COSIMPLICIAL RESOLUTIONS AND HOMOTOPY SPECTRAL SEQUENCES IN MODEL CATEGORIES A.K. BOUSFIELD Abstract.We develop a general theory of cosimplicial resolutions, homotop* *y spec- tral sequences, and completions for objects in model categories, extendin* *g work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based o* *n a generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolu* *tion model categories, and we are able to construct our homotopy spectral sequences * *and com- pletions using very flexible weak resolutions in the spirit of relative h* *omological algebra. We deduce that our completion functors have triple structures an* *d pre- serve certain fiber squares up to homotopy. We also deduce that the Bende* *rsky- Thompson completions over connective ring spectra are equivalent to Bousf* *ield-Kan completions over solid rings. The present work allows us to show, in a su* *bsequent paper, that the Bendersky-Thompson homotopy spectral sequences over arbit* *rary ring spectra have well-behaved composition pairings. Contents 1. Introduction 2 2. Homotopy spectral sequences of cosimplicial objects * * 4 3. Existence of resolution model categories * * 8 4. Examples of resolution model categories * * 16 5. Derived functors, completions, and homotopy spectral sequences * * 19 6. Weak resolutions are sufficient * * 22 7. Triples give weak resolutions * *26 8. Triple structures of completions * * 29 9. Comparing different completions 31 10. Bendersky-Thompson completions of fiber squares * * 34 11. p-adic K-completions of fiber squares * * 38 12. The unpointed theory 42 References 45 __________ Date: January 15, 2001. 2000 Mathematics Subject Classification. Primary: 55U35; Secondary: 18G55, 55* *P60, 55T15. The author was partially supported by the National Science Foundation. 1 2 1.Introduction In [18] and [19], Bousfield-Kan developed unstable Adams spectral sequences a* *nd completions of spaces with respect to a ring, and this work was extended by Bendersky-Curti* *s-Miller [3] and Bendersky-Thompson [7] to allow a ring spectrum in place of a ring. In the pre* *sent paper, we develop a much more general theory of cosimplicial resolutions, homotopy spectr* *al sequences, and completions for objects in model categories. Among other things, this provides * *a flexible approach to the Bendersky-Thompson spectral sequences and completions, which is especial* *ly needed because the original chain level constructions of pairings and products [20] do not rea* *dily extend to that setting. We rely heavily on a generalized cosimplicial version of the Dwyer-Kan-Stover* * [24] theory of resolution model categories (or E2 model categories in their parlance). This pr* *ovides a simplicial model category structure c CG on the category c C of cosimplicial objects over * *a left proper model category C with respect to a chosen class G of injective models (see Theorems 3* *.3 and 12.4). Of course, our cosimplicial statements have immediate simplicial duals. Other more* * specialized versions of the simplicial theory are developed by Goerss-Hopkins [28] and Jardine [33] * *using small object arguments which are not applicable in the duals of many familiar model categori* *es. When C is discrete, our version reduces to a variant of Quillen's model category structur* *e [39, IIx4] on cC, allowing many possible choices of "relative injectives" in addition to Quillen'* *s canonical choice (see 4.3 and 4.4). However, we are most interested in examples where C is the catego* *ry of pointed spaces and where G is determined by a ring spectrum (4.9) or a cohomology theory (4.6)* *. In the former case, the model category provides Bendersky-Thompson-like [7] cosimplicial reso* *lutions of spaces with respect to an arbitrary ring spectrum, which need not be an S-algebra. In general, a cosimplicial G-injective resolution, or G-resolution, of an obj* *ect A 2 C consists of a trivial cofibration A ! ~Aoin c CG to a fibrant target ~Ao. By applying th* *e constructions of [18] and [21] to G-resolutions, we obtain right derived functors RsT(A) = ßsT(A* *~o), G-completions ^LGA = Tot~Ao, and G-homotopy spectral sequences {Es,tr(A; M)}r 2= {Es,tr(A~o; * *M)}r 2abutting to [M, ^LGA]* for A, M 2 C (see 5.5, 5.7, and 5.8). We proceed to show that the* * G-resolutions in these constructions may be replaced by weak G-resolutions, that is, by arbitrar* *y weak equivalences in c CG to termwise G-injective targets (see Theorems 6.2 and 6.5). This is con* *venient since weak G-resolutions are easy to recognize and arise naturally from triples on C. The * *Bendersky-Thompson resolutions are clearly examples of them. 3 We deduce that the G-completion functor ^LGbelongs to a triple on the homotop* *y category Ho C (see Corollary 8.2), and we introduce notions of G-completeness, G-goodnes* *s, and G-badness for objects in HoC. This generalizes work of Bousfield-Kan [18] on the homotopi* *cal R-completion functor R1 for pointed spaces. We dicuss an apparent error in the space-level a* *ssociativity part of the original triple lemma [18, p.26] for R1 , but we note that this error does * *not seem to invalidate any of our other results (see 8.9). We also develop criteria for comparing dif* *ferent completion functors, and we deduce that the Bendersky-Thompson completions with respect to* * connective ring spectra are equivalent to Bousfield-Kan completions with respect to solid rings* * (see Theorem 9.7), even though the associated homotopy spectral sequences may be very different. Finally, we show that G-completion functors preserve certain fiber squares up* * to homotopy (see Theorem 10.9), and we focus particularly on the Bendersky-Thompson K-completion* *s and the co- homological ^K*-completions, where K and ^Kare the spectra of nonconnective p-l* *ocal and p-adic K-theory at a prime p. In particular, we find that the K-completion functor pre* *serves homotopy fiber squares when their K*-cobar spectral sequences collapse strongly and thei* *r spaces have free K*-homologies, while the ^K*-completion functor preserves homotopy fiber square* *s when their K=p*- cobar spectral sequences collapse strongly and their spaces have torsion-free ^* *K*-cohomologies (see Theorems 10.12 and 11.7). In general, the completions and homotopy spectral seq* *uences with re- spect to K are very closely related to those with respect to ^K*(see Theorem 11* *.4), though the latter may have better technical properties. For instance, the homotopy spectral seque* *nces with respect to ^K*seem especially applicable to spaces whose p-adic K-cohomologies have Ste* *enrod-Epstein-like U(M) structures as in [13]. In much of this paper, for simplicity, we assume that our model categories ar* *e pointed. However, as in [28], this assumption can usually be eliminated, and we offer a brief acc* *ount of the unpointed theory in Section 12. We thank Paul Goerss for suggesting such a generalization. In a sequel [16] to this paper, we develop pairings and products for our homo* *topy spectral se- quences and discuss the E2-terms from the standpoint of homological algebra. Th* *is extends the work of [20], replacing the original chain-level formulae over rings by more ge* *neral constructions. It applies to give composition pairings for the Bendersky-Thompson spectral sequen* *ces. Although we have long been interested in the present topics, we were prompted* * to write this paper and its sequel by Martin Bendersky and Don Davis who are using some of ou* *r results in [4] and [5], and we thank them for their questions and comments. We also thank Assa* *f Libman for his 4 comments. Throughout this paper, we assume a basic familiarity with Quillen mod* *el categories and generally follow the terminology of [18], so that "space" means "simplicial set* *." 2.Homotopy spectral sequences of cosimplicial objects We now introduce the homotopy spectral sequences of cosimplicial objects in m* *odel categories, thereby generalizing the constructions of Bousfield-Kan [18] for cosimplicial s* *paces. This general- ization is mainly due to Reedy[41], but we offer some details to establish nota* *tion and terminology. We first consider 2.1. Model categories. By a model category we mean a closed model category in Q* *uillen's original sense [39]. This consists of a category with three classes of maps called weak * *equivalences, cofibrations, and fibrations, satisfying the usual axioms labeled MC1-MC5 in [25, pp.83-84]. * * We refer the reader to [25], [29], [30], and [31] for good recent treatments of model catego* *ries. A model category is called bicomplete when it is closed under all small limits and colimits. It * *is called factored when the factorizations provided by MC5 are functorial. We note that most interestin* *g model categories are bicomplete and factored or factorable, and some authors incorporate these c* *onditions into the axioms (see [30] and [31]). 2.2. Cosimplicial objects. A cosimplicial object Xo over a category C consists * *of a diagram in C indexed by the category of finite ordinal numbers. More concretely, it con* *sists of objects Xn 2 C for n 0 with coface maps di: Xn ! Xn+1 for 0 i n + 1and codegenera* *cy maps si: Xn+1! Xn for 0 j n satisfying the usual cosimplicial identities (see [1* *8, p.267]). Thus a cosimplicial object over C corresponds to a simplicial object over Cop. The cat* *egory of cosimplicial objects over C is denoted by c C, while that of simplicial objects is denoted b* *y s C. When C is a model category, there is an induced model category structure on c* * C = s(Cop) due to Reedy [41]. This is described by Dwyer-Kan-Stover [24], Goerss-Jardine [29],* * Hirschhorn [30], Hovey [31], and others. For an object Xo 2 c C, consider the latching maps LnXo* * ! Xn in C for n 0 where LnXo = colimXk `:[k]![n] with ` ranging over the injections [k] ! [n] in for k < n, and consider the m* *atching maps Xn ! MnXo in C for n 0 where MnXo = lim Xk OE:[n]![k] 5 with OE ranging over the surjections [n] ! [k] in for k < n. A cosimplicial m* *ap f : Xo ! Y o2 c C is called: (i)a Reedy weak equivalence when f : Xn ! Y nis a weak equivalence in C for n 0; ` (ii)a Reedy cofibration when Xn LnXoLnY o! Y nis a cofibration in C for n 0; (iii)a Reedy fibration when Xn ! Y nxMnY oMnXo is a fibration in C for n 0. Theorem 2.3 (Reedy).If C is a model category, then so is c C with the Reedy wea* *k equivalences, Reedy cofibrations, and Reedy fibrations. Example 2.4.Let S and S*denote the categories of spaces (i.e. simplicial sets) * *and pointed spaces with the usual model category structures. Then the Reedy model category structu* *res on c S and c S* reduce to those of Bousfield-Kan [18, p.273]. Thus a map Xo ! Y oin c S or* * c S* is a Reedy weak equivalence when it is a termwise weak equivalence, and is a Reedy cofibra* *tion when it is a termwise injection such that a(Xo) ~=a(Y o) where a(Xo) = {x 2 X0 | d0x = d1x} * *is the maximal augmentation. 2.5. Simplicial model categories. As in Quillen [39, II.1], by a simplicial cat* *egory, we mean a category C enriched over S, and we write map(X, Y ) 2 S for the mapping space o* *f X, Y 2 C. When they exist, we also write X K 2 C and hom(K, X) 2 C for the tensor and cotens* *or of X 2 C with K 2 S. Since there are natural equivalences Hom S(K, map(X, Y )) ~= HomC(X K, Y ) ~= HomC(X, hom(K, Y )), any one of the three functors, map, , and hom, determines the other two unique* *ly. As in Quillen [39, II.2], by a simplicial model category, we mean a model category C which is* * also a simplicial category satisfying the following axioms SM0 and SM7 (or equivalently SM70): SM0: The objects X K and hom(K, X) exist for each X 2 C and each finite K 2 S. SM7: If i : A ! B 2 C is a cofibration and p : X ! Y 2 C is a fibration, then the map map(B, X) ---! map(A, X) xmap(A,Ym)ap(B, Y ) is a fibration in S which is trivial if either i or p is trivial. 6 SM70: If i : A ! B 2 C and j : J ! K 2 S are cofibrations with J and K finite, then the map a (A K) (B J) ---! B K A J is a cofibraton in C which is trivial if either i or j is trivial. Theorem 2.6.If C is a simplicial model category, then so is the Reedy model cat* *egory c C with (Xo K)n = Xn K and hom(K, Xo)n = hom(K, Xn) for Xo 2 c C and finite K 2 S. Proof.The simplicial axiom SM70follows easily using the isomorphisms Ln(Xo K) * *~=LnXo K for n 0. |__| To construct our total objects and spectral sequences, we need 2.7. Prolongations of the mapping functors. Let C be a bicomplete simplicial mo* *del category. Then the objects X K 2 C and hom(K, X) 2 C exist for each X 2 C and each K 2 * *S, without finiteness restrictions. For A 2 C, Y o2 c C, and Jo 2 c S, we define map(A, Y* * o) 2 c S and A Jo 2 c C termwise, and we let hom(Jo, -) : c C ! C denote the right adjoint o* *f - Jo : C ! c C. It is not hard to show that the functor : C x c S ! c C satisfies the analogu* *e of SM70, and hence the functors map : Copx c C ! c S and hom : (c S)opx c C ! C satisfy the analog* *ues of SM7. 