T-MODEL STRUCTURES
HALVARD FAUSK AND DANIEL C. ISAKSEN
Abstract.For every stable model category M with a certain extra structur*
*e,
we produce an associated model structure on the pro-category pro-M and
a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequenc*
*e,
with reasonably good convergence properties for computing in the homotopy
category of pro-M. Our motivating example is the category of pro-spectra.
The extra structure referred to above is a t-model structure. This is*
* a
rigidification of the usual notion of a t-structure on a triangulated ca*
*tegory.
A t-model structure is a proper simplicial stable model category M with a
t-structure on its homotopy category together with an additional factori*
*zation
axiom.
1.Introduction
Recent efforts to understand the homotopy theory of pro-objects have resulted
in several different model structures on pro-categories, such as the strict mod*
*el
structure [18], the ss*-model structure on pro-spaces [16], and the ss*-model s*
*tructure
on pro-spectra [19]. The arguments required for establishing these model struct*
*ures
are similar, yet the published proofs are distinct and have an ad hoc flavor. I*
*n the
accompanying paper [9] we develop a general framework of filtered model categor*
*ies
for giving model structures on pro-categories.
In this paper we explore a particular class of filtered model structures on s*
*ta-
ble model categories. These filtered model structures arise from t-structures *
*on
the homotopy category of the stable model category (recall that such a homotopy
category is a triangulated category).
More precisely, we work with a t-model structure. This is a proper simplicial
stable model category with a t-structure on its homotopy category together with*
* a
certain kind of lift of the t-structure to the model category itself. This "rig*
*idifica-
tion" of the t-structure is expressed in terms of an additional factorization a*
*xiom
for the model category.
If M is a t-model category, then we produce a model structure on the category
pro-M (see Theorem 6.3). We show that the associated homotopy category P of
pro-M has a t-structure (see Proposition 9.4). One important property of this t-
structure is that an object lies in P n for all n if and only if it is contract*
*ible (see
Proposition 9.4 again). This property has at least two important consequences.
First, it allows us to construct an Atiyah-Hirzebruch type spectral sequence wi*
*th
reasonable convergence properties (see Theorem 10.3). Second, it allows us to p*
*rove
results for detecting weak equivalences analogous to the Whitehead theorem (see
Theorem 9.13).
____________
Date: October 31, 2005.
1991 Mathematics Subject Classification. Primary 55P42; Secondary 18E30, 55U*
*35.
1
2 HALVARD FAUSK AND DANIEL C. ISAKSEN
Although our main interest in t-model structures is to produce model structur*
*es
on pro-categories, the notion of a t-model structure is likely to be useful in *
*other
contexts. Motivic homotopy theory [22] is a combination of ideas from homotopy
theory and from motivic algebraic geometry. Since model categories are important
in homotopy theory and since t-structures are important in the study of motives,
the interaction between these two notions is probably relevant in motivic homot*
*opy
theory. See [21] for a possible example.
One specific example of a model structure on a pro-category obtained from a
t-model structure is the ss*-model structure for pro-spectra. It is obtained f*
*rom
any reasonable model category of spectra, where the t-structure on the homotopy
category of spectra is given by Postnikov sections. The original motivation for*
* this
paper was an extension of this model structure to a category of pro-G-spectra w*
*hen
G is a profinite group. That extension is treated in a separate paper [10].
Another example is the category of pro-cochain complexes of modules over a
unital ring. It is obtained from the projective model structure on the category*
* of
cochain complexes equipped with the standard t-structure on its derived categor*
*y.
In both of these examples, the t-structure on the homotopy category lifts to a
t-model structure.
1.1. ss*-model structure on pro-spectra. Our main application is the ss*-model
structure on pro-spectra. Since the main theorems of this paper can be notation*
*ally
confusing when applied to spectra, we present here a brief summary of our resul*
*ts
for pro-spectra. We remind the reader that the theorems that appear later are
much more general. The pro-categorical terminology is established in Section 5.
Definition 1.1. A map f of pro-spectra is a ss*-weak equivalence if:
(1) f is an essentially levelwise m-equivalence for some m, and
(2) ssnf is a pro-isomorphism of pro-groups for all integers n.
We acknowledge that the first condition above appears unnatural at first glan*
*ce.
However, we believe that it is not possible to construct a model structure on p*
*ro-
spectra if this condition is not included. In fact, the definition that appears*
* later
is different (see Definition 6.2). Here we have given a more concrete equivale*
*nt
reformulation (see Theorem 9.13).
Theorem 1.2 (Theorem 6.3). There is a model structure on the category of pro-
spectra in which the weak equivalences are the ss*-weak equivalences.
The cofibrations in this model structure are the essentially levelwise cofibr*
*ations.
The fibrations are more complicated to describe. Section 6 contains a reasonably
concrete characterization of the fibrations.
One of the key observations about this model structure is that the map X !
{PnX} from a spectrum to its Postnikov tower is a ss*-weak equivalence. In fact,
Postnikov towers are the key ingredient in constructing fibrant replacements.
One of the main uses of the previous theorem is the construction of an Atiyah-
Hirzebruch spectral sequence for pro-spectra.
Theorem 1.3 (Theorem 10.3). Let X and Y be pro-spectra. There is a spectral
sequence with
Ep,q2= Hp(X; ss-qY ).
The spectral sequence converges conditionally to [X, Y ]p+q if:
T-MODEL STRUCTURES 3
(1) X is uniformly bounded below (i.e., there exists an integer N such that
ssnXs = 0 for all n N and all s), or
(2) X is essentially levelwise bounded below (i.e., for each s, there exists*
* an
integer N such that ssnXs = 0 for n N) and Y is a constant pro-spectru*
*m.
In the previous theorem, the notation [X, Y ]p+q refers to weak homotopy clas*
*ses
of maps of degree p + q from X to Y in the homotopy category of pro-spectra.
The E2-term Hp(X; ss-qY ) is singular cohomology of the pro-spectrum X with
coefficients in the pro-abelian group ss-qY . Recall that the pth cohomology gr*
*oup
Hp(X; A) of a pro-spectrum X with coefficients in an abelian group A is defined*
* to
be colimsHp(Xs; A). The definition in the general case is obtained from Definit*
*ion
2.13 and Propositions 8.4 and 9.11.
1.2. Summary. We summarize the contents of the paper by section.
We give a short review of the basic properties of t-structures in Section 2, *
*as-
suming no prior knowledge of t-structures. In Section 3, starting with a t-stru*
*cture
on the homotopy category of a stable model category M, we produce a filtration *
*on
the class of morphisms in M. We reformulate some basic properties of t-structur*
*es
in this language. In Section 4 we introduce t-model categories.
The rest of the paper is concerned with pro-categories. The basic theory of
pro-categories is briefly reviewed in Section 5. We also review the strict mod*
*el
structure and discuss its mapping spaces. In Section 6 we show that a t-model
category gives rise to a filtered model category, and we use this to give a mod*
*el
structure on its pro-category. For reasons that will be apparent later, a model
structure on a pro-category obtained in this way is called an H*-model structur*
*e.
We describe the cofibrations and fibrations of H*-model structures in some deta*
*il
and discuss Quillen equivalences between pro-categories with H*-model structure*
*s.
