Generalized Andr{e}-Quillen Cohomology

GENERA LI ZED ANDR ´ E-QUILLEN COHOMOLOGY D A VID BLANC Abstract. W e explain ho w the approach of Andr´ e and Quillen to deﬁning cohomol- ogy a nd homo logy as s uitable der ived functors extends to g eneralized (co)homolo gy theories, and how this identiﬁcation may b e used to s tudy the rela tionship betw een them. Intr oduction After the cohomology of top ological spaces w as disco v ered in the 1930’s, the concept w as expanded to groups, and later to a sso ciativ e, comm utativ e, and Lie algebras, in the 1940’s and early 1950’s. In the follo wing decade the ﬁrst generalized cohomology theories for spaces app ear ed (see [Mc2, Mas ]). All these examples started out in the form o f explicit constructions, and only lat er w ere their theoretical underpinnings pro- vided: in particular, cohomology for general algebraic categories w as described b y Bec k and others in terms of triples (see [Be], and compare [D1]), and then b y Andr´ e a nd Quillen in terms of (non-ab elian) deriv ed functors (see [An, Q1]). In the latter v ersion, cohomology g roups are the derive d functors of Hom in t o a ﬁxed a b elian g roup ob ject (and homology groups are the deriv ed functors of a b elianization). Ho wev er, for top ological spaces the only ab elian group ob jects are (pro ducts o f ) Eilenb erg - Mac Lane spaces, whic h represen t ordinar y cohomology . Th us w e need a diﬀeren t framew ork to describ e generalized (co)homolog y: this is provided b y stable homotop y theory (cf. [Br, Wh]). Our go a l here is to provide a uniform deﬁnition for homology and cohomology en- compassing the theories men tioned ab o ve, as w ell as some new ones. As a side b eneﬁt, w e clarify exactly what assumptions on an (a lgebraic) category C are needed in order for the a ppro ac h of Andr ´ e and Quillen to w or k. (This is the reason for the somewhat tec hnical Section 3 .) The approac h given here applies, inter a lia , to: (a) Homology and cohomolo gy o f groups and v arious t yp es of alg ebras; (b) V ersions of the ab ov e with lo cal co eﬃcien ts ( § 4.1 -4.2); (c) Unstable generalized (co)homology of spaces ( § 5.7-5.10); (d) Generalized (co)homolog y of sp ectra and spaces ( § 2.18); (e) Cohomology of op erads, and of alg ebras ov er an op erad ( § 4.1 5); (f ) Cohomology of dia grams of spaces or algebras ( § 4.7). The last tw o hav e applicatio ns to deformation theory (see [Mar1, MS2] and [GS1, GGS], r esp ective ly). The cohomology of shea ve s has a dual deﬁnition to the one presen ted here here (see § 4.17). Of course, there are other concepts of cohomology whic h do not ﬁt in to our Date : August 10, 2007; revised: F ebruar y 10, 20 08 . 1 2 DA VID BLANC framew ork; most nota bly , a num b er of v ersions of the cohomology of categories (see § 4.16). 0.1 . Representing cohomology . In order to deﬁne a coho mo lo gy theory in a category C , w e need a represen ting ob ject G ∈ C , as w ell as a suitable mo del categor y structure on the category s C = C ∆ op of simplicial o b jects ov er C (see § 2.7). How ev er, in this generalit y Hom C ( − , G ) will tak e v alues in sets, and applying this functor t o a simplicial resolution V • → X in s C just yields a cosimplicial set, fo r whic h w e ha ve no a ppro priate mo del category . It t urns out that in order to get an intere sting cohomology theory , tw o ingredien ts a re g enerally needed: • The category C must b e enric hed o v er a symmetric monoidal category V ; • The represen ting ob j ect G mus t hav e additional “a lg ebraic” structure. W e shall use the concept of a sketch – a straigh tforward generalization of La wv ere’s concept of a the ory – to describ e this additional s tructure (see § 1.1). In this language, w e sa y that G is a Φ-algebra in C , for a suitable FP-sk etc h Φ. W e also use sk etc hes to describe the kind o f a lg ebraic categories t o whic h our approach applies: this will allo w us to t r eat op erads and their algebras, for example, uniformly with the usual univers al algebras. • Note that the functor Hom C ( − , G ) now tak es v alues in the category D of (cosimplicial) Φ-algebras in V . Our ﬁna l requiremen t is that the ab o v e t w o ingredien ts m ust com bine to mak e D in to a (semi-) triangulated mo del category (see § 2.2). The question we consider here is in some sens e dual to that of Bro wn Represen tability in t r iangulated categories (cf. [CKN, F, K, N]): rather than asking whic h cohomology functors a re represen table, we seek conditions for a represen table functor to b e a co- homology theory . 0.2. Examp l e s. In the catego ry of groups (where V = S et ), w ith a n ab elian group G as t he co eﬃcien ts, the mo del category w e consider is that of simplicial groups. The total left deriv ed functor of Hom( − , G ) then tak es v alues in the semi-triangulated category of cosimplicial ab elian gr o ups (equiv alen tly , co c ha in complexes). On the other hand, for p o in ted simplicial sets or top o lo gical space s (where V = S ∗ ), w e may ta k e Φ = Γ, and Hom( − , G ) ta k es v alues in Γ-spaces – again, a sem i- triangulated category . Note that the category of sp ectra is tria ng ulated ( a nd enric hed o v er it self ), so we can t a k e an y sp ectrum G as co eﬃcien ts. Our original motiv ation for creating a join t setting for algebraic and generalized top ological (co)homology theories w as to try to gain a b etter understanding of the relationship b et ween homology and cohomology . This is pro vided b y a univ ersal co- eﬃcien ts spectral sequence (see Theorem 6 .1 2 b elo w). W e o btain a similar result for homology ( Prop osition 6.14), as w ell as “rev erse Adams sp ectral seq uences” (Theorems 6.17 a nd 6 .1 8) relating homot op y to (co)homology . 0.3 . Not ation and con v en tions. The category of top o logical spaces is denoted b y T , and that of p ointed connected to p ological spaces b y T ∗ . The category of groups is denoted by G p , t hat of ab elian gro ups b y A bg p , and t ha t of p ointed sets b y S et ∗ . F or a ny category C , gr S C denotes the category of S -graded ob jects o v er C GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 3 (i.e., diagrams indexed by the discrete catego r y S ), s C that of simplicial ob jects ov er C , and c C that o f cosimplicial ob jects ov er C . The category o f simplical sets will b e denoted b y S , that of reduced simplicial sets by S ∗ , and that o f simplicial groups b y G . F or an y Z ∈ C , w e write c ( Z ) • for the constan t simplicial ob ject determined by Z , and c ( Z ) • for the constan t cosimplicial ob ject. If A is an y ab elian category , w e denote the category of c hain complexes ov er A by C h ( A ) ; ho w eve r, w e write C h R for C h ( R - M od ), and similarly c C h R for co c hain complexes of R -mo dules. 0.4 . Organization. Section 1 provide s bac kground material on sk etc hes, the ories, and algebras ov er them. In Section 2 we give our abstract deﬁnition of homology and cohomology , in the con text of suitable mo del categories. Ab elian group ob jects in sk etc hable categories are describ ed in Section 3, and these are used in Section 4 to deﬁne the ( co) ho mology of Θ-a lg ebras. Section 5 explains how generalized cohomologies ﬁt in to our framew o rk, using Γ-spaces. Finally , the theory is applied in Section 6 to construct univ ersal co eﬃcien t and rev erse Adams sp ectral sequences in this general framew ork. 0.5. A cknow le dgements. This pap er is an outgro wth o f jo int w o rk with Georg e P esc hk e, in [BP], and I would lik e to thank him fo r man y useful discussions and insights. I also thank the referee for his or her helpful commen ts, and the Institut Mittag-Leﬄer (Djur- sholm, Sw eden) for its hospitalit y during the p erio d when this pap er was completed. 1. Algebras and theories As Lawv ere observ ed (cf. [La]) , ‘v arieties of univ ersal algebras’ in the sense of Mac Lane (cf. [Mc1, V,6]) can b e corepresen ted b y functors out of a ﬁxed category Θ. This idea w as later generalized b y Ehresmann to sk etch es (see [BE]), whic h turn out to b e the most conv enien t language to describ e b oth the algebraic categories w e w ork in, and the represen ting ob jects for cohomology . 1.1. Deﬁnition. A sketch h Θ , P , I i is a small category Θ with distinguished sets P of (limit) cones a nd I of (colimit) co cones. In particular, a ﬁnite pr o duct (FP-)sketch is a sk etc h in whic h P consists o nly of ﬁnite pro ducts (and I = ∅ ). A the ory is an FP-sk etc h Θ con taining a zero ob ject, for which P consists of al l ﬁnite pro ducts. W e think of a map f : ϑ 1 × . . . × ϑ n → θ in Θ as corepresen ting a (p ossibly graded) n -a r y o p eration. A theory Θ is sorte d by a set S ⊆ Ob j Θ if ev ery o b ject in Θ is uniquely isomorphic to a ﬁnite pro duct of o b jects from S (see [Bo r, § 5.6]). La wv ere o r ig inally considered o nly theories sorted b y { 1 } , so that Ob j(Θ) = N , with n ∼ = Q n i =1 1 for n ≥ 0. If Θ is an FP-sk etc h and C is a n y p ointed category , a Θ -alge b r a in C is a p ointe d functor X : Θ → C whic h preserv es all pro ducts in P . More generally , if Θ is an y sk etc h, a Θ-a lgebra X : Θ → C is required to preserv e all distinguished limits (in P ) and colimits (in I ). The category o f Θ-a lgebras in C is denoted by Θ- C ; a Θ-algebra in S et ∗ will b e called simply a Θ -algebr a , and we write Θ- A lg for Θ- S et ∗ . W e call a catego ry D sketchab le if it is equiv alen t to Θ- A l g , and sa y that Θ sketche s D . Such categories are accessible , in the sense of mo del theory , as well as b eing lo cally presen table (see [AR, Cor. 2.61 & 1.52]). A map of the ories (or of sk etc hes) ψ : Θ → Θ ′ is a functor which preserv es a ll pro ducts (resp ectiv ely , all distinguished limits and colimits). Suc h a map ψ induces a functor ψ ∗ : Θ ′ - A l g → Θ- A l g . 