2.8. Total objects. Now let C be a pointed bicomplete simplicial model category* *, and let Xo 2 c C be Reedy fibrant. The total object TotXo = hom( o, Xo) 2 C is defined using the* * cosimplicial space o 2 c S of standard n-simplices n 2 S for n 0. It is the limit of the Tot t* *ower {TotsXo}s 0 with TotsXo = hom(sks o, Xo) 2 C where sks o 2 c S is the termwise s-skeleton o* *f o. Since o is Reedy cofibrant and its skeletal inclusions are Reedy cofibrations, TotXo* * is fibrant and {TotsXo}s 0is a tower of fibrations in C by 2.7. For M, Y 2 C and n 0, let ßn(Y ; M) = [M, Y ]n = [ nM, Y ] denote the group or set of homotopy classes from nM to Y in the homotopy categ* *ory HoC. Note that ßn(Y ; M) = ßnmap (M~, ~Y) where M~ is a cofibrant replacement of M and ~Y* *is a fibrant replacement of Y . 7 2.9. The homotopy spectral sequence. As in [18, pp.258,281], the Tot tower {Tot* *sXo}s 0 now has a homotopy spectral sequence {Es,tr(Xo; M)} for r 1 and t s 0, ab* *utting to ßt-s(TotXo; M) with differentials dr: Es,tr(Xo; M) ---! Es+r,t+r-1r(Xo; M) and with natural isomorphisms Es,t1(Xo; M) ~= ßt-s(FibsXo; M) ~= Nsßt(Xo; M) Es,t2(Xo; M) ~= ßt-s(FibsXo; M)(1)~=ßsßt(Xo; M) for t s 0 involving the fiber FibsXo of TotsXo ! Tots-1Xo, the normalizatio* *n Ns(-), the cou- ple derivation (-)(1), and the cosimplicial cohomotopy ßs(-) (see [11, 2.2] and* * [18, p.284]). This is equivalent to the ordinary homotopy spectral sequence of the cosimplicial space* * map(M~, Xo) 2 c S*, and its basic properties follow immediately from earlier work. We refer the rea* *der to [18, pp.261- 264] and [11, pp.63-67] for convergence results concerning the natural surjecti* *ons ßi(TotXo; M) ! limsQsßi(TotXo; M) for i 0 where Qsßi(TotXo; M) denotes the image of ßi(TotXo* *; M) ! ßi(TotsXo; M) and concerning the natural inclusions Es,t1+(Xo; M) Es,t1(Xo; M* *) where Es,t1+(Xo; M) denotes the kernel of Qsßt-s(TotXo; M) ! Qs-1ßt-s(TotXo; M) and where Es,t1(Xo* *; M) = T s,t o r>sEr (X ; M). As in [11], the spectral sequence may be partially extended be* *yond the t s 0 sector, and there is an associated obstruction theory. Finally, in preparation * *for our work on reso- lution model categories, we consider 2.10. The external simplicial structure on c C. For a category C with finite li* *mits and colimiits, the category c C = s(Cop) has an external simplicial structure as in Quillen [3* *9, II.1.7] with a mapping space mapc(Xo, Y o) 2 S, a cotensor homc(K, Xo) 2 c C, and a tensor Xo cK 2 c C* * for Xo, Y o2 c C and finite K 2 S. The latter are given by homc(K, Xo)n = hom(Kn, Xn) (Xo cK)n = Xo (K x n) for n 0,using the coend over and letting hom(S, Xn) and Xn S respectively d* *enote the product and coproduct of copies of Xn indexed by a set S. When C is a model category, t* *he external simplicial structure on c C will usually be incompatible with the Reedy model category str* *ucture. However, it will satisfy the weakened version of SM70obtained by replacing "either i or * *j is trivial" by "i is trivial" (see [29, p. 372]). Moreover, as suggested by Meyer [37, Theorem 2.4],* * we have 8 Lemma 2.11. Suppose C is a bicomplete simplicial model category. Then for Y o2 * *c C and K 2 S, there is a natural isomorphism Tothomc(K, Y o) ~= hom(K, TotY o) 2 C. Proof.It suffices adjointly to show, for A 2 C and K 2 S, that there is a natur* *al isomorphism (A o) cK ~=(A K) o2 c C. This follows from the isomorphisms (A o) (K x n) ~= A (K x n) 2 C in codimensions n 0, obtained by applying A - to o (K x n) ~=K x n 2 S* *. |__| 2.12. The external homotopy relation. In a general simplicial category, two map* *s f, g : X ! Y are simplicially homotopic when [f] = [g] in ß0map(X, Y ). In cC, to avoid ambi* *guity, we say that two maps f, g : Xo ! Y oare externally homotopic or cosimplicially homotopic (w* *rittenf c~g) when [f] = [g] in ß0mapc(X, Y ). For homomorphisms ff, fi : Ao! Bo of cosimplicial a* *belian groups, the relation ff c~fi corresponds to the chain homotopy relation for Nff, Nfi : NAo * *! NBo by Dold- Puppe [21, Satz 3.31], and hence ff c~fi implies ff* = fi* : ßsAo ! ßsBo for s * * 0. Likewise for homomorphisms ff, fi : Ao! Bo of cosimplicial groups (or pointed sets), the rel* *ation ff c~fi implies ff*= fi*: ßsAo! ßsBo for s = 0, 1 (or s = 0). Over a bicomplete simplicial model category C, we now have Proposition 2.13.If f, g : Xo ! Y o2 c C are maps of Reedy fibrant objects wit* *h f c~g, then Totf, Totg : TotXo ! TotY oare simplicially homotopic. Moreover, when C is poin* *ted, f*= g*: ß*(TotXo; M) ! ß*(TotY o; M) and f*= g*: Es,tr(Xo; M) ! Es,tr(Y o; M) for M 2 C* *, t s 0, and 2 r 1+. Proof.Totf and Totg are simplicially homotopic since Totpreserves strict homoto* *pies Xo ! homc( 1, Y o) by Lemma 2.11. The proposition now follows by 2.12. * * |__| 3.Existence of resolution model categories We now turn to the resolution model category structures of Dwyer-Kan-Stover [* *24] on the category c C = s(Cop) of cosimplicial objects over a model category C. These have more w* *eak equivalences than the Reedy structures and are much more flexible since they depend on a spe* *cified class of injective models in HoC. Moreover, they are compatible with the external simpli* *cial structure on c C. Our version of this theory is more general than the original one, and we h* *ave recast the proofs accordingly. We must assume that our model category C is left proper, meaning t* *hat each pushout 9 of a weak equivalence along a cofibration is a weak equivalence. As explained i* *n [30, 11.1], this condition holds for most familiar model categories including those where every * *object is cofibrant as assumed in [24]. For simplicity, we now also assume that C is pointed, and post* *pone the unpointed generalization until Section 12. 3.1. G-injectives. Let C be a left proper pointed model category, and let G be * *a class of group objects in the homotopy category HoC. A map i : A ! B in HoC is called G-monic * *when i* : [B, G]n ! [A, G]n is onto for each G 2 G and n 0, and an object Y 2 HoC is ca* *lled G-injective when i* : [B, Y ]n ! [A, Y ]n is onto for each G-monic map i : A ! B in HoC. Fo* *r instance, the objects nG 2 HoC are G-injective for G 2 G and n 0. The classes of G-monic m* *aps and of G-injective objects in HoC clearly determine each other. We say that HoC has en* *ough G-injectives when each object of HoC is the source of a G-monic map to a G-injective target,* * and we then call G a class of injective models in HoC. We always assume that a class of injectiv* *e models consists of group objects in the homotopy category. We say that an object of C is G-inje* *ctive when it is G-injective in HoC, and say that a map in C is G-monic when it is G-monic in Ho* *C. In Lemma 3.7 below, we show that a fibrant object F 2 C is G-injective if and only it the fi* *bration F ! * has the right lifting property for the G-monic cofibrations in C. Extending this co* *ndition, we say that a fibration in C is G-injective when it has the right lifting property for the G-* *monic cofibrations in C. A more explicit characterization of G-injective fibrations is given later in Le* *mma 3.10. 3.2. The G-resolution model structure on c C. Recall that a homomorphism in the* * category sGrp of simplicial groups is a weak equivalence or fibration when its underlyin* *g map in S is one. For a map f : Xo ! Y oin c C, we say: (i)f is a G-equivalence when f* : [Y o, G]n ! [Xo, G]n is a weak equivalence * *in sGrp for each G 2 G and n 0; (ii)f is a G-cofibration when f is a Reedy cofibration and f* : [Y o, G]n ! [Xo, G]n is a fibration in sGrp for each G 2 G and n 0; (iii)f is a G-fibration when f : Xn ! Y nxMnY oMnXo is a G-injective fibration in C for n 0. We let cCG denote the category c C with weak equivalences defined as G-equivale* *nces, with cofi- brations defined as G-cofibrations, with fibrations defined as G-fibrations, an* *d with the external simplicial structure (2.10). 10 Theorem 3.3 (after Dwyer-Kan-Stover).If C is a left proper pointed model catego* *ry with a class G of injective models in HoC, then c CG is a left proper pointed simplicial mod* *el category. We call c CG the G-resolution model category and devote the rest of Section 3* * to proving this theorem. Since the proof is somewhat long and technical, the reader might wish * *to proceed directly to Section 4 for a discussion of the result with some general examples. We star* *t by noting Proposition 3.4.The limit axiom MC1, the weak equivalence axiom MC2, and the re* *traction axiom MC3 hold in cCG. To go further, we must study G-monic cofibrations and G-injective fibrations * *in C, and we start with a lemma due essentially to Dan Kan (see [30, 11.1.16]). It applies to a co* *mmutative diagram A~--u--!A ----! X ?? ? ? y~i ?yi ?yp ~B--v--!B ----! Y in a left proper model category C such that u and v are weak equivalences, ~ian* *d i are cofibrations, and p is a fibration. Lemma 3.5. If the combined square has a lifting ~B! X, then the right square ha* *s a lifting B ! X. Proof.Using a lifting ~B! X, we break the right square into ` A ----! A ~A~B----!X ?? ? ? y ?y ?y B ----! B ----! Y` Since C is left proper, the maps ~B! A ~A~B! B are weak equivalences, and the * *second map ` factors into a trivial cofibration A ~A~B! E and trivial fibration E ! B. Thus* * the original right square has a lifting B ! E ! X. |_* *_| Henceforth, we assume that C and G satisfy the hypotheses of Theorem 3.3. Sin* *ce each cofibration A ! B in C can be approximated by a cofibration ~A! ~Bbetween cofibrant objects* *, Lemma 3.5 implies Lemma 3.6. A fibration in C is G-injective if and only if it has the right lift* *ing property for each G-monic cofibration between cofibrant objects. This easily implies 11 Lemma 3.7. A fibrant object F 2 C is G-injective in HoC if and only if the fibr* *ation F ! * is G-injective. The classes of G-monic cofibrations and of G-injective fibrant objects (or G-* *injective fibrations) in C now determine each other by Lemma 3.8. A cofibration i : A ! B in C is G-monic if and only if i* : Hom C(B* *, F) ! Hom C(A, F) is onto for each G-injective fibrant object F 2 C. Proof.For the if part, it suffices to show that i* : [B, nG] ! [A, nG] is ont* *o for each G 2 G and n 0. Since C is left proper, each map A ! nG 2 HoC can be represented b* *y a map f : A ! F 2 C for some G-injective fibrant object F with F ' nG. Since f is in* * the image of i*: HomC(B, F) ! HomC(A, F), the if part follows easily, and the only if part i* *s trivial. |__| Lemma 3.9. A map f : A ! B in C can be factored into a G-monic cofibration f0: * *A ! E and a G-injective fibration f00: E ! B. Proof.Since C is left proper and HoC has enough G-injectives, we may choose a G* *-monic cofibration fl : A ! F to a G-injective fibrant object F 2 C. We factor (f, fl) : A ! B x F* * as the composite of a cofibration f0: A ! E and a trivial fibration q : E ! B x F. This gives the d* *esired factorization f = f00f0where f00is the composite of q with the projection B x F ! B. * * |__| As suggested by Paul Goerss, this leads to a fairly explicit characterization* * of the G-injective fibrations in C. A map E ! Y in C is called G-cofree if it may be expressed as * *a composition of a trivial fibration E ! Y x F and a projection Y x F ! Y for some G-injective fib* *rant object F. Lemma 3.10. A map X ! Y in C is a G-injective fibration if and only if it is a * *retract of some G-cofree map E ! Y . Proof.For the only if part, we assume that X ! Y is a G-injective fibration, a* *nd we factor it as a composition of a G-monic cofibration X ! E and a G-cofree map E ! Y x F ! Y as * *above. Since X ! Y has the right lifting property for the G-monic cofibration X ! E, it must* * be a retract of the G-cofree map E ! Y as required. This gives the only if part, and the if par* *t is trivial. |__| Finally, consider a push-out square in C: A ----! C ?? ? yi ?yj B ----! D 12 Lemma 3.11. Suppose i is a G-monic cofibration in C. Then so is j, and the func* *tor [-, G]n carries the square to a pullback of groups for each G 2 G and n 0. Proof.The first conclusion follows by Lemma 3.8, while the second follows homot* *opically since C is left proper and each G 2 G is a group object in HoC. * * |__| Our next goal is to describe the G-cofibrations of c C in terms of the G-moni* *c cofibrations of C using 3.12. Partial latching objects. For Xo 2 c C and a finite K 2 S, we obtain an o* *bject Xo K = (Xo cK)0 2 C as in 2.10. This gives the latching object LnXo = Xo @ n as we* *ll as Xn = Xo n for n 0. We now let LnkXo = Xo Vknfor n k 0 where Vkn n is the k-horn spanned by di'n for all i 6= k. More generally, for a subset oe {0* *, 1, . .,.n}, we let LnoeXo = Xo Fnoewhere Fnoe n is spanned by di'n for all i 2 oe. Thus, LnkX* *o = LnoeXo for oe = {0, . .,.