We then introduce functorial towers of truncation functors in Section 7. They
are used to form fibrant replacements and also to construct the Atiyah-Hirzebru*
*ch
spectral sequence. We next describe the weak homotopy type of mapping spaces in
H*-model structures in Section 8. This is used in Section 9 to give a t-structu*
*re on
the homotopy category of a H*-model structure. Under reasonable assumptions,
we identify the heart of this t-structure. In Section 10 we construct an Atiya*
*h-
Hirzebruch spectral sequence for triangulated categories with a t-structure. T*
*he
spectral sequence has reasonably good convergence properties when applied to the
t-structure on the homotopy category of an H*-model structure.
The last two sections of the paper are devoted to multiplicative structures on
pro-categories. One reason for including this material is to warn the reader th*
*at
tensor structures on pro-categories do not have all the properties one might ho*
*pe
for. In Section 11 we give some basic results concerning the interaction of ten*
*sor
structures and pro-categories. In Section 12 we discuss tensor model categories
from our point of view. The H*-model structures on pro-categories do not behave
well with respect to tensor structures. We get a partially defined tensor produ*
*ct for
some objects in the homotopy category. At the very end, we consider multiplicat*
*ive
structures on the Atiyah-Hirzebruch spectral sequence constructed in Section 10.
1.3. Conventions. We assume that the reader is familiar with model categories.
The reference [14] is particularly relevant because of it emphasis on stable mo*
*del
categories, but [13] is also suitable.
4 HALVARD FAUSK AND DANIEL C. ISAKSEN
We also assume that the reader has a certain practical familiarity with pro-
categories. Although a brief review is given in Section 5, see [2], [7], or [1*
*6] for
additional background.
We use cohomological grading when working with triangulated categories and t-
structures because this is the usual convention in homological algebra [11]. We*
* warn
the reader that this is somewhat unnatural from the perspective of stable homot*
*opy
theory, where the usual grading is homological. Thus, a translation is required
when applying the general results of this paper to the specific example of spec*
*tra.
For example, the usual notion of an "n-equivalence" of spectra corresponds to o*
*ur
notion of a cohomological (-n)-equivalence. Also, the usual notion of a "bounded
above" spectrum corresponds to our notation of a bounded below object.
Throughout the paper, M is a proper simplicial stable model category. We al-
ways assume that M has functorial factorizations, even though the model structu*
*re
on pro-M does not have functorial factorizations. We let D stand for the homoto*
*py
category of M because the notation Ho(M) is too cumbersome for our purposes.
Finally, P stands for the homotopy category of pro-M.
The simplicial assumption on M is probably not really necessary for any of
our results, but it is a very convenient hypothesis and does not really restrict
the generality. On the other hand, the properness assumption on M is absolutely
essential for the existence of model structures on pro-M; see [18] for an expla*
*nation.
If C is any category containing objects X and Y , then C(X, Y ) denotes the s*
*et
of morphisms from X to Y . Occasionally we will use the notation [X, Y ] for th*
*e set
of morphisms in a homotopy category; in this case, the context makes the precise
meaning clear.
2.t-structures
In this section we give a short review of the theory of t-structures on trian*
*gulated
categories. We only discuss the aspects that are relevant for our work. The ori*
*ginal
source for this material is [3], but we refer to [11] whenever possible. We ass*
*ume
the reader has a working knowledge of triangulated categories.
Let D be a triangulated category, and let denote the shift functor so that
distinguished triangles are of the form A ! B ! C ! A.
Definition 2.1. A t-structure on D consists of two strictly full subcategories
D 0 and D 0 such that
(1) D 0 is closed under , and D 0 is closed under -1.
(2) For every object X in D, there is a distinguished triangle
X0! X ! X00! X0
such that X02 D 0 and X002 -1D 0.
(3) D(X, Y ) = 0 whenever X 2 D 0 and Y 2 -1D 0.
Recall that a subcategory is strictly full if it is full and if it is closed *
*under
isomorphisms.
In any t-structure, there are two Z-graded families of strictly full subcateg*
*ories
defined by D n = -n D 0 and D n = -n D 0 for any integer n. By part (1) of
Definition 2.1, the categories D n are an increasing filtration in the sense th*
*at D n
is contained in D n+1 , and the categories D n are a decreasing filtration in t*
*he
sense that D n+1 is contained in D n . Note that parts (2) and (3) of Definition
2.1 can now be rewritten in terms of D 1 instead of -1D 0.
T-MODEL STRUCTURES 5
Example 2.2. Let D be the derived category of cochain complexes of modules
over a ring. The standard t-structure on D [11, IV.4.3] is given by
D n = {X | Hi(X) = 0 fori > n}
D n = {X | Hi(X) = 0 fori < n}.
The shift functor is defined by ( X)n = Xn+1, and the differential ( X)n-1 !
( X)n is equal to the negative of the differential Xn ! Xn+1.
Example 2.3. Let D be the homotopy category of spectra. The Postnikov t-
structure on D is given by
D n = {X | ssk(X) = 0, k < -n}
D n = {X | ssk(X) = 0, k > -n}.
The proof of this fact is classical stable homotopy theory. The main point is *
*to
show that [X, Y ] = 0 when X is (-1)-connected (i.e., sskX = 0 for k < 0) and Y
is 0-coconnected (i.e., sskY = 0 for k 0). One proof of this can be found in *
*[20,
Prop. 3.6].
At first glance, Examples 2.2 and 2.3 may appear rather different. However,
the main difference is that the second example is homologically graded rather t*
*han
cohomologically graded. In other words, the nth stable homotopy group functor
ssn for spectra corresponds to the (-n)th cohomology functor H-n for cochain
complexes.
The two strictly full subcategories D n and D n+1 of a t-structure determine
each other as follows.
Lemma 2.4. An object X is in D n if and only if D(X, Y ) = 0 for all Y in
D n+1 , and an object Y is in D n+1 if and only if D(X, Y ) = 0 for all X in D *
*n .
Proof.We prove the first claim; the proof for the second claim is similar. One
direction follows immediately from part (3) of Definition 2.1.
For the other direction, suppose that D(X, Y ) = 0 for all Y in D n+1 . Part *
*(2) of
Definition 2.1 says that we can find a distinguished triangle X0! X ! X00! X0
such that X0 belongs to D n and X00belongs to D n+1 . Now apply D(-, X00) to
obtain a long exact sequence. Since X0 and X0 are both in D n , it follows that
D(X, X00) and D(X00, X00) are isomorphic. But our assumption implies that the
first group is zero, so the second group is also zero. Thus X00is isomorphic to*
* 0,
and X0! X is an isomorphism.
The next corollary follows immediately from Lemma 2.4.
Corollary 2.5.
(1) The zero object 0 belongs to both D n and D n for every n.
(2) An object that is both in D n and in D n+1 is isomorphic to 0.
(3) The subcategories D n and D n of D are closed under retract.
Corollary 2.6. Let X ! Y ! Z ! X be a distinguished triangle.
(1) If X and Z belong to D n , then so does Y .
(2) If X and Z belong to D n , then so does Y .
(3) If X belongs to D n+1 and Y belongs to D n , then Z belongs to D n .