4 DA VID BLANC More generally , if Θ is a theory (or FP-sk etch), a Θ -algebr a in any symmetric monoidal category hV , ⊗ , I i (cf. [Bor, § 6.1]) is a functor X : Θ → V taking the (distinguished) pro ducts in Θ to ⊗ -pro ducts in V , with X ( ∗ ) = I . 1.2. R em ark. Since we can t hink of a Θ-algebra X in C as a certain kind of diagram in C (with sp eciﬁed pro ducts), w e see that Hom C ( − , X ) tak es v alues in Θ- A l g . More generally , if C is enric hed o ver a symmetric monoidal catego r y hV , ⊗ , I i via map C (cf. [Bor, § 6.2]), and map C ( A, − ) ta k es pro ducts to ⊗ , then map C ( − , X ) tak e v a lues in Θ- V . 1.3. Examples. (a) The categor y of g r oups is ske tc hed by a theory G , with µ : 2 → 1 represen ting the group op eration, ρ : 1 → 1 the in ve rse, and e : 0 → 1 the identit y (satisfying the o b vious relations). Similarly , the category of ab elian groups is sk etc hed b y a theory A (with the same maps, satisfying a further relation) and the inclusion i : G ⊂ A induces the inclusion of categories A bg p ⊆ G p . (b) An op erad Γ = (Γ( n )) ∞ n =0 is an O -algebra in a symmetric monoida l categor y hV , ⊗ , I i , where O is a “univ ersal” theory for op erads. Similarly , an algebra ov er the op erad Γ (see [Ma y2, § 14]) is just a Θ Γ -algebra in h V , ⊗ , I i , where the theory Θ Γ is obtained from Γ in the ob vious w ay (replacing ⊗ with × ). The same applies more generally to PR O P’s, colored op erads, and other v arian ts (see [MSS] fo r a surv ey on op erads, esp ecially in the algebraic con text). (c) G iv en a top ological space X , let U denote t he directed set o f non-empty op en sets in X , with inclusions – so that U op sk etc hes preshea ves of sets. By adding arbitrary formal copro ducts ` α ∈ A U α for any collection { U α } α ∈ A in U , w e obta in a category ˆ U , in whic h the diagram; (1.4) ` ( α,β ) ∈ A × A U α ∩ U β i / / j / / ` α ∈ A U α κ / / S α ∈ A U α is a co equalizer (if the ﬁrst term is empt y , κ is an isomorphism). If w e no w let Θ U := ˆ U op (sorted b y U ), with P consisting of t he opp osites of the formal copro ducts a nd of all the co equalizers (1.4) (a nd I = ∅ ), w e obtain a sk etc h whose alg ebras F : Θ U → S et are shea v es of sets on X . F urthermore, for any V ∈ U , if: C V ( U ) := ( {∗} if U ⊆ V ∅ if U 6⊆ V , there is a natural isomorphism Hom Θ U - A lg ( C V , F ) = F ( V ). 1.5. Deﬁnition. Given a theory X , an X - the ory (or sk etc h) Θ is one equipp ed with a map o f theories (or sk etche s) ψ : ` S X → Θ whic h is bijectiv e on ob jects, where the copro duct is tak en in the category of theories (or ske tc hes) o v er some index set S . If X is sorted b y { 1 } , an X -structur e at an ob j ect c in a category C is an X - a lgebra ρ : X → C with ρ ( 1 ) = c . A theory Θ sorted by S is an X -theory if and only if it is equipped with an X - structure at each s ∈ S . If a ll other maps o f Θ comm ute with t ho se coming from ψ , we call Θ a str ong X -the ory (or sk etc h). GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 5 1.6. Example. If Θ is a G -theory , then t he map of theories ψ : ` S G → Θ induces an “underlying S -graded group” functor ψ ∗ , whic h w e denote b y V : Θ- A l g → G p S = ` S G - A l g . Θ is a strong G -theory if all the op erations in Θ a re homomorphisms of the underlying g r aded g r o up. 1.7 . F ree Θ -algebras. F or an y theory Θ, let Θ δ denote the discrete theory with the same o b jects (and pro ducts) as Θ. If Θ is sorted b y S , Θ δ sk etc hes the category of S -gra ded sets, and the inclusion I : Θ δ ֒ → Θ induces the forgetful functor U = U Θ : Θ- A l g → Θ δ - A l g . As usual, there is a f r e e functor F = F Θ : Θ δ - A l g → Θ- A l g left adjoin t to U Θ . W e denote by F Θ the full sub category of Θ- A l g whose ob jects are free (tha t is, in the imag e of F Θ ). Since a ll limit-ske tc hable categories are lo cally presen t able, t hey are complete (see, e.g., [AR, Theorem 1.4 6]) a nd cocomplete. Th us for any theory Θ, the category Θ - A l g of Θ- algebras has all limits and colimits. 1.8 . Sk etc hing Φ -algebras in Θ - A lg . If Θ is a theory (sorted by S ) and Φ is another theory (singly sorted, for simplicit y), the category Φ-Θ- A l g of Φ-algebras in Θ- A l g is sk etc hed b y a theory Φ(Θ) (sorted b y S ), deﬁned as follo ws: (a) W e ﬁrst add an S -graded cop y of Φ to Θ, setting Θ Φ := Θ ∪ S ` S Φ, so that we now ha v e eac h op erat ion of Φ acting on each θ ∈ S . The inclusion i : Θ ֒ → Θ Φ induces a fo rgetful functor i ∗ : Θ Φ - A l g → Θ- A lg . (b) Next, w e for ce all op erations of Θ to comm ute with the new op erations - that is, fo r eac h f : θ 1 → θ 2 in Θ and g : n → k in Φ, we require t ha t θ n 1 g / / f n   θ k 1 f k   θ n 2 g / / θ k 2 comm ute, so we obtain a quotien t of theories q : Θ Φ → → Φ(Θ). By construction Φ(Θ)- A l g ∼ = Φ-Θ- A lg . Note tha t q ∗ and i ∗ comm ute with the underlying S -graded set functors U Θ , U Θ Φ , and U ΦΘ , which create all limits in their resp ectiv e categor ies, so q ∗ and i ∗ comm ute with all (small) limits. Th us by [Bor, Theorem 5.5.7 ] eac h has a left adjoin t. The adjoint of the comp osite i ∗ ◦ q ∗ : Φ-Θ- A l g → Θ- A lg will b e called the Φ -lo c alization of Θ- A lg , and denoted by L Φ : Θ- A lg → Φ-Θ- A l g . 1.9. R emark. Note that giv en G in Φ-Θ- A l g , b y Remark 1.2 Hom Θ- A lg ( − , G ) has a natural structure of a Φ-algebra. F urthermore, if i ∗ ◦ q ∗ is a faithful embedding of categories (whic h will happ en if Θ is a Φ-theory , for example), then L Φ is idemp oten t and an y Φ-algebra in Θ- A lg is in the image of L Φ , up to natural isomorphism. Thu s Hom Φ-Θ- A lg ( − , − ) has a natural structure of a Φ-alg ebra, in this case. By mimic king the construction of A × B → A ⊗ B for a b elian groups, one can then mak e Φ-Θ - A l g in to a closed symmetric monoidal category (see [Bo r, § 6.1.3]). 2. Gene ralized homology and cohomology W e are no w able to giv e a deﬁnition of homology and cohomology for mo del cate- gories, somewhat mo r e general than Quillen’s original approach (cf. [Q1, I I, § 5 ]) : 6 DA VID BLANC 2.1 . T riangulated categories. The target o f a cohomology functor should b e a mo del category whose homot op y category is triangulated. There are a n umber of v a rian ts of this concept, originally due to G rothendiec k. F o r our purp oses, a triangulate d c ate gory is an additiv e categor y C equipped with an automorphism T : C → C (called the tr an s lation f unctor), and a collection D of distinguishe d triangles o f the for m h X f − → Y g − → Z h − → T X i , satisfy ing the four axioms of [Ha, § 1] (whic h co dify t he prop erties of coﬁbration sequences in p oin ted mo del categories – see [Q1, I, § 3]) . 2.2. Deﬁnition. A semi- triangulate d category is an additiv e categor y ˆ C equipp ed with a collection D of distinguishe d triangles of satisfying the ab ov e four axioms, as w ell as a translatio n functor T : ˆ C → ˆ C whic h is an isomorphism on to its image. In all cases of in terest, T can b e formally in v erted to yield a full triangulated category C = ˆ C [ T − 1 ] with ˆ C a s a full sub category; how ever, this prop ert y is not needed in what f ollo ws. A set P of cogr o up ob jects in ˆ C will b e called a set o f gener ators if the collection of functors { Hom ˆ C ( T i P , − ) } P ∈P ,i ≥ 0 detects a ll isomorphisms in ˆ C . 2.3. Example. Typic ally , (semi-)triangulat ed categories app ear as the homotop y cate- gory of a suitable (semi-)stable mo del cat ego ry , as deﬁned a xiomatically in [Ho , Ch. 7] (see also [HPS]). Th us, the motiv ating example of a triang ula ted catego r y is the homotop y category of (un b ounded) chain complexes o ver an ab elian category A . An- other example is pro vided by Bo ardman’s stable homotop y category ho Spec (cf. [V]), where there a r e a n um b er of diﬀeren t underlying stable mo del categories (see [HSS], [Sc1], or [EKMM]). The sub category ˆ C of non-nega t iv ely graded c hain complexes is semi-triang ula ted; if A ha s a pro jectiv e generator P , then K ( P , 0) (t he c hain complex with P concen t rated in degree 0) is a g enerator for ˆ C . Similarly , the homotopy c ategory of connectiv e sp ectra, ho Sp ec (0) , is semi-triangu- lated (with generator S 0 ). 2.4 . Cohomology . In order to deﬁne cohomolog y f unctor s on a mo del category E , w e assume that E is equipp ed with: (a) An FP-sk etc h Φ and a category V suc h that V and Φ- V are symmetric monoidal, E is enriched ov er V via map E ( − , − ) : E op × E → V , and Φ- E is enric hed ov er Φ- V via Hom ( − , − ) : (Φ- E ) op × Φ- E → Φ- V . (b) An FP- ske tc h Φ and a mo del catego r y structure on Φ- V for whic h ho Φ- V is semi-triangulated. Then for any G ∈ Φ- E , we deﬁne the c ohomolo gy of X ∈ E with c o eﬃcients in G to b e the tot a l left deriv ed f unctor L map E ( − , G ) of map E ( − , G ), applied to X . Recall t hat total left derive d functor of a “left exact” functor F : C → D b et w een mo del categories is deﬁned b y applying F to a coﬁbrant r eplacemen t of X (see [Q1, I, § 4] or [Hi , 8.4]). If ho Φ- E has a set o f generators P , then the P - graded group H n ( X ; G ) := [ T n P , ( L map E ( − , G )) X ] P ∈P is called the n -th c o h omolo gy g r oup of X with coeﬃcien ts in G . 