^k, . .,.n}, although usually LnkXo 6= Ln{k}Xo. For a cofibration* * J ! K of finite objects in S and a Reedy cofibration Xo ! Y oin c C, we note that the map a (Xo K) (Y o L) ---! Y o K Xo L is a cofibration in C since a (Xo cK) (Y o cL) ---! Y o cK Xo cL is a Reedy cofibration in c C. Proposition 3.13.Let f : Xo ! Y obe a Reedy cofibration in c C. Then: ` (i)f is a G-cofibration if and only if the cofibration Xn LnkXoLnkY o! Y nis G-monic whenever n k 0; ` (ii)f is a G-trivial cofibration if and only if the cofibration Xn LnXoLnY o! Y nis G-monic whenever n 0. Proof.For G 2 G, oe {0, 1, . .,.n}, and n 0, we obtain a square [Y n, G]* -Id---! [Y n, G]* ?? ? y ?y ` [Xn LnoeXoLnoeY o,-G]*---![Xn, G]*xMoen[Xo,G]*Moen[Y o, G]* 13 where Moenis the matching functor, dual to Lnoe, for simplicial groups. Each of* * the statements in (i) (resp. (ii)) asserts the surjectivity of a vertical arrow in this square for oe* * of cardinality |oe| = n (resp. |oe| = n + 1). The proposition now follows inductively using our next le* *mma. |__| ` Lemma 3.14. Given n 1, suppose that the cofibration Xm LmoeXoLmoeY o! Y mis * *G-monic for each m < n and each oe {0, 1, . .,.m} with |oe| = m (resp. |oe| = m + 1). The* *n the map ` [Xm LmoeXoLmoeY o, G]*- --![Xm , G]*xMoem[Xo,G]*Moem[Y o, G]* is an isomorphism for each G 2 G, each m n, and each oe {0, 1, . .,.m} with* * |oe| m (resp. |oe| m + 1). ` Proof.We first claim that the cofibration Xm LmoeXoLmoeY o! Y mis G-monic for * *each m < n and each oe {0, 1, . .,.m} with |oe| m (resp. |oe| m + 1). This follows by in* *ductively applying the first part of Lemma 3.11 to the pushout squares ` ` Xm-1 Lm-1oeXoLm-1oeY-o---!Xm LmoeXoLmoeY o ?? ? y ?y ` Y m-1 ----!Xm LmøXoLmøY o where oe = {i1, . .,.ik-1} and ø = {i1, . .,.ik} for 0 i1 < . .<.ik m with * *m < n. The lemma now follows by inductively applying the pullback part of Lemma 3.11 to th* *ese squares with m n. |__| Proposition 3.13 combines with Lemma 3.8 to give Corollary 3.15.Let f : Xo ! Y obe a Reedy cofibration in c C. Then f is a G-cof* *ibration (resp. G-trivial cofibration) if and only if f* : HomC(Y o, F) ! HomC(Xo, F) is a fibr* *ation (resp. trivial fibration) in S for each G-injective fibrant object F 2 C. The G-trivial cofibration condition on a map Xo ! Y oin c C now reduces to th* *e G-monic cofi- ` bration condition on each Xn LnXoLnY o! Y n, just as the G-fibration condition* * reduces to the G-injective fibration condition on each Xn ! Y nxMnY oMnXo. Hence the model cat* *egory axioms pertaining to these conditions now follow easily. Proposition 3.16.The lifting and factorization axioms MC4(ii) and MC5(ii) (for * *fibrations and trivial cofibrations) hold in c CG. 14 Proof.This follows by Reedy's constructions [41] since the G-injective fibratio* *ns have the right lifting property for G-monic cofibrations, and since the maps in C may be factored as i* *n Lemma 3.9. |__| Using the external simplicial structure (2.10) on c C, we now also have the s* *implicial axiom SM70 by Proposition 3.17.If i : Ao ! Bo 2 c C is a G-cofibration and j : J ! K 2 S is a* * cofibration of finite objects, then the map a (Ao cK) (Bo cJ) ---! Bo cK Ao cJ is a G-cofibration in c C which is trivial if either i or j is trivial. Proof.Since this map is a Reedy cofibration by 2.10, the result follows from Co* *rollary 3.15 by an adjunction argument using the isomorphism HomC(Ao cK, F) ~=map(K, HomC(Ao, F)) * *in S for F 2 C. |__| To prove the factorizaton axiom MC5(i) (for G-cofibrations and G-trivial fibr* *ations), we need Lemma 3.18. The G-cofibrations and G-trivial cofibrations in c C are closed und* *er pushouts. Proof.This follows from Corollary 3.15. * * |__| Since the G-cofibrant objects of c C are the same as the Reedy cofibrant ones* *, we may simply call them cofibrant. Lemma 3.19. A map f : Xo ! Y oof cofibrant objects in c C can be factored into * *a G-cofibration i : Xo ! Mf and a G-equivalence q : Mf ! Y o. Proof.Let Mf be the mapping cylinder ` o o 1 a o` Mf = (Xo c 1) XoY = (X c ) (Y Xo) Xo` Xo ` Then the natural map i : Xo ! Mfis a G-cofibration by Lemma 3.18 since Xo Xo !* * Xo c 1is a G-cofibration by Proposition 3.17. Likewise, the natural map j : Y o! Mf is a G* *-trivial cofibration, and its natural left inverse q : Mf ! Y ois a G-equivalence. This gives the req* *uired factorization f = qi. |__| We can now prove MC5(i). 15 Proposition 3.20.A map f : Xo ! Y oin c C can be factored into a G-cofibration * *i : Xo ! Nf and a G-trivial fibration p : Nf ! Y o. Proof.First take Reedy cofibrant replacements to give a map ~f: ~Xo! ~Yaond use* * Lemma 3.19 to factor ~f. Then use a pushout of ~fto factor f into a G-cofibration j : Xo ! Eo* * and a G-equivalence r : Eo ! Y o. Finally apply Proposition 3.16 to factor r into a G-trivial cofib* *ration s : Eo ! Nf and a G-trivial fibration p : Nf ! Y o, and let i = sj. * * |__| To prove the lifting axiom MC4(i) (for G-cofibrations and G-trivial fibration* *s), we need several preliminary results. Lemma 3.21. If a map f in c C has the right lifting property for G-cofibrations* * (resp. G-trivial cofibrations), then f is a G-trivial fibration (resp. G-fibration). Proof.This follows by first using Proposition 3.20 (resp. Proposition 3.16) to * *factor f, and then using the given right lifting property to express f as a retract of the appropr* *iate factor. |__| Lemma 3.22. For a G-fibrant object Fo 2 c C and a cofibration (resp. trivial co* *fibration) L ! K of finite objects in S, the induced map homc(K, Fo) ! homc(L, Fo) 2 cC has the rig* *ht lifting property for G-trivial cofibrations (resp. G-cofibrations). Proof.This follows by Propositions 3.16 and 3.17. * * |__| We now let PFo 2 c C be the standard path object given by PFo = homc( 1, Fo) xFo * = homc( 1, Fo) xFoxFo Fo . Lemma 3.23. For a G-fibrant object Fo 2 c C, the natural map PFo ! Fo (resp. PF* *o ! *) has the right lifting property for G-trivial cofibrations (resp. G-cofibrations) in* * c C. Proof.This follows from Lemma 3.22 since right lifting properties are preserved* * by pullbacks. |__| Lemma 3.24. If Fo ! * is a G-trivial fibration with Fo cofibrant, then Fo ! * h* *as the right lifting property for G-cofibrations. Proof.The G-fibration PFo ! Fo has a cross-section by Proposition 3.16, and Fo * *! * has the right lifting property for G-cofibrations since PFo ! * does by Lemma 3.23. * * |__| 16 Lemma 3.25. If f and g are maps in c C such that gf is a G-cofibration and f is* * a Reedy cofibration, then f is a G-cofibration. Proof.This follows since a simplicial group homomorphism G ! H is a fibration i* *f and only if it induces surjections of Moore normalizations NqG ! NqH for q > 0 (see [39, IIx3]* *). |__| We can now prove MC4(i). Proposition 3.26.A G-trivial fibration f : Xo ! Y oin c C has the right lifting* * property for G- cofibrations. Proof.First suppose that Xo is cofibrant. By Proposition 3.20, the map Xo ! * f* *actors into a G-cofibration OE : Xo ! Fo and a G-trivial fibration Fo ! *, and the map (f, OE* *) : Xo ! Y ox Fo factors into a Reedy cofibration Xo ! Eo and a Reedy trivial fibration Eo ! Y o* *x Fo. Then the map Eo ! Y ois a G-trivial fibration with the right lifting property for G-cofi* *brations by Lemmas 3.21 and 3.24. Hence, Xo ! Eo is a G-equivalence and a G-cofibration by Lemma * *3.25. Thus Xo ! Y ois a retract of Eo ! Y oby Proposition 3.16, and Xo ! Y oinherits the r* *ight lifting property for G-cofibrations. In general, by Lemma 3.5 (applied in Reedy's c C),* * it suffices to show that Xo ! Y ohas the right lifting property for each G-cofibration of cofibrant* * objects Co ! Do. This follows since a map Co ! Xo factors into a Reedy cofibration Co ! ~Xoand R* *eedy trivial fibration ~Xo! Xo, where the composed map ~Xo! Xo ! Y omust have the right lift* *ing property for G-cofibrations since it is a G-trivial fibration with ~Xocofibrant. * * |__| This completes the proof that c CG is a model category, and Theorem 3.3 will * *follow from Proposition 3.27.The G-resolution model category c CG is left proper. Proof.By [15, Lemma 9.4], it suffices to show that a pushout of a G-equivalence* * f : Ao! Y oalong a G-cofibration Ao! Bo of cofibrant objects is a G-equivalence. We may factor f i* *nto a G-equivalence OE : Ao ! ~Ywoith ~Ycoofibrant and a Reedy weak equivalence q : ~Y!oY o. The pr* *oposition now follows since the pushout of OE is a G-equivalence by [41, Theorem B], and the * *pushout of q is a Reedy weak equivalence. |__| 4.Examples of resolution model categories If C is a left proper pointed model category with a class G of injective mode* *ls in HoC, then Theorem 3.3 gives the G-resolution model category c CG. In this section, we dis* *cuss some general examples of these model categories. 17 4.1. Dependence of c CG on G. As initially defined, the G-resolution model stru* *cture on c C seems to depend strongly on G. However, by Proposition 3.13, the G-cofibrations and G* *-trivial cofibrations in c C are actually determined by the G-monic maps in HoC. Hence, the G-resolut* *ion model structure on c C is determined by the class of G-monic maps, or equivalently by the class* * of G-injective objects in HoC. 4.2. A refinement of Theorem 3.3. Adding to the hypotheses of Theorem 3.3, we s* *uppose that the model category C is factored (2.1) and that the class G of injective models* * is functorial, meaning that there exists a functor : C ! C and a transformation fl : 1C ! (X) such * *that fl : X ! (X) is a G-monic map to a G-injective object (X) for each X 2 C. Then the model ca* *tegory c CG is also factored by the constructions in our proof of Theorem 3.3. Of course, if C is b* *icomplete, then c CG is also bicomplete. 4.3. Constructing c CG for discrete C. Let C be a pointed category with finite * *limits and colimits, and give C the discrete model category structure in which the weak equivalences* * are the isomor- phisms, and the cofibrations and fibrations are arbitrary maps. Then HoC = C wi* *th [X, Y ]0 = Hom C(X, Y ) and with [X, Y ]n = * for X, Y 2 C and n > 0. Now let G be a class* * of group objects in C. If C has enough G-injectives, then we have a simplicial model category c * *CG by Theorem 3.3. This provides a dualized variant of Quillen's Theorem 4 in [39, IIx4], allowing* * many possible choices of "relative injectives" in addition to Quillen's canonical choice. For instanc* *e, we consider 4.4. Abelian examples. Let C be an abelian category, viewed as a discrete model* * category, and let G be a class of objects in C such that C has enough G-injectives. Recall that c* * C is equivalent to the category Ch+C of nonnegatively graded cochain complexes over C by the Dold-Kan * *correspondence (see e.g. [21] or [29]). Thus the G-resolution model category c CG corresponds * *to a model category Ch+CG. For a cochain map f : X ! Y in Ch+CG, a careful analysis shows that: (i)f is a G-equivalence when f* : HnHom (Y, G) ~=HnHom (X, G) for each G 2 G and n 0; (ii)f is a G-cofibration when f : Xn ! Y nis G-monic for n 1; (iii)f is a G-fibration when f : Xn ! Y nis splittably epic with a G-injective kernel for n 0. For example, when C has enough injectives and G consists of them all, we recove* *r Quillen's model category Ch+CG [39, IIx4] where: (i) the G-equivalences are the cohomology equi* *valences; (ii) the G-cofibrations are the maps monic in positive degrees; and (iii) the G-fibratio* *ns are the epic maps 18 with injective kernels in all degrees. For another example, when G consists of * *all objects in C, we obtain a model category Ch+CG where: (i) the G-equivalences are the chain homot* *opy equivalences; (ii) the G-cofibrations are the maps splittably monic in positive degrees; and * *(iii) the G-fibrations are the maps splittably epic in all degrees. In this example, all cochain compl* *exes are G-fibrant and G-cofibrant. 4.5. Constructing c CG for small G. Let C be a left proper pointed model catego* *ry with arbitrary products, and let G be a (small) set of group objects in HoC. Then HoC has enou* *gh G-injectives, since for each X 2 HoC, there is a natural G-monic map Y Y Y X ---! nG G2G n 0 f:X! nG to a G-injective target, where f ranges over all maps X ! nG in HoC. Thus we h* *ave a simplicial model category c CG by Theorem 3.3. Note that an object X 2 HoC is G-injective * *if and only if X is a retract of a product of terms nG for various G 2 G and n 0. Also note* * that if C is factored, then the class G is functorial by a refinement of the above construct* *ion, and hence the model category c CG is factored by 4.2. 