(4) If X belongs to D n+1 and Y belongs to D n , then Z belongs to D n .
(5) If Y belongs to D n and Z belongs to D n-1 , then X belongs to D n .
6 HALVARD FAUSK AND DANIEL C. ISAKSEN
(6) If Y belongs to D n and Z belongs to D n-1 , then X belongs to D n .
Proof.The first two claims follow immediately from Lemma 2.4. The other claims
follow from the first two; we illustrate with the fifth statement.
We have an exact triangle
-1Z ! X ! Y ! Z
in which both -1Z and Y belong to D n . The result now follows from the first
claim.
We now give a key lemma about t-structures.
Lemma 2.7. Let X0 ! X ! X00! X0 and Y 0! Y ! Y 00! Y 0be two
distinguished triangles with X0 and Y 0in D n and X00and Y 00in D n+1 . Let
f : X ! Y be a map in D. Then there are unique maps f0 : X0 ! Y 0and
f00: X00! Y 00such that there is a commutative diagram
X0 _____//X____//_X00___// X0
f0|| f|| f00|| ||f0
fflffl| fflffl| fflffl| fflffl|
Y 0_____//Y____//_Y_00__// Y 0.
Proof.By part (3) of Definition 2.1, we have that D(X0, Y 00) = 0. Hence there
exists a map f0 : X0 ! Y 0making the left square commute, and then also a map
f00: X00! Y 00such that we get a map of distinguished triangles.
Now we have to show that f0 and f00are unique. Let j be the map Y 0! Y .
There is an exact sequence
j* 0
D(X0, -1Y 00)___//_D(X0, Y_0)//_D(X , Y ).
The left group is trivial by part (3) of Definition 2.1 since -1Y 00belongs to*
* D n+1 .
Therefore, j* is injective, so f0 is unique. A similar argument involving the m*
*ap
X ! X00shows that f00is also unique.
The importance of Lemma 2.7 is expressed in the following proposition.
Proposition 2.8. There are truncation functors o n : D ! D n and o n
: D ! D n together with natural transformations ffln : o n ! 1, jn : 1 ! o n , *
*and
o n+1 ! o n such that
o n X ! X ! o n+1 X ! o n X
is a distinguished triangle for all X. Up to canonical isomorphism, these prope*
*rties
determine the truncation functors uniquely.
Notation 2.9. We usually write X n and X n for o n X and o n X respectively.
The functors o n and o n enjoy many useful properties. Most of the claims in
the next two paragraphs are proved in [11, Sec. IV.4]; the rest follow easily. *
*In any
case, they are easily verifiable directly for Examples 2.2 and 2.3.
The functors o n and o m commute (up to natural isomorphism) for all n and
m. If m n, then o n o m and o m o n are both naturally isomorphic to o n ,
while o n o m and o m o n are both naturally isomorphic to o m . Also, o 0 n
is naturally isomorphic to no n , and o 0 n is naturally isomorphic to no n .
Both o n+1 o n and o n o n+1 are isomorphic to the zero functor.
T-MODEL STRUCTURES 7
2.1. Hearts and cohomology.
Definition 2.10. The heart H(D) of a t-structure D is the full subcategory
D 0 \ D 0 of D.
For any t-structure, the heart H(D) is an abelian category [11, Sec. IV.4].
Definition 2.11. The nth cohomology functor Hn associated to a t-structure
is defined to be the functor o 0 o 0 n.
The cohomology functor Hn is a covariant functor D ! H(D). The following
lemma is proved in [11, Thm. IV.4.11a].
Lemma 2.12. Let X ! Y ! Z ! X be a distinguished triangle. There is a long
exact sequence
. .!.HkX ! HkY ! HkZ ! Hk+1X ! . . .
in the abelian category H(D).
Definition 2.13. Let E be an object in the heart H(D). The nth cohomology
functor Hn (-; E) with E-coefficients is D(-, nE).
The functor Hn(-; E) is a contravariant functor from D to the category of
abelian groups. The following lemma follows immediately from formal properties
of triangulated categories.
Lemma 2.14. For any E in H(D), the functor Hn(-; E) takes distinguished tri-
angles to exact sequences.
Example 2.15. Let D be the triangulated category of cochain complexes with
the standard t-structure (see Example 2.2). The heart of D is isomorphic to the
category of abelian groups. The nth cohomology HnX of a cochain complex X is
the usual nth cohomology
ker(Xn ! Xn+1)=im(Xn-1 ! Xn)
of X. For any abelian group E, the nth cohomology with E-coefficients of a coch*
*ain
complex X is the nth hyperext group Extn(X, E).
Example 2.16. Let D be the category of spectra with the Postnikov t-structure.
The heart of this t-structure is the full subcategory of Eilenberg-Mac Lane spe*
*ctra
(in degree 0). This category is equivalent to the category of abelian groups. T*
*he
nth cohomology functor is ss-n . When E is an abelian group, nth cohomology with
E-coefficients is nth singular cohomology with coefficients in E.
The following lemma says that the cohomology functors are the layers in the
towers obtained from the two sequences of truncation functors.
Lemma 2.17. For every integer n and every X in D, there are distinguished
triangles
X n-1 ! X n ! -n HnX ! X n-1
and
-n HnX ! X n ! X n+1 ! -n+1HnX.
8 HALVARD FAUSK AND DANIEL C. ISAKSEN
Proof.We construct the first distinguished triangle. The construction of the se*
*cond
one is similar.
Start with the object X n of D. There is a distinguished triangle
o n-1 o n X ! o n X ! o n o n X ! o n-1 o n X.
This is equal to the distinguished triangle
o n-1 X ! o n X ! -n HnX ! o n-1 X.
The following lemma tells us when cohomology theories detect trivial objects.*
* It
is also proved in [11, IV.4.11(b)].
Lemma 2.18. Let D be a triangulated category equipped with a t-structure such
that \nD n consists only of the objects isomorphic to 0. Assume that X is in D m
for some m. Then the following are equivalent:
(1) X is isomorphic to 0.
(2) Hn(X) = 0 for all n.
(3) Hn(X, E) = 0 for all n and all E in H(D).
Proof.Condition (1) implies conditions (2) and (3), so we need to show that eit*
*her
condition (2) or (3) implies condition (1).
Since X is in D m , the natural map X n ! X is an isomorphism whenever
n m. Assuming either condition (2) or (3), we prove by induction on n that
X n ! X is an isomorphism for all n. Our assumption on \nD n then implies
condition (1). Suppose for induction that the map X n ! X is an isomorphism.
Condition (2) and the first part of Lemma 2.17 gives that the composition
X n-1 ! X n ! X is an isomorphism.
On the other hand, condition (3) and our inductive assumption implies that
D(X n , -n HnX) is zero. Now apply the functor D(-, -n HnX) to the first
triangle in Lemma 2.17 and use part 3 of Definition 2.1 to conclude that
D( -n Hn(X), -n Hn(X))
is zero. It follows that Hn(X) is isomorphic to 0, and as for condition (2) th*
*is
implies that X n-1 ! X is an isomorphism.
