e use M := {A M}M2M as models in -C (x2.12), with T n as the suspension (x2.1), to define the resolution model category structure on s -C. As in the proof of* * [BS , Theorem 4.2], let Vo ! A Z be a free simplicial resolution in s -C, so that by* * (6.8) the natural map diagVo ! A Z is a weak equivalence. 24 DAVID BLANC If we set Eo := Hom__(Vo, G) (a cosimplicial -algebra in C), then by (6.* *10) and (2.14): TotEo = Hom__(diag Vo, G) ' Hom__(A Z, G) ~=map (Z, G) = L map (-, G)* *(Y ) ao ssM ,t-s(Tot Eo) = ssM ,t-smap (Z, G) = Ht-s(Y ; G) by Definition 2.15. On the other hand, since each Vn is cofibrant: ssM ,*En = ssM ,*Hom_(Vn, G) = ~T(ssM ,*Vn) and since Vo ! A Z is a cofibrant replacement, ssM ,*Vo ! ssM ,*A Z =: H*Y* * is a free resolution in ( -C)- -Alg, so: sssssM ,*Eo = sss(T~(ssM ,*Vo)) = sssLT~(H*Y ) = Ls~T(H*Y ) , as claimed. Note that for generalized cohomology of spaces this takes the familar form (* *cf. [Ad ] and [EKMM , IV, x4]): 6.13. Corollary. For any special G 2 -S* and K 2 S* there is a second quad* *rant spectral sequence with: Es,*2~= Extsss-Mod(ssS*K, G) =) Hs-t(K; G). There is also a version for homology: 6.14. Proposition. Let be a complete cohomological setting, * *and let G be a -algebra in C. Then for any Y 2 C there is a natural first quadrant s* *pectral sequence with ____ (6.15) E2s,t~= Tors,t(H*Y, G) =) Ht+s(Y ; G) , ____ where Tors,*(C, G) := (Ls~T(C)) for any C 2 ( -C)- -Alg , and T := - G. Proof. This generalization of [BS , Theorem 4.4] for the composite functor: A - G M --! -C ---! -C is proven like Theorem 6.12, with (6.8) replacing (6.9). For generalized homology this takes the form: 6.16. Corollary. For any special G 2 -S* and K 2 S* there is a natural fir* *st quadrant spectral sequence with: E2s,t~= Torss-Mods,t(ssS*K, G) =) Ht+s(K; G). Finally, we have the following two generalizations of [B1 ]: 6.17. Theorem. Let be a complete cohomological setting, and* * let G be a -algebra in C. Then for any Y 2 C there is a natural first quadrant s* *pectral sequence with E2s,t~= Ls~T(ssM,*Y )t =) Ht+s(Y ; G) , where where T := A (-) G. 'E-QUILLEN COHOMOLOGY 25 GENERALIZED ANDR Proof. Similar to the proof of Theorem 6.12, except that here we start with a f* *ree simplicial resolution Vo ! Y in sC, and note that in this case ssM,*Vo ! s* *sM,*Y is a free simplicial resolution in the category sC- -Alg . In [Se1, Prop. 5.1], Segal produced a stable version of this spectral sequen* *ce for any generalized homology theory k* (converging strongly to k*X if k* is conne* *ctive). 6.18. Theorem. For Y and G as above, there is a natural second quadrant spectr* *al sequence with: s t-s Es,t2~= dExtt(ssM,*Y, G) =) H (X; G) , s where dExt(-, G) := Ls~T for T := map C(-, G). Note that Schwede, in [Sc2, x5.5], also defined a spectral sequence relating* * the stable homotopy of a -algebra to Quillen homology. References [AR] J.V. Ad'amek & J. Rosick'y, Locally presentable and accessible categori* *es, Lond. Math. Soc. Lec. Notes Ser. 189, Cambridge U. Press, Cambridge, UK, 1994. [Ad] J.F. Adams, "Lectures on generalized homology theories", in P.J. 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Whitehead, "Generalized homology theories", Trans. AMS 102 (1962),* * pp. 227-283. Dept. of Mathematics, Faculty of Sciences, Univ. of Haifa, 31905 Haifa, Isra* *el E-mail address: blanc@math.haifa.ac.il