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