3.n-equivalences and co-n-equivalences
From now on, we no longer consider just triangulated categories but rather
proper simplicial stable model categories M (with functorial factorization). We
write D for the homotopy category of M, which is automatically a triangulated
category because M is stable [14, 7.1]. We briefly review the main properties of
stable model categories that we need. See [14, Ch. 7] for more details.
Recall that a stable model category M is pointed. In addition to unreduced
tensors X K and cotensors Map (K, X), a pointed simplicial model category also
has reduced tensors X ^ K and reduced cotensors Map *(K, X) for any
pointed simplicial set K and any object X of M.
The suspension of any object X of M is defined to be X ^ S1. Note that this
construction is homotopically correct only if X is cofibrant. In general, one m*
*ust
first take a cofibrant replacement for X. In a simplicial stable model category*
*, sus-
pension is a left Quillen functor that induces an automorphism on D. Its associ*
*ated
T-MODEL STRUCTURES 9
right Quillen functor is reduced cotensor with S1, which is also known as loops
[14, 6.1.1].
The homotopy category of a stable model category is a triangulated category
whose shift functor is the left derived functor of suspension. We use the sym*
*bol
for the right derived functor of loops. Note that induces the inverse shift*
* -1
on D. Homotopy cofiber sequences and homotopy fiber sequences are the same,
and they induce the distinguished triangles in the homotopy category.
Since M has functorial factorizations, there are functorial constructions of *
*ho-
motopy fibers and homotopy cofibers in M. We write hocofibf and hofibf for the
functorial homotopy cofiber and homotopy fiber of a map f. Using the properness
assumption, there are natural maps hofibf ! X and Y ! hocofibf for any map
f : X ! Y . In D, hocofibf is isomorphic to hofibf.
These constructions induce functors on the homotopy category D of M. The
functoriality of these constructions is one of the chief advantages of working *
*with
stable model categories rather than just triangulated categories, where homotopy
cofibers and homotopy fibers are only defined up to non-canonical isomorphism.
We now lift the full subcategories given by a t-structure on D to full subcat*
*egories
on M.
Definition 3.1. Let M be a proper simplicial stable model category whose homo-
topy category D is equipped with a t-structure. Let M n be the full subcategory
of M consisting of those objects whose weak homotopy types belong to D n , and
let M n be the full subcategory of M consisting of those objects in M whose
weak homotopy types belong to D n .
Many properties of D n and D n from Section 2 carry over to the classes M n
and M n . For example, M n is closed under , and and M n is closed under
. Also, an object X belongs to M n if and only if nX belongs to M 0 (and
similarly for M n ). Finally, if X belongs to M n and Y belongs to M n+1, then
D(X, Y ) is zero.
To explore the interaction between a stable model structure and a t-structure
on its homotopy category, we find it more convenient to work with subclasses of
morphisms associated to the t-structure rather then the subclasses of objects g*
*iven
by the t-structure.
Definition 3.2. Let M be a proper simplicial stable model category whose homo-
topy category D is equipped with a t-structure. The class of n-equivalences in
M is
Wn = {f | hofibf 2 M n } = {f | hocofibf 2 M n-1.}
The class of co-n-equivalences in M is
coWn = {f | hocofibf 2 M n } = {f | hofibf 2 M n+1}.
Example 3.3. Consider the standard t-structure on cochain complexes from Ex-
ample 2.2. A map is an n-equivalence if and only if it induces a cohomology iso*
*mor-
phism in degrees greater than n and it induces a surjection in degree n. Simila*
*rly,
a map is a co-n-equivalence if and only if it induces a cohomology isomorphism *
*in
degrees less than n and it induces an injection in degree n.
Example 3.4. Let M be a model category of spectra with the Postnikov model
structure on its homotopy category (see Example 2.3). A map is a (cohomological*
*ly
graded) n-equivalence if and only if it induces isomorphisms in homotopy groups
10 HALVARD FAUSK AND DANIEL C. ISAKSEN
below dimension -n and it induces a surjection in dimension -n. Similarly, a map
is a (cohomologically graded) co-n-equivalence if and only if it induces isomor*
*phisms
in homotopy groups above dimension -n and it induces an injection in dimension
-n.
This terminology unfortunately conflicts with the usual notion of n-equivalen*
*ces
and co-n-equivalences of spectra. This is a result of our use of cohomological *
*grad-
ing, rather than the homological grading that is standard for spectra.
We now translate many of the results from Section 2 into properties of n-
equivalences and co-n-equivalences.
Lemma 3.5.
(1) The class W nis closed under , and the class coW nis closed under .
(2) The class W nis contained in W n+1, and the class coW n+1is contained in
coW n.
(3) The classes W nand coW ncontain all weak equivalences.
(4) If a map is both an n-equivalence and a co-n-equivalence, then it is a w*
*eak
equivalence.
(5) The classes W nand coW nare closed under retract.
Proof.The first two claims follow from the properties of M n and M n stated
after Definition 3.1 and the fact that the functors and commute with the
functors hofiband hocofibup to weak equivalence.
The next two claims follow immediately from parts (1) and (2) of Corollary 2.*
*5,
together with the fact that a map is a weak equivalence if and only if its homo*
*topy
fiber is contractible.
For the fifth claim, let g be a retract of a map f. Then hofibg is a retract *
*of
hofibf because homotopy fibers are functorial. Now we just need to use part (3)*
* of
Corollary 2.5.
Lemma 3.6. Let f : X ! Y and g : Y ! Z be two maps.
(1) If f and g both belong to W n, then so does gf.
(2) If f and g both belong to coW n, then so does gf.
(3) If f belongs to W n+1and gf belongs to W n, then g belongs to W n.
(4) If f belongs to coW n+1and gf belongs to coW n, then g belongs to coW n.
(5) If g belongs to W n-1and gf belongs to W n, then f belongs to W n.
(6) If g belongs to coW n-1and gf belongs to coW n, then f belongs to coW n.
Proof.We have a distinguished triangle
hofibf ! hofibgf ! hofibg ! hofibf
in the homotopy category D of M. Corollary 2.6 gives the desired results.
Lemma 3.7. The classes Wn and coWn are both closed under base changes along
fibrations and cobase changes along cofibrations.
Proof.In a proper model structure, the homotopy fiber of a map is weakly equiv-
alent to the homotopy fiber of its base change along a fibration. Similarly, t*
*he
homotopy cofiber of a map is weakly equivalent to the homotopy cofiber of its
cobase change along a cofibration.
The next result gives a general setting in which cohomology theories detect w*
*eak
equivalences in M.
T-MODEL STRUCTURES 11
Theorem 3.8. Assume that \nW nis equal to the class of weak equivalences. Let
f : X ! Y be an m-equivalence for some m. The following are equivalent:
(1) f is a weak equivalence.
(2) Hn(f) is an isomorphism in the heart H(D) for all n.
(3) Hn(Y ; E) ! Hn(X, E) is an isomorphism for all n and all E in H(D).
Applied to the Postnikov t-structure on spectra from Example 2.3, this is the
usual Whitehead theorem for detecting weak equivalences with stable homotopy
groups or with ordinary cohomology.
Proof.Note first that \nW nis equal to the class of weak equivalences if and on*
*ly
if \nD n consists only of contractible objects. This follows immediately from t*
*he
definitions and the fact that a map is a weak equivalence if and only if its ho*
*motopy
fiber is contractible.
