NEW MODEL CATEGORIES FROM OLD DAVID BLANC Revised: January 25, 1995 Abstract.We review Quillen's concept of a model category as the proper se* *tting for defining derived functors in non-abelian settings, explain how one ca* *n transport a model structure from one category to another by mean of adjoint functor* *s (un- der suitable assumptions), and define such structures for categories of c* *osimplicial coalgebras. 1.Introduction Model categories, first introduced by Quillen in [Q1 ], have proved useful in* * a num- ber of areas - most notably in his treatment of rational homotopy in [Q2 ], a* *nd in defining homology and other derived functors in non-abelian categories (see * *[Q3 ]; also [BoF , BlS, DwHK , DwK , DwS , Goe, ScV]). From a homotopy theorist's po* *int of view, one interesting example of such non-abelian derived functors is the E2* *-term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identi* *fy this E2-term as a sort of Ext in the category CA of unstable coalgebras over * *the mod p Steenrod algebra (see x7.4). The original purpose of this note was to provide an element in this identific* *ation which appears to be missing from the literature: namely, an explicit model cat* *egory structure for the category cCA of cosimplicial coalgebras as above. What one would really like is a model category for arbitrary categories of cosimplicial * *universal coalgebras, analogous to Quillen's treatment of simplicial universal algebras i* *n [Q1 , II, x4]. This treatment is based on Quillen's "small object argument" (see Proposi* *tion 4.8 below), and implicitly uses a procedure for transfering model category stru* *ctures by means of adjoint functors (in the direction of the left adjoint). The proce* *dure is made explicit in Theorem 4.15 below. ___________ 1991 Mathematics Subject Classification. Primary: 18G55; Secondary:55U35. Key words and phrases. model categories, homotopical algebra, derived functor* *s, adjoint functors, cosimplicial coalgebras. I would like to thank the Department of Mathematics at the Hebrew University * *of Jerusalem for its hospitality during the period when this paper was written 1 2 DAVID BLANC Unfortunately, Quillen's procedure cannot be dualized, in the categorical sen* *se. The reason is essentially set-theoretic: more can be said about maps into a seq* *uential colimit of sets than about maps out of a sequential limit (and thus, for exampl* *e, colim is exact, for R-modules, while lim is not). Therefore, for our purposes we describe, in Theorem 4.14, alternative (and le* *ss elegant) conditions for using adjoint functors to create new model category str* *uctures. The dual version, Theorem 7.6, then allows us to define model category structur* *es for certain categories of cosimplicial universal coalgebras - including cCA * * (see Proposition 7.7). 1.1. notation and conventions. For any category C, we denote by grC the category of non-negatively graded objects over C, by gr+C the category of posit* *ively graded objects, by sC the category of simplicial objects over C (cf. [May , x2]* *), and by cC the category of cosimplicial objects over C. For an abelian category M* *, we let c*M denote the category of chain complexes over M (in non-negative degrees); similarly c*M is the category of cochain complexes. The category of sets will be denoted by Set, that of topological spaces by * * Top, that of groups by Gp, and that of simplicial sets by S (rather than sSet). F* *or any ring R, the categories of left (respectively, right) R-modules are denoted by * *R-Mod (resp. Mod-R). Fp denotes the field with p elements. We have tried to be consi* *stent in using A for a category of universal algebras (x3.6 below), B for a category* * of universal coalgebras (x7.3), and M for an abelian category. Throughout we shall use "dual" to refer to the categorical dual (cf. [Mac1 , * *II,x1]); other duals (such as the vector space dual) will be called by other names (e.g.* *, "con- jugate"). For any functor F : I ! C we denote the (inverse) limit of F simply by limF or limIF , (rather than lim ), and the colimit (i.e., direct limit) by col* *imF . In particular, sequential limits of type are limits indexed by an (infinite) ordi* *nal : lim< X , and similarly for colimits. An initial object (in any category C) wi* *ll be denoted by *I, and a terminal object by *T. 1.2. organization. In section 2 we review the definition of model categories and some related concepts, as well as their relevance to derived functors. In sect* *ion 3 we make explicit the relation between adjoint functors and limits, and in secti* *on 4 we explain their relation to defining new model category structures. In sectio* *ns 5 & 6 respectively we discuss simplicial and cosimplicial objects over abelian cate* *gories. Finally, in section 7 we describe the "universal coalgebras" we are interested * *in, and apply our results to define a model category structure on such categories of cosimplicial coalgebras. Acknowledgements 1.3.I would like to thank the referee for many useful comments, and in particular for suggesting Theorem 4.15 in its present generality. NEW MODEL CATEGORIES FROM OLD 3 I understand that in [CaG ], Cabello and Garzon have also given conditions for defining model category structures by means of adjoint functors. 2. Model categories We begin with an exposition of Quillen's theory of model categories, in a form suited to our (algebraic) purposes: Definition 2.1.A class W of morphisms in a category C will be called a class of quasi-isomorphisms if there is a functor fl : C ! D such that f 2 W , fl(f* *) is an isomorphism in D. Definition 2.2.A map f : X ! Y is called a retract of a map g : K ! L if there are maps k; `; r; s making the following diagram commute: idX X ________________-XXX: | kXXz K r | | | | | | |f |g |f | |? | | | |? ` : LX X Xs |? Y ________________-YXz idY Figure 1 Note that any class of quasi-isomorphisms is closed under retracts (i.e., g 2* * W ) f 2 W in Figure 1). 