2.5 . Homology . T o deﬁne homology , w e need also a homoto py functor A Φ : E → Φ- E equipped with a natural isomorphism map E ( E , X ) ∼ = − → Hom ( A Φ E , X ) in Φ- V (cf. GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 7 § 1.2) for E ∈ E and X ∈ Φ- E . W e then deﬁne the homolo gy of X ∈ E to b e the total left deriv ed functor o f A Φ , applied to X ( § 2.4). Aga in the n -th homolo gy g r o up of X is: H n X := [ T n A Φ P , ( L A Φ ) X ] P ∈P . If Φ- E is a sy mmetric monoidal mo del catego ry (see [Ho, § 4.2.6]), with Hom ( − , Y ) righ t adjoin t (o ve r Φ - V ) to − ⊗ Y , the n for an y G ∈ Φ- E , homolo gy with c o eﬃcients in G is the total left deriv ed functor of A Φ ( − ) ⊗ G (assuming A Φ E is alw a ys coﬁbran t). The homology groups H n ( X ; G ) are deﬁned as ab o v e. Compare [BB, I]. 2.6. Example. If E = V = S ∗ (or T ∗ ) a nd Φ = A , then Φ- C ∼ = Φ- V ∼ = s A bg p and G is a (generalized) Eilenberg-Mac La ne space, so w e hav e ordinary cohomology . The functor A Φ : E → Φ- C is the usual ‘ab elianization’ X 7→ Z X , whic h yields o rdinary (singular) homology . 2.7 . Resolution mo del categories. T o prov ide a uniform treatmen t of the v arious kinds of (co)ho mo lo gy it will b e con ve nien t to use a framew ork originally c onceiv ed by Dwy er, Kan a nd Stov er in [DKS] under the name of “ E 2 mo del categories”, and later generalized b y Bo usﬁeld (see [Bou, J]. Recall that the concept of a mo d el c ate g ory w a s introduced by Quillen in [Q1] to allo w application o f the metho ds and constructions of homotop y theory (of to p ological spaces) in more general contexts . This is a category C , equipp ed with three distin- guished classes of morphisms – we ak equiv alences, coﬁbratio ns, and ﬁbrations – satisfying certain axioms (a nalogous to those whic h ho ld for the corresp onding classes in T ). See [Hi] or [Ho] for further details. Let C b e a p oin t ed coﬁbran tly generated r igh t prop er mo del category ( cf. [Hi, 7.1, 11.1]), equipp ed with a set M of coﬁbran t homotopy cogro up ob jects in C , called m o dels (pla ying t he role of the spheres in T ∗ ). Let Π M denote the smallest full sub category of C con taining M a nd closed under copro ducts, a nd susp ensions (cf. [Q1, I, § 3]). F or an y X ∈ C , M ∈ M , and k ≥ 0, set π M ,k X := [Σ k M , X ′ ], where X → X ′ is a ﬁbran t replacemen t. W e write π M ,k X f o r the M -gr a ded group ( π M ,k X ) M ∈M . 2.8. Deﬁnition. A map f : V → Y in s C is homotopic al ly M -fr e e if for eac h n ≥ 0, there is: a) a coﬁbrant ob ject W n ∈ Π M , and b) a map ϕ n : W n → Y n in C inducing a trivial coﬁbratio n ( V n ∐ L n V L n Y ) ∐ W n → Y n , where the n -th latching obje ct f or Y is L n Y := ` 0 ≤ i ≤ n − 1 Y n − 1 / ∼ , with s j 1 s j 2 . . . s j k x ∈ ( Y n − 1 ) i is equiv a len t to s i 1 s i 2 . . . s i k x ∈ ( Y n − 1 ) j whenev er s i s j 1 s j 2 . . . s j k = s j s i 1 s i 2 . . . s i k . The r esolution mo del c ate g ory structur e on s C determined by M is no w deﬁned by declaring a ma p f : X → Y to b e: (i) a we ak e quiva l e nc e if π M ,k f is a w eak equiv a lence of M -graded simplicial groups for each k ≥ 0; (ii) a c oﬁbr ation if it is a retract of a homotopically M -free map; (iii) a ﬁbr ation if it is a Reedy ﬁbration (cf. [Hi , 15.3 ]) and π M ,k f is a ﬁbration of simplicial groups for each M ∈ M and k ≥ 0. 8 DA VID BLANC 2.9. R emark. The resolution mo del category s C is simplicial (cf. [Q 1 , I I, § 1], and is itself endo we d with a set of mo dels, of the form ˆ M := { S n ⊗ M | M ∈ M , n ∈ N } , where S n ∈ S is the simplicial sphere. 2.10. Ex a mples. T ypical resolution mo del categories include the f o llo wing: (i) When C = G p , let M := { Z } , so Π M is the sub category of all free groups. The resulting resolution mo del category structure o n the category G = s G p of simplicial groups is the usual one (see [Q1, I I, § 3]). (ii) More generally , if Θ is a G -theory ( § 1.5) , let M := F ′ Θ denote the collection of all monogenic free Θ -algebras F Θ ( s ) in F Θ , with s a singleton in Θ δ - A lg (i.e., a graded set, indexed by t he discrete sk etc h Θ δ , consis ting of a single elemen t in some degree). In this case Π M ∼ = F Θ , and the mo del category on s Θ- A lg is tha t of [Q1, I I, § 4]) . (iii) F or C = T ∗ , let M := { S 1 } , s o that Π M is t he homotop y category of w edges of spheres. I n this case the mo del category of simplicial spaces is the original E 2 -mo del category of Dwy er, Ka n and Stov er (cf. [DKS]). 2.11. R emark. The ab ov e discussion is also v alid if we w ork in the comma catego ry Θ- A l g /X (cf. [Mc1, I I,6]), for a G -theory Θ and some ﬁxed Θ- algebra X . In fact, an y p : F Θ → X in F Θ /X is determined by its a dj o in t ˜ p : T → U Θ X – in other w ords, b y the U Θ X -gr a ded set { p − 1 ( x )] } x ∈ U Θ X . Therefore, Θ- A l g / X can b e sk etche d by a theory Θ /X , sorted by U Θ X = { φ x | x ∈ U Θ X } . Note that Θ /X is a G -sketch over X in the sense that it has G -structures of the form: m ( x 1 ,x 2 ) : φ x 1 × φ x 2 → φ m θ ( x 1 ,x 2 ) for eve ry θ ∈ Θ and x 1 , x 2 ∈ U Θ X θ (and similarly for other morphisms in Θ). Equiv alen tly , w e can equate t he discrete theory Θ / X δ with Θ δ - A l g /U Θ X , and use the adj o in tness of ( F Θ , U Θ ) to deﬁne an adjoint pair: Θ /X - A l g = Θ- A l g /X F Θ ⇋ U Θ Θ δ - A lg /U Θ X = Θ /X δ . W e can then t a k e the monogenic free Θ-algebras F ′ Θ /X (cf. § 2.1 0(ii)) as our mo dels, and obtain a resolution mo del category structure on s (Θ- A lg / X ). In particular, an y free resolution V • → X in s Θ- A lg is also a resolution (coﬁbran t replacemen t) in s (Θ- A l g / X ). 2.12 . A simplicial ve rsion of (co) homology. In order to mak e the abstract description o f (co)homology giv en in § 2.4-2.5 more concrete, it is conv enient to fo r ma lize the ingredien ts needed in the fo llo wing: 2.13. D eﬁnition. A c o h omolo gic al setting hC , M , V , Φ , A Φ i consists of: (1) A mo del catego ry C , enriche d via map C ( − , − ) o v er a symmetric monoidal category V . (2) A set of mo dels M for C . (3) An FP-sk etc h Φ, suc h that : (i) h Φ- C , ⊗ , I , Hom i is a closed symme tric monoidal category (with Hom( G, − ) righ t adjoint to − ⊗ G ). (ii) c Φ- V ha s a model category structure for whic h ho c Φ- V semi-triangula t ed. GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 9 (4) A homotop y functor A Φ : Π M → Φ- C , equipp ed with a natural isomorphism: (2.14) ν : map C ( F , G ) ∼ = Hom ( A Φ F , X ) for F ∈ Π M and G ∈ Φ- C . 2.15. Deﬁnition. Give n a cohomological setting hC , M , V , Φ , A Φ i , take E := s C , with the resolution mo del category structure deﬁned by M . Then for an y ob ject X and Φ-a lg ebra G in C , the c ohomolo gy of X with c o eﬃc i e nts in G is the total left deriv ed functor of map C ( − , G ), applied to X . The n -th c ohomolo gy gr o up of X with c o eﬃcients in G is the M -g raded group: H n ( X ; G ) := [ T n c ( A Φ M ) • , ( L map C ( − , G )) X ] M ∈M . 2.16. Deﬁnition. F or hC , M , V , Φ , A Φ i as ab ov e, no te that A Φ M is a homotop y cogroup ob ject in Φ- C for eac h M ∈ M , so we hav e a resolution mo del category structure on s Φ- C determined b y the set of mo dels M Φ := { A Φ M } M ∈M . Deﬁne the homolo gy of X to b e the total left derived functor o f A Φ applied to X . The n -th homolo gy gr oup of X ∈ C is the M -gr a ded group: H n X := π M Φ ,n L A Φ X (cf. § 2.9). (F or this par t of the deﬁnition w e o nly require that Φ- C b e enriche d ov er itself via Hom – w e do not need the symmetric mo no idal structure.) If G ∈ Φ- C , w e deﬁne the n -th homolo gy gr o up of X wi th c o eﬃcients in G to b e: H n ( X ; G ) := π M Φ ,n ( L ( A Φ ( − ) ⊗ G )( X ) 2.17. Example. The simplest example is when C = G p (with M = { Z } as on § 2.10(i)), Φ = G ( or A ), a nd V = S et , so Φ- C ∼ = Φ- V ∼ = A bg p . In this case Φ- C ∼ = A bg p , so the categor y c Φ- C of cosimplicial Φ-alg ebras in C is equiv alen t to the category of co c hain complexes. T h us K ( Z , n ) (a co c hain complex concen trated in degree n ) corepresen ts the n -th cohomology group of a co c hain comple x ( n ∈ N ). This yields the usual cohomology groups o f a group X with co eﬃcien ts in an ab elian group G (as a trivial X -mo dule). The functor A Φ : Π M → Φ- C is the a b elianization Ab : G p → A bg p , and the closed symmetric monoidal structure h A bg p, ⊗ , Z , Hom A bg p i yields the usual homology of gro ups. 2.18. Example. Another simple example is provide d by a sy mmetric mono idal category of sp ectra, suc h as the symmetric sp ectra of [HSS], o r the S - mo dules of [EKMM]. In the latter v ersion, for example, w e take E = M S , with the symmetric monoidal smash pro duct ∧ S , and t he in ternal f unction complexes F S ( − , − ) ∈ V = E (cf. [EKMM, I I, 1.6]) . Since ho M S is the usual stable homo t o p y category , it is trian- gulated, with generator S . Th us w e can t a k e Φ = ∗ to b e the trivial FP theory , an y S -mo dule M yields a cohomolog y theory F S ( − , M ), and A Φ : E → Φ- E is the iden tity . Similar ly if E = M R for some S -algebra R . 2.19. R e mark. These deﬁnitions ma y app ear somewhat conv oluted; they hav e b een set up to describ e b ot h the algebraic and (generalized) top olog ical theories in a uniform w ay , as appropriate deriv ed functors. Note that in general the total homology and cohomology functors, as w ell as the homology and cohomology groups, take v alues in diﬀeren t cat ego ries. 10 DA VID BLANC 3. Theories and Abelianiza tion In this section we describe the necessary back ground for deﬁning (co)homology in a category C = Θ- A l g of Θ-algebras. Most of it should b e familiar fro m the case C = G p , and the generalizations of Bec k and Quillen f or algebras (see [Be, Q3]); ho w ev er, it seems t ha t the literat ure lac ks a full description in this generalit y . W e start with the concept of (ab elian) group o b jects, whic h a re to play the ro le of Φ-algebras in C . 