4.6. A homotopical example. Let Ho*= HoS*be the pointed homotopy category of sp* *aces, and recall that a cohomology theory E* is representable by spaces E_n2 Ho*with ~EnX* * ~=[X, E_n] for X 2 Ho*and n 2 Z. For G = {E_n}n2Z, we obtain a G-resolution model category c S* *G*by 4.5. Note that the G-equivalences in c S* are the maps inducing ßs~E*-isomorphisms for s * * 0. Also note that c SG*is factored by 4.5. Our next example will involve 4.7. Quillen adjoints. Let C and D be left proper pointed model categories, and* * let S : C ø D : T be Quillen adjoint functors, meaning that S is left adjoint to T and the * *following equivalent conditions are satisfied: (i) S preserves cofibrations and T preserves fibratio* *ns; (ii) S preserves cofibrations and trivial cofibrations; and (iii) T preserves fibrations and tri* *vial fibrations. Then by [39] or [25, Theorem 9.7], S has a total left derived functor LS : HoC ! HoD, a* *nd T has a total right derived functor RT : HoD ! HoC, where LS is left adjoint to RT. Moreover,* * LS preserves homotopy cofiber sequences and suspensions, while RT preserves homotopy fiber s* *equences and loopings. 4.8. Construction c CG from Quillen adjoints. Let S : C ø D : T be Quillen adjo* *ints as in 4.7, and let H be a class of injective models in HoD. Then we obtain a class G = {(R* *T)H | H 2 H} of 19 injective models in HoC, and obtain Quillen adjoints S : c CG ø c DH : T. We no* *te that if C and D are factored and H is functorial, then G is also functorial and hence c CG and * *c DH are factored. 4.9. Another homotopical example. Let Sp be the model category of spectra in th* *e sense of [17] (see also [32]), and let Hos= Ho(Sp) be the stable homotopy category. The infin* *ite suspension and 0-space functors S*ø Sp are Quillen adjoints, and their total derived functors * *are the usual infinite suspension and infinite loop functors 1 : Ho*ø Hos: 1 . Let S 2 Hosbe the sph* *ere spectrum, and suppose that E 2 Hosis a ring spectrum, meaning that it is equipped with a * *multiplication E ^E ! E 2 Hosand unit S ! E 2 Hossatisfying the identity and associativity pro* *perties in Hos. Let H be the class of E-module spectra in Hosand note that Hoshas enough H-inje* *ctives since the unit maps X ! E ^ X are H-monic with H-injective targets. Thus by 4.8, we o* *btain a class G = { 1 N | N 2 H} of injective models in Ho*, and we have resolution model cat* *egories c SpH and c SG*by Theorem 3.3. Various alternative choices of G will lead to the same* * G-injectives in Ho* and hence to the same resolution model category c SG*. For instance, we could e* *quivalently let G be { 1 (E ^ 1 X) | X 2 Ho*} or { 1 (E ^ Y ) | Y 2 Hos}. These resolution model ca* *tegories are factored. 5.Derived functors, completions, and homotopy spectral sequences Let C be a left proper pointed model category with a class G of injective mod* *els in HoC. We now introduce G-resolutions of objects in C and use them to construct right derived* * functors, completions, and the associated homotopy spectral sequences. In Section 6, we shall see that* * a weaker sort of G-resolution will suffice for these purposes. 5.1. G-resolutions in C. A G-resolution (= cosimplicial G-injective resolution)* * of an object A 2 C consists of a G-trivial cofibration ff : A ! ~Aoto a G-fibrant object ~Aoin c C* *, where A is considered constant in c C. This exists for each A 2 C by MC5 in c CG, and exists functori* *ally when c CG is factored. In general, G-resolutions are natural up to external homotopy (2.12) * *by Lemma 5.2. If ff : A ! Io is a G-trivial cofibration in c CG, and if f : A ! Jo* * is a map to a G-fibrant object Jo 2 c CG, then there exists a map OE : Io ! Jo with OEff = f * *and OE is unique up to external homotopy. Proof.This follows since ff* : mapc(Io, Jo) ! mapc(A, Jo) is a trivial fibratio* *n in S by SM7 in c CG. |__| 20 The terms of a G-resolution are G-injective by Lemma 5.3. If an object Io 2 c C is G-fibrant, then In is G-injective and fibra* *nt in C for n 0. Proof.More generally, if f : Xo ! Y ois a G-fibration in c C, then f : Xn ! Y n* *xMnY oMnXo is a G-injective fibration for n 0 by definition, and hence each f : Xn ! Y nis a * *G-injective fibration in C by Corollary 2.6 of [29, p.366]. * * |__| Conseqently, the terms In are H-spaces in HoC by Lemma 5.4. If J is a G-injective object in HoC, then J admits a multiplication * *with unit. Proof.The coproduct-to-product map J _ J ! J x J is G-monic since nG is a grou* *p object of Ho C for each G 2 G and n 0. Hence, the folding map J _ J ! J extends to a ma* *p J x J ! J giving the desired multiplication. * * |__| 5.5. Right derived functors. Let T : C ! M be a functor to an abelian category * *M. We define the right derived functor RsGT : C ! M for s 0, with a natural transformation* * ffl : T ! R0GT, by setting RsGT(A) = ßsTA~o = Hs(NTA~o) for A 2 C, where A ! ~Ao2 c C is a G-re* *solution of A and NTA~ois the normalized cochain complex of TA~o2 cM. This is well-defined * *up to natural equivalence by 2.12 and 5.2. Similarly, let U : C ! Grp and V : C ! Set* be fun* *ctors to the categories of groups and pointed sets. We define the right derived functors R0G* *U : C ! Grp and R1GU, R0GV : C ! Set* by setting RsGU(A) = ßsUA~oand RsGV (A) = ßsV ~Aoas above* *. Since the G-fibrant objects in c C are termwise G-injective by Lemma 5.3, these derived f* *unctors depend only on the restrictions of T, U, V to the full subcategory of G-injective objects i* *n C. Thus they may be defined for such restricted functors. 5.6. Abelian examples. Building on 4.4, suppose C is an abelian category with a* * class G of injective models, and suppose T : C ! M is a functor to an abelian category M. Then a G-r* *esolution of A 2 C corresponds to an augmented cochain complex A ! ~Ao2 Ch+C where ~Anis G-i* *njective for n 0 and where the augmented chain complex Hom(A~o, G) ! Hom(A, G) is acyclic * *for each G 2 G. When T is additive, we have RsGT(A) = HsTA~ofor s 0, and we recover the usual* * right derived functors RsGT : C ! M of relative homological algebra [26]. In general, we obta* *in relative versions of the Dold-Puppe [21] derived functors. Now suppose that the model category C is simplicial and bicomplete. 21 5.7. G-completions. For an object A 2 C, we define the G-completion ff : A ! ^L* *GA 2 HoC by setting ^LGA = Tot~Aowhere A ! ~Ao2 c C is a G-resolution of A. This determine* *s a functor ^LG: C ! HoC which is well-defined up to natural equivalence by 5.2 and 2.13. I* *n fact, by Corollary 8.2 below, the G-completion will give a functor ^LG: HoC ! HoC and a natural tr* *ansformation ff : Id! ^LGbelonging to a triple on HoC. When C is factored and G is functori* *al (4.2), the G-completion is canonically represented by a functor ^LG: C ! C with a natural * *transformation ff : Id! ^LG. 5.8. G-homotopy spectral sequences. For objects A, M 2 C, we define the G-homot* *opy spectral sequence {Es,tr(A; M)G}r 2of A with coefficients M by setting Es,tr(A; M)G = Es* *,tr(A~o; M) for 0 s t and 2 r 1+ using the homotopy spectral sequence (2.9) of ~Aofor a* * G-resolution A ! ~Ao. Since this is the homotopy spectral sequence of a pointed cosimplicial* * space map(M~, ~Ao), composed of H-spaces by 5.4, we see that Es,tr(A; M)G is a pointed set for 0 * *s = t r - 2 and is otherwise an abelian group by [11, Section 2.5]. The spectral sequence is fring* *ed on the line t = s as in [18], and the differentials dr: Es,tr(A; M)G ---!Es+r,t+r-1r(A; M)G are homomorphisms for t > s. It has Es,t2(A; M)G = ßsßt(A~o; M) = RsGßt(A; M) for 0 s t by 2.9 and 5.5, and it abuts to ßt-s(^LGA; M) with the usual conv* *ergence properties which may be expressed using the natural surjections ßi(^LGA; M) ! limsQsßi(^LG* *A; M) for i 0 and the natural inclusions Es,t1+(A; M)G Es,t1(A; M)G as in 2.9. The spectral* * sequence is well- defined up to natural equivalence and depends functorially on A, M 2 C by 5.2 a* *nd 2.13. 5.9. Immediate generalizations. The above notions extend to an arbitrary object* * Ao 2 c C in place of A 2 C. A G-resolution of Aostill consists of a G-trivial cofibration f* *f : A ! ~Aoto a G-fibrant object ~Ao2 c C. A functor T : C ! M to an abelian category M still has right d* *erived functors RsGT : c C ! M with RsGT(Ao) = ßsTA~o2 M for s 0. Moreover, Ao still has a G-* *homotopy spectral sequence {Es,tr(Ao; M)G}r 2with coefficients M 2 C, where Es,tr(Ao; M)* *G = Es,tr(A~o; M) for 0 s t and 2 r 1+. This has Es,t2(Ao; M)G = ßsßt(A~o; M) = RsGßt(Ao; M) for t s 0 and abuts to ßt-sTotGAo where TotGAo = Tot~Ao2 HoC (see 8.1). It * *retains the properties described above in 5.8. 22 6.Weak resolutions are sufficient Let C be a left proper pointed model category with a class G of injective mod* *els in HoC. We now introduce the weak G-resolutions of objects in C and show that they may be * *used in place of actual G-resolutions to construct right derived functors, G-completions, and G-* *homotopy spectral sequences. This is convenient since weak G-resolutions arise naturally from tri* *ples on C (see Section 7) and are generally easy to recognize. Definition 6.1.A weak G-resolution of an object A 2 C consists of a G-equivalen* *ce A ! Y oin c C such that Y nis G-injective for n 0. Such a Y ois called termwise G-injective. Any G-fibrant object of c C is termwise G-injective by Lemma 5.3, and hence a* *ny G-resolution is a weak G-resolution. As our first application, we consider the right derived fu* *nctors of a functor T : C ! N where N is an abelian category or N = Grp or N = Set*. We suppose tha* *t T carries weak equivalences in C to isomorphisms in N. Theorem 6.2.If A ! Y o2 c C is a weak G-resolution of an object A 2 C, then the* *re is a natural isomorphism RsGT(A) ~=ßsTY ofor s 0. It is understood that s = 0, 1 when N = Grp and that s = 0 when N = Set*. Thi* *s theorem will be proved in 6.14, and we cite two elementary consequences. Corollary 6.3.If A 2 C is G-injective, then ffl : T(A) ~=R0GT(A) and RsGT(A) = * *0 for s > 0. Proof.This follows using the weak G-resolution Id: A ! A. * * |__| A map f : A ! B in C is called a G-equivalence when f* : [B, G]n ~=[A, G]n fo* *r G 2 G and n 0, or equivalently when f is a G-equivalence of constant objects in c C. Corollary 6.4.If f : A ! B is a G-equivalence in C, then f*: RsGT(A) ~=RsGT(B) * *for s 0. Proof.This follows since f composes with a weak G-resolution of B to give a wea* *k G-resolution of A. |__| To give similar results for G-completions and G-homotopy spectral sequences, * *we suppose that C is simplicial and bicomplete. Theorem 6.5.Suppose A ! Y ois a weak G-resolution of an object A 2 C. Then ther* *e is a natural equivalence ^LGA ' TotY_o2 HoC for a Reedy fibrant replacement Y_oof Y o, and t* *here are natural 23 isomorphisms Es,tr(A; M)G ~=Es,tr(Y_o; M) and Qsßi(^LGA; M) ~=Qsßi(TotY_o; M) f* *or M 2 C, 0 s t, 2 r 1+, and i 0. This will be proved later in 6.19 and partially generalized in 9.5. It has th* *e following elementary consequences. Corollary 6.6.Suppose A 2 C is G-injective. Then ^LGA ' A 2 HoC and ( Es,tr(A; M)G ~= ßt(A; M)when s = 0 0 when 0 < s t for M 2 C and 2 r 1+. Corollary 6.7.If f : A ! B is a G-equivalence in C, then f induces ^LGA ' ^LGB * *and Es,tr(A; M)G ~= Es,tr(B; M)G for M 2 C, 0 s t, and 2 r 1+. In particular, the G-completion ^LG: C ! HoC carries weak equivalences to equ* *ivalences and induces a functor ^LG: HoC ! HoC. To prepare for the proofs of Theorems 6.2 and* * 6.5. we need 6.8. The model category c(c CG). Let c(c CG) be the Reedy model category of cos* *implicial objects Xoo= {Xno}n 0over the G-resolution model category c CG. Its structural maps are* * called Reedy G-equivalences, Reedy G-cofibrations, and Reedy G-fibrations. Thus a map f : X* *oo ! Y oois a Reedy G-equivalence if and only if f : Xno ! Y nois a G-equivalence for each n * * 0. Moreover, if f : Xoo! Y oois a Reedy G-cofibration (resp. Reedy G-fibration), then f : Xn* *o ! Y nois a G-cofibration (resp. G-fibration) for each n 0 by [29, Corollary VII.2.6]. Le* *t diag: c(c CG) ! c CG be the functor with diagY oo= {Y nn}n 0. Lemma 6.9. If f : Xoo! Y oois a Reedy G-equivalence, then diagf : diagXoo! diag* *Y oois a G-equivalence. Proof.For each G 2 G, the bisimplicial group hommorphism f* : [Y oo, G]* ! [Xoo* *, G]* restricts to a weak equivalence [Y no, G]* ! [Xno, G]* for n 0, and thus restricts to a* * weak equivalence [diagY oo, G]*! [diagXoo, G]* by [17, Theorem B.2]. * * |__| Lemma 6.10. If f : Xoo! Y oois a Reedy G-fibration, then diagf : diagXoo! diagY* * oois a G-fibration. Proof.For Xoo2 c(c CG), we may express diagXooas an end Z diagXoo ~= homc( n, Y no), [n]2 24 and hence interpret diagXooas the total object (2.8) of the cosimplicial object* * Xooover c CG. The lemma now follows by 2.7. |__| 6.11. Special G-fibrant replacements. For an object Y o2 c CG, we let conY o2 c* *(c CG) be the vertically constant object with (conY o)n,i= Y nfor n, i 0. We choose a * *Reedy G-trivial o * * oo cofibration ff : conY o! ~Y ooto a Reedy G-fibrant target ~Y oo, and we let ~~Y* *= diag~Y. This o ~o induces a G-equivalence ff : Y o! ~~Ywith ~YG-fibrant by Lemmas 6.9 and 6.10. W* *ith some work, o we can show that this special G-fibrant replacement ff : Y o! ~~Yis actually