Now Lemma 2.18 gives the desired result, using the long exact sequences of
Lemmas 2.12 and 2.14.
Remark 3.9. In Theorem 3.8, an additional assumption on the t-structure allows
one to avoid the assumption that X ! Y is an m-equivalence. Namely, if both
\nW nand \ncoW nare equal to the class of weak equivalences, then a map f is a
weak equivalence in M if and only if Hn(f) is an isomorphism for all n; the pro*
*of
of [11, IV.4.11] can be easily adapted to show this. Both the standard t-struct*
*ure
on cochain complexes from Example 2.2 and the Postnikov t-structure on spectra
from Example 2.3 satisfy this condition.
Our assumptions in Theorem 3.8 are dictated by the t-structures we consider on
homotopy categories of pro-categories. In that case we have that \nW nis equal
to the class of weak equivalences, while \ncoW nis practically never equal to t*
*he
class of weak equivalences. See Lemma 9.5 for more details.
Remark 3.10. Consider the situation of a t-structure on a triangulated category
D that is not associated to a stable model category. Even though homotopy fibers
and homotopy cofibers are not well-defined in D, one can still define classes of
n-equivalences and co-n-equivalences in D as in Definition 3.2. The point is th*
*at
homotopy fibers and homotopy cofibers are well-defined up to non-canonical iso-
morphism, and that is good enough for the purposes of Definition 3.2. All of the
lemmas of this section remain true except for Lemma 3.7, which does not make
sense without a model structure.
4.t-model structures
We continue to work in a proper simplicial stable model category M whose
homotopy category D has a t-structure. We need to assume that the t-structure
on D can be rigidified in a certain sense.
Definition 4.1. A t-model structure is a proper simplicial stable model category
M equipped with a t-structure on its triangulated homotopy category D together
with functorial factorizations of maps in M into n-equivalences followed by co-*
*n-
equivalences.
There is a t-model structure on the category of cochain complexes that induces
the standard t-structure of Example 2.2. More importantly for our applications,
all reasonable model categories of spectra have t-model structures that induce *
*the
12 HALVARD FAUSK AND DANIEL C. ISAKSEN
Postnikov t-structure on the stable homotopy category. To factor a map into an
n-equivalence followed by a co-n-equivalence, apply the small object argument to
the set of maps consisting of all generating acyclic cofibrations and also gene*
*rating
cofibrations whose cofibers are spheres of dimension greater than -n.
Lemma 4.2. The truncation functors o n : D ! D n and o n : D ! D n can
be lifted to functors o n : M ! M n and o n : M ! M n . Similarly, the
natural transformations ffln : o n ! 1 and jn : 1 ! o n can be lifted to natu*
*ral
transformations on M such that ffln is a natural co-(n + 1)-equivalence, jn is a
natural (n - 1)-equivalence, and o n X ffln!X jn+1!o n+1 X is a natural homotopy
fiber sequence in M.
Proof.Given any object X of M, factor the map * ! X functorially into an (n+1)-
equivalence * ! X0 followed by a co-(n + 1)-equivalence X0 ! X. Define o n X
to be X0, and ffln(X) to be the natural map X0 ! X. Define o n X to be the
homotopy cofiber of ffln-1(X) and jn(X) to be the map X ! o n X. We have that
X0 belongs to M n since * ! X0 is an (n + 1)-equivalence, and o n X belongs to
M n since it is the homotopy cofiber of a co-n-equivalence.
Definition 4.3. Let M be a t-model structure. A map in M is an n-cofibration
if it is both a cofibration and an n-equivalence. A map in M is a co-n-fibration
if it is both a fibration and a co-n-equivalence.
Lemma 4.4. The classes of n-cofibrations and co-n-fibrations are closed under
composition and retract.
Proof.This follows immediately from the fact that the classes of cofibrations, *
*fibra-
tions, n-equivalences, and co-n-equivalences are all closed under composition a*
*nd
retract by part (5) of Lemma 3.5 and parts (1) and (2) of Lemma 3.6.
Lemma 4.5. There is a functorial factorization of maps in M into n-cofibrations
followed by co-n-fibrations.
Proof.We construct a factorization explicitly. Let f : X ! Y be a map in M. We
have a natural diagram
X ______u_____//__@Z___v_____//_OO@Y??>>
@@ ~ ""??" @@~@ "">>"
@@@ "" @@@ ""
@__ "" ~ OO""
A OO@@________//B????"
@@ """
~ @@OO@"""~
C
obtained as follows. First, factor f into an n-equivalence u : X ! Z followed b*
*y a
co-n-equivalence v : Z ! Y . Next, factor u into a cofibration X ! A followed by
an acyclic fibration A ! Z, and factor v into an acyclic cofibration Z ! B foll*
*owed
by a fibration B ! Y . Now the composition A ! Z ! B is a weak equivalence, so
it can be factored into an acyclic cofibration A ! C followed by an acyclic fib*
*ration
C ! B.
The composition X ! A ! C is a cofibration because it is a composition of two
cofibrations, and it is an n-equivalence by Lemma 3.6.
Similarly, the composition C ! B ! Y is a co-n-fibration.
T-MODEL STRUCTURES 13
We next prove that the classes of n-cofibrations and co-n-fibrations determine
each other via lifting properties.
Lemma 4.6. A map is an n-cofibration if and only if it has the left lifting pro*
*perty
with respect to all co-n-fibrations. A map is a co-n-fibration if and only if i*
*t has
the right lifting property with respect to all n-cofibrations.
Proof.Let i be an n-cofibration and p a co-n-fibration. We use the abstract ob-
struction theory of [6] to show that there exists a lift B ! X in the diagram
A _____//X
i|| |p|
fflffl|fflffl|
B _____//Y.
A lift exists in the diagram if the obstruction group D(hofibi, hofibp) vanishe*
*s [6,
8.4]. By definition, hofibi belongs to M n , and hofibp belongs to M n+1. Hence
the obstruction group vanishes because there are only trivial maps in D from ob*
*jects
in M n to objects in M n+1.
Now suppose that a map i has the left lifting property with respect to all co*
*-n-
fibrations. Lemma 4.5 allows us to apply the retract argument and conclude that
i is a retract of an n-cofibration. But n-cofibrations are preserved by retrac*
*t by
Lemma 4.4, so i is an n-cofibration.
A similar argument shows that if p has the right lifting property with respec*
*t to
all n-cofibrations, then p is a co-n-fibration.
Corollary 4.7. The class of n-cofibrations is closed under arbitrary cobase cha*
*nge.
The class of co-n-fibrations is closed under arbitrary base change.
Proof.This follows immediately from Lemma 4.6 together with the facts that
cobase changes preserve left lifting properties and base changes preserve right*
* lifting
properties.
Lemma 4.8. Every acyclic fibration is a co-n-fibration. Every acyclic cofibrati*
*on
is an n-cofibration. If m n, then every n-cofibration is an m-cofibration, a*
*nd
every co-m-fibration is a co-n-fibration.
Proof.This follows from part (2) and (3) of Lemma 3.5.