3.1 . Group ob jects. In general, for a sk etc hable category C = Θ - A l g we do not exp ect any enric hmen t b ey ond V = S et ; so the natural c ho ice for a cohomological setting is Φ = A . Recall that an (ab elian) gr oup o bje ct structur e o n an ob ject G in a category C is a natural (ab elian) group structure on Hom C ( X , G ) for all X ∈ C – in other w ords, a lifting of the functor Hom C ( − , G ) from S et to G p (or A bg p ); this is equiv alen t to a G - (resp., A -) structure at G . 3.2. R emark. No te that if C = Θ- A l g for some G -theory Θ, an y group ob ject structure on G comm utes with the underlying (graded) G -structure, so that the tw o necess arily agree and are commutativ e. In particular, in this case a Θ-algebra can hav e at most one (necessarily a b elian) group ob ject structure. This is of course not true for general C (as is sho wn by the example of sets). 3.3 . Ab elianization of Θ -algebras. If Θ is any theory ( sorted by S ), the category of ab elian group ob jects in Θ- A lg is sk etche d b y the theory Θ ab := A Θ o f § 1.8. W e call the A -lo calization L A : Θ- A lg → Θ ab - A l g the ab elianiza tion functor for Θ, and denote it by A Θ . Note that A Θ ( F Θ T ) = F Θ ab T . 3.4. Examples. (a) When Θ is a G -t heory , Θ ab := G (Θ), b y Remark 3.2, and w e can take Θ G := Θ in § 1.8, so q : Θ → → Θ ab is a quotien t of theories, and q ∗ is simply the inclusion of the full sub category of ab elian Θ-algebras in Θ- A l g (cf. [BP, § 2.8]). Note that b y Remark 1.9 w e can then mak e Θ ab in to a closed symmetric monoidal categor y . (b) On the other hand, if Θ = Θ δ , then Θ ab = Θ A sk etc hes S -g r a ded ab elian groups, q ∗ : Θ ab - A lg → Θ- A lg is the forgetful functor U : gr S A bg p → gr S S et , and its left adjoint A Θ is the free graded a b elian group functor. 3.5 . Θ -algebras ov er X . W e now show how the ab o ve discussion extends to the category Θ- A l g /X of Θ- algebras o ver a ﬁxed ob ject X (see § 2.11 ). F irst, we need a: 3.6. Deﬁnition. If Θ is any theory a nd X ∈ Θ- A l g , then: (a) An X -algebr a is an ob ject K in Θ- A l g equipped with maps ˆ f : K ( ϑ ) × X ( ϑ ) → K ( ϑ ′ ) for eac h f : ϑ → ϑ ′ in Θ , satisfying: ˆ g ( ˆ f ( k , x ) , X ( f )( x )) = [ g ◦ f ( k , x ) for eve ry ( k , x ) ∈ K ( ϑ ) × X ( ϑ ), and g : ϑ ′ → ϑ ′′ , with ˆ f ( k , 0) = K ( f )( x ). (b) The semi-dir e ct pr o duct of a Θ-alg ebra X b y an X -algebra K is the Θ- algebra K ⋊ X o v er X give n by: (i) ( K ⋊ X )( ϑ ) := K ( ϑ ) × X ( ϑ ) (as sets); GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 11 (ii) F or each f : ϑ → ϑ ′ in Θ , ( K ⋊ X )( f )( k , x ) := ( ˆ f ( k , x )) , X ( f ) ( x )). If w e w a n t K ⋊ X t o b e a gro up ob ject in Θ- A lg /X , w e m ust require more. F rom no w on, let Θ b e a G -theory (sorted by S ) , and X a ( ﬁxed) Θ-a lgebra. 3.7. Deﬁnition. An X -mo dule is an X - algebra K whic h is an ab elian g r o up ob ject in Θ- A l g , suc h that for each ﬁxed x ∈ X ( ϑ ), each ˆ f ( − , x ) : K ( ϑ ) → K ( ϑ ′ ) is additiv e (in the sense that it comm utes with the giv en ab elian group structure). The category of X -mo dules will b e denoted b y X - M od (see [Be, § 3]). 3.8. R ema rk. In t his case the underlying S -graded group V K is a n S - graded V X - mo dule in the tra ditional sense (a mo dule ov er the graded group ring Z [ V X ]), and the g r o up op erat io n at each θ ∈ Θ is give n b y m θ (( k , x ) , ( ℓ, y ) = ( k + x · ℓ, xy ), as usual. 3.9. Deﬁnition. Assume that p : Y → X is a map of Θ- algebras, and K is an X -mo dule. A function ξ : Y → K ( preserving the pro ducts of Θ) will b e called a derivation with r esp e ct to p if ξ ( Y ( f )( y )) = ˆ f ( ξ ( y ) , p ( y )) fo r any f : ϑ → ϑ ′ in Θ. The set o f a ll suc h will b e denoted by Der p ( Y , K ). In particular, a deriv ation with resp ect to Id X will b e called simply a derivation , a nd Der( X , K ) := Der Id ( X , K ). 3.10. R emark . Note that this holds in particular f or f = m θ : θ × θ → θ , so that by Remark 3.8: ξ ( m θ ( y 1 , y 2 ))) = ˆ m θ (( ξ ( y 1 ) , p ( y 1 )) , ( ξ ( y 2 ) , p ( y 2 )) = ξ ( y 1 ) + p ( y 1 ) · ξ ( y 2 ) . Th us ξ is a deriv atio n (crossed homomomorphism) with resp ect to the G -structure. F urthermore, Der p ( Y , K ) is an ab elian group (under the addition of K ), and an y map of X -mo dules α : K → L induces a homomorphism α ∗ : D er p ( Y , K ) → Der p ( Y , L ). The following results do not app ear in this form in the literature, but their pro ofs are straightforw ard generalizations o f the corresp onding (classical) results f o r groups (see, e.g., [Be , § 3-4 ] and [R, § 11.1]). 3.11. Pr op osition. A ny gr oup ob je ct structur e on p : Y → X in Θ - A l g /X is ne c essarily ab eli a n. Mor e ov e r, K := Ker( p ) is a n X -mo dule, with Y ∼ = K ⋊ X , and for some derivation ξ : X → K , the gr o up op er ation m ap µ : Y × X Y → Y is given (under the id e ntiﬁc ation U Y = U K × U X ) by µ ( k , k ′ , x ) = ( k + k ′ + ξ ( x ) , x ) , the zer o m ap by ( k , x ) 7→ ( − ξ ( x ) , x ) , and the inverse by ( k , x ) 7→ ( − k − 2 ξ ( x ) , x ) . Conversely, for an y X -mo dule K and deriv a tion ξ : X → K , the ab ove fo rm ulas make K ⋊ X into an ab elian gr oup obje ct over X . 3.12. Cor ollary . T h er e is an e quivalenc e of c ate gories ℓ ∗ G - (Θ - A l g /X ) → A - (Θ - A lg /X ) , induc e d b y the quotient m a p ℓ : G ֒ → A . 3.13. Lemma. Any homomorp h i sm φ : K ⋊ X → L ⋊ X b etwe en gr oup obje cts over X (with gr oup op er ations determine d by σ ∈ D er( X , K ) and τ ∈ Der( X , L ) , r e sp e ctively) is of the form φ ( k , x ) = ( α ( k ) + ξ ( x ) , x ) , wher e α : K → L is a homomorphism o f X -mo dules and ξ := α ◦ σ − τ . In particular, a n y tw o group ob ject structures ov er X on the semi-direct pro duct K ⋊ X are canonically isomorphic, so we deduce: 12 DA VID BLANC 3.14. Prop osition. The functor λ : X - M od → A - (Θ - A l g /X ) , deﬁn e d λ ( K ) := K ⋊ X (with the g r o up op er ation m ap d e termine d by the zer o de rivation), is an e quivalenc e of c ate gories, with inve rse κ : A - (Θ - A l g /X ) → X - M od which assigns to an ab elian gr oup obj e c t p : Y → X the X -mo dule Ker( p ) . 3.15. R emark. Since the forgetful functor U = U Θ : Θ- A l g → Θ δ - A l g is faithful, for an y Θ-algebra Y and semi-direct pro duct K ⋊ X ∈ Θ- A l g we ha ve: Hom Θ- A lg ( Y , K ⋊ X ) U ֒ → Hom Θ δ - A l g ( U Y , U ( K ⋊ X )) = Hom Θ δ - A lg ( U Y , U K × U X ) = Hom Θ δ - A l g ( U Y , U K ) × Hom Θ δ - A l g ( U Y , U X ) . (3.16) Th us giv en p : Y → X , w e can write any map φ : Y → K ⋊ X o v er X in the form φ ( y ) = ( α ( y ) , p ( y )), and the requiremen t that φ b e a map o f Θ-algebras means that α : F Θ T → K is a deriv ation with resp ect to p ( § 3.9), so in fact: (3.17) Hom Θ- A l g/X ( Y , K ⋊ X ) ∼ = Der p ( Y , K ) as a b elian gro up (o nce we c ho ose a ﬁxed group structure on K ⋊ X ). Three sp ecial cases should b e no t ed: (a) F o r p = Id : X → X , w e see that Der( X , K ) is the space of sections for K ⋊ X , as usual. (b) If Y = L ⋊ X for some L ∈ X - M od , then b y Prop osition 3.14: Hom X - M od ( L, K ) λ = Hom Θ- A l g/X ( L ⋊ X, K ⋊ X ) = Der p ( L ⋊ X, K ) . On the other hand, b y Lemma 3.13 an y map of X - mo dules α : L → K induces a homo mo r phism of group ob jects φ = λ ( α ) : L ⋊ X → K ⋊ X (where w e use the zero deriv ation to deﬁne the group structures o n the semi-direct pro ducts). Th us in fact: (3.18) Hom A -(Θ- A lg /X ) ( L ⋊ X, K ⋊ X ) = Der π 2 ( L ⋊ X, K ) as a b elian groups. (c) If Y = F Θ T is free, then b y adjointnes s w e actually hav e equalities of sets: Hom Θ- A l g ( F Θ T , K ⋊ X ) = Hom Θ δ - A l g ( T , U K ) × Hom Θ δ - A lg ( T , U X ) in (3.16), so for p : F Θ T → X in F Θ /X , we hav e: (3.19) Hom Θ- A lg /X ( F Θ T , K ⋊ X ) ∼ = Hom Θ δ - A lg ( T , U K ) ∼ = Hom Θ- A l g ( F Θ T , W K ) , where W : X - M od → Θ- A l g is the fo r g etful functor. In particular: Der p ( F Θ T , K ) ∼ = Hom Θ- A lg ( F Θ T , W K ) as sets (though this iden tiﬁcation is no t natural in t he full sub category F Θ in Θ- A lg ). 