a * *G-resolution, but that will not be needed. Let T : C ! M be a functor to an abelian category M such that T carries weak * *equivalences to isomorphisms. Lemma 6.12. If Y o2 c CG is Reedy fibrant and termwise G-injective, then the ab* *ove map ff : Y o! ~~Yoinduces an isomorphism ff*: ßsTY o~=ßsTY~~ofor s 0. Proof.Since ff : Y n! ~Y nois a G-resolution of the G-injective fibrant object * *Y n, we have ßsTY~no= 0 for s > 0 and ß0TY~no~=TY n. Hence, Tff : T(conY o) ! TY~oorestricts to ß*-eq* *uivalences of all vertical complexes, and must therefore restrict to a ß*-equivalence of the * *diagonal complexes * * o by the Eilenberg-Zilber-Cartier theorem of Dold-Puppe [21]. Hence, ff : T(Y o)* * ! T(~~Y) is a ß*-equivalence. |__| Lemma 6.13. If Y o, Zo 2 c CG are termwise G-injective and f : Y o! Zo is a G-e* *quivalence, then f*: ßsTY o~=ßsTZo for s 0. * * o Proof.After replacements, we may assume that Y oand Zo are Reedy fibrant. Let f* *f : Y o! ~~Y * * o ~o and fi : Zo ! ~~Zobe special G-fibrant replacements as in 6.11 with an induced * *map ~~f: ~~Y! ~Z such that ~~fff = fif. Then ff and fi are ß*T-equivalences by Lemma 6.12. After* * Reedy cofibrant replacements, ~~fbecomes a G-equivalence of G-fibrant cofibrant objects and hen* *ce a cosimplicial homotopy equivalence. Thus ~~fis also a ß*T-equivalence, and hence so is f. * * |__| 6.14. Proof of Theorem 6.2. Consider the case of T : C ! M as above. Given a * *weak G- resolution ff : A ! Y o, we choose G-resolutions u : A ! A~oand v : Y o! ~Y,oan* *d choose ~ff: ~Ao! ~Ywoith ~ffu = vff. Then RsGTA ~= ßsTA~o~= ßsTY~o ~=ßsTY o for s 0 by Lemma 6.13 as required. The remaining cases of T : C ! Grp and T :* * C ! Set* are similarly proved. |__| 25 To prepare for the proof of Theorem 6.5, we let M be a bicomplete simplicial * *model category. For an object Moo2 c(c M), we define TotvMoo2 c M by (TotvMoo)n = Tot(Mno) for * *n 0. Lemma 6.15. For Moo2 c(c M), there is a natural isomorphism TotTotvMoo~=Totdiag* *Moo. Proof.The functor Tot: c M ! M preserves inverse limits and gives Tothomc(K, No* *) ~=hom(K, TotNo) for No 2 c M and K 2 S by Lemma 2.11. Hence, the induced functor Totv= c(Tot) :* * c(c M) ! c M respects total objects (2.8), and we have TotTotvMoo ~=TotTotMoo ~=TotdiagMoo with TotMoo~=diagMooby the proof of Lemma 6.10. |* *__| Using the Reedy and Reedy-Reedy model category structures (2.3) on c M and c(* *c M), we have Lemma 6.16. If Moo! Noois a Reedy-Reedy fibration in c(c M), then TotvMoo! Totv* *Noois a Reedy fibration in c M. Proof.This follows since Tot: c M ! M preserves fibrations and inverse limits. * * |__| For Theorem 6.5, we also need the following comparison lemma of [11, 6.3 and * *14.4] whose hypotheses are expressed using notation from xx2, 14 of that paper. Lemma 6.17. Let f : V o! Wo be a map of pointed fibrant cosimplicial spaces suc* *h that: (i)f*: ß0ß0V o~=ß0ß0Wo; (ii)f induces an equivalence Totßgd1V o~=Totßgd1Wo of groupoids; (iii)f*: ß*ßt(V o, b) ~=ß*ßt(Wo, fb) for each vertex b 2 Tot2V oand t 2. Then f induces an equivalence TotV o~=TotWo and isomorphisms QsßiTotV o~=QsßiTo* *tWo and Es,trV o~=Es,trWo for 0 s t, 2 r 1+, and i 0. This leads to our final preparatory lemma. * * o Lemma 6.18. If Y o2 c CG is Reedy fibrant and termwise G-injective, then ff : Y* * o! ~~Y(as in 6.11) o o * *~o induces an equivalence TotY o' Tot~~Yand isomorphisms Qsßi(TotY ; M) ~=Qsßi(Tot* *~Y; M) and o Es,tr(Y o; M) ~=Es,tr(~~Y; M) for a cofibrant M 2 C, 0 s t, 2 r 1+, and* * i 0. Proof.Since Y nis G-fibrant, the G-resolution ff : Y n! ~Y nois a cosimplicial * *homotopy equivalence such that Y nis a strong deformation retract of ~Y nofor n 0. Thus ff : Y o! * *Totv~Y oois a Reedy weak equivalence of Reedy fibrant objects by Proposition 2.13 and Lemma 6.16, a* *nd o TotY o-! TotTotv~Y oo~=Tot~~Y 26 is an equivalence by Lemma 6.15 as desired. For the remaining conclusions, it * *suffices to show o that map(M, Y o) ! map(M, ~~Y) satisfies the hypotheses (i)-(iii) of Lemma 6.17* *. This follows by double complex arguments since map(M, Y n) ! map(M, ~Y)nois a cosimplicial homo* *topy equiva- lence such that map(M, Y n) is a strong deformation retract of map(M, ~Y)nofor * *n 0, and hence this homotopy equivalence induces: (i) a ß0ß0-isomorphism; (ii) a Totßgd1-equiv* *alence; and (iii) a ß*ßt(-, b)-isomorphism for each vertex b 2 Tot2map(M, Y o) and t 2. In (iii) * *we note that the vertex b determines a map b : sk2 n ! map(M, Y n) which provides a sufficiently* * well defined basepoint for map(M, Y n) since the space sk2 n is simply connected. * * |__| 6.19. Proof of Theorem 6.5. The proof of Theorem 6.2 is easily adapted to give * *Theorem 6.5 using Lemma 6.18 in place of Lemma 6.12. * *|__| 6.20. Immediate generalizations. In 5.9, we explained how the notions of G-reso* *lution, right derived functor, and G-homotopy spectral sequence apply not merely to objects A* * 2 C but also to objects Ao2 c C. Similarly, we may now define a weak G-resolution of an object * *Ao2 c C to be a G- equivalence Ao! Y osuch that Y ois termwise G-injective. Then the results 6.2-6* *.7 have immediate generalizations where: A, B 2 c C are replaced by Ao, Bo 2 c C; G-injective is * *replaced by termwise G-injective; and ^LGA is replaced by TotGAo. 7.Triples give weak resolutions We now explain how weak G-resolutions may be constructed from suitable triple* *s, and give some examples. We can often show that our weak G-resolutions are actual G-resolution* *s, but that seems quite unnecessary. 7.1. Triples and triple resolutions. Recall that a triple or monad < , j, OE> o* *n a category M consists of a functor : M ! M with transformations j : 1M ! and ~ : ! * * satisfying the identity and associativity conditions. For an object M 2 M, the triple resoluti* *on ff : M ! oM 2 c M is the augmented cosimplicial object with ( oM)n = n+1M and di = ij n-i+1: ( oM)n ! ( oM)n+1 si = i~ n-i: ( oM)n+1! ( oM)n for n -1. The augmentation map ff : M ! oM 2 c M is given by d0: M ! ( oM)0.* * An object I 2 M is called -injective if j : I ! I has a left inverse. 27 Lemma 7.2. For a triple < , j, ~> on M and object M 2 M, the triple resolution * *ff : M ! oM induces a weak equivalence ff*: Hom( oM, I) ! Hom(M, I) in S for each -injecti* *ve I 2 M. Proof.Since I is a retract of I, it suffices to show that ff*: Hom( oM, I) ! * *Hom(M, I) is a weak equivalence. This follows by Lemma 7.3 below since the augmented simplicial set* * Hom ( oM, I) admits a left contraction s-1 with s-1f = ~( f) for each simplex f. * * |__| For an augmented simplicial set K with augmentation operator d0: K0! K-1, a l* *eft contraction consists of functions s-1 : Kn ! Kn+1for n -1 such that, in all degrees, ther* *e are identities d0s-1= 1, di+1s-1= s-1difor i 0, and sj+1s-1= s-1sjfor j -1. As shown in [2* *9, p.190], we have Lemma 7.3. If K admits a left contraction, then the augmentation map K ! K-1 is* * a weak equivalence in S. Now suppose that C is a left proper pointed model category with a given class* * G of injective models in HoC. Theorem 7.4.Let < , j, ~> be a triple on C such that j : A ! A is G-monic with* * A G-injective for each A 2 C. If : C ! C preserves weak equivalences, then the triple resol* *ution : A ! oA is a weak G-resolution for each A 2 C. Proof.Since ( oA)n = n+1A is G-injective for n 0, it suffices to show that f* *f : A ! oA induces a weak equivalence ff* : [ oA, tG] ! [A, tG] in S for each G 2 G and t 0. T* *his follows by Lemma 7.2 since < , j, ~> gives a triple on HoC such that each tG is -injecti* *ve. |__| Various authors including Barr-Beck [2], Bousfield-Kan [18], and Bendersky-Th* *ompson [7] have used triple resolutions to define right derived functors, completions, or homot* *opy spectral sequences, and we can now fit these constructions into our framework. Starting with a trip* *le, we shall find a compatible class of injective models giving 7.5. An interpretation of triple resolutions. Let M be a left proper pointed mo* *del category, and let < , j, ~> be a triple on M such that preserves weak equivalences. The* *n there is an induced triple on HoM which is also denoted by < , j, ~>. For each X 2 HoM, we suppose: (i) X is a group object in HoM; (ii) X is -injective in HoM. 28 Now G = { X | X 2 HoM} is a class of injective models in HoM, and we can interp* *ret the triple resolution ff : A ! oA of A 2 M as a weak G-resolution by Theorem 7.4. 7.6. The discrete case. Suppose M is a pointed category with finite limits and * *colimits, and suppose < , j, ~> is a triple on M such that X is a group object in M for each* * X 2 M. The above discussion now applies to the discrete model category M and allows us to * *interpret the triple resolution ff : A ! oA of A 2 M as a weak G-resolution where G = { X | X 2 M}.* * Thus if T : C ! N is a functor to an abelian category N or to N = Grp or to N = Set*, t* *hen we obtain RsGT(A) = ßsT( oA) thereby recovering the right derived functors of Barr-Beck [* *2] and others. 7.7. The Bousfield-Kan resolutions. For a ring R, there is a triple o* *n the model category S* of pointed spaces where (RX)n is the free R-module on Xn modulo the* * relation [*] = 0. This satisfies the conditions of 7.5, so that we may interpret the Bou* *sfield-Kan resolution ff : A ! RoA 2 c S* as a weak G-resolution of A 2 S* where G = {RX | X 2 Ho*} o* *r equiva- lently G = { 1 N | N is an HR-module spectrum} as in 4.9. Thus we recover the B* *ousfield-Kan R-completion R1 X ' ^LGX and the accompanying homotopy spectral sequence. More * *generally, we consider 7.8. The Bendersky-Thompson resolutions. For a ring spectrum E, there is an obv* *ious triple on Ho*carrying a space X to 1 (E ^ 1 X). In [7, Proposition 2.4], Bendersky a* *nd Thompson suppose that E is represented by an S-algebra [27], and they deduce that the ab* *ove homotopical triple is represented by a topological triple, and hence by a triple * *on S*. This triple satisfies the conditions of 7.5, so that we may interpret the Bendersky-Thompson resoluti* *on A ! EoA 2 c S* as a weak G-resolution of A 2 S*, where G = {EX | X 2 Ho*} or equivalently (see* * 4.9) where G is the class { 1 N | N is an E-module spectrum} or the class { 1 (E^Y ) | Y 2 Hos}. Th* *us we recover the Bendersky-Thompson E-completion ^XE' ^LGX and the accompanying homotopy spectra* *l sequence {Es,tr(A; M)E} = {Es,tr(A; M)G} over an arbitrary ring spectrum E which need no* *t be an S-algebra. As pointed out by Dror Farjoun [22, p.36], Libman [34], and Bendersky-Hunton [6* *], this generality can also be achieved by using restricted cosimplicial E-resolutions without cod* *egeneracies. However, we believe that codegeneracies remain valuable; for instance, they are essentia* *l for our constructions of pairings and products in these spectral sequences [16]. We remark that these* * various alternative constructions of homotopy spectral sequences over a ring spectrum all produce e* *quivalent E2-terms and almost surely produce equivalent spectral sequences from that level onward.* * Finally we consider 29 7.9. The loop-suspension resolutions. For a fixed integer n 1, we let < , j, * *~> be a triple on S*representing the n-th loop-suspension triple n n on Ho*. This satisfies the * *conditions of 7.5, so that we may interpret the n-th loop-suspension resolution A ! oA 2 c S* as a w* *eak G-resolution of A 2 S* where G = { n nX | X 2 Ho*} or equivalently where G = { nY | Y 2 Ho*}* *. The n-th loop-suspension completion of A is now given by Tot_oA_' ^LGA, and will be iden* *tified in 9.9. 