2.3. axioms for model categories. Let C be a category with three distinguished classes of morphisms: W, C, and F. Consider the following two axioms: Axiom 1. For any morphism f : A ! B in C p (i) there is a factorization A -i! C -! B (f = p O i) with i 2 C \ W and p 2 F. 0 p0 (ii) Moreover, if also f = p0Oi0 with i02 C\W and p02 F for A -i! C0- ! B, then there is a map h : C ! C0 making the following diagram commute: * CHp p i pppHH ppp H Hj AH ppph B H H ppp * i0H Hj|0? p0 C Figure 2 4 DAVID BLANC Note that if W is a class of quasi-isomorphisms, necessarily h 2 W - so that* * (ii) says the factorization in (i) is unique up to quasi-isomorphism. Axiom 2. For any morphism f : A ! B in C p (i) there is a factorization A -i! C -! B with i 2 C and p 2 F \ W. (ii) If f = p0O i0 is another such factorization, there is an h making the d* *iagram in Figure 2 commute. Definition 2.4.Let C be a category and W, C and F classes of morphisms in C. Assume that W is a class of quasi-isomorphisms and C and F are each closed under compositions. Then (i) If C has all finite limits, and satisfy Axiom 1, we call t* *his a right model category (RMC) structure on C, or say that C is an RMC. (ii) If C has all finite colimits, and satisfy Axiom 2, we call * *this a left model category (LMC) structure on C. (iii) If both hold, is called a model category. Remark 2.5. In order to "do homotopy theory" in C one requires the full force o* *f a model category; in fact, it is often convenient to have additional structure, s* *uch as simplicial Hom___-objects (cf. [Q1 , II, x1]), properness (cf. [BoF , Def. 1.2* *]), and so on (see [Bau , I] and [He , II] for more general treatments). However, for the pu* *rposes of "homotopical algebra" - i.e., homological algebra in non-abelian categories * *- it is enough to have an RMC or an LMC (see xx2.14-2.16 below). Example 2.6. The original motivating example of a model category is the catego* *ry Top of topological spaces, with W the class of homotopy equivalences, C the c* *lass of cofibrations, and F the class of (Hurewicz) fibrations (cf. [St]). An alter* *native model category structure on Top is given in [Q1 , II,x3]. However, for our purposes the basic example of a model category will be the c* *at- egory S of simplicial sets, with W the class of weak equivalences (maps inducin* *g an isomorphism in ss*(-)), C the class of one-to-one maps, and F the class of K* *an fibrations (cf. [May , x7]). See sections 5 & 6 below for further examples. Remark 2.7. Given satisfying axioms 1 and 2, in order for W to b* *e a class of quasi-isomorphisms it suffices that: (a) W include all isomorphisms, (b) W be closed under retracts, and f g (c) Given A -! B -! C with two out of {f; g; g O f} in W, the third is, t* *oo. In this situation Quillen constructs in [Q1 , I,x1] a localization of C with * *respect to W, which comes with a functor fl : C ! HoC such that f 2 W , fl(f) is an NEW MODEL CATEGORIES FROM OLD 5 isomorphism in HoC. However, in almost all known examples of model categories W is given to begin with as a class of quasi-isomorphisms. Definition 2.8.We call the closure of F under retracts (x2.2) the class of fibr* *ations in C; similarly, the closure of C under retracts is called the class of cofibr* *ations. A fibration which is in W will be called a trivial fibration, and a cofibrati* *on in W will be called a trivial cofibration. Remark 2.9. In Quillen's definition no distinguished subclasses F, C of the c* *lasses of (co)fibrations appear (nor are right or left model categories mentioned). * *But such classes occur naturally in many examples (just as free R-modules form a di* *stin- guished subclass of projective R-modules), and allow a convenient simplificatio* *n of the axioms. There is in fact no loss of generality in our definition, in light of the fol* *lowing facts: Definition 2.10.Given any commutative square ff A ________-X | ppp| | ppp | i| pp |f | ppp | | pp | |?pp fi |? B ________Y- we say that f has the right lifting property (RLP) with respect to i - or equiv* *alently, that i has the left lifting property (LLP) with respect to f - if a dotted arr* *ow exists making the diagram commute. Lemma 2.11. If is an RMC, any fibration in C has the RLP with respect to any trivial cofibration; dually, if is an LMC, any t* *rivial fibration in C has the RLP with respect to any cofibration. Proof.Let C be a right model category, and assume given a diagram as in Figure * *1. First, if f 2 F and i 2 W \ C, one can factor ff and fi using Axiom 1(i): ff A ________________-_-XXX: |@ XXXzX0 p | | @ k pSp| | | pp6 | i|j O@i pppShf0O |fp | @ pp S | | @R0 qS | |? j :B X X XSw |? B ________________-YXz fi Thus we have h0 : B0 ! X0 by Axiom 1(ii) (since C and F are closed under compositions, and (f O p) O k and q O (j O i) are two factorizations of f O ff * *= fi O i), and h = p O h0O j is the required lifting. 6 DAVID BLANC Next, assume that f is a fibration - i.e., a retract of a map g 2 F - so we h* *ave a commutative diagram: idX A __________-ffX______________-XXX: | | XkXzK r | | | | | | | | |i |f |g |f | | |? | | | | |? |? ` : L X XXs |? B __________-fiY______________-YXz idY and by the first case (for g 2 F) there is a lifting k O ff A ________-K | ppp| | ppp | i| hpp |g | ppp | | pp | |?p`pO fi |? B ________L- so r O h : B ! X is the required lifting for the i with respect to the fibrat* *ion f. The case where i is any trivial cofibration is dealt with similarly; and the ca* *se of a left model catgeory is of course dual. ___|_|_ Fact 2.12. ([Q2 , II, Prop. 1.1]). The fibrations of a right model category are* * precisely those morphisms having the RLP with respect to all i 2 C \ W, and conversely, the trivial cofibrations are those morphisms which have the LLP with respect to* * all f 2 F. The cofibrations of a left model category are characterized by having t* *he LLP with respect to all f 2 F \ W, and the trivial fibrations are those morphisms* * which have the RLP with respect to all i 2 C. Proof.If f : A ! B has the RLP with respect to all i 2 C \ W, use Axiom 1 to p factor f as A -i! C -! B with i 2 C \ W; the lifting r : X ! A which exi* *sts by hypothesis shows that f is a retract of p 2 F, so a fibration. Similarly * *for the other cases. ___|_|_ Corollary 2.13. The (trivial) fibrations of an RMC (resp. LMC) are preserved un- der base change, products, and sequential limits - that is, (a) If p : X ! Y is a (trivial) fibration, f : X ! Z is any map, and W is t* *he pushout of (q; f), with structure maps q : Z ! W , g : Y ! W , then q is a (trivial) fibration. Q Q Q (b) If {pff: Xff! Yff}ff2Aare all (trivial) fibrations, so is ffpff: ffXff! * * ffYff. (c) If (p : X+1 ! X )< is a sequence of (trivial) fibrations, the map q :* * X = lim< X ! X is a (trivial) fibration for each < . NEW MODEL CATEGORIES FROM OLD 7 Similarly, the (trivial) cofibrations of a LMC (resp., RMC) are preserved und* *er cobase change, coproducts, and sequential colimits. Proof.The constructions (a)-(c) all preserve the lifting property with respect * *to any (fixed) map. ___|_|_ We now recall how model