3.20 . A b elianization o ver a Θ -algebra. Recall fr o m § 2.11 that for a ﬁxed Θ- algebra X , Θ- A l g /X can b e sk etc hed by Θ /X (sorted by U Θ X ). Similarly , A -(Θ- A l g / X ) can b e sk etc hed b y A Θ /X , obtained from Θ /X as in § 1.8 b y adding: (a) a section – i.e., constan ts in eac h φ x (in the nota tion of § 2.11); (b) group structure maps µ : φ x × φ x → φ x and ρ : φ x → φ x , GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 13 satisfying the obvious identitites. Again the map of theories i : Θ / X ֒ → A Θ /X induces the for g etful functor i ∗ : A -(Θ- A lg /X ) → Θ- A lg /X , with an adjoin t A Θ /X : Θ- A l g /X → A -(Θ- A l g / X ) called the ab elian ization of Θ- A l g /X . This is needed in order to deﬁne homolog y for Θ-algebras (see § 4.2 b elo w). Note that the category X - M od can also b e sk etc hed b y an A -theory Θ X , obtained from Θ ab ( § 3.3) b y adding op era t io ns x · ( − ) : θ → θ for eac h x ∈ U Θ X , satisfying the o b vious iden titites. The inclusion j : Θ ab ֒ → Θ X induces the fo rgetful functor j ∗ : Θ X - A lg → Θ ab - A lg . If w e deﬁne κ : Θ- A lg /X → Θ- A lg as in Prop osition 3.14, w e obta in the comm uta tiv e outer diagram: (Θ- A lg /X ) ab i ∗ / / κ   Θ- A l g /X κ   ˆ A Θ /X s s g g g g g g g g g g g g X - M od = Θ X - A l g λ ∼ = O O j ∗ / / Θ ab - A l g q ∗ / / Θ- A lg in whic h the horizon tal a rro ws are forgetful functors (a nd q ∗ , i ∗ ha ve adjoints A Θ , A Θ /X , resp ectiv ely , with ˆ A Θ /X := κ ◦ A Θ /X : Θ- A lg /X → X - M od ). Note that b y (3.19), the ab elianization functor ˆ A Θ /X tak es any f r ee Θ-algebra p : F Θ T → X o v er X to the corresp o nding free X -mo dule F Θ X T ∈ Θ X - A l g = X - M od . Moreo ver, for any ϕ ∈ D er p ( F Θ T , K ) (determined b y ϕ ( t i ) = k i ∈ K for t i ∈ T ), the corresp onding ˆ ϕ ∈ Hom X - M od ( F Θ X T , K ) is also determined b y requiring that ˆ ϕ ( t i ) = k i . Now assume giv en a map ψ : F Θ T ′ → F Θ T in F Θ /X , determined by the condition that, fo r eac h t ′ ∈ T ′ , ψ ( t ′ ) = f ′ ∗ ( t i 1 , . . . , t i n ) for some f ′ in Θ . Then: ( ψ ∗ ϕ )( t ′ ) = ˆ f ′ (( t i 1 , . . . , t i n ) , ( p ( t i 1 ) , . . . , p ( t i n ))) ∈ K . 3.21. R emark. Eviden tly , t he discussion of ab elian g r o up ob jects and ab elianizatio n o ver a Θ-algebra X extends the absolute case of § 3.1 ﬀ. , taking X = 0. More generally , K will b e called a trivial X -mo dule if ˆ f ( k , x ) = f ( k ) for ev ery f ∈ Θ ( § 3.6) – so that K is simply an ab elian Θ- algebra, K ⋊ X is t he pro duct in Θ- A l g , a nd a deriv ation in t o K is just a map o f Θ-alg ebras. 4. (Co)homology of Θ -alge bras Andr ´ e (in [An ]) and Quillen (in [Q1, I I, § 5] and [Q3, § 2]) deﬁned homolog y and cohomology gro ups in categor ies of univers al algebras. Quillen also sho w ed ho w this generalized the earlier deﬁnition of triple cohomology (see [Be , § 2]). W e now indicate brieﬂy ho w this deﬁnition ﬁts in to the setup of § 2.4. 4.1 . Cohomology of Θ -algebras. Let Θ b e a G -theory , and C := Θ- A l g (or Θ- A l g /X for a ﬁxed Θ-algebra X ), with the resolution mo del category structure on s C describ ed in § 2.10(ii) ( o r § 2.11). As in Example 2.17, here V = S et , so w e m ust tak e Φ = A (or equiv alen tly , by Corollary 3.12: Φ = G ), s ince cosimplicial sets do not ha v e any useful mo del category structure (see how ev er [Bou]). Th us if G is an a b elian group ob ject in C , and V • → Y is a f r ee simplicial resolution (coﬁbra nt replacemen t in s C ), then the cosimplicial ab elian group W • := Hom C ( V • , G ) corresp onds under the Dold-Kan equiv alence (cf. [DP, § 3] and [W e , 9.4]) to a co chain complex W ∗ , and the category c C h Z of non- negativ ely graded cochain complexes o f ab elian groups em b eds in the category C h Z of un b o unded (co)c hain complexes, whic h is a stable mo del category (cf. [Ho, Ch. 7]. 14 DA VID BLANC Susp ensions of g := K ( Z , 0) detect homology in c C h Z (or C h Z ), so c A bg p ∼ = c C h Z is semi-triangulated in the sense of § 2.2, and in fact [ T i g , W ∗ ] = H i ( Y ; G ) is the i - t h Andr ´ e-Quillen cohomology group of Y . Remark 3 .1 5 show s that t hese can b e though t of as usual a s the deriv ed functors o f Der( − , G ), in the case C = Θ- A l g /X , and as Ext i ( Y , G ) in t he case C = Θ- A l g ( § 3.21). This iden tiﬁcation has b een the basis for a n um b er of deﬁnitions of cohomology in v ario us top ological settings - see, e.g., [MS2], and the surve y in [BR]. 4.2 . Homology of Θ -algebras. In this situatio n one can deﬁne the homology of a Θ-algebra Y as the total left deriv ed functor of ab elianization A Θ : Θ- A lg → Θ ab - A lg ( § 3.3), whic h tak es v alues in the category s Θ ab - A l g of simplicial Θ ab -algebras (a s usual, w e only need to ev aluate A Θ on F Θ , so L A Θ actually tak es v alues in s F Θ ab ). Since Θ ab - A l g is an ab elian category (with enough pro j ectiv es, na mely: F Θ ab ), s Θ ab - A lg is equiv alen t t o the stable mo del categor y C h (Θ ab ) of c hain complexes o ver Θ ab , and the homology groups [ T i K ( F Θ ab s, 0) , A Θ V • ] = H i Y (for s an S -graded singleton) are themselv es Θ ab -algebras. The same holds for Y ∈ Θ- A lg /X : using § 3.20, we may deﬁne H i ( Y /X ) as the i -th deriv ed functor of A Θ X : F Θ /X → (Θ- A lg /X ) ab , taking v alues in (Θ- A l g /X ) ab – or equiv alen tly (Prop osition 3.14) in X -mo dules. F o r groups, H ∗ ( G/G ) is the homology of G with co eﬃcien ts in Z [ G ]. F or a p ointed connected space X with G = π 1 ( X , x ), H ∗ ( X/B G ) is the homolo gy o f X with co eﬃcien ts in t he lo cal system Z [ G ]. 4.3. Deﬁnition. T o deﬁne homology of Y → X with co eﬃcien ts in a n ar bit r a ry X -mo dule G , we need a monoidal structure on X - M od ∼ = (Θ- A lg / X ) ab , induced via the a dj o in t pair Θ X - A lg F Θ X ⇋ U Θ X Θ δ - A l g from the usual monoidal structure (Θ δ - A lg , × ) of Cartesian pro ducts of graded sets. More precisely , deﬁne ⊗ : F Θ X × F Θ X → F Θ X b y F Θ X T ⊗ F Θ X S := F Θ X ( T × S ). The 0-th deriv ed functor in the second v aria ble deﬁnes F Θ X T ⊗ G for any Θ X -algebra ( X -mo dule) G ; and the n -th left deriv ed functor of A Θ X ( − ) ⊗ G (in the ﬁrst v ariable) is by deﬁnition H n ( Y /X ; G ). 4.4. Example. When Θ = G , a free simplicial resolution V • of a group G in s G p is actually a coﬁbra nt mo del for the classifying space B G (in S ∗ ). Applying the functor ˆ A Θ /X of § 3.20 to V • dimension wise yields a mo del for the c hains on the univ ersal contractible G -space E G (since conv ersely , ta king the free Z -mo dule on the bar construction mo del for E G and dividing out by the free G -action yields Z B G , so Z E G ≃ Z [ G ] V • ). T aking homotop y groups of Z [ G ] V • is the same as taking the homology of the ch ain complex corresp onding to Z E G , whic h is just H ∗ ( G ; Z [ G ]). 4.5. R ema rk . Note that the previous discuss ion actually deﬁnes homology and coho- mology for any simplicial Θ-alg ebra Y • , not only for the constant ones. Moreo v er, if Θ = G , the adjo int pairs of functors: (4.6) T ∗ |−| ⇋ S S ∗ G ⇋ ¯ W G = s G p induce equiv alences of the homotopy categories of p oin ted connected to p ological spaces, reduced simplicial sets, and simplicial g roups. Here | − | is the geometric GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 15 realization functor, S is the singular se t functor, ¯ W is the E ilen b erg-Mac Lane classify- ing space functor, and G is Ka n’s lo op functor (cf. [May1, § 26.3] and [Q 1, I,4 & I I,3]). Th us Quillen’s a pproac h provides an alg ebraic description of ordinary homology and cohomology of spaces (with lo cal co eﬃcien ts). Note, ho wev er, the shift in indexing: in particular, w e lose H 0 , since w e can deal only with connected spaces from this p o int of view. There is also an algebraic mo del for not- necessarily-connected spaces due to D wy er and Kan, using simplicial group oids (see [GJ, V, § 5]) , and Quillen’s approac h, as w ell as muc h of the discussion here, carries ov er to that setting (compare [D2]). Ho w ev er, in or der to a v oid further complicating the description, we restrict atten tio n here to simplicial groups. 4.7 . Diagrams of Θ -algebras. If D is a small category and Θ is a G -theory , there is a model category structure on the functor category s Θ- A l g D , and the ob jectwis e descriptions of a b elian group ob jects and ab elianization (for eac h d ∈ D ) prov ide deﬁnitions of (co)homolo g y fo r diagr a ms of Θ- algebras, to o (see [BJT, § 4] for the details). Moreo ver, ev en for C = Θ- A l g or Θ- A l g / X , w e can allow our co eﬃcien ts to b e diagr ams G : D → A -Θ- A l g of ab elian group ob jects (or X - mo dules). This enables us to treat a map suc h a s Z → → Z /p ( r eduction mo d p ), sa y , as the co eﬃcien ts for a cohomology theory (rather than a nat ura l transformation). In pa r ticular, w e can apply an y general machinery , suc h as univ ersal co eﬃcien t theorems, t o H ∗ ( − ; G ), to o. 4.8 . Spherical mo del categories. When C = Θ- A l g for some G -theory Θ, the resolution mo del category s C ( a nd the mo dels M = F ′ Θ - cf. § 2.10(ii)) will ha v e additional useful structure whic h is familar to us from top olog ical spaces: 1. F or an y n ≥ 1, π M ,n ( − ) is naturally an ab elian group o b ject o v er π M , 0 ( − ). 2. Eac h V • ∈ s C has a functorial Postnikov tower of ﬁbrations: . . . → P n V • p ( n ) − − → P n − 1 V • p ( n − 1) − − − → · · · → P 0 V • , as we ll as a w eak equiv alence r : V • → P ∞ V • := lim n P n V • and ﬁbra t ions P ∞ V • r ( n ) − − → P n V • suc h that r ( n − 1) = p ( n ) ◦ r ( n ) for all n , and ( r ( n ) ◦ r ) # : π M ,k V • → π M ,k P n V • is an isomorphism for k ≤ n , and zero for k > n . 3. F or ev ery Θ- a lgebra X , there is a classifying obje ct B X with B X ≃ P 0 B X and π M , 0 B X ∼ = X , unique up to homotop y . 4. Giv en a Θ-algebra X and an X -mo dule G , there is an extend e d G -Eilenb er g- Mac L a n e obje ct E = E X ( G, n ) in s C /X for eac h n ≥ 1, unique up to homotop y , equipp ed with a section s for p (0) : E → P 0 E ≃ B X , suc h that κπ M ,n E ∼ = G as X -mo dules; and π M ,k E = 0 for k 6 = 0 , n . If G is a trivial X -mo dule ( § 3.21), we write simply E ( G, n ). An y resolution mo del cat ego ry with this additional structure (as w ell as functorial k -inv arian ts) is called a spheric al mo del c ate gory . See [B3, § 1 - 2] fo r the details. 