Lemma 4.9. Let f be a cofibration. Then f is an n-cofibration if and only if f *
*^S1
is an (n - 1)-cofibration.
Proof.First, - ^ S1 preserves cofibrations because the model structure on M is
simplicial. Let C be the cofiber of f; this is also the homotopy cofiber of f b*
*ecause
f is a cofibration. Note that C is cofibrant. Now C ^ S1 is the cofiber (and al*
*so
the homotopy cofiber) of f ^ S1 because the model structure on M is simplicial.
Since C is cofibrant, C ^ S1 is homotopically correct and is a model for C in *
*D.
Now C belongs to M n-1 if and only if C ^ S1 belongs to M n-2.
We next show that n-cofibrations interact appropriately with the simplicial s*
*truc-
ture. This will be needed to show that our later constructions behave well simp*
*li-
cially.
14 HALVARD FAUSK AND DANIEL C. ISAKSEN
Proposition 4.10. Suppose that f : A ! B is an n-cofibration and i : K ! L is
a cofibration of simplicial sets. Then the map
g : A L qA K B K ! B L
is also an n-cofibration.
Proof.The map i is a transfinite composition of cobase changes of maps of the f*
*orm
@ [j] ! [j]. Therefore, the map g is a transfinite composition of cobase chang*
*es
of maps of the form
A [j] qA @ [j]B @ [j] ! B [j].
Since n-cofibrations are characterized by a left lifting property (see Lemma 4.*
*6),
n-cofibrations are preserved by cobase changes and transfinite compositions. Th*
*ere-
fore, we may assume that i is the map @ [j] ! [j].
Since M is a simplicial model category and f is a cofibration, g is also a co*
*fi-
bration. We need only show that g is an n-equivalence.
Let C be the cofiber of f, so C belongs to M n-1. Then the cofiber of g is
C ^ Sj, where the simplicial set Sj is the sphere [j]=@ [j] based at the image
of @ [j]. We need to show that C ^ Sj also belongs to M n-1. But C ^ Sj is a
model for jC in D because C is cofibrant, so C ^ Sj belongs to M n-1 because
M n-1 is closed under .
Note that the reduced version of Proposition 4.10 also holds. Namely, if f : *
*A !
B is an n-cofibration and i : K ! L is a cofibration of pointed simplicial sets*
*, then
the map
A ^ L qA^K B ^ K ! B ^ L
is also an n-cofibration. The proof is identical.
Corollary 4.11. Let A ! B be an n-cofibration, and let X ! Y be a co-n-
fibration. The map
f : Map(B, X) ! Map(A, X) xMap(A,Y )Map(B, Y )
is an acyclic fibration of simplicial sets.
Proof.This follows from the lifting property characterization of acyclic fibrat*
*ions,
adjointness, and Proposition 4.10.
4.1. Producing t-model categories. We give some elementary results for con-
structing t-model structures.
Lemma 4.12. Assume that D is the homotopy category of a proper simplicial
stable model category M. Let D 0 be a strictly full subcategory of D that is cl*
*osed
under . Define D n to be -n D 0. Let W nbe defined as in Definition 3.2, and
set F n= injC \ W n. Let D n+1 be the full subcategory of D whose objects are
isomorphic to hofib(g) for all g in F n. If there is a functorial factorization*
* of any
map in M as a map in C \ W nfollowed by a map in F n, then D 0, D 0 is a
t-structure on D, and hence we get a t-model structure on M.
Proof.We verify that D 0 and D 0 satisfy the three axioms of a t-structure on D
given in Definition 2.1. Axiom 1 holds since D 0 is closed uner . The factoriz*
*ation
applied to * ! X (or X ! *) gives a natural triangle fulfilling axiom 2 for a t-
structure.
T-MODEL STRUCTURES 15
Now assume that X 2 D 0 and that Y 2 D 1. We can assume that X is
cofibrant. Factor X ! * into a cofibration g : X ! Z followed by an acyclic
fibration Z ! *. We have that g is in C \W nsince g is a cofibration with homot*
*opy
cofiber in D -1 . By our assumption Y is weakly equivalent to the pullback Y 0of
a fibration p : E ! B with fibrant target having the right lifting property with
respect to g. For any map f : X ! Y 0there are commutative squares
X _____//Y_0___//E
g || || |p|
fflffl|fflffl|~fflffl|
Z _____//_*____//_B.
We get that the left square lifts by our assumptions. Hence any map f : X ! Y 0
factors through a contractible object. Since X is cofibrant and Y 0is fibrant w*
*e get
that D(X, Y 0) = 0. Hence we conclude that D(X, Y ) = 0 whenever X 2 D 0 and
Y 2 D 1.
Proposition 4.13. Let M be a proper simplicial stable cofibrantly generated mod*
*el
category with homotopy category D. Let I be a set of generating cofibrations and
let J be a set of generating acyclic cofibrations. Let Kn be subsets of J for n*
* 2 Z.
Let C(n) be the class of retracts of relative I [ Kn-cell complexes. Let W n be
the corresponding class of n-equivalences defined as the class of maps that is *
*the
composite of a map in C(n) followed by an acyclic fibration.
If W nis equal to -n W 0for all n, then the structure defined above is a t-m*
*odel
structure and C(n) = C \ W n.
Proof.We have functorial factorization of any map as a map in C(n) followed by a
map in inj-C(n) [13, 10.5, 11.1.2]. The result follows from Lemma 4.12 by letti*
*ng
D 0 be the full subcategory of D consisting of objects isomorphic to the homoto*
*py
fibers of maps in W 0.
5.Review of pro-categories
We give a brief review of pro-categories. This section contains mostly standa*
*rd
material on pro-categories [1] [2] [7].
Definition 5.1. For any category C, the category pro-C has objects all cofilter*
*ing
diagrams in C, and
pro-C(X, Y ) = limtcolimsC(Xs, Yt).
Composition is defined in the natural way.
A constant pro-object is one indexed by the category with one object and one
(identity) map. Let c : C ! pro-C be the functor taking an object X to the cons*
*tant
pro-object with value X. Note that this functor makes C a full subcategory of p*
*ro-C.
The limit functor lim : pro-C ! C is the right adjoint of c.
A level map X ! Y is a pro-map that is given by a natural transformation
(so X and Y must have the same indexing category); this is a very special kind *
*of
pro-map. Up to pro-isomorphism, every map is a level map [2, App. 3.2].
Let M be a collection of maps in C. A level map g in pro-C is a levelwise M-
map if each gs belongs to M. A pro-map is an essentially levelwise M-map if
it is isomorphic to a levelwise M-map.
16 HALVARD FAUSK AND DANIEL C. ISAKSEN
We say that a level map is directed (resp., cofinite directed) if its indexin*
*g cate-
gory is a directed set (resp., cofinite directed set). Recall that a directed i*
*ndexing
set S is cofinite if for all s 2 S, the set {t 2 S | t < s} is finite.
Definition 5.2. A map in pro-C is a special M-map if it is isomorphic to a
cofinite directed levelwise map f = {fs}s2S with the property that for each s 2*
* S,
the map
Msf : Xs ! limt 0. As far as we know, it is not
possible to define T nfor all n so that the desired natural transformations be*
*tween
these functors exist.