categories are used to define derived functors in non- abelian categories. Let be a model category. Definition 2.14.The homotopy category HoX of any model category X is obtained from it by localizing with respect to the weak equivalences, with fl : X ! HoX the localization functor. Quillen shows that HoX is equivalent to the catego* *ry ss(Xcf), whose objects are those objects X 2 X for both fibrant (i.e., X ! * **T is a fibration) and cofibrant (i.e., *I ! X is a cofibration), and whose morphism* *s are homotopy classes of maps (cf. [Q1 , I, x1]). Under this equivalence of HoX and ss(Xcf), the localization functor is determ* *ined by the choice, for each object X 2 X , of a cofibrant and fibrant object A wi* *th a weak equivalence A ! X. This is called a resolution of X, and all such are homotopy equivalent. However, we can sometimes make do with less: Definition 2.15.If H : X ! Y is a functor between model categories which preserves weak equivalences between cofibrant objects, the total left derived f* *unctor of H is the functor L H = "HO fl : X ! HoY, where H" : HoX ! HoY is induced by H on Xc (the subcategory of cofibrant objects in X ). Remark 2.16. In fact we need only a left model category structure on X in order* * for L to be defined. Of course right derived functors are defined analogously in an* *y right model category. In the particular case where Y = sC is a category of simplicial objects over* * some concrete category C, the usual n-th derived functor of any T : X ! sC, denoted LnT , assigns to an object X 2 C the object (LnT )X = ssn(L T )X = ssnT A, whe* *re A ! X is any resolution. If also X = sD for some D, and T : C ! D is prolonged to a functor sT : sC ! sD (by applying it dimensionwise to simplicial objects), then for C 2 C we * *have (LnT )C = ssn(sT )Ao, where Ao is a resolution of the constant simplicial ob* *ject which is equal to C in each dimension. When T is an additive functor between abelian categories with enough projectives, this reduces to the usual definitio* *n of derived functors (see also [Bo2 , x7], [DoP ], [EM2 ], & [Hu ]). We have avoided the question of when a functor will in fact preserve weak equ* *iv- alences between cofibrant objects. This depends on the specific model categori* *es in question (see Remark 7.8 below). 8 DAVID BLANC 3. adjoint functors and limits We next recall some general facts about limits and adjoint functors: Let C U* *FD be a pair of adjoint functors (i.e., F is (left) adjoint to U), with the natural a* *djunction isomorphism # : HomC(F D; C) ~= HomD(D; UC). We denote #-1(idUC ) : F UD ! D by "D. Remark 3.1. It is not hard to see that U preserves all limits which exist in C,* * and dually, F preserves all colimits which exist in D (cf. [Mac1 , V, x6]). F evidently preserves projectivity, so if D is a category in which all object* *s are projective (e.g., D = Set) and "D is always an epimorphism, then im(F ) consis* *ts of projectives and "D : F UD ! D is a functorial projective cover. Definition 3.2.Given a diagram S : I ! C, we say that a functor T : C ! D creates the limit limIS in C (cf. [Mac1 , V,1]) if limIT O S exists in D, * *limIS exists in C, and T (limIS) = limI(T S). Similarly for creation of colimits. Definition 3.3.Given adjoint functors C UFD and a diagram S : I ! C, we say the pair (U; F ) produces the colimit colimS 2 C if colimI(US) 2 D exists* *, and colimIS 2 C exists and is obtained as follows: Let L0 = colimI(US), and L1 = colimI(UF US). There are two natural mor- phisms F L1 ! F L0, namely: d0, induced by the natural transformation "FU(* *j) for every j 2 I, and d1, induced by F U("j). We require that colimS be * *the coequalizer (in C) of d0 and d1 (see [L] or [Mac1 , X, x1]; there seems to * *be no accepted name for this procedure). In order for this construction to be of use, we need some information on coeq* *ualizers in C - at least for those which appear here. Such coequalizers, called split* * (or contractible), have a map s : F L0 ! F L1 such that d0s = id, d1sd0 = d1sd1.* * In the cases of interest to us split coequalizers are created by U, so the definit* *ion makes sense (cf. [BaW , 3.3, Prop. 3]). Remark 3.4. Since we know that F preserves all colimits - so that the colimit* * of a diagram in C which factors through F is determined by the corresponding colimit* * in D - in our situation (see x3.6) the left model category structure we shall de* *fine on sC will allow us to identify any colimit in C as the 0-th left derived functor* * (x2.16) of the same colimit defined on the image of F . This is analogous to viewing the usual tensor product of R-modules, say, as t* *he 0-th derived functor of the more naturally defined functor of tensor product of* * free R-modules on specified sets of generators X, Y : Def R R = R: One may dually define "the pair (F; U) produces the limit limS in D", (w* *ith equalizers replacing coequalizers, etc.). NEW MODEL CATEGORIES FROM OLD 9 Example 3.5. Let GpUFSet denote the adjoint "underlying set" and "free group" functors between the categories of groups and sets respectively. Then U create* *s all limits in Gp - i.e., an inverse limit of a diagram of groups is just the corr* *esponding limit for the underlying sets, endowed with a natural group structure. Likewis* *e, the adjoint pair (U; F ) produces all colimits in Gp. For instance, the coproduct* * ("free product") G q H of two groups is obtained by choosing sets of generators X, Y * *for G, H respectively - say, X = U(G), Y = U(H) - and setting G q H = F (X [ Y )= ~ ; where [a][a0] ~ [aa0], [a]-1 ~ [a-1] for a; a0 both in X or both in Y - w* *hich is precisely the coequalizer of the two obvious homomorphisms F (UF U(G) [ UF U(H)) !! F (U(G) [ U(H)): Definition 3.6.Recall that in a category of universal algebras (or variety of a* *lge- bras) the objects are sets X, together with an action of a fixed set of n-ary o* *perators W = [1n=0{! : Xn ! X}, satisfying a set of identities E; the morphisms are functions on the sets which commute with the operators. (We can slightly modify the definition to cover the case where the set X is p* *ointed, non-negatively graded, and so on.) Example 3.7. Categories of universal algebras include: The category Gp of groups; the category R-Mod of left R-modules for any r* *ing R, as well as that of (commutative or associative) algebras over R; the catego* *ry of Fp-algebras over the mod-p Steenrod algebra; and the categories of Lie rings, * *or of restricted Lie algebras over Fp. All the above examples, and many others (though not the