4.9. R ema rk. The homotopy groups π M ,n in the resolution mo del catego r y s Θ- A l g are corepresen ted b y S n ⊗ F Θ ( s ) for M = F Θ ( s ) ∈ F ′ Θ , s ∈ S ⊆ Θ (cf. § 2.9). Thus 16 DA VID BLANC b y a djoin t ness fo r any V • ∈ s Θ- A lg w e ha ve: π M ,n V • = [ S n ⊗ F Θ ( s ) , V • ] ∗ = [ S n , ( U Θ V • ) s ] = π n ( U Θ V • ) s , so that the g roup π M ,n X (induced by the ho mo t o p y cogroup structure of S n ) is the usual n -th simplicial homotop y group of t he gr a ded simplicial group U Θ V • in the appropriate degree. This works also in s Θ- A l g /X : more pr ecisely , π M ,n V • as deﬁned ab ov e is an ab elian g roup ob ject in Θ- A lg /π M , 0 V • , and applying κ of Prop osition 3.14 yields a π M , 0 V • -mo dule, whose underlying S - graded set is π n U Θ V • (see [BP, § 4.14 ]) . 4.10 . Cohomology in s Θ - A lg . It may appear more natural to take as a represen ting ob ject an ab elian group ob ject in the mo del category s Θ- A l g itse lf. In most cases this will yield no new cohomolo gy g roups, but it will enable us to deﬁne, and in some cases compute, the primary cohomology op er ations – as w e do for top olo gical spaces (see, e.g., [P ]). The ob vious examples a re those of the form E ( G , n ) as ab ov e (or E X ( G, n ) in s Θ- A l g /B X , if we w ant lo cal co eﬃcien ts). In most cases o f in terest – including T ∗ , S ∗ , G = s G p – the o nly ob jects in A - s Θ- A l g are pro ducts of the a b o v e. F urthermore, since E ( − , n ) : A -Θ- A lg → s Θ- A l g is a functor, w e can deﬁne an Eilen b erg-Mac Lane diagram E ( G, n ) for an y diagra m G : D → A -Θ- A l g as in § 4.7. Th us for an y coﬁbrant W • in s Θ- A l g and co eﬃcien ts M ∈ A -Θ- A lg D , for eac h n ≥ 1 w e deﬁne the n -th c o h omolo gy gr oup of W • with co eﬃcien ts in G , denoted by H n ( W • ; G ), to b e the set of comp onen ts of map( W • , E ( G, n )) (whic h is a D -diag ram of simplicial ab elian groups, so the comp onen ts constitute a D -diagr am of ab elian groups). Again, there is also a lo cal v ersion, for G in Θ- A l g /X or M : D → A -Θ- A l g /X , yielding: H n ( W • /X ; G ) := π 0 map s Θ- A l g /X ( W • , E X ( G, n )) for each n ≥ 1 . 4.11. P rop osition. If Θ a G -the ory, X is a Θ -algebr a, and G is in A - (Θ - A lg /X ) , then c o h omolo gy with c o eﬃcients in G as deﬁne d in § 4.1 is na tur al ly isomorphic to that deﬁne d in § 4.10. Compare [D 1, § 3]. Pr o o f . Let K b e the X - mo dule corresp onding to G = K ⋊ X , so E • := E X ( G, n ) is o f the form E ( K, n ) ⋊ X , where E ( K , n ) is obta ined from the analogous c hain complex ( ov er X - M od ) b y the D old-Kan equiv a lence (cf. [Ma y1, p. 95]). Thus : (4.12) E i =          X for 0 ≤ i < n K ⋊ X i = n ( L n j =0 s j K ) ⋊ X i = n + 1 M i E • i ≥ n + 2 , (where M i E • is the i -th matchin g obje ct – see [BK, X, § 4.5] or [BJT , § 2.1]), with the diﬀerential: (4.13) ∂ n +1 ( x, λ )) := ( n +1 X i =0 d i x, λ ) for eve ry ( x, λ ) ∈ E n +1 . GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 17 Let W • b e a free simplicial ob ject in s C , with ε : W 0 → X inducing π 0 W • ∼ = X (for example, W • could b e a resolution of X ). F rom (4.12) and (4.13) w e see that Hom s C /X ( W • , E • ) is naturally isomorphic to the subgroup o f Ho m C /X ( W n , K ⋊ X ) consisting of maps f : W n → K ⋊ X (o ve r X ) for whic h f ◦ d i is the pro jection to X (the zero of Hom( C /X )( W n +1 , K ⋊ X ) for each 0 ≤ i ≤ n + 1. Here W n maps to X b y ε ◦ d 0 ◦ · · · ◦ d 0 . Again b y the Dold-Kan equiv alence, there is a path ob ject E I • for E • in s Θ- A l g /X with (4.14) E I i =          X for 0 ≤ i < n − 1 K ⋊ X i = n − 1 ( K ⊕ K ⊕ L n − 1 j =0 s j K ) ⋊ X i = n M i E • i ≥ n + 1 , with d 0 the iden t it y on the ﬁrst copy of K ⋊ X in E I n , and min us the iden tity on the second copy . There are tw o ob vious pro jections p 0 , p 1 : E I • → E • , a nd a homotopy b et w een t wo maps f 0 , f 1 : W • → E • o ver X is a map F : W • → E I • with p i ◦ F = f i ( i = 0 , 1), whic h in turn corresp onds to a map F ′ : W n − 1 → K ⋊ X ov er X for whic h F ′ ◦ d 0 represen ts f 0 , f 1 resp ectiv ely on the tw o copies of K ⋊ X . Th us w e see that H n ( W • /X , M ) := [ W • , E • ] s C /X is canonically isomorphic to the n -th coho mo t o p y gro up of t he cosimplicial ab elian g r oup Hom ( C / X )( W • , K ⋊ X ), as claimed.  4.15 . Cohomology of op erads and their algebras. As noted in § 1.3(b) , our deﬁnition of sk etc hable categories co vers b oth the categor y of op erads, O - A lg , and that of a lg ebras ov er a giv en o p erad P . Of course, O is no t a G -theory; how ev er, essen tia lly all known applications are to op erads of (connected) top ological spaces or of ch ain complexes (see [MSS]). In the ﬁrst case, w e can use (4.6) to replace T ∗ b y G , so that in b o th cases w e ma y assume, without loss of generalit y , that our op erad ta kes v alue in s Θ- A l g for some G - t heory Θ. Not e that the category o f O -a lgebras in s Θ- A l g is equiv alen t to s ˜ Θ- A l g , where ˜ Θ = O × Θ ( pro duct of FP-sk etc hes) is no w an G -theory (see § 1.8). Thus the deﬁnition of § 4.10 ( a pplied to ˜ Θ) is v alid for o p erads of spaces o r c hain complexes . The same applies t o algebras ov er a ﬁxed op erad P t a king v alues in T ∗ or C h k for some ﬁeld k (see [May2, § 2]), as we ll as to t he coho mology of a k - linear category (tha t is, algebras ov er a k -linear PROP ) considered in [Mar2]. W e should observ e, how eve r, t hat the v arious cohomology theories constructed – in the conte xt 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 sp eciﬁc diﬀerential graded resolution. T o show that t hese agree with our general deﬁnition requires a generalization of Quillen’s equiv alence b et wee n simplicial and diﬀerential g raded Lie algebras o v er Q (see [Q2, I, § 4], and compar e [D P, § 3]). One can exp ect suc h an equiv alence only for suitable k - linear categories ov er a ﬁeld k of characteristic 0. 4.16. R emark. W e should p oin t out that a diﬀerent deﬁnition o f (co)homolo gy for Θ-algebras, based o n the Baues-Wirsc hing and Ho chs c hild-Mitche ll cohomolog ies o f categories (cf. [BW, Mit]), is giv en by Jibladze and Pirash vili in [JP]. See [Sc2, Theorem 6.7] for an equiv alen t fo rm ulatio n in terms o f the top ological Ho c hsc hild (co)homology of suitable ring sp ectra. 18 DA VID BLANC 4.17 . Cohomology of shea ve s. W e hav e assume d so far that Θ w as a G -theory . This is necessary for the approach describ ed here at t wo points: in order to iden tif y the (ab elian) group o b jects in Θ- A l g (see Section 3), and to deﬁne the mo del category structure on s Θ- A l g (see § 2.1 0(ii)). This is a resolution mo del category (induced b y t he adjoint pair ( F Θ , U Θ ) of § 1.7) only with some suc h additional assumption (cf. [B2]): otherwise the free Θ -algebras are not necessarily cogroup ob jects. One obv ious example where this fa ils is the category of sets, where w e apparen tly ha ve no meaningful concept of coho mo lo gy . A more in teresting case is the category of shea v es on a top ological space X , sk etc hed b y Θ U (see § 1.3). Note that there is no free/forgetful adjoin t pair b etw een Θ δ U - A l g and Θ U - A lg or Θ ab = A -Θ U ∼ = Θ U - A bg p , since sheav es of a b elian groups r a rely hav e any pro jectiv es (e.g., Z C U in § 1.3 (c) is not generally a sheaf ). How ev er, they do hav e enough injectiv es, so if w e replace left deriv ed functors b y righ t deriv ed f unctors in § 2.4, with E = Θ U - A lg , V = S et , and Φ = A , w e may deﬁne H n ( X ; F ), for any F ∈ Φ- E , to b e the righ t derive d functors of Hom E ( C X , − ), applied to F . This also explains wh y our deﬁnition of homology do es not mak e sense for shea ve s. 5. Generalized cohomology F or simplicial Θ- algebras o v er a G -theory Θ – and thu s for simplicial sets or top ological spaces – the only strict ab elian group o b jects are g eneralized Eilen b erg- Mac Lane ob jects (cf. [Mo o, 19.6]). Of course, in an y mo del category D , an y ab elian group ob j ect G in ho D deﬁnes a functor [ − , G ] : ho D → A bg p ; but suc h functors do not usually satisfy the axioms of a coho mo lo gy theory . F ro m our p oint o f view, this is b ecause the structure maps on the higher pro ducts G k ( k ≥ 3) whic h are needed to mak e G an G - or A -algebra in D are not uniquely deﬁned. One w a y to deal with this problem would b e to require that G hav e a n E ∞ -op erad acting on it (cf. [May2, § 14]). If D = T ∗ (or S ∗ ), by a result o f Boardman and V og t, under mild top olog ical restrictions an y E ∞ H -space is homotopy equiv alen t to a strict ab elian monoid in D ( cf. [BV, Theorem 4.58]. 5.1 . Γ -spaces. Ho mo t o p y-coherent a b elian monoids ma y b e conv enien tly describ ed in t erms o f a lax v ersion of A , represen ting Γ-spaces (cf. [Se2 ]): Let Γ denote the category of ﬁnite p oin ted sets, and choose a set n + = { 0 , . . . , n } (with basep oint 0) fo r each n ∈ N . A Γ -obje ct in a p ointed category C is a p oin ted functor G : Γ → C ; the category of all suc h will b e denoted b y Γ- C . Note that if C is co complete, w e can extend G to all of S et ∗ b y a ssuming it commutes with arbitrary colimits. A Γ-space G – tha t is, an o b ject in Γ- S ∗ (or Γ- T ∗ ) – is called sp e cial if for A, A ′ ∈ Γ, the nat ural map G ( A ∨ A ′ ) → G ( A ) × G ( A ′ ) is a weak equiv alence. This implies tha t f o r each n ∈ N , t he obvious map (5.2) G ( n + ) → G ( 1 + ) × . . . × G ( 1 + ) | {z } n is a we ak equiv a lence. Such a G is called very sp e cia l if in addition π 0 G ( 1 + ) is an ab elian group under the induced monoid structure. 