Note also that T ntakes values in fibrant objects, and the natural map T nX*
* !
T n+1X is a co-(n - 1)-fibration.
To define T nX for n -1, first recall that maps in M have functorial
homotopy fibers. Then T nX is defined to be the homotopy fiber of the map
X ! T n+1X. As before, we end up with a tower
. .!.T -3X ! T -2X ! T -1X,
and each T nX belongs to M n .
The following lemma shows that T nX and T nX have the desired homotopy
types.
Lemma 7.1. The objects T nX and o n X of M are weakly equivalent. Similarly,
the objects T nX and o n X are weakly equivalent.
Proof.There is a homotopy fiber sequence T n-1X ! X ! T nX such that
T n-1X belongs to M n-1 and T nX belongs to M n . On D, o n-1 and o n
are the unique functors with this property (see Lemma 2.7). Therefore, T n-1
induces o n-1 on D, which means that T n-1X and o n-1 X are weakly equivalent
for all X. Similarly, T n induces o n on D, so T nX and o n X are weakly
equivalent.
22 HALVARD FAUSK AND DANIEL C. ISAKSEN
Lemma 7.2. Let Y be a pro-object indexed by a cofiltered category I. Consider t*
*he
pro-object Z indexed on I x N such that Zs,n= T nYs. The natural map Y ! Z
is an H*-weak equivalence.
Proof.For each s and n, the map Ys ! Zs,nhas homotopy fiber T n-1Ys, so this
map is an (n - 1)-equivalence. This shows that the map Y ! Z is an essentially
levelwise k-equivalence for all k.
The next result shows how the functors T n are of tremendous value in con-
structing H*-fibrant replacements.
Proposition 7.3. Let Y be a pro-object indexed by a cofiltered category I. Con-
sider the pro-object Z indexed on I x N such that Zs,n= T nYs. A strict fibrant
replacement for Z is an H*-fibrant replacement for Y .
Proof.In order to construct an H*-fibrant replacement for Y , Lemma 7.2 says th*
*at
we may construct an H*-fibrant replacement for Z instead. Finally, Corollary 6.*
*10
says that a strict fibrant replacement for Z is the desired H*-fibrant replacem*
*ent.
8. Homotopy classes of maps of pro-spectra
We continue to work in a t-model structure M. Recall that D is the homotopy
category of M. Let P be the H*-homotopy category of pro-M.
The mapping space Map (X, Y ) is related to homotopy classes in the following
way [14, 6.1.2]. For every cofibrant X and H*-fibrant Y , P(X, Y ) is isomorphi*
*c to
ss0Map (X, Y ).
Lemma 8.1. When pro-M is equipped with the H*-model structure, the constant
pro-object functor c : M ! pro-M and the limit functor lim: pro-M ! M are a
Quillen adjoint pair.
Proof.Note that c preserves cofibrations and acyclic cofibrations.
Proposition 8.2. The right derived functor Rlim of lim : pro-M ! M is given
by RlimY = holims,nT nYs.
Proof.We may assume that Y is indexed by a cofinite directed set I. Let Z be the
pro-object indexed by I x N such that Zs,nequals T nYs. Recall from Lemma 7.2
that the natural map Y ! Z is an H*-weak equivalence.
Let ^Zbe a strict fibrant replacement for Z. Corollary 6.10 says that ^Zis an
H*-fibrant replacement for Y , so RlimY is equal to lim^Z. As observed in Secti*
*on
5.1, lim^Zis the same as holimZ.
Corollary 8.3. There is a natural isomorphism
P(cX, Y ) ~=D(X, holimt,nT nYt)
for all X in M and all Y in pro-M. There is a natural isomorphism
P(cX, cY ) ~=D(X, holimn!-1 T nY )
for all X and Y in M.
Consequently, if Y ! holimn!-1 T nY is a weak equivalence for all Y in
M, then the homotopy category of M embeds into the H*-homotopy category
on pro-M.
T-MODEL STRUCTURES 23
Proposition 8.4. Let ^Xbe cofibrant replacement of X, and let ^Ybe a strict fib*
*rant
replacement of {T nYt} in pro-M. The homotopically correct mapping space of
maps from X to Y in the H*-model structure is weakly equivalent to the space
holimt,ncolimsMap(Xs, T nYt).
Proof.This follows by Propositions 5.3 and 7.3.
Let X and Y be objects in pro-M. In general the group P(X, Y ) is quite diffe*
*rent
from pro-D(X, Y ). There is not even a canonical map from one to the other. The
next Lemma says that under some strong conditions the homsets in P and pro-D
agree. We choose to use conceptual proofs rather than the higher derived limit
spectral sequence relating P(X, Y ) to higher derived limits of the inverse sys*
*tem
{colimsD(Xs, Yt)}t of abelian groups.
Lemma 8.5. Let X and Y be two pro-objects such that Xs is in M n for all s
and Yt is in M n for all t. Then P(X, Y ) is isomorphic to limtcolimsD(Xs, Yt).
Proof.We may assume that X is cofibrant. By taking a levelwise fibrant replace-
ment, we may assume that each Ys is fibrant. Corollary 6.10 and Proposition
5.3 imply that the homotopically correct mapping space of maps from X to Y is
holimtcolimsMap(Xs, Yt), and we want to compute ss0 of this space.
For k 1, the only map kXs ! Yt in D is trivial because kXs belongs to
D n-k while Yt belongs to D n . Therefore, ss0Map ( kXs, Yt) = sskMap (Xs, Yt) *
*is
trivial. We have just shown that Map (Xs, Yt) is a homotopy-discrete space.
Filtered colimits preserve homotopy-discrete spaces; moreover, they commute
with ss0. Similarly, homotopy limits preserve homotopy-discrete spaces and resp*
*ect
ss0 in the sense that the set of components of a homotopy limit is the ordinary
limit of the sets of components of each space. In our situation, this implies t*
*hat
ss0holimtcolimsMap(Xs, Yt) is equal to limtcolimsss0Map (Xs, Yt), which is the *
*de-
sired result.
Corollary 8.6. Let X and Y be two pro-objects such that Xs is in M n for all s
and Yt is in M n+1 for all t. Then P(X, Y ) is trivial.
Proof.This follows from Lemma 8.5 and part (3) of Definition 2.1.
Corollary 8.7. If Y is a bounded below object in M and X is any object in pro-M,
then P(X, cY ) is isomorphic to colimsD(Xs, Y ).
Proof.The proof is nearly the same as the proof of Lemma 8.5. We need to compute
ss0colimsMap(Xs, Y ). We just need to observe that ss0 commutes with filtered
colimits.
Since pro-M is a model category, one may consider homotopy limits internal to
pro-M. In other words, given a diagram of pro-objects, one can form the homotopy
limit of this diagram and obtain another pro-object. We will need the following*
* basic
result about homotopy limits of countable towers later when we discuss converge*
*nce
of spectral sequences.
Lemma 8.8. Let M be a simplicial model category, and let D be its homotopy
category. Let X belong to M, and let
. .!.Y 2! Y 1! Y 0
24 HALVARD FAUSK AND DANIEL C. ISAKSEN
be a countable tower in M. There is a natural short exact sequence
lim1kD( X, Y k) ! D(X, holimkY k) ! limkD(X, Y k).