categories of monoids, semigroups, and semirings), have another convenient feature: their objects hav* *e the underlying structure of a (possibly graded) group. Remark 3.8. For each category C of universal algebras there is a pair of adjoin* *t func- tors C UFSet, with U(A) the underlying set of A 2 C, and F (X) the free algebra on the set of generators X (cf. [Co , III.5 & IV.2] Thus in particular every category of universal algebras has enough projectives (x3.1). In fact, the functor U : C ! Set is monadic in the sense that the ca* *tegory C can be reconstructed from the monad (=triple) UF : Set ! Set (compare x7.1 below; cf. [BaW , 3.3, Prop. 4] or [Co , VIII.3] for the precise statement). * *Moreover: Proposition 3.9. For any category of universal algebras C, the functor F create* *s all limits and sequential colimits of monomorphisms in C, and the pair (U; F ) pr* *oduces all colimits in C. 10 DAVID BLANC Proof.Just as in example 3.5 above. The statement on colimits is due to Linton ([L]; see also [BaW , 9.3]). ___|_|_ 4. adjoint functors and model categories We now explain how adjoint functors C UFD can be used to transfer an existi* *ng model category structure on D to the category C. Convention 4.1.To simplify the statements of our results, we assume that in all* * (left or right) model categories discussed in this section, all cofibrations are in p* *articular monomorphisms. This need not hold in general (see Proposition 6.4 below), but it will hold in all situations we are interested in. It should be clear from the p* *roofs how the statements must be modified without this assumption. First, we require the following somewhat ad hoc Definition 4.2.We say that a left model category has canonical * * j factorizations of type for Axiom 2 (for some ordinal ) if the factorization X * *-! p Z -! Y of Axiom 2(i) for any f : X ! Y in C is obtained as follows: (a) There is a sequence of commuting diagrams j(0) ___________-j() Z(-1)= X _____-XXZ(0)qqqqZ() Z(+1) qqqq X XXPP P p(0)Q () j(+1) (-1) X XXPP Q p j p p = f X XzPPqQQs jj+ Y for < , such that Z = colim< Z() and p and j are induced by the maps (p())< and (j())< respectively. (b) For each -1, the object Z(+1) is constructed as a pushout: g V ______-Z() | | | | () () i| ___|j_ P O | |PO||__ |? |? W ______-Z(+1) (c) i : V ! W is in turn constructed functorially as a coproduct: ` a iff a V = Vff __________- Wff= W ff2K ff2K where the set of maps {iff}ff2K depends functorially on p() (i.e., this is a fu* *nctor on the comma category of maps in C), and each iff is in CC. NEW MODEL CATEGORIES FROM OLD 11 (d) There is a set of maps hff: Wff! Y (ff 2 K ), also depending functorially on` p(), such that p(+1) : Z(+1) ! Y is induced by p() : Z() ! Y and ( ff2Khff) : W ! Y . (e) For each limit ordinal we have Z() = colim< Z(). Remark 4.3. Note that because each iff is a cofibration, the maps j() : Z() ! Z(+1), as well as the structure maps Z() ! Z (and thus j : X ! Z itself) a* *re cofibrations by Corollary 2.13. Note also that canonical factorization implies in particular functoriality in* * Axiom jf pf 2(i) - that is, any f : X ! Y may be factored X -! Zf -! Y (with jf 2 C and pf 2 F \ W), in such a way that, given maps f0 : X0 ! Y 0, x : X ! X0 and y : Y ! Y 0 such that y O f = f0O x, there is a map z : Zf ! Zf0 such that z O if = if0O x and y O pf = pf0O z. While this functoriality is not pa* *rt of Quillen's original definition, it is a useful property (which is in fact enjoye* *d by almost all model categories). For examples of canonical factorizations, see Example 4.6 and Remark 5.3(ii) * *below. The most common situation is when = !. The above is of course simply a partial axiomatization of Quillen's "small ob* *ject argument" construction (see [Q1 , II, 3.3-3.4]). Were we not interested in a du* *alizable version (see sections 6 and 7 below) - we could have started with a full axioma* *tization of Quillen's construction, as follows: Definition 4.4.If is a left model category, we say that a set* * of cofibrations {ifl: Vfl! Wfl}fl2 is a collection of -compact test cofibrations * *for C if: (a) any map f : X ! Y in C which has the RLP with respect to each ifl(fl 2 ) is a trivial fibration, and (b) the domain Vflof each test cofibration is -compact in C - that is, HomC(Vfl* *; -) commutes with sequential colimits of monomorphisms of type . (When = !, such objects are called (sequentially) small - cf. [Mit, II, x16]). Remark 4.5. Note that if C is a concrete category, any object C is -compact for* * any ordinal of cofinality greater than the cardinality |C| of C (cf. [TZ , 10.5* *1]), so (b) above is automatically satisfied for = (supfl2{|Vfl|})+, the successor c* *ardinal of the supremum of the cardinalities of (the underlying sets of) all objects Vf* *l. (The idea of thus eliminating the requirement of "smallness" in Quillen's constructi* *on is due to Bousfield - cf. [Bo1 , x11].) It will also hold in other cases - for example, any finite simplicial set (* *i.e., one with finitely many non-degenerate simplices), or finitely generated R-module, * *is !-compact. 12 DAVID BLANC Example 4.6. The motivating example of test cofibrations is the model category* * of simplicial sets S (x2.6): by [Q1 , II, x2, Prop. 1], a map f : X ! Y is a tr* *ivial o fibration if it has the RLP with respect to all the cofibrations ik : [k] ! * * [k] o (k 0), where [k] is the standard simplicial k-simplex, and ik : [k] ,!* * [k] is the inclusion of its (k - 1)-skeleton. Definition 4.7.If {ifl: Vfl! Wfl}fl2 is a set of morphisms in some category C, * *the j * * p associated Quillen construction of type is a functorial factorization X -! Z * *-! Y of any morphism f : X ! Y in C, constructed`as in x4.2, where for each < , the set K (in 4.2(c)) is K = ff2AD;ff, with D;ff= the set all commutat* *ive diagrams (d) of the form: gd Vff________-Z() | | () if|f |p (d) , |? hd |? Wff_______Y- with iff in the given collection {ifl}fl2. We set hff= hd in 4.2(d). This is the ingredient needed to make Quillen's small object argument [Q1 , I* *I, 3.4] work: Proposition 4.8. If is a left model category with a collectio* *n of - compact test cofibrations {ifl: Vfl! Wfl}fl2, then the associated Quillen con* *struc- tion of type yields canonical factorizations (which we shall call canonical Qu* *illen factorizations) for Axiom 2. j p Proof.For any f : X ! Y in C, let X -! Z -! Y be the Quillen construction associated to the set of test cofibrations; then j is a cofibration by Corollar* *y 2.13. To see that p is a trivial fibration, we must show