8. Triple structures of completions Let C be a left proper, bicomplete, pointed simplicial model category with a * *class G of injective models in HoC. We now show that the G-completion functor ^LG: HoC ! HoC and tra* *nsformation ff : 1 ! ^LGbelong to a triple on HoC, and we introduce notions of G-completene* *ss, G-goodness, and G-badness in HoC. This generalizes work of Bousfield-Kan [18] on the R-comp* *letion functor R1 : Ho*! Ho*where R is a ring. By 2.7 and 2.8, the functor o - : C ! c C is left adjoint to Tot: c C ! C, * *and these functors become Quillen adjoint (4.7) when c C is given the Reedy model category structu* *re. This remains true when c C is given the G-resolution model category structure by Proposition 8.1.The functors o - : C ! c CG and Tot: c CG ! C are Quillen adj* *oint. Proof.For a cofibration (resp. trivial cofibration) A ! B in C, it suffices by* * Corollary 3.15 to show that the Reedy cofibration A o ! B o induces a fibration (resp. tri* *vial fibration) Hom C(B o, F) ! HomC(A o, F) in S for each G-injective fibrant object F 2* * C. This follows from the axiom SM7 on C, since this fibration is just map(B, F) ! map(A, F). * * |__| The resulting adjoint functors L(- o) : HoC ø Ho(c CG) : R Tot will be denoted by con: HoC ø Ho(c CG) : TotG. Thus, for A 2 HoC and Xo 2 Ho(c CG), we have con(A) ' A 2 Ho(c CG) and TotGXo '* * Tot~Xo where Xo ! ~Xois a G-fibrant approximation to Xo. Corollary 8.2.The G-completion functor ^LG: HoC ! HoC and transformation ff : 1* * ! ^LGbelong to a triple <^LG, ff, ~> on HoC. Proof.We easily check that ^LGand ff belong to the adjunction triple of the abo* *ve functors conand TotG. |__| 30 Definition 8.3.An object A 2 HoC is called G-complete if ff : A ' ^LGA; A is ca* *lled G-good if ^LGA is G-complete; and A is called G-bad if ^LGA is not G-complete. A G-injective object of HoC is G-complete by Corollary 6.6, and a G-complete * *object is clearly G-good. To study these properties, we need Lemma 8.4. For a map f : A ! B in HoC, the following are equivalent: (i)f : A ! B is a G-equivalence (see 6.4); (ii)^LGf : ^LGA ' ^LGB; (iii)f* : [B, I] ~=[A, I] for each G-complete object I 2 HoC. Proof.We have (i) ) (ii) by Corollary 6.7. To show (ii) ) (iii), note that a ma* *p u : A ! I extends to a map ff-1I(^Lu)(^Lf)-1ffB : B ---! I so f* is onto; and note that if u : A ! I extends to a map r : B ! I, then r = ff-1I(^Lr)ffB = ff-1I(^Lr)(^Lf)(^Lf)-1ffB = ff-1I(^Lu)(^Lf)-1ffB so f* is monic. To show (iii) ) (i), note that nG is G-complete for each G 2 G* * and n 0, since it is G-injective. |_* *_| Proposition 8.5.An object A 2 HoC is G-good if and only if ff : A ! ^LGA is a G* *-equivalence. Proof.If either of the maps ff, ^Lff : ^LA ! ^L^LA is an equivalence in HoC, th* *en so is the other since they have the same left inverse ~ : ^L^LA ! ^LA. The result now follows from Le* *mma 8.4. |__| Thus the G-completion ff : A ! ^LGA of a G-good object A 2 HoC may be interpr* *eted as the localization of A with respect to the G-equivalences (see [9, 2.1]), and the G-* *completion functor is a reflector from the category of G-good objects to that of G-complete objects in * *HoC. In contrast, for G-bad objects, we have Proposition 8.6.If an object A 2 HoC is G-bad, then so is ^LGA. Proof.Using the triple structure <^L, ff, ~>, we see that the map ff : ^LA ! ^L* *^LA is a retract of ff : ^L^LA ! ^L^L^LA. Hence, if the first map is not an equivalence, then the s* *econd is not. |__| 31 8.7. The discrete case. Let M be a bicomplete pointed category, viewed as a dis* *crete model category (4.3), with a class G of injective models in HoM = M, and let I M be* * the full subcategory of G-injective objects in M. By Lemma 8.8 below for A 2 M, there i* *s a natural isomorphism ^LGA ~= lim I 2 M f:A!I where f ranges over the comma category A # I, and the G-completion ff : A ! ^LG* *A is the canonical map to this limit. Hence, ^LG: M ! M is a right Kan extension of the inclusion * *functor I ! M along itself, and may therefore be viewed as a codensity triple functor (see [3* *6, X.7]). We have used Lemma 8.8. For A 2 M, there is a natural isomorphism ^LGA ~=limf:A!II where f r* *anges over A # I. Proof.Let ff : A ! Jo be a G-resolution of A in M. Then Jn 2 I for n 0 by Lem* *ma 5.3, and ff*: Hom(Jo, I) ! Hom(A, I) is a trivial fibration in S for each I 2 I by Corol* *lary 3.15. Thus the maps ff : A ! J0 and d0, d1 : J0 ! J1 satisfy the conditions: (i) J0, J1 2 I; (* *ii) d0ff = d1ff; (iii) if f : A ! I 2 I, then there exists ~f: J0 ! I with ~fff = f; and (iv) if g0, g* *1 : J0 ! I 2 I and g0ff = g1ff, then there exists ~g: J1 ! I with ~gd0 = g0 and ~gd1 = g1. Hence, * *limf:A!II is the equalizer of d0, d1: J0 ! J1, which is isomorphic to TotJo ~=^LGA. * * |__| 8.9. The Bousfield-Kan case with an erratum. By 7.7 and Corollary 8.2, the Bous* *field-Kan R-completion ff : X ! R1 X belongs to a triple on Ho*. However, we no longer be* *lieve that it belongs to a triple on S* or S, as we claimed in [18, p.26]. In that work, we c* *orrectly constructed functors Rs: S*! S* with compatible transformations 1 ! Rs and RsRs! Rs satisfy* *ing the left and right identity conditions for 0 s 1, but we now think that our transfor* *mation RsRs! Rs is probably nonassociative for s 2, because the underlying cosimplicial pairi* *ng c in [18, p.28] is nonassociative in cosimplicial dimensions 2. The difficulty arises because ou* *r "twist maps" do not compose to give actual summetric group actions on the n-fold composites R . .R.* *for n 3. The partial failure of our triple lemma in [18] does not seem to invalidate any of * *our other results, and new work of Libman [35] on homotopy limits for coaugmented functors shows that * *the functors Rs must all still belong to triples on the homotopy category Ho*. 9. Comparing different completions We develop machinery for comparing different completion functors and apply it* * to show that the Bendersky-Thompson completions with respect to connective ring spectra are * *equivalent to 32 Bousfield-Kan completions with respect to solid rings, although the associated * *homotopy spectral sequences may be quite different. We continue to let C be a left proper, bicomp* *lete, pointed simplicial model category with a class G of injective models in HoC. In addition, we suppo* *se that C is factored and that G is functorial, so that the model category c CG is also factored by 4* *.2. Thus the G- completion functor ^LGis defined on C (not just HoC) by 5.7. We start by expre* *ssing the total derived functor TotG: Ho(c CG) ! HoC of 8.2 in terms of the prolonged functor ^* *LG: c C ! c C with (^LGXo)n = ^LG(Xn) for n 0 and the homotopical Totfunctor Tot_: c C ! C with * *Tot_Xo = TotX_o, where X_ois a functorial Reedy fibrant replacement of Xo 2 c C. Theorem 9.1.For Y o2 c C, there is a natural equivalence TotGY o' Tot_(^LGY o) * *in HoC. Proof.As in 6.11, let con(Y o) ! ~Y oobe the functorial Reedy G-resolution of * * con(Y o). This o ~o induces a G-equivalence of diagonals Y o! ~~Ywith ~YG-fibrant and therefore ind* *uces o oo v oo v oo TotGY o' Tot~~Y= Totdiag~Y ' TotTot ~Y ' Tot_Tot~Y in HoC by Lemma 6.15. Now let Y n! ~Y nobe the functorial G-resolution of Y nfo* *r n 0, and functorially factor con(Y o) ! ~Y oointo a Reedy G-trivial cofibration con(Y o)* * ! Kooand a Reedy G-fibration Koo! ~Y.ooNext choose a map Koo! ~Y ooextending con(Y o) ! ~Y.ooSin* *ce the maps Y n! ~Y no, Y n! ~Y no, and Y n! Kno are G-resolutions for n 0, the maps ~Y n* *o Kno! ~Y no are Tot-equivalences, and we obtain Tot_-equivalences Totv~Y oo----TotvKoo---! Totv~Y oo' ^LGY o which combine to give Tot_Totv~Y'ooTot_(^LGY o) in HoC. This completes our chai* *n of equivalences from TotGY oto Tot_(^LGY o). |__| Corollary 9.2.A G-equivalence Xo ! Y oin c C induces an equivalence Tot_(^LGXo)* * ' Tot_(^LGY o) in HoC. This follows immediately from Theorem 9.1 and specializes to give Corollary 9.3.For an object A 2 C, each G-equivalence A ! Y oin c C induces an * *equivalence ^LGA ' Tot_(^LGY o) in HoC. Definition 9.4.A G-complete expansion of an object A 2 C consists of a G-equiva* *lence A ! Y oin c C such that Y nis G-complete for n 0. Each weak G-resolution of A is a G-complete expansion of A, and the completio* *n part of Theorem 6.5 now generalizes to 33 Theorem 9.5.If A ! Y ois a G-complete expansion of an object A 2 C, then there * *is a natural equivalence ^LGA ' Tot_Y oin HoC. Proof.By Corollary 9.3, the maps ^LGA -! Tot_^LGY o- Tot_Y oare weak equivalen* *ces in C. |__| By this theorem, any functorial G-complete expansion of the objects in C give* *s a G-completion functor on C which is "essentially equivalent" to ^LGsince it is related to ^LG* *by natural weak equivalences. The following theorem will show that different choices of G may g* *ive equivalent G- completion functors even when they give very different G-homotopy spectral sequ* *ences. Theorem 9.6.Suppose G and G0are classes of injective models in HoC. If each G-i* *njective object is G0-injective and each G0-injective object is G-complete, then there is a nat* *ural equivalence ^LGA ' ^LG0A for A 2 C. Proof.Let A ! Jo be a G0-resolution of A. Then A ! Jo is a G-trivial cofibratio* *n by Corollary 3.15, and Jo is termwise G0-injective. Hence A ! Jo is a G-complete expansion o* *f A, and ^LGA ' Tot_Jo ' ^LG0A by Theorem 9.5. |* *__| For example, consider the Bendersky-Thompson completion A ! ^AEof a space A w* *ith respect to a ring spectrum E as in 7.8. Suppose E is connective (i.e., ßiE = 0 for i < * *0), and suppose the ring ß0E is commutative. Let R = core(ß0E) be the subring R = {r 2 ß0E | r 1 = 1 r 2 ß0E ß0E}, and recall that R is solid (i.e., the multiplication R R ! R is an isomorphis* *m) by [8]. Theorem 9.7.If E is a connective ring spectrum with commutative ß0E, then there* * are natural equivalences ^AE' (ß0E)1 A ' R1 A for A 2 Ho*where R = core(ß0E). Proof.Let G0(resp. G) be the class of all 1 N 2 Ho*for E-module (resp. Hß0E-mo* *dule) spec- tra N. Then G G0 since each Hß0E-module spectrum is an E-module spectrum via * *the map E ! Hß0E, and hence each G-injective space is G0-injective. If N is an E-module* * spectrum, then (ß0E)1 1 N ' 1 N by [18, II.2]. Hence each G0-injective space J is G-complete* *, since it is a retract of 1 N for N = E ^ 1 J. Consequently, ^AE' ^LG0A ' ^LGA ' (ß0E)1 A by* * Theorem 9.6, and (ß0E)1 A ' R1 A by [18, p.23]. |* *__| 34 9.8. Examples of E-completions. In [8] and [10, 6.4], we determined all solid r* *ings R, and they are: (I) R = Z[J-1] for a set J of primes; (II) R = Z=n for n 2; (III) R = Z[* *J-1] x Z=n for n 2 Q and a set J of primes including the factors of n; and (IV) R = core(Z[J-1] x p* *2KZ=pe(p)) for infinite sets K J of primes and positive integers e(p). In [18, I.9], we show* *ed that the completions R1 X in cases (I)-(III) can be expressed as products of their constituent compl* *etions Z[J-1]1 X and (Z=p)1 X for the prime factors p of n, and we extensively studied these bas* *ic completions. We found that a nilpotent space X is always R-good in cases (I) and (II), but is ü* * sually" R-bad in cases (III) and (IV). For instance K(Z, m) for m 1 is R-bad in cases (III) and (IV)* *. These results are now applicable to the completion ^XEof a space X 2 Ho*with respect to a connect* *ive ring spectrum E with ß0E commutative. For instance, we have ^XE' Z1 X for E = S and ^XE' Z(p)* *1X for E = BP. 9.9. The loop-suspension completions. We may also apply Theorem 9.6 to reprove * *the result that loop-suspension completions of spaces are equivalent to Z-completions. In * *more detail, for a fixed integer n 1, we consider the n-th loop-suspension completion (7.9) of a* * space A 2 S* given by ^LGA where G = { nY | Y 2 Ho*}, and we compare it with the Bousfield-Kan Z-c* *ompletion ^LHA ' Z1 A where H = { 1 N | N is an H-module spectrum}. Since the G-injective* * spaces are the retracts of the n-fold loop spaces, they have nilpotent components and are * *H-complete. Thus, since H G, Theorem 9.6 shows ^LGA ' ^LHA, and the nth loop-suspension complet* *ion of A is equivalent to Z1 A. 10.Bendersky-Thompson completions of fiber squares Let C remain a left proper, bicomplete, pointed simplicial model category wit* *h a class G of injective models in HoC. Also suppose that C is factored and G is functorial so that the * *G-completion functor ^LGis defined on C (not just HoC) by 5.7. In this section, we show that ^LGpres* *erves fiber squares whose "G-cohomology cobar spectral sequences collapse strongly,ä nd we special* *ize this result to the Bendersky-Thompson completions (see Theorems 10.11 and 10.12). We need a we* *ak assumpton on 10.1. Smash products in HoC. For A, B 2 HoC, let A ^ B 2 HoC be the smash produ* *ct rep- resented by the homotopy cofiber of the coproduct-to-product map A _ B ! A x B * *for cofibrant- fibrant objects A, B 2 C. We assume that the functor - ^ B : HoC ! HoC has a ri* *ght adjoint (-)B : HoC ! HoC. This holds as usual in HoS*= Ho*, and it is easy to show 35 Lemma 10.2. For an object B 2 HoC, the following are equivalent: (i)if a map X ! Y in HoC is G-monic, then so is X ^ B ! Y ^ B; (ii)if an object I 2 HoC is G-injective, then so is IB ; (iii)for each G 2 G and i 0, the object ( iG)B is G-injective. Definition 10.3.An object B 2 HoC will be called G-flat (for smash products) wh* *en it satisfies the equivalent conditions of Lemma 10.2. An object B 2 C (resp. Bo 2 c C) will * *also be called G-flat when B (resp. each Bn) is G-flat in HoC. Lemma 10.4. If f : Xo ! Y oand g : Bo ! Co are G-equivalences of termwise fibra* *nt objects in c C such that Y oand Bo are G-flat, then f x g : Xox Bo ! Y ox Co is also a G-e* *quivalence. Proof.Working in c(Ho C) instead of c C, we note that f ^Bn : Xo^Bn ! Y o^Bn is* * a G-equivalence for n 0 by Lemma 10.5 below, since ( iG)Bn 2 HoC is a G-injective group objec* *t for each G 2 G and i 0. Hence, f ^ Bo : Xo ^ Bo ! Y o^ Bo is a G-equivalence as in the proof* * of Lemma 6.9. Similarly Y o^g : Y o^Bo ! Y o^Co is a G-equivalence, and hence so is f ^g : Xo* *^Bo ! Y o^Co. Thus the ladder Xo_ Bo ----! Xox Bo ----!Xo^ Bo ?? ? ? yf_g ?yfxg ?yf^g Y o_ Co----! Y ox Co----! Y o^ Co is carried by [-, G]* to a ladder of short exact sequences of simplicial groups* * such that (f _ g)* and (f ^ g)* are weak equivalences. Consequently (f x g)* is a weak equivalence. * * |__| We have used Lemma 10.5. If f : Xo ! Y o2 c C is a G-equivalence and I 2 HoC is a G-injectiv* *e group object, then f* : [Y o, I]*! [Xo, I]* is a weak equivalence of simplicial groups. Proof.The class of G-monic maps in HoC is clearly the same as the class of G0-m* *onic maps for G0= G[{I}. Hence, G and G0give the same model category structure on c C by 4.1,* * and f : Xo ! Y o is a G0-equivalence in c C. * *|__| Theorem 10.6.Suppose the G-injectives in HoC are G-flat. If A, B, M 2 HoC are o* *bjects with A or B G-flat, then there is a natural equivalence ^LG(AxB) ' ^LGAxL^GB and a nat* *ural isomorphism Es,tr(A x B; M)G ' Es,tr(A; M)G x Es,tr(B; M)G for 2 r 1+ and 0 s t. 36 Proof.We may suppose A and B are fibrant in C and take G-resolutions A ! ~Aoand* * B ! ~Boin c C. Then the product A x B ! ~Aox ~Bois a weak G-resolution by Lemma 10.4, and* * the result follows from Theorem 6.5. |__| We now study the action of ^LGon a commutative square C ----! B ?? ? (10.7) y ?y A ----! of fibrant objects in C using 10.8. The geometric cobar construction. Let B(A, , B)o 2 c C be the usual geom* *etric cobar construction with B(A, , B)n = A x x . .x. x B for n 0 where the factor occurs n times (see [40]). It is straightforward t* *o show that B(A, , B)o is Reedy fibrant with TotB(A, , B)o ~=P(A, , B) where P(A, , B) is the double mapping path object defined by the pullback P(A, , B)----!hom( 1, ) ?? ? y ?y A x B ----! x . Thus P(A, , B) represents the homotopy pullback of the diagram A ! B (see * *[25, x10]), and (10.7)is called a homotopy fiber square when the map C ! P(A, , B) is a weak e* *quivalence. Our main fiber square theorem for G-completions is Theorem 10.9.Suppose the G-injectives in HoC are G-flat. If (10.7)is a square o* *f G-flat fibrant objects such that the augmentation C ! B(A, , B)o is a G-equivalence, then ^LG* *carries (10.7)to a homotopy fiber square. Proof.Since C ! B(A, , B)o is a G-equivalence, it induces an equivalence ^LGC * *' Tot_^LGB(A, , B)o by Corollary 9.3, and there are equivalences Tot_^LGB(A, , B)o ' Tot_B(^LGA, ^LG , ^LGB)o ' P(^LGA, ^LG , ^LGB) by Theorem 10.6. Hence, ^LGC is equivalent to the homotopy pullback of ^LGA ! ^* *LG ^LGB. |__| 37 The hypothesis that the augmentation C ! B(A, , B)o is a G-equivalence may b* *e reformulated to say that the G-cohomology cobar spectral sequences collapse strongly for (10* *.7), although we shall not develop that viewpoint here. 10.10. The Bendersky-Thompson case. For a commutative ring spectrum E, we consi* *der the Bendersky-Thompson E-completion A ! ^AE= ^LGA of a space A 2 S* with respect to* * the class G of all 1 N 2 Ho*for E-module spectra N as in 7.8. All spaces in Ho*are now G-f* *lat, and Theorem 10.9 will apply to the square (10.7)provided that the N*-cobar spectral sequenc* *e collapses strongly for each N in the sense that ( *C for s = 0 ßsN*B(A, , B)o ~= N 0 for s > 0. Here we may assume that N is an extended E-module spectrum since any N is a hom* *otopy retract of E ^ N. To eliminate N from our hypotheses, we suppose: (i)E satisfies the Adams UCT condition that the map N*X ! Hom*E*(E*X, ß*N) is an isomorphism for each X 2 Ho*with E*X projective over E* and each extended E-module spectrum N; (ii)E*A, E* , E*B and E*C are projective over E*. Condition (i) holds for many common ring spectra E, including the p-local ring * *spectrum K and arbitrary S-algebras by [1, p.284] and [27, p.82]. Condition (ii) implies that E*B(A, , B)n ~=E*A E*E* E*. . .E*E*B is projective over E* for n 0, and we say that the E*-cobar spectral sequence* * collapses strongly when E*C ! E*B(A, , B)o is split exact as a complex over E*. Now Theorem 10.9 * *implies Theorem 10.11.Suppose E is a commutative ring spectrum satisfying the Adams UCT* *-condition. If the spaces of (10.7)have E*-projective homologies and the E*-cobar spectral * *sequence collapses strongly, then the Bendersky-Thompson E-completion functor carries (10.7)to a h* *omotopy fiber square. Specializing this to E = K, we suppose that the spaces of (10.7)have K*-free * *homologies, and we say that the K*-cobar spectral sequence collapses strongly if ( CotorK*s(K*A, K*B) = K*C for s = 0 0 for s > 0. Now Theorem 10.11 reduces to 38 Theorem 10.12.If the spaces of (10.7)have K*-free homologies and the K*-cobar s* *pecral sequence collapses strongly, then the Bendersky-Thompson K-completion functor carries (1* *0.7)to a homotopy fiber square. This result is applied by Bendersky and Davis in [5]. 11.p-adic K-completions of fiber squares Working at an arbitrary prime p, we now consider a p-adic variant of the Bend* *ersky-Thompson K- completion of spaces and establish an improved fiber square theorem for it. We * *also briefly consider the associated homotopy spectral sequence which seems especially applicable to * *spaces whose p-adic K-cohomologies have Steenrod-Epstein-like U(M) structures as in [13]. We first * *recall 11.1. p-completions of spaces and spectra. For a space A 2 S*, we let ^A= AH=pb* *e the p- completion given by the H=p*-localization of [9]. When A is nilpotent, this is * *equivalent to the p-completion (Z=p)1 A of [18]. For a spectrum E, we likewise let ^E= ES=pbe the* * p-completion given by the S=p*-localization of [10]. Thus, when the groups ß*E are finitely * *generated, we have ß*^E= ß*E ^Zpusing the p-adic integers ^Zp. We now introduce 11.2. The p-adic K-completion. For a space A 2 S*, let ^AK^= ^LGK^A be the ^K*-* *completion obtained using the class of injective models GK^ = {K^_n}n2Z = {U^, ^Zpx dBU} Ho* representing the p-adic K-cohomology theory ^K*(-) = K*(-; ^Zp) as in 4.6. Als* *o consider the associated homotopy spectral sequence {Es,tr(A; M)K^} = {Es,tr(A; M)^G} with re* *spect to ^K*for A, M 2 S*. 11.3. Comparison with the Bendersky-Thompson K-completion. For the p-local ring* * spec- trum K and a space A 2 S*, let ^AK= ^LGKA be the Bendersky-Thompson K-completio* *n obtained using the class of injective models GK = { 1 N | N is a K-module spectrum} Ho*. or using the triple K(-) : S* ! S* as in 7.8. Also consider the associated hom* *otopy spectral sequence {Es,tr(A; M)K } = {Es,tr(A; M)GK} for A, M 2 S*. Since GK^ GK , ther* *e is a natural map ^AK! ^AK^constructed as follows for a space A 2 S*. First take a GK -resolu* *tion A ! Io of A and then take a GK^-resolution Io ! Jo of Io in c S*. Since the composed map * *A ! Jo is a GK^-resolution of A, the map Io ! Jo induces the desired map ^AK' TotIo ! TotJo* * ' ^AK^. It also 39 induces a map {Es,tr(A; M)K } ! {Es,tr(A; M)K^} of homotopy spectral sequences * *for A, M 2 S*. The following theorem will show that these maps are ä lmost p-adic equivalences* *" when K*(A; ^Zp) is torsion-free. For a space Y 2 Ho*, let Y 2 Ho* be the n-connected sectio* *n of Y , and let Y <~n> 2 Ho*be the section with 8 ><ßiY for i > n ßiY <~n> = > (ßnY )~for i = n : 0 for i < n where (ßnY )~is the divisible part of ßnY assuming n 2. Theorem 11.4.If A, M 2 S*are spaces with K*(A; ^Zp) torsion-free and with ~H*(M* *; Q) = 0, then: (i)^AK^<2> is the p-completion of ^AK<~2>; (ii)[M, ^AK]*~=[M, ^AK^]*; (iii)Es,tr(A; M)K ~=Es,tr(A; M)K^for 0 s t and 2 r 1+. This will be proved in 11.18. For a space A as above, we may actually constru* *ct the ^K*-completion of A and the associated homotopy spectral sequence quite directly from the Bend* *ersky-Thompson triple resolution A ! KoA of 7.8. We simply apply the p-completion functor to g* *ive a map A ! [KoA in c S* and obtain Theorem 11.5.If A 2 S* is a space with K*(A; ^Zp) torsion-free, then A ! [KoAis* * a weak GK^- resolution of A. Hence ^AK^' Tot_([KoA) and Es,tr(A; M)K^~=Es,tr([KoA; M) for M* * 2 S*, 0 s t, and 2 r 1+. This will be proved in 11.17. We now turn to our fiber square theorem for the* * ^K*-completion. For a commutative square C ----! B ?? ? (11.6) y ?y A ----! we say that the K*(-; Z=p)-cobar spectral sequence collapses strongly when ( CotorK*(s;Z=p)(K*(A; Z=p), K*(B; Z=p)) = K*(C; Z=p)for s = 0 0 otherwise. Theorem 11.7.If the spaces in (11.6)have torsion-free K*(-; ^Zp)-cohomologies a* *nd the K*(-; Z=p)- cobar spectral sequence collapses strongly, then the p-adic K-completion functo* *r carries (11.6)to a homotopy fiber square. 40 This will be proved below in 11.12 using our general fiber square theorem 10.* *9. It applies to a broader range of examples than its predecessor Theorem 10.12 for the Bendersk* *y-Thompson K- completion, and we remark that its strong collapsing hypothesis holds automatic* *ally by [12, Theorem 10.11] whenever the spaces are connected and the coalgebra map K*(B; Z=p) ! K*(* * ; Z=p) belongs to an epimorphism of graded bicommutative Hopf algebras (with possibly artifici* *al multiplications). We devote the rest of this section to proving the above theorems, and we must f* *irst show that the GK^-injectives in Ho*are GK^-flat. Lemma 11.8. If B 2 Ho*is a space with K*(B; ^Zp) torsion-free, then B is GK^-fl* *at. Proof.A map f : X ! Y is GK^-monic if and only if f* : ~K*(Y ; ^Zp) ! ~K*(X; ^Z* *p) is onto, and this property is inherited by f ^ B : X ^ B ! Y ^ B since K*(B; ^Zp) is torsion-free* *. Thus B is GK^-flat by Definition 10.3. |__| Lemma 11.9. If E 2 Hosis a spectrum with K*(E; ^Zp) torsion-free, then K*( 1 E;* * ^Zp) is also torsion-free. Proof.This follows by [12, Theorem 8.3]. * * |__| Lemma 11.10. If E 2 Hos is a K-module spectrum with ß*E torsion-free, then K*(E* *; ^Zp) is torsion-free. Proof.We have K*(E; ^Zp) = K*(øpE; ^Zp) where øpE = E ^ S-1Zp1 is the p-torsion* * part of E, and we note that øpE is a K-module spectrum with ß*øpE divisible p-torsion. Hence ß* **øpE is a direct sum of groups Zp1, and øpE is a wedge of suspensions of øpK. Since K*(øpK; ^Zp)* * = K*(K; ^Zp) is torsion-free, we conclude that K*(E; ^Zp) is torsion-free. * * |__| Lemma 11.11. The GK^-injectives in Ho*are GK^-flat. Proof.This follows from Lemmas 11.8-11.10 since each GK^-injective in Ho*is a r* *etract of 1 E for some product E of spectra ^Kand K^. |* *__| 11.12. Proof of Theorem 11.7. By Lemmas 11.8 and 11.11, the GK^-flatness hypoth* *eses of The- orem 10.9 are satisfied, and it suffices to show that C ! B(A, , B)o is a GK^-* *equivalence. We know that the functors K*(-; Z=p) and K*(-; Z=p) carry C ! B(A, , B)o to acyclic co* *mplexes. Hence, the torsion-free complex K*(B(A, , B)o; ^Zp) ! K*(C; ^Zp) is also acyclic, and* * C ! B(A, , B)o is a GK^-equivalence. |__| 41 To prepare for the proof of Theorem 11.5, we recall that the stable p-complet* *ion functor (b-) of 11.1 is right adjoint to the stable p-torsion part functor øp of 11.10 with nat* *ural isomorphisms [øpX, øpY ]* ~=[øpX, Y ]* ~=[X, ^Y]* ~=[X^, ^Y]* for X, Y 2 Hosby [14, 3.5 and 7.3]. Moreover, a spectrum Y is p-complete if and* * only if the groups ß*Y are Ext-p-complete. Lemma 11.13. If f : X ! Y 2 Hos is a map of spectra with f* : K*(Y ; ^Zp) ! K*(* *X; ^Zp) onto, then f* : [Y, M]* ! [X, M]* is onto for each p-complete K-module spectrum* * M with ß*M torsion-free. Proof.Since øpM is a K-module spectrum with ß*øpM divisible p-torsion, there is* * a natural iso- morphism [X, M]* ~=[X, ^M]* ~=[øpX, øpM]* ~=Hom *K*(K*øpX, ß*øpM) for X 2 Hos. The lemma now follows since ß*øpM is K*-injective and since f* : K* **øpX ! K*øpY is monic because it is Pontrjagin dual to the epic map f* : K*+1(Y ; ^Zp) ! K*+* *1(X; ^Zp) by [14, 10.1]. |__| Lemma 11.14. If M 2 Hos is a p-complete K-module spectrum with ß*M torsion-free* *, then 1 M 2 Ho*is GK^-injective and splits as a product 1 M ' K(ß0M, 0) x K(ß1M, 1) x K(ß2M, 2) x 1 M<2>. Proof.As in 4.5, we obtain a map f : M ! E 2 Hoswhere E is a product of spectra* * ^Kand K^, and where f* : K*(E; ^Zp) ! K*(M; ^Zp) is onto. Then M is a retract of E in Hos* *by Lemma 11.13, and the lemma holds for M since it holds for E by [38, Lemma 2.1]. In more deta* *il, we obtain the desired splitting of 1 M by constructing a left inverse of 1 M ! 1 M for m 2 using a left inverse of 1 E ! 1 E. * * |__| Lemma 11.15. If N 2 Hosis a K-module spectrum with ß*^Ntorsion-free, then the s* *pace " 1 Nis GK^-injective. Proof.The map ßi" 1 N! ßi 1 ^Nis an isomorphism for i 2 and a monomorphism to* * a direct summand for i = 1. Thus by Lemma 11.14 with M = ^N, the space " 1 N<0> is a ret* *ract of 1 ^N<0>, and both spaces are GK^-injective. The lemma now follows since " 1 N' " 1 N<0> * *x K(ß0N, 0) and since K(ß0N, 0) is also GK^-injective because it is discrete. * * |__| 42 Lemma 11.16. If A 2 Ho* is a space with K*(A; ^Zp) torsion-free, then the space* * dKA is GK^- injective. Proof.Recall that KA ' 1 (K ^ 1 A). The groups ß*øp(K ^ 1 A) are divisible p* *-torsion since they are Pontrjagin dual to K*+1(A; ^Zp). Hence, the groups ß*(K "^ 1)Aare tor* *sion-free, and the lemma now follows by Lemma 11.15 with N = K ^ 1 A. * * |__| 11.17. Proof of Theorem 11.5. Since A ! KoA is a GK -equivalence, it is also a * *GK^-equivalence, and hence so is A ! [KoA. Since K*(A; ^Zp) is torsion-free, so is each K*(K . .