Proof.We may assume that X is cofibrant, that each Y kis H*-fibrant, and that
each map Y k! Y k-1is an H*-fibration. We have that Map (X, limkYk) is isomor-
phic to limkMap(X, Yk) since Map (X, -) is right adjoint to tensoring with X. S*
*ince
- X sends acyclic cofibrations of simplicial sets to acyclic cofibrations in *
*M, we
get that the tower Map (X, Yk) is a tower of fibrations between fibrant simplic*
*ial
sets.
Hence Map (X, limkYk) is equivalent to holimkMap(X, Y k). The claim now fol-
lows by the lim1short exact sequence for simplicial sets [5, IX.3.1].
9. t-model structure for pro-categories
We now define a t-structure on the H*-homotopy category P of pro-M.
Definition 9.1. Let (pro-M) 0 be the full subcategory of pro-M on all objects
that are H*-weakly equivalent to a pro-object X such that each Xs belongs to
M 0. Let (pro-M) 0 be the full subcategory of pro-M on all objects that are
H*-weakly equivalent to a pro-object X such that each Xs belongs to M 0.
We define (pro-M) n and (pro-M) n to be the subcategories -n (pro-M) 0
and -n (pro-M) 0 respectively. Recall that here refers to the levelwise susp*
*en-
sion functor on pro-objects.
Lemma 9.2. The subcategory (pro-M) n is the full subcategory of pro-M on all
objects that are H*-weakly equivalent to a pro-object X such that each Xs belon*
*gs
to M n . Similarly, the subcategory (pro-M) n is the full subcategory of pro-M
on all objects that are H*-weakly equivalent to a pro-object X such that each Xs
belongs to M n .
Proof.We prove the first claim. The proof of the second claim is similar.
First suppose that X is a pro-object such that each Xs belongs to M n . Since
n takes M n to M 0, it follows that nX belongs to M 0 levelwise. Therefore
nX belongs to (pro-M) 0, and X belongs to -n (pro-M) 0.
Now suppose that Y belongs to -n (pro-M) 0. It follows that nY belongs
to (pro-M) 0, so it is H*-weakly equivalent to a pro-object X such that each
Xs belongs to M 0. Note that -n X belongs to M n levelwise. But -n X is
H*-weakly equivalent to Y . This is the desired result.
Definition 9.3. Let P n be the full subcategory of P on all objects whose H*-
weak homotopy types belong to (pro-M) n . Let P n be the full subcategory of
P on all objects whose H*-weak homotopy types belong to (pro-M) n .
Proposition 9.4. The classes P 0 and P 0 are a t-structure on the H*-homotopy
category P of pro-M. Moreover, \nP n consists only of contractible objects.
Proof.We verify the axioms in Definition 2.1.
For part (1), suppose that X belongs to P 0. We may assume that each Xs
belongs to M 0. Now each Xs belongs to M 0, so X belongs to M 0 levelwise.
Thus X lies in P 0. To show that P 0 is closed under -1, use the dual argumen*
*t.
Lemma 9.2 implies that -1P 0 is the full subcategory of P on all objects that
are H*-weakly equivalent to an object X such that each Xs belongs to M 1. This
will be needed in parts (2) and (3) below.
T-MODEL STRUCTURES 25
For part (2), let X be any object in pro-M. Apply the truncation functors o 0
and o 1 to obtain a levelwise homotopy cofiber sequence o 0 X ! X ! o 1 X.
Finally, observe that levelwise homotopy cofiber sequences are homotopy cofiber
sequences in the H*-model structure because levelwise cofibrations are cofibrat*
*ions.
Note that we need Lemma 9.2 to conclude that o 1 X belongs to P 1.
Part (3) is Corollary 8.6, again using Lemma 9.2 to identify P 1.
For the last claim, suppose that X belongs to \nP n . Fix a value of n. Then
we may assume that each Xs belongs to M n , so the map X ! * is a levelwise
n-equivalence. Thus X ! * is an H*-weak equivalence, so X is contractible.
The subcategory \nP n contains only contractible objects even if \nD n con-
tains non-contractible objects. On the other hand , \nP n contains only con-
tractible objects isomorphic to 0 if and only if D = D 0.
Lemma 9.5. If all the objects of \nP n are contractible, then D is equal to D 0.
Proof.Assume that there are noncontractible elements`Xm 2 M m in each degree
m. Define a pro-object {Yn} by letting Yn = m n Xm and letting the map
Yn-1 ! Yn be the canonical map (M has a zero object). We have that {Yn} is in
\n(pro-M) n , but {Yn} is noncontractible in pro-M: If there is a weak equivale*
*nce
between {Yn} and * in P, then for every n there are integers n0and m such that *
*in
the homotopy category D of M the map Yn0m ! Yn m is the zero map. This gives
a contradiction since Yn0m is not contractible in D for any m.
We can now identify the heart H(P) of the t-structure from Proposition 9.4.
Lemma 9.6. The category P 0 \ P 0 is the H*-homotopy category of the subcat-
egory pro-(M 0 \ M 0) of pro-M.
Proof.Let X be an object of P 0 \ P 0. We need to show that X is H*-weakly
equivalent to an object Y of pro-(M 0 \ M 0).
We may assume that X belongs to M 0 levelwise. If we apply the functors
o -1 and o 0 to X levelwise, we obtain a levelwise homotopy cofiber sequence
o -1 X ! X ! o 0 X. This is a homotopy cofiber sequence in pro-M because
cofibrations are defined to be levelwise cofibrations. Therefore,
o -1 X ! X ! o 0 X ! o -1 X
is a distinguished triangle in P. The object o -1 X belongs to both P -1 and to
P 0, so it is contractible. This means that the map X ! o 0 X is an H*-weak
equivalence.
Finally, o 0 X belongs to M 0 levelwise by definition of o 0 , and it also be*
*longs
to M 0 levelwise because X belongs to M 0 levelwise. Thus, o 0 X is the desired
pro-object Y .
As in Definition 3.2, we define an n-equivalence (resp., co-n-equivalence)
in pro-M to be a map whose homotopy fiber belongs to (pro-M) n (resp., homo-
topy cofiber belongs to pro-M) n ). By definition, the n-equivalences in pro-M
are different than the levelwise n-equivalences. A similar warning applies to *
*co-
n-equivalences. However, we will show below in Lemma 9.9 that actually they
coincide.
26 HALVARD FAUSK AND DANIEL C. ISAKSEN
Unfortunately, we cannot conclude that pro-M has a t-model structure. Al-
though we can factor any map in pro-M into an n-equivalence followed by a co-n-
equivalence, it does not seem to be possible to make this factorization functor*
*ial.
Absence of functorial factorizations is a general problem with pro-categories. *
*How-
ever, we will prove a slightly weaker result below in Proposition 9.8.
Lemma 9.7. If f is an essentially levelwise n-equivalence in pro-M, then f is an
n-equivalence in pro-M. If f is an essentially levelwise co-n-equivalence in pr*
*o-M,
then f is a co-n-equivalence in pro-M.
Proof.Let f be a levelwise n-equivalence. Homotopy cofibers of pro-