that it has the RLP with res* *pect to all test cofibrations ifl: Vfl! Wfl- i.e., we must produce liftings "hfor * *any h, g as below: g Vfl_______-Z | pp | | ppp | if|l "pph |p | ppp | | pp | |?pp h |? Wfl_______B- But by the -compactness of Vfl, any map g : Vfl! Z = colim< Z() factors through ^g: Vff! Z() for some < , and the diagram NEW MODEL CATEGORIES FROM OLD 13 ^g Vfl ________-Z() | | | | () ifl | |p | | | | |? h |? Wfl _______B- is one of the diagrams (d) used to construct Z(+1) in x4.7, so the structure* * map Wfl! Z(+1) defines the required lifting for the original g and h. ___|_|_ Remark 4.9. The same definitions are possible for a right model category - alth* *ough contrary to what one might expect, the construction is not dual to the above: We say that a right model category has canonical factorizatio* *ns if the factorization f = pOj (j 2 C\W, p 2 F) of Axiom 1(i) is obtained functorial* *ly precisely as in Definition 4.2, except that the maps iff of 4.2(c) are require* *d to be trivial cofibrations. Of course, Proposition 4.8 also holds for right model ca* *tegories, with the Quillen construction using test trivial cofibrations. Example 4.10. Let V (n; k) [n] (0 k n) denote the simplicial set generated by all the (n - 1)-dimensional faces of [n], except for the k-th* * one. The inclusions in;k: V (n; k) ,! [n] are the test trivial cofibrations for S * * (cf. [Q1 , II, x2, Prop. 2]), so S has canonical Quillen factorizations (of type !) for Ax* *iom 1, too. Definition 4.11.Let C UFD be adjoint functors, and D a left model category with canonical factorizations (of type ) as in Definition 4.2. The derived factoriz* *ation q () of any f : A ! B in C is A -i! C -! B, where C = colim< C , and the maps i : A ! C and q : C ! B, are obtained from a commutative diagram i(0) (0) () ___________-i()(+1) A _____-XXCqqqq C C qqqq XX XPPP q(0)Q () j(+1) f X X XPP Q q jq X XzPPqQQs jj+ B defined as follows: apply the construction in D to the map U(f) : U(A) ! U(B) to get the pushout diagram P O(-1)of x4.2; then C(0)is the pushout of the adjoi* *nt diagram "P O(-1): 14 DAVID BLANC ^g F V ______-A | | | | (0) " (-1) F i | ___|i_ P O | |PO||__ |? |? F W _____-C(0) in which ^g: F (V ) ! A is the adjoint of g : V ! UA, and q(0): C(0)! B is induced by f and the adjoint ^h: F (W ) ! B of the coproduct of the maps hff: Wff! UB (see x4.2). More generally, for each < , since the diagram P O() depends functorially only on the map Up() : UC() ! UB, we may define its adjoint "P O()precisely as for P O(-1), and let C(+1) be its pushout. Setting C() = colim< C() * *for each limit ordinal < completes the construction. Note that if the pair (U; F ) produces the colimits in C (x3.3), applying U* * to the above factorization yields, for any map of the form U' : UA ! UB, a construction Up of UA -Ui!UC -! UB in D which can be described purely in terms of colimits in D and the monad (or triple) UF : D ! D (which in fact determines the category C - see [Mac1 , VI, x2]). Example 4.12. If A UFS are adjoint functors, with A a category of universal algebras (x3.6), the derived factorization A ! C ! B of any ' : A ! B in A * *is constructed as in [Q1 , II, x4, Prop. 3]. Definition 4.13.Let be a left model category with canonical factorizations, and C UFD a pair of adjoint functors. We say that the pair * *(U; F ) creates a left model category structure if (i) f 2 FC , Uf 2 FD; (ii) f 2 WC , Uf 2 WD; (iii) CC = {i = i1O : :O:in | each ik is the first factor of some derived facto* *rization}. Theorem 4.14. Let be a left model category with canonical fa* *c- torizations of type , and C UFD a pair of adjoint functors. Assume that U cr* *eates sequential colimits of monomorphisms of type , that C has all finite colimits, * *and that p (*) the derived factorization A -!i C - ! B for any f in D satisfies Up 2 FD \ WD. Then (U; F ) creates a left model category structure with canonical factorizat* *ions. Proof.For Axiom 2(i) use the derived factorization of x4.11. For Axiom 2(ii), it suffices to show that any i : A ! C constructed by the derived factorization * *of some f : A ! B has the LLP with respect to any trivial fibration in C; but if NEW MODEL CATEGORIES FROM OLD 15 j : V ! W is a cofibration in D, then F j : F V ! F W has the LLP with respe* *ct to any trivial fibration in C, by Def. 4.13(i)-(ii). We then see that the i's * *constructed in x4.11 have the LLP by Corollary 2.13. ___|_|_ Hypothesis (*) of Theorem 4.14 may seem hard to verify, but it seems unavoida* *ble for our purpose (that is, for categorical dualizing). However, for other purpos* *es the following version of the theorem, using the Quillen construction, may be more u* *seful: Theorem 4.15. Let be a left model category with a set {ifl}fl2* * of -compact test cofibrations (and thus canonical Quillen factorizations), let C U* *FD be a pair of adjoint functors, and assume that U creates sequential colimits of mo* *nomor- phisms of type , and that C has all finite colimits. Then (U; F ) create a le* *ft model category structure on C with a set {F ifl}fl2 of -compact test cofibrations (an* *d thus canonical Quillen factorizations). Proof.Since U creates sequential colimits of monomorphisms of type , its left a* *djoint F preserves -compactness, and by Definition 4.13, F preserves the property of b* *eing test cofibrations. Thus {F ifl}fl2 is a set of -compact test cofibrations for * *C. Alternatively, one could show directly (as in the proof of Proposition 4.8) t* *hat for p any derived factorization A -i! C -! B, the map Up : UC ! UB has the RLP with respect to all test cofibrations - and then apply Theorem 4.14. Of cour* *se, the derived factorizations are just the canonical Quillen factorizations with r* *espect to the new set of test cofibrations. ___|_|_ Again, the analogous theorem holds for right model categories, so that in fac* *t we have a way to transport full model category structures using adjoint functors. Example 4.16. When C = sA for some category of universal algebras A, the functors sA UFS allow us to transfer the left model category structure of S t* *o sA, by Theorem 4.15. This yields the left model category structure on C described * *in [Q1 , II, x4], in which CC is the class of free maps of simplicial algebras (ib* *id.). This is actually a full model category structure on sA, with the derived factoriza* *tion obtained from the construction of x4.10 serving for Axiom 1(i). 