5.3. Deﬁnition. A sp ecial Γ-space G has a class ifying Γ -sp ac e B G , whic h is itself sp ecial, de ﬁned by setting ( B G )( n + ) := G ( n + × n + ), with the diagona l structure maps GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 19 (see [Se2, 1.3] and compare [Mil]) . By iterating the functor B we obtain a Ω-sp ectrum B G := h ( B i G )( 1 + ) i ∞ i =0 . Th us G ( 1 + ) itself is a n inﬁnite lo op space (with a sp eciﬁed H - space structure) if and only if G is v ery sp ecial. 5.4 . The Γ + -construction. F or an y p ointed simplicial set K ∈ S ∗ , Barratt deﬁn es the free sim plicial monoid Γ + K to b e ` n ≥ 1 K n × Σ n W Σ n / ∼ , w here ∼ is generated b y the ob vious inclusions K n ֒ → K n +1 and Σ n ֒ → Σ n +1 (cf. [Ba, § 4]). Then Γ + K is actually a Γ-space (see [A1, § 8 ]) . T o av oid confusion in the notation we shall denote this functor by γ + : S ∗ → Γ- S ∗ . The (dimensionw ise) group completion γ K := Ω B γ + K is a very sp ecial Γ-space, whic h mo dels t he inﬁnite lo op space Ω ∞ Σ ∞ K . The functor γ : S ∗ → Γ- S ∗ is left adjoint to G 7→ G ( 1 + ). If K is connected, then γ + K ≃ γ K (cf. [Ba, Theorem 6.1]). Note that w e can think of S := γ S 0 as the inclusion functor Γ → S ∗ (cf. [Ly, 2.7 ]). 5.5 . The mo del category of Γ -spaces. In [BF, § 3], Bousﬁeld and F riedlander deﬁne a prop er simplicial mo del category structure on Γ- S ∗ as a diagram category with Σ n - action on eac h G ( n + ), which they call the strict mo del categor y: a map f : G → G ′ is a weak equiv alence if f ( n + ) : G ( n + ) → G ′ ( n + ) is a Σ n -equiv ariant weak equiv alence for eac h n ≥ 1, and it is a (co)ﬁbration if it is a Σ n -Reedy (co)ﬁbration (see [Hi, § 15.3]). They sho w t ha t the homotop y category of v ery sp ecial Γ-spaces is equiv alent to that of connectiv e sp ectra (see [BF, Theorem 5.1]), with Quillen equiv alences pro vided b y iterations of the functor B and its adj o in t. They then deﬁne a stable w eak equiv alence of Γ-spaces to b e a map inducing a w eak equiv alence o f the corresp onding sp ectra, and so obtain a new simplicial mo del categor y structure o n Γ- S ∗ (with the same coﬁbrations, but few er ﬁbrations), whose homot o p y catego ry is aga in equiv alent to the usual stable cat ego ry o f connectiv e sp ectra (see [BF, Theorem 5.8]). V ariants o n these tw o mo del category structures (with the same w eak equiv a lences) are pro vided in [Sc1, App. A]. 5.6 . Γ -simplicial groups. In view of (4.6) , it is natural to think of the category Γ- G of Γ- simplicial groups as represen ting connected inﬁnite lo op spaces; note that ev ery sp ecial Γ-ob ject here is trivially very special, b ecause of the shift in indexing fo r homotop y groups. A Γ-simplicial group G also known as a c h a in functor (cf. [A2, § 1]), since one can asso ciate to it a g eneralized ho mology theory b y setting H n ( X ; G ) := π n ( G • X ) for eac h X ∈ S ∗ , where the simplicial g roup G • X is deﬁned b y G n X := G ( X n ) n . Here eac h G ( X n ) ∈ G is deﬁned as ab o ve b y extending G from Γ to S et ∗ , so that G • X is actually the diag o nal of a bisimplicial group. Equiv alen tly , giv en a Γ-space G ∈ Γ- S ∗ , extend it via colimits from Γ to S et ∗ and th us via the diagonal to a functor ˜ G : S ∗ → S ∗ , whic h in fact ta k es a (pre)sp ectrum ( X n ) n ∈ N to a (pre)sp ectrum ( ˜ GX n ) n ∈ N using: S 1 ∧ ˜ G ( X n ) → ˜ G ( S 1 ∧ X n ) → ˜ G ( X n +1 ) . Th us for each X ∈ S ∗ , one may ev aluate t he homolo g y theory asso ciated t o G on X b y: H n ( X ; G ) ∼ = π S n ˜ G ( S ∧ X ) = colim k → ∞ π n + k ˜ G ( S k ∧ X ) , 20 DA VID BLANC where S := h S n i ∞ n =0 is the sphere sp ectrum. Note that if G is v ery sp ecial, then ˜ G ( S ∧ X ) is the Ω-sp ectrum corresp onding to Anderson’s G • X (see [BF, § 4]. 5.7 . Generalized cohomology . W e now explain ho w the deﬁnitions of § 2.4 apply in this context: ﬁrst, note that the usual mo del category structure o n E = S ∗ is symmetric monoidal and enric hed o v er V = S ∗ (cf. [Q1, I I, § 3]). No w for Φ = Γ, Lydakis (in [Ly]) deﬁned a smash pro duct of Γ- spaces making Φ- V = Γ- S ∗ , to o, in to a symmetric monoidal catego ry , with unit S . He also deﬁnes internal function complexes Hom Γ- S ∗ ( G, H ) ∈ Γ- S ∗ for G, H ∈ Γ- S ∗ b y setting: (5.8) Hom Γ- S ∗ ( G, H )( n + ) := map Γ- S ∗ ( G, H ( n + ∧ − )) , where H ( n + ∧ − ))( k + ) := H ( n + ∧ k + ) and map Γ- S ∗ ( − , − ) ∈ S ∗ is the usual simplicial function complex. Th us Φ- E = Γ- S ∗ is indeed enrich ed ov er Φ- V (cf. [Ly, 2.1]). Moreo v er, Φ- V is semi-triangulated, with the delo oping B : Γ- S ∗ → Γ- S ∗ ( § 5.3) as the “susp ension automorphism” T of § 2.3. The delo opings of the 0- sphere { B n S } ∞ n =0 corepresen t homotop y groups in ho Γ- S ∗ , since its homoto py category is equiv alen t to that of connectiv e sp ectra, with generator S (corresp onding to S 0 ). No w for an y Γ-space G ∈ Φ- E and an y p ointe d simplicial set K ∈ E , Hom E ( K , G ) is a ﬁbra nt Γ-set ( § 1.2), so the S ∗ -function complex M := map ∗ ( K , G ) is a Γ-space. If G is (very ) sp ecial, so is M , since map ∗ ( K , − ) has ho mo t o p y meaning and preserv es pro ducts. Moreo ver, applying Barra t t ’s functor yields a sp ecial Γ-space γ K , and the adjunc- tion isomorphism: (5.9) M = map ∗ ( K , G ) ∼ = − → Hom Φ- E ( γ K , G ) induces an isomorphism b et w een the homotopy groups of M and those of Hom Φ- E ( γ K , G ) (corepresen ted b y S and its susp ensions). Therefore, f or sp ecial G the homotop y g r oups of M are determined by those of M ( 1 + ) = map ∗ ( K , G ( 1 + )), whic h are b y deﬁnition H ∗ ( K ; G ), the generalized coho- mology groups asso ciated t o t he Ω-sp ectrum f o r G . 5.10 . Generalized homology . Barratt’s functor γ : E → Φ- E is the required functor A Φ , by (5.9), so its left derived functors are π ∗ γ K (since ev ery K is coﬁbran t). These turn out to b e the stable homoto p y gro ups of K , and a re by deﬁnition the homology groups of K in this con text. Finally , since the smash pro duct o f (coﬁbrant) Γ- spaces is tak en to the smash pro duct of sp ectra under the equiv alence of homotop y categories (see [Ly, Lemma 5.16 ]), w e see that the groups H ∗ ( K ; G ) of § 2.5 are just the generalized homology groups associated to the Ω- sp ectrum for G . 5.11 . The (co)simplicial v ersion. W e next show how these deﬁnitions can b e made to ﬁt the description in § 2.1 2: First, note that s S , as w ell as s T ∗ and s G (cf. § 4.5 ), ha ve resolution mo del category structures with M = { S 1 } – this is the o riginal E 2 -mo del category o f [DKS, § 5.10], whic h w as constructed precisely so that if V • is a r esolution of X ∈ S , then the diagio nal diag V • (or equiv alently , the realization o f the corresp onding simplicial GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 21 space) is w eakly equiv a lent to X . Moreo v er, S , as w ell as T ∗ and G , are enric hed ov er V := S with its usual closed symmetric monoidal structure. W e also need a suitable model category structure on the category c Γ- S ∗ of cosimpli- cial Γ-spaces – namely , t he dual o f Mo erdijk’s mo del category of bisimplicial sets (cf. [Mo e, § 1]), in whic h a map f : X • → Y • of cosimplicial Γ-spaces is a we ak eq uiv alence (resp., coﬁbration) if T ot f is a w eak equiv alence (resp., coﬁbration) o f Γ-spaces. This implies that T ot : c Γ- S ∗ → Γ- S ∗ induces a n equiv a lence of homotopy catego r ies, so for all practical purp oses we can a void w o rking with cosimplicial o b jects altogether (but see Theorem 6.18 b elo w). The inv erse equiv alence c Γ- S ∗ → c Γ- S ∗ is deﬁned by Φ 7→ c (Φ) • (the constan t cosimplicial ob ject). Th us ho( c Γ- S ∗ ) (with this structure) is equiv alent to the stable category of connectiv e sp ectra, whic h is semi-triangulated, with c ( B ) • ◦ T ot : c Γ- S ∗ → c Γ- S ∗ ( § 5.3) as the susp ension automor phism T , and c ( S ) • as g enerator. No w, given a sp ecial Γ-space G ∈ Γ- S ∗ and a free simplicial resolution V • → X in the orig inal resolution mo del category s S , for any simplicial set Y – in particular, for Y = G ( 1 + ) – w e ha v e: (5.12) map ∗ (diag V • , Y ) ∼ = T ot map ∗ ( V • , Y ) (see [BK, XI I, § 4.3]). Th us in our case the cosimplicial Γ-space map ∗ ( V • , G ) is weakly equiv alen t to the (constan t cosimplicial) space c (map ∗ ( X , G ( 1 + ))) • , whose homotopy groups are H ∗ ( K , G ) ( § 5.7). Finally , note that Barratt’s functors γ + and γ are deﬁned dimension wise on a simplicial set K , so that diag γ V • = γ dia g V • for any bisimplicial set V • . Th us we may deﬁne A Φ : E → Φ- E to b e γ , and its total left deriv ed functor is naturally equiv alen t to γ (in Mo erdijk’s mo del category s S ∗ ), since diag V • ≃ − → K for an y free simplicial resolution V • → K . Th us again the (unadorned) homolog y groups a re t he stable homotop y g roups o f K , and H ∗ ( K ; G ) are the generalized homolo g y gr oups asso ciat ed to the Ω-sp ectrum for G . 6. The sp ectral seque nces W e now w an t to use this machine ry t o try to understand relationships among the v ar io us homolo g y and cohomology theories. First, w e shall need a preliminary notion: 6.1. Deﬁnition. If M is a set of mo dels in a mo del category C (with Π M ⊆ C as in § 2.7), then C -Π := (ho Π M ) op is a G -theory , whic h ske tc hes the category C -Π- A lg of C - Π -algeb r a s (cf. [BS, § 3]). 