* *K.A; ^Zp) by Lemma 11.9. Thus the terms of [KoAare GK^-injective by Lemma 11.16, and A ! [KoAis a * *weak GK^- resolution. The final statement follows from Theorem 6.5. * * |__| 11.18. Proof of Theorem 11.4. For 0 s 1, we obtain a homotopy fiber square Tot_s(KoA)----! Tot_s([KoA) ?? ? y ?y Tot_s(KoA)(0)----!Tot_s([KoA)(0) by applying Tot_sto the termwise arithmetic square [23] of KoA. Since the lowe* *r spaces of the square are HQ*-local [9, p.192], the upper map has an HQ*-local homotopy fiber * *and induces an equivalence map*(M, Tot_s(KoA)) ' map*(M, Tot_s([KoA)) Thus by Theorem 11.5, the map ^AK! A^^Khas an HQ*-local homotopy fiber and indu* *ces an equivalence map*(M, ^AK) ' map*(M, ^AK^). The theorem now follows easily. * * |__| 12.The unpointed theory As in [28], much of the preceding work can be generalized to unpointed model * *categories. In this section, we develop such a generalization of the existence theorem for G-r* *esolution model cate- gories (Theorem 12.4), and then briefly discuss the resulting unpointed theory * *of G-resolutions, right derived functors, and G-completions. This leads, for instance, to unpointed Ben* *dersky-Thompson completions of spaces. We start with preliminaries on loop objects in unpointed* * model categories. Let C be a model category with terminal object e, and let C*= e # C denote th* *e associated pointed model category whose weak equivalences, cofibrations, and fibrations are the ma* *ps having these properties when basepoints are forgotten. The forgetful functor C*! C is a Quil* *len right adjoint of ` the functor C ! C*sending X 7! X e and has a total right derived functor HoC*!* * HoC (see 4.7). We let J : HoC*! (Ho C)* be the associated functor to the pointed category (Ho * *C)*= [e] # HoC. 43 Lemma 12.1. For a left proper model category C, the isomorphism classes of obje* *cts in HoC* correspond to the isomorphism classes of objects in (Ho C)* via the functor J. Proof.We first choose a trivial fibration ~e! e in C with ~ecofibrant. Then an * *object X 2 (Ho C)* is represented by a cofibration ~e! X in C, and the map X ! X=~eis a weak equiv* *alence since C is left proper. Hence X ' J(X=~e) in (Ho C)*. For objects W1, W2 2 HoC* with J(* *W1) ' J(W2), we may choose fibrant representatives W1, W22 C* and factor each ~e! e ! Wiinto* * a cofibration ~e! ~Wiand a trivial fibration ~Wi! Wiin C. Using the homotopy extension theore* *m [39, I.1.7] and the equivalence J(W1) ' J(W2), we obtain a weak equivalence ~W1! ~W2under ~* *e. Hence W1' ~W1=~e' ~W2=~e' W2 in HoC*. |__| 12.2. Loop objects in (Ho C)*. For a left proper model category C and n 0, th* *e ordinary n-fold loop functor n : HoC* ! HoC* now determines an n-fold loop operation n on the* * isomorphism classes of objects in (Ho C)*via the correspondence of Lemma 12.1. Thus for eac* *h object Y 2 (Ho C)*, we obtain an object nY 2 (Ho C)*defined up to isomorphism, where 0Y = Y . We * *note that nY admits a group object structure in HoC for n 1, which is abelian for n 2, s* *ince it comes from an n-fold loop object of HoC* via a right adjoint functor HoC*! HoC. For X 2 Ho* *C, we let [X, Y ]n ~=[X, nY ] ~= HomHoC(X, nY ) be the resulting homotopy set for n = 0, group for n = 1, or abelian group for * *n 2. When the original category C is pointed, we can identify C*with C, and our constructions* * give the usual objects nY 2 HoC and sets or groups [X, Y ]n. 12.3. The G-resolution model category. For a left proper model category C, let * *G be a class of group objects in HoC. Then each G 2 G, with its unit map, represents an object * *of (Ho C)* and thus has an n-fold loop object nG 2 HoC giving an associated homotopy functor * *[-, G]n on HoC for n 0. A map i : A ! B in HoC is called G-monic when i*: [B, G]n ! [A, G]n * *is onto for each G 2 G and n 0, and an object Y 2 HoC is called G-injective when i* : [B, Y ] * *! [A, Y ] is onto for each G-monic map i : A ! B in HoC. We retain the other definitions in 3.1 a* *nd 3.2, and we obtain a structured simplicial category c CG. This leads to our most general ex* *istence theorem for resolution model categories. Theorem 12.4 (after Dwyer-Kan-Stover).If C is a left proper model category with* * a class G of injective models in HoC, then c CG is a left proper simplicial model category. 44 The proof proceeds exactly as in 3.4-3.22, but thereafter requires some sligh* *t elaborations which we now describe. To introduce path objects in the unpointed category c C, we fi* *rst choose a Reedy trivial fibration ~eo! e with ~eocofibrant in c C. Then, for an object Fo 2 c C* * with a map ff : ~eo! Fo, we let PffFo 2 c C be the path object given by PffFo = homc( 1, Fo) xFo~eo= homc( 1, Fo) xFoxFo(~eox Fo) with the natural maps ~eo! PffFo ! Fo factoring ff. We now replace Lemma 3.23 by Lemma 12.5. For a G-fibrant object Fo 2 c C with a map ff : ~e! Fo, the natural* * map PffFo ! Fo (resp. PffFo ! e) has the right lifting property for G-trivial cofibrations (re* *sp. G-cofibrations) in c C. Proof.This follows easily from Lemma 3.22 since the map ~e! e has the right lif* *ting property for G-cofibrations. |__| We likewise replace Lemma 3.24 by Lemma 12.6. If Fo ! e is a G-trivial fibration with a G-trivial cofibration ff * *: ~eo! Fo, then Fo ! e has the right lifting property for G-cofibrations. Proof.The G-fibration PffFo ! Fo has a cross-section since it has the right lif* *ting property for the G-trivial cofibration ff : ~eo! Fo by Proposition 3.16. Hence Fo ! e has the ri* *ght lifting property for G-cofibrations since PffFo ! e does by Lemma 12.5. * * |__| We now retain Lemma 3.25 but replace Proposition 3.26 by Proposition 12.7.A G-trivial fibration f : Xo ! Y oin c C has the right lifting* * property for G- cofibrations. ` Proof.First suppose Xo is cofibrant. By Proposition 3.20, the map Xo ~eo! e fa* *ctors into a G- ` cofibration OE : Xo ~eo! Fo and a G-trivial fibration Fo ! e, and the map (f, * *OE) : Xo ! Y ox Fo factors into a Reedy cofibration Xo ! Eo and a Reedy trivial fibration Eo ! Y o* *x Fo. Then the map Eo ! Y ois a G-trivial fibration with the right lifting property for G-cofi* *brations by Lemmas 3.21 and 12.6. The proof now proceeds as in 3.26. * * |__| We retain Proposition 3.27, and thereby complete the proof of Theorem 12.4. 45 12.8. The unpointed theory. Our main definitions and results pertaining to G-re* *solutions, right derived functors, and G-completions in Sections 4-9 are now easily generalized * *to an unpointed model categories. However, the main results in Sections 10-11 must be slightly * *modified since the G-flatness condition for smash products (Definition 10.3) must be replaced by a* * suitable G-flatness condition for ordinary products. This is easily accomplished when C = S and, mo* *re generally, when the functor - x B : HoC ! HoC has a right adjoint (-)B : HoC ! HoC with ( nY )B* * ' n(Y B) for each B 2 HoC and Y 2 (Ho C)*. We finally consider a general example leading* * to unpointed Bendersky-Thompson completions. 12.9. A general unpointed example. Let C be a left proper model category with a* * class H of injective models in the associated pointed homotopy category HoC*. As in 4.8, t* *he forgetful functor Ho C*! HoC now carries H to a class G of injective models in HoC, and we obtain* * simplicial model categories c CG and c CH*together with Quillen adjoints c CG ø c CH*. For an ob* *ject A 2 C* with an H-resolution A ! ~Aoin c C*, we easily deduce that A ! ~Aorepresents a weak G-r* *esolution of A in c C. Thus, when C is bicomplete and simplicial, the H-completion ^LHA 2 HoC** * represents the G-completion ^LGA 2 HoC, and we may view ^LGas an unpointed version of ^LH. 12.10. The unpointed Bendersky-Thompson completions. The above discussion appli* *es to give unpointed versions of the Bendersky-Thompson E-completions for ring spectr* *a E (7.8) and of cohomological completions such as the p-adic K-completion (11.2). References [1]J.F. Adams, Stable Homotopy and Generalized Homology, University of Chicago* * Press, 1974. [2]M. Barr and J. Beck, Homology and standard constructions, Lecture Notes in * *Mathematics, vol. 80, Springer-Verlag, 1969, pp. 245-335. [3]M. Bendersky, E.B. Curtis, and H.R. Miller, The unstable Adams spectral seq* *uence for gen- eralized homology, Topology 17(1978), 229-248. [4]M. Bendersky and D.M. Davis, Compositions in the v1-periodic homotopy group* *s of spheres, Forum Math., to appear. [5]M. Bendersky and D.M. Davis, A stable approach to an unstable homotopy spec* *tral sequence, preprint. [6]M. Bendersky and J.R. Hunton, On the coalgebraic ring and Bousfield-Kan spe* *ctral sequence for a Landweber exact spectrum, preprint. [7]M. Bendersky and R.D. Thompson, The Bousfield-Kan spectral sequence for per* *iodic homol- ogy theories, Amer. J. Math. 122(2000), 599-635. [8]A.K. Bousfield, The core of a ring, J. Pure Appl. Algebra 2(1972), 73-81. C* *orrection in 3(1973),409. [9]A.K. Bousfield, The localization of spaces with respect to homology, Topolo* *gy 14(1975), 133- 150. [10]A.K. Bousfield, The localization of spectra with respect to homology, Topol* *ogy 18(1979), 257-281. [11]A.K. Bousfield, Homotopy spectral sequences and obstructions, Israel J. Mat* *h. 66(1989), 54- 104. 46 [12]A.K. Bousfield, On ~-rings and the K-theory of infinite loop spaces, K-Theo* *ry 10(1996), 1-30. [13]A.K. Bousfield, On p-adic ~-rings and the K-theory of H-spaces, Math. Zeits* *chrift(1996), 483-519. [14]A.K. Bousfield, The K-theory localizations and v1-periodic homotopy groups * *of H-spaces, Topology 38(1999), 1239-1264. [15]A.K. Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. M* *ath. Soc. 353(2000), 2391-2426. [16]A.K. Bousfield, Pairings of homotopy spectral sequences in model categories* *, in preparation. [17]A.K. Bousfield and E.M. Friedlander, Homotopy theory of -spaces, spectra, * *and bisimplicial sets, Lecture Notes in Mathematics, vol. 658, Springer-Verlag, 1978, pp. 80-* *130. [18]A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations* *, Lecture Notes in Mathematics, vol. 304, Springer-Verlag, 1972. [19]A.K. Bousfield and D.M. Kan, The homotopy spectral sequence of a space with* * coefficients in a ring, Topology 11(1972), 79-106. [20]A.K. Bousfield and D.M. Kan, Pairings and products in the homotopy spectral* * sequence, Trans. Amer. Math. Soc. 177(1973), 319-343. [21]A. Dold and D. Puppe, Homologie nicht-additiver Funktoren; Anwendungen, Ann* *. Inst. Fourier (Grenoble) 11(1961), 201-312. [22]E. Dror Farjoun, Two completion towers for generalized homology, Contempora* *ry Math. 265(2000), 27-39. [23]E. Dror, W. Dwyer, and D. Kan, An arithmetic square for virtually nilpotent* * spaces, Illinois J. Math. 21(1977), 242-254. [24]W.G. Dwyer, D.M. Kan, and C.R. Stover, An E2model category structure for po* *inted sim- plicial spaces, J. Pure and Applied Algebra 90(1993), 137-152. [25]W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbo* *ok of Alge- braic Topology, North Holland, Amsterdam, 1995, pp. 73-126. [26]S. Eilenberg and J.C. Moore, Foundations of relative homological algebra, M* *em. Amer. Math. Soc. 55(1968). [27]A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, Rings, Modules, and Algebr* *as in Stable Homotopy Theory Mathematical Surveys and Monographs, vol. 47, American Mathe* *matical Society, 1997. [28]P.G. Goerss and M.J. Hopkins, Resolutions in model categories, preprint. [29]P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathe* *matics, vol. 174, Birkhauser-Verlag, 1999. [30]P.S. Hirschhorn, Localization of Model Categories, preprint availabl* *e from http://math.mit.edu/~psh. [31]M. Hovey, Model Categories, Mathematical Surveys and Monographs, vol. 63, A* *merican Mathematical Society, 1998. [32]M. Hovey, B. Shipley, J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13(2* *000), 149-208. [33]J.F. Jardine, E2model structures for presheaf categories, preprint. [34]A. Libman, Universal spaces for homotopy limits of modules over coaugmented* * functors, preprint. [35]A. Libman, Homotopy limits of triples, preprint. [36]S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Ma* *thematics, vol. 5, Springer-Verlag, 1971. [37]J.-P. Meyer, Cosimplicial homotopies, Proc. Amer. Math. Soc. 108(1990), 9-1* *7. [38]G. Mislin, Localization with respect to K-theory, J. Pure Appl. Algebra 10(* *1977), 201-213. [39]D.G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, S* *pringer-Verlag, 1967. [40]D.L. Rector, Steenrod operations in the Eilenberg-Moore spectral sequence, * *Comm. Math. Helv. 45(1970), 540-552. [41]C.L. Reedy, Homotopy theory of model categories, 1973 preprint available * *from http://math.mit.edu/~psh. Department of Mathematics, University of Illinois at Chicago, Chicago, Illino* *is 60607 E-mail address: bous@uic.edu