5. simplicial object over abelian categories If M is any abelian category, there are adjoint functors sM NKc*M which are equivalences between the categories of (respectively) the simplicial objects an* *d the chain complexes over M (cf. [Do , Thm 1.9]). In order to define a model category structure on sM it thus suffices to do so for the more familiar category of c* *hain complexes, as Quillen does in [Q1 , II, 4.11-4.12]: Definition 5.1.Let M be an abelian category. Define Wc*M to be the class of homology isomorphisms, Fc*M the class of maps f : A* ! B* which are surjective in positive degrees, and Cc*M the class of one-to-one maps whose cok* *ernel 16 DAVID BLANC is projective in each dimension (if M = R-Mod, we may require the cokernel to be dimensionwise free). Proposition 5.2. If M is an abelian category with enough projectives, then is a model category. Proof.For notational simplicity we consider the case where every A* 2 c*M has* * a functorial projective cover "A*: F A* ! A*. For Axiom 1(i), let L : grM ! c*M denote the left adjoint of the forgetful functor V : c*M ! grM, with natural transformation #LB: LV B* ! B*, and use the factorization: iA* (f; #LBO "LV B*) A* _____-A*q F LV (B*)________________-B* For Axiom 2(i), we wish to construct a sequence of commuting diagrams: j(0) (0) (n-1)__________-j(n)(n) A*_____-XC*qqqq C* C* qqqq X X XP P (0)Q (n-1) j f X X XPPp Q p j p(n) X XPXzPPqQQs jj+ B* where (i) each j(n) is a cofibration; (ii) each p(n) is a fibration; (iii) p(n) induces an epimorphism p(n)*: HiC(n)*!! HiB* for all i; (iv)(n) p(n)*: HiC(n)*! HiB* is a monomorphism for i < n. and then set C* = colimC(n)*. j(0) (0)p(0) To get a factorization A* -! C* -! B* of the given f satisfying conditio* *ns (i), (ii), & (iii), let T : grM ! c*M be the left adjoint of the functor Z : Def (0) c*M ! grM defined Z(A*)n = ZAn = Ker{@n : An ! An-1}, and set C* = A* q F T Z(B*) q F LV (B*). For the inductive step, assume given f : A* ! B* satisfying conditions (ii* *), j p (iii), & (iv)(n) above; we wish to construct a factorization A* -! C* -! B* o* *f f satisfying conditions (i)-(iv)(n+1): i q Let Kn = Ker{fn : An ! Bn} \ ZAn,! An and Kn!! Qn = Kn=(Im{@n+1\ Kn), and let L denote the set of liftings : F Qn ! Kn in FqQn qq || qq ||"Qn q || |?|? Kn ______-_-qQn : NEW MODEL CATEGORIES FROM OLD 17 ` Let E* equal A* in degrees n, En+1 = 2L (F Qn) , and Ei= 0 for i > n+1, with @En+1: En+1 ! An equal to i O on (F Qn) . We have a map g : E* ! B* equal to f in dimensions n and 0 elsewhere, and define C* to be the pushout * *of A* - onA* ,! E* (where onA* is A* truncated above degree n). The lifting properties of Axioms 1(ii) & 2(ii) follow from those of projectiv* *e objects in M in a straightforward manner. ___|_|_ Remark 5.3. We have given the proof because we have not seen it elsewhere; fur- thermore, it provides an illustration of the various types of model categories * *and factorizations which may occur: (i) If M has enough projectives, we merely get a model category structure on c** *M, as stated. (ii) If M has functorial projective covers, the construction given in the proof* * shows the (left and right) model categories c*M have canonical factorizations as in* * Def. 4.2. Thus if C UFM are adjoint functors satisfying suitable hypotheses, then* * the (left) model category structure for c*M, and thus on sM, can be used to defi* *ne a (left) model category structure on sC (in addition to the existence of suit* *able colimits in C, we require that UF X be projective in M for any X 2 M). (iii) Of course, if M = R-Mod then c*M ~=sM has canonical Quillen factoriza- tions - by 4.16, since then M is a category of universal algebras. Note however that for categories of universal algebras over R-Mod (in which the objects have the underlying structure of a R-module, and all operations are R-linear), the construction given in the proof above, combined with Theorem 4.1* *4, yields a simpler description of the factorization of Axiom 2 - and thus of "pro* *jective resolutions" - than that provided for arbitrary universal algebras by Theorem 4* *.15 and x4.16. (iv) The situation is of course greatly simplified when all objects in M are pr* *ojec- tive, (e.g., for M = F-mod where F is a field), since then the fibrations are* * just epimorphims. In that case, if we let ^B*denote the (shifted) cone on B* - i* *.e., B^n= Bn+1Bn, with @^n(b; b0) = (@Bn+1b0-b; @Bnb) - and p the projection B^*!! B* **, then a functorial factorization of f for Axiom 2(i) is then given by: (id;0) (f;p) A* -! A* ^B*!! B*: Nevertheless, this is not a canonical factorization in the sense of Def. 4.2,* * so it will not be suitable for the purposes of Theorem 4.14. 6. cosimplicial object over abelian categories The definitions and results of section 5 are readily dualized to cosimplicial* * objects, as follows: 18 DAVID BLANC Definition 6.1.Recall that a cosimplicial object Xo over any category C is a sequence of objects X0; X1; : :;:Xn; : :i:n C equipped with coface and codegene* *racy maps di: Xn ! Xn+1, sj : Xn+1 ! Xn (0 i; j n) satisfying the cosimplicial identities (cf. [BoK1 , X, x2.1]). We denote the category of cosimplical objects over C by cC. If M is an abel* *ian category, we denote by c*M the category of cochain complexes over M. Dual to [Do , Thm. 1.9] (noted in the beginning of section 5) we have: Proposition 6.2. For any abelian category M there is a natural isomorphism of categories cM ~=c*M. Proof.GivenTCo 2 cM, the functor N : cM ! c*M is defined by Nn = (NCo)n = n-1 j n n-1 n n n+1 P n i i j=0Ker{s : C ! C }, with ffi : N ! N equal to i=0(-1) (d |\n-1j=0K* *ersj). Given A* 2 c*M, the inverse functor L : c*M ! cM is defined LA* = Co, where Co may be described explicitly in a manner dual to [May , p. 95] or [* *Bl, 5.2.1]: For each n 0 and 0 n, let In denote the set of all sequences I = (i1; : :;:i ) of |I| = integers such that 0 i1 < i2 < : :<:i n; l* *et sI = si1O . .O.si be the corresponding -fold codegeneracy map. (We allow = 0, with the corresponding s; = 0). Then Def Y Y n- (6.3) Cn = A(I): 0n I2In We write ss(I): Cn!!An-|I|(I)for the projection onto the copy of An-|I|indexe* *d by I. For each 0 n and 0 k n - 1 there is a one-to-one function skn: I-1n-1! In, where skn(I) = J is defined by the requirement that sI O s* *k = sJ under the cosimplicial identities. The codegeneracy map sk : Cn ! Cn-1 is then defined to be the composite: Y Y (skn)-1* Y Y n- Cn ____-_- An-(sknI)______- A(I) = Cn-1 : 1n I2im(skn)In 0n-1 I2In-1 The coface map dj : Cn ! Cn+1 is determined by the requirement that ss(;)Od0 = ffin : An ! An+1 and ss(;)O dj = 0 for j > 0, and by the cosimplicial ident* *ities - that