6.2. R em ark. If w e think of M and its susp ensions as corepresen ting homotopy groups in C (cf. § 4.9 ), then C -Π-algebras are graded g roups equipp ed with an action of the corr espo nding primary homotopy op erations - the motiv ating example b eing π M , ∗ X for any X ∈ s C . This notion ma y b e extended to an y concrete category C b y the conv en tions of [BS, § 3.2.2 ], and ma y also b e dualized as in [Bou] by taking C -Π := ho Π M , rather than the opp osite category (cf. [BP, § 1.13]). Note that the derive d functors of an y functor in to C actually tak e v alues in C -Π- A l g . 6.3. Examples. (a) If C has a trivial mo del category structure, and M consists of (enough) pro jectiv e generators – e.g., if C = Θ- A lg and M = F ′ Θ – then C -Π- A l g ∼ = C . 22 DA VID BLANC (b) If C = s D or c D for some ab elian category D , and M again consists of (enough) pro jective generator s – e.g., for C = s Θ X and M as ab ov e – then C -Π- A l g ∼ = gr N D (where w e use lo w er or upp er indices for the grading according to the usual con ven tion). (c) F or C = T ∗ or S ∗ , with M = { S 1 } , then C -Π- A l g ∼ = Π- A l g is the category of ordinary Π-algebras, mo deling the usual homotop y groups of top ological spaces. (d) If C = Γ- S ∗ and M = { S } , then C -Π- A lg is equiv alen t to the categor y of graded connected π -mo dules f o r π = π S ∗ S 0 (homotop y groups of the sphere sp ectrum), since π M , ∗ G a r e just the stable homotop y groups of the Ω-sp ectrum corresp onding to G ∈ Γ- S ∗ . Using the Quillen equiv alence of (4.6), we see that when C = s Θ- A l g w e often ha ve intere sting categories of C - Π-algebras (see, e.g., [BS, § 3.2.1]). W e shall also need the followin g v ersion o f [BS, Prop. 3.2.3]: 6.4. P rop osition. Any c ontr a v ariant functor T : C → c B fr om a mo del c ate gory C (e quipp e d with a set of m o d e ls M ) to a c oncr ete c ate gory B induc es a g r a d e d functor ¯ T ∗ : s C - Π - A l g → s B - Π - A lg by setting ¯ T k ( π M , ∗ V • ) := π k ( T V • ) for c oﬁbr ant V • ∈ s C , and extending by taking 0 -th deri v e d functor. Pr o o f . Since π M , ∗ : ho Π M → F C -Π is an equiv alence of categor ies (onto the free C -Π- algebras), in part icular π M , ∗ V • ∼ = π M , ∗ W • ⇔ V • ≃ W • for coﬁbrant V • , W • ∈ s C , so ¯ T ∗ is we ll-deﬁned on free C -Π-algebras.  6.5 . A general setting. In Sections 3-5 the algebraic and top ological v ersions of homolo g y and cohomology ha ve b een treated separately . W e no w show how the Pro crustean framew ork of § 2.12 ma y be us ed in order to obtain a uniform description of v arious relations b etw een them. 6.6. Examples. W e wish to concen trate on the following list of cohomo lo gical settings (Deﬁnition 2.13), discusse d a b o v e: (a) hC = Θ- A lg , M = F ′ Θ , V = S et, Φ = A , A Φ = A Θ i for some G -theory Θ; (b) More generally , h C = Θ- A l g /X , M = F ′ Θ /X , V = S et, Φ = A , A Φ = A Θ X i for some G -theory Θ and ﬁxed X ∈ Θ- A l g . (c) hC = s Θ- A lg /X , M = { c ( F Θ ( s )) • | F Θ ( s ) ∈ F ′ Θ } , V = S ∗ , Φ , A Φ i where Φ is some strong A -sk etc h. (d) hC = S ∗ , M = { S 1 } , V = S ∗ , Φ = Γ , A Φ = γ i (with the sym metric monoidal struc- ture o n Γ- S ∗ of § 5.7) . In all these examples w e ha v e additional prop erties which w e shall require in our applications, whic h w e may formalize as follo ws: 6.7. Deﬁnition. A cohomological setting hC , M , V , Φ , A Φ i is c omplete if if it is equipped with: (1) A left adjoin t diag : s C → C to the inclusion c ( − ) • : C → s C , which induces diag : s Φ- C → Φ- C , as w ell as a con vergen t ﬁrst- quadran t sp ectral sequen ce with: (6.8) E 2 s,t ∼ = π s π M ,t V • = ⇒ π M ,s + t (diag V • ) , for each V • ∈ s C and M ∈ M ; GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 23 (2) A righ t adjo in t T ot : c V → V to the inclusion c ( − ) • : V → c V , whic h induces T ot : c Φ- V → Φ- V , as w ell as a second-quadran t sp ectral sequence with: (6.9) E s,t 2 ∼ = π s π M ′ ,t X • = ⇒ π M ′ ,t − s (T ot X • ) , for each X • ∈ c V and M ′ ∈ M Φ (w e do not address questions of con ve r- gence); (3) A natural “ Φ- C -adjointness ” isomorphism: (6.10) T ot(Hom ( V • , G ) ∼ = − → Hom (diag V • , G ) for any V • ∈ s Φ- C and G ∈ Φ- C . 6.11. P rop osition. Each of the exa mples of § 6.6 is a c omplete c oho m olo gic al setting. Pr o o f . Since ( a) and (b) are instances of (c), we ha v e only tw o cases to consider: (1) Assume C = s Θ- A l g /X for some G -theory Θ sorted by S . Then V • ∈ s C is a bisimplicial Θ-a lg ebra (ov er X ), and let diag V • b e the usual diagonal (with (diag V • ) n := ( V n ) n ). Note that U Θ V • is just an an S -graded bisimplicial set, with U Θ diag V • = diag U Θ V • (ev en thoug h colimits are not generally preserv ed by U Θ ). By Remark 4.9 w e see that the Bousﬁeld-F riedlander sp ectral sequence for U Θ V • in eac h degree (cf. [BF, Theorem B.5]) has the form (6 .8). Similarly , given a cosimplicial ob ject X • ∈ c ( s Φ-Θ- A l g /X ), the usual T o t fo r the ( S -graded) cosimplicial simplicial set U Θ X • is deﬁned to b e the simplicial set T • with T n := Ho m c S e t ( ∆ • ⊗ ∆[ n ] , X • ), and this has a natural structure o f a Φ- algebra in Θ- A l g /X by Remarks 1.2 and 2.1 1 and § 1.8. Th us T ot U Θ X • lifts to T ot X • ∈ s Θ- A lg . The homotopy sp ectral sequence for the cosimplicial space U Θ X • , with: E s,t 2 = π s π t U Θ X • = ⇒ π t − s (T ot U Θ X • ) , (see [BK, X, 6.1 & 7.2]) giv es (6.9) (tho ug h it do es not necess arily conv erge!). Finally , (6.1 0) follows from (5.12). (2) F or C = S ∗ w e can use the usual diagonal a nd T ot and the original spectral sequence s for (co)simplicial spaces. F or (6.1 0), consider the cosimplicial Γ-space E • := Hom Γ- S ∗ ( V • , G ): D eﬁnition (5.8) of Hom Γ- S ∗ in terms of the simplicial function complex map Γ- S ∗ sho ws that T ot E • ∼ = Hom Γ- S ∗ (diag V • , G ) again, b y (5.12).  With this at hand, w e can describe sev eral sp ectral sequences connecting the v arious functors we ha v e deﬁned so fa r. First, a univ ersal co eﬃcien ts theorem fo r cohomology: 6.12. Theorem. L et hC , M , V , Φ , A Φ i b e a c ompl e te c o h omolo gic al setting, and let G b e a Φ -algebr a in C . Then for any Y ∈ C ther e is a natur al c ohomolo gic al sp e ctr al se quenc e with E s,t 2 ∼ = Ext s,t ( H ∗ Y , G ) = ⇒ H t − s ( Y ; G ) , wher e Ext s,t ( C , G ) := ( L s ¯ T ( C )) t for any C ∈ (Φ - C ) - Π - A l g , and T := Hom( − , G ) . Pr o o f . Let Z → Y b e a coﬁbrant replacemen t in C , and assume G is ﬁbra n t. W e use M Φ := { A Φ M } M ∈M as mo dels in Φ- C ( § 2.12), with T n as the suspension ( § 2.1), to deﬁne the resolution mo del category structure o n s Φ- C . As in the pro of o f [BS, Theorem 4.2], let V • → A Φ Z b e a free simplic ial resolution in s Φ- C , so that b y (6.8) the nat ur a l map diag V • → A Φ Z is a w eak equiv alence. 24 DA VID BLANC If w e set E • := Ho m( V • , G ) (a cosimplicial Φ-algebra in C ), then b y (6.10) and (2.14): T ot E • = Hom(diag V • , G ) ≃ Hom( A Φ Z , G ) ∼ = map( Z , G ) = L map( − , G )( Y ) ao π M Φ ,t − s (T ot E • ) = π M Φ ,t − s map( Z , G ) = H t − s ( Y ; G ) by Deﬁnition 2.15. On the ot her hand, since eac h V n is coﬁbrant: π M Φ , ∗ E n = π M Φ , ∗ Hom ( V n , G ) = ¯ T ( π M Φ , ∗ V n ) and since V • → A Φ Z is a coﬁbrant replacemen t, π M Φ , ∗ V • → π M Φ , ∗ A Φ Z =: H ∗ Y is a free resolution in (Φ- C )-Π- A l g , so: π s π M Φ , ∗ E • = π s ( ¯ T ( π M Φ , ∗ V • )) = π s L ¯ T ( H ∗ Y ) = L s ¯ T ( H ∗ Y ) , as claimed.  Note that for generalized cohomology of spaces this ta k es the familar form (cf. [Ad] and [EKMM, IV, § 4]): 6.13. Corollary . F or any sp e cial G ∈ Γ - S ∗ and K ∈ S ∗ ther e is a se c ond quadr ant sp e ctr al se quenc e with: E s, ∗ 2 ∼ = Ext s π - M od ( π S ∗ K, G ) = ⇒ H s − t ( K ; G ) . There is also a v ersion f o r homolog y: 6.14. Proposit ion. L et h C , M , V , Φ , A Φ i b e a c omplete c ohom olo gic al setting, and let G b e a Φ -algeb r a in C . The n fo r any Y ∈ C ther e is a natur al ﬁrs t q uad r an t sp e ctr al se quenc e with (6.15) E 2 s,t ∼ = T or s,t ( H ∗ Y , G ) = ⇒ H t + s ( Y ; G ) , wher e T or s, ∗ ( C , G ) := ( L s ¯ T ( C )) for any C ∈ (Φ - C ) - Π - A lg , and T := − ⊗ G . Pr o o f . This g eneralization o f [BS, Theorem 4.4] for the comp osite f unctor: Π M A Φ − − → Φ- C −⊗ G − − − → Φ- C is prov en like Theorem 6.12, with (6.8) replacing (6.9).  F or generalized homology this tak es t he form: 6.16. Corollary . F or any sp e cial G ∈ Γ - S ∗ and K ∈ S ∗ ther e is a natur al ﬁrst quadr ant sp e ctr al se quenc e w ith: E 2 s,t ∼ = T or π - M od s,t ( π S ∗ K, G ) = ⇒ H t + s ( K ; G ) . Finally , w e ha v e the follow ing t wo generalizatio ns of [B1]: 6.17. Theorem. L et hC , M , V , Φ , A Φ i b e a c ompl e te c o h omolo gic al setting, and let G b e a Φ -algeb r a in C . The n fo r any Y ∈ C ther e is a natur al ﬁrs t q uad r an t sp e ctr al se quenc e with E 2 s,t ∼ = L s ¯ T ( π M , ∗ Y ) t = ⇒ H t + s ( Y ; G ) , wher e wher e T := A Φ ( − ) ⊗ G . GENERALIZED AND R ´ E-QUILLEN COHOMOLOGY 25 Pr o o f . Similar to the pro of o f Theorem 6.12, except that here w e start with a f r ee simplicial resolution V • → Y in s C , and note that in this case π M , ∗ V • → π M , ∗ Y is a free simplicial resolution in the category s C -Π- A l g .  In [Se1, Prop. 5 .1], Segal pro duced a stable v ersion of this sp ectral sequence for any generalized homology theory k ∗ (con verging strongly to k ∗ X if k ∗ is connectiv e). 6.18. Theorem. F o r Y and G as ab ov e , ther e is a natur al se c ond quadr ant sp e ctr al se quenc e with: E s,t 2 ∼ = d E xt s t ( π M , ∗ Y , G ) = ⇒ H t − s ( X ; G ) , wher e d E xt s ( − , G ) := L s ¯ T for T := map C ( − , G ) . 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