is, given J 2 In+1, use the cosimplicial identities to write sJ O dj* * = OE O SI (where |I| = |J| + ffl - 1, and either OE = id, ffl = 0 or OE = di, ffl * *= 1). Then ss(J)O dj : Cn ! An+1-|J|(J)is the composite ss(I)n-|I| n-|I| OE n-|I|+ffl n+1-|J| Cn -! A(I) ~=A(;) -! A(;) ~=A(J) : NEW MODEL CATEGORIES FROM OLD 19 ___ |_|_ One thus has a model category structure on cM, induced by the following dual of Proposition 5.2: Proposition 6.4. If M is an abelian category with enough injectives, there is a model category structure on c*M with Wc*M the class of cohomology isomorphisms, Cc*M the maps which are one-to-one in positive degrees, and Fc*M the surject* *ive maps with injective kernel. Proof.The proof is precisely dual to the case of chain complexes. For convenien* *ce of reference below we briefly recapitulate the factorization for Axiom 1(i): p * Given f : A* ! B* in c*M, we want A* -i!C* -! B with i 2 Cc*M\Wc*M and p 2 Fc*M (f = p O i), again under the assumption that every A* 2 c*M has a functorial injective envelope "A* : A* ,! IA*. As in x4.2 we wish to constru* *ct a sequence of commuting diagrams: A*X X |HH X X XX j(n) j(n-|1) HHj(0) X XX X f ||? H HHj XX X Xz qqqqC* ss_________-* qqqq * ____________-* (n) p(n) C(n-1) C(0) p(0) B where each p(n) is a fibration, and each j(n) is a cofibration which is monic* * in cohomology (in all dimensions), and epic in cohomology through dimension n - 1. We then set C* = limC*(n). In this case the forgetful functor V : c*M ! grM has a right adjoint R : grM ! c*M, and the functor C : c*M ! grM, defined by: CA*n= Coker(ffin-1A), has a Def * * * right adjoint T : grM ! c*M. Thus if we set C*(0)= B x IT C(A ) x IRV (A ), we find that the map j(0): A* ,! C*(0)is a cofibration which is monic in cohomo* *logy, and the projection ssB* : C*(0)!! B* is a fibration. For the inductive step, assume given a cofibration f : A* ! B* which is monic* * in in cohomology, and epic in cohomology through dimension n-1. Let P n= ZnB[An ,! Bn) and Qn = P n=(An [ Im(ffin-1B)), with tn : P n! IQn the obvious compos* *ite map. Let N denote the set of extensions : Bn ! IQn ( O in = tn), and define the cochain complex E* toQbe equal to B* in dimensions n, zero above dimension n + 1, with En+1 = 2N IQn() and ffin+1E: Bn ! En+1 determined by the 's. Finally let C* be the pullback of B* ! onB* E*, (where onB* is again the j * p * truncated complex), so Ci = Bi for i n. The obvious maps A* -! C -! B give a factorization with j a cofibration which is monic in cohomology, and epi* *c in cohomology through dimension n. 20 DAVID BLANC Note that since the maps p(n)are isomorphisms in degrees n, there is no lim1 in calculating HiC* = Hi(limC*(n)) (cf. [Mil]). This problem did not arise i* *n the dual case (x4.2 and Proposition 5.2), since colim is exact. ___|_|_ Note that M = R-Mod has functorial injective envelopes (constructed as in [Mac2 , III, 7.4]). The construction given here is of course an example of dual* * canonical factorizations (defined dually to Def. 4.2). Remark 6.5. The explicit factorization of any f : Ao ! Bo in the category cM of cosimplicial objects over M is easily obtained from the proof of Proposition* * 6.4 using Proposition 6.2. It should be pointed out that in place of the truncated* * cochain complex onA*, which vanished above dimension n, we must use the n-th coskeleton Def n o csknAo = L(o NA ), which is defined in cosimplicial dimensions > n by (6.3) (compare the dual description in [Bl, 5.3.4]). Similarly for the construction * *of E* from onA* (cf. [Bl, 5.3.2]). 7. Cosimplicial coalegbras In order to define right derived functors over a category of coalgebras, one * *would obviously like to dualize the constructions of section 4. However, there seems * *to be no reasonable (right) model category structure on the category of cosimplicial set* *s. Thus our approach here is more restricted. First, we recall the definition of a cat* *egory of coalgebras: Definition 7.1.(i) A comonad (or cotriple) S on a category C consists of a func* *tor S : C ! C equipped with two natural transformations: " : S ! idC and : S ! S2 satisfying: S" O = "S O = idC and S O = S O : S3 ! S (cf. [EM1 , x2]). (ii) A coalgebra over a comonad S = is an object C 2 C togeth* *er with a morphism ' : C ! SC such that S' O "C = idC and S' O ' = O ' : C ! S3C. The category of coalgebras over S (with the obvious morphisms) will be denoted by CS. Remark 7.2. For every comonad S = there is a pair of adjoint functors C VGCS such that S = V G (and conversely, every pair of adjoint functo* *rs yield a comonad). V : CS ! C is the faithful "underlying C-object" functor, a* *nd GC = (C; ). The relation between the categorical definition and the more concrete analogu* *e of Definition 3.6 is more problematic, since we need the underlying C-object to be* * an object in an abelian category, and not just a set. (This is in order to make us* *e of the model category structure on cM defined in section 6, since we do not have one* * on cSet, as noted above). Thus we specialize to the case where C is a monoidal ab* *elian category (see [Mac1 , VII,1] for the definition; the only example we s* *hall actually need being M = R-Mod and = - R -): NEW MODEL CATEGORIES FROM OLD 21 Definition 7.3.A category of universal coalgebras over is a category B, whose objects are objects A 2 M, together with an action of a fixed set of n-ar* *y co- operators W = [1n=0{! : A ! An }, satisfying a set of identities E; the morphis* *ms are functions on the sets which commute with the co-operators. Example 7.4. Categories of universal coalgebras include: The category CR of coalgebras over a ring R ([Sw , 1.0]) ; the category * *CCR of cocommutative coalgebras over R ([Sw , 3.2]) ; and for each prime p, the cate* *gory CAp of (graded) unstable coalgebras over the mod p Steenrod algebra (see [BoK2* * , x11.3]). More generally, let A be a category of universal algebras (see x3.7), in which U : A ! Set factors through U0 : A ! R-Mod, for some ring R. We may then define a conjugate category A? of universal coalgebras as follows: Since the n-ary operators of A are in one-to-one correspondence with the elem* *ents of the set UF Xn, where Xn is a set with n elements, we let An = U0F Xn, * *and define the n-ary co-operators of A? to be the elements of the R-module conjuga* *te Def (or R-dual): A?n = HomR(An; R) 2 Mod-R. We assume that An is a finitely generated R-module for each n (of finite type, if R is a graded ring). The relations among the co-operators correspond to the elements of HomR(A?n; * *A?m), just as the relations in A are determined by HomSet(UF Xn; UF Xm ). In some cases the functor G : M ! B, which is right adjoint to the "underly* *ing object" functor V : B ! M, has a description as a "cofree coalgebra" functor. This is true for B = CF, where F is a field; see [Sw , 6.4.1] for an explicit d* *escription. Similarly for B = CCF (see [Sw , 6.4.1,6.4.4]). For B = CAp, we have 1Y GX* = H*( K(Xn; n); Fp) n=1 where M = gr+Fp-Mod (see [BoK2 , 11.4]). It is clear that Proposition 6.2 and Theorem 4.14 (as well as their proofs, * *and xx4.1, 4.2, 4.7, 4.11, and 4.13) may be dualized to yield: Proposition 7.5. For any category of universal coalgebras B over M = R-Mod, the functor G creates all colimits and sequential colimits of epimorphisms in B* *, and the pair (G; V ) produces all limits in B. Theorem 7.6. Let be a right model category with dual canonic* *al factorizations of type , and C GVD a pair of adjoint functors. Assume that V creates sequential limits of epimorphisms of type , that D has all finite limi* *ts, and that p (*) the derived factorization A -i! C -! B for any f in D satisfies V i 2 CC\* *WC. 22 DAVID BLANC Then (G; V ) create a right model category structure . In order to see when hypothesis (*) of the Theorem applies, let us consider* * the case where C = cB and D = cM are both categories of cosimplicial objects, over a category B of universal coalegbras and an abelian category M, respectively, and* * the adjoint functors C D have been prolonged (x2.16) from some pair M GVB. Now the derived factorization of a map f : Ao ! Bo in C = cB, as given by the proof of Proposition 6.4, may be described as follows: Def o o o We start with Co(0)= A x GIT C(B ) x GIRV (B ), where G : cM ! cB is as above, I : cM ! cM is the (prolonged) injective envelope functor, and grM CTc*M are the adjoint pair of Proposition 6.4. (Here we identify cM wi* *th c*M by Proposition 6.2). In general, Co(n)is the pullback of Co(n-1)! GcsknCo(n) GEo (see remark 6.5), so we see that Co(n)agrees with Co(n-1)through dimension n, Cn+1(n)= Cn+1(n-1)x GEn+1, and Ci(n)(i > n) is determined by (6.3). Thus it* * is clear that Ao ! Co will be a cofibration. To verify that the inductive cohomol* *ogy conditions hold for the derived construction in cB, note that they hold in c*M * *~=cM because the composite map n n n () n+1 H ,! P n-i! B = V C(n-1)-! E is a monomorphism. Here H ~=HnB*=Im(f*) ~=Hn(V Co(n-1))=Im(j(n-1))*. In the derived factorization we need to know that the composite: V f() n+1 H ,! V Cn(n-1)-!V GE is monic, where g(): Cn(n-1)! GEn+1 is adjoint to (). This follows because f* *or any f : V X ! Y in M we have j O V "f= f (for j : V GY ! Y the adjoint of idGY ). Thus even though Proposition 4.8, and thus Theorem 4.15, do not dualize usefu* *lly to our situation (because Quillen's small object argument does not dualize to l* *imits), we have the following simplified situation where (*) of Theorem 7.6 holds: Proposition 7.7. Let cM be the category of cosimplicial objects over an abeli* *an category M with functorial injective envelopes, endowed with the model category structure given by Propositions 6.2 & 6.4, and let M GVB be a pair of adjoint* * func- tors such that V is faithful. Then the factorization given in the proof of Prop* *osition 6.4 satisfies hypothesis (*) of Theorem 7.6. NEW MODEL CATEGORIES FROM OLD 23 Remark 7.8. Theorem 7.6 provides a right model category structure on cB for a* *ny category of universal coalgebras B over an abelian category M, since the hypoth* *eses of Proposition 7.7 will in fact hold for such a B (compare [BaW , 3.3, Thm. 9* *]). However, for the purposes of "homotopical algebra", further assumptions may be needed. In particular, in order for the "triple derived functors" (cf. [BaB ]) of T t* *o coincide with the right derived functors (as defined in x4.11), we would want GA to be* * an injective in B for any A 2 M. This will be true, for example, if all objects * *in M are injective (e.g., if M = F-Mod for some field F ). For any B 2 B one has a cosimplicial coalgebra Co 2 cB obtained by the "dual standard construction" (cf. [BaB ] or [God , App., x3]), with Cn = (GV )n+1B * * and the coface and codegeneracy maps determined by the comonad structure maps " and of x7.1. Moreover, the coaugmentation " : B ! C0 defines a map i : c(B)o ! Co (where c(B)o is the constant cosimplicial object which is B in each dimension). Under the hypothesis that GA is always an injective, it is readily verified* * that i : c(B)o ! Co is a trivial cofibration: it is always a weak equivalence, and* * it is a cofibration by Fact 2.12 and the extension properties of injective objects. A s* *imilar argument shows that if T : B ! B0 is a functor between such categories of unive* *rsal algebras (possibly trivial - that is, simply abelian categories), its prolong* *ation cT : cB ! B0 will preserve weak equivalences between cofibrant objects (compare [Mac1 , III, Thm 3.1]), so that its right derived functors are defined (x2.16).* * Thus in particular we have the following Fact 7.9.In the right model category structure on cCAp defined by Theorem 7.6 and Proposition 6.4, we may identify the E2-term of the mod p Bousfield-Kan spe* *ctral sequence as the right derived functors of HomCAp(B; -), as in [BoK2 , Thm. 12* *.1]. Remark 7.10. It should perhaps be observed that the situation for an abelian ca* *tegory M, in which both left and right derived functors may be defined, is anomolous: * *it arises because M may be viewed either as a category of universal algebras or as* * a category of universal coalgebras, over itself. In general, most algebraic cate* *gories will support either left or right derived functors, but not both. 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Quillen, "On the (co-)homology of commutative rings", In Applicatio* *ns of categorical algebra, Proc. Symp. Pure Math. 17, AMS, Providence, RI, 1970, pp. 65-87. [ScV] R. Schw"anzl & R.M. Vogt, "The categories of A1 - and E1 -monoids and ri* *ng spaces as closed simplicial and topological model categories", Arch. Math., 56 (19* *91), pp. 405-411. [St] A. Strom, "The homotopy category is a homotopy category", Arch. Math. 23* * (1972), pp. 435-441. [Sw] M.E. Sweedler, Hopf Algebras, W.A. Benjamin, New York, 1969. [TZ] G. Takeuti & W.M. Zaring, Introduction to Axiomatic Set Theory, Springer* *-Verlag Grad. Texts in Math. 1, Berlin-New York, 1971. Dept. of Mathematics and Computer Science, University of Haifa, Haifa 31905 Israel E-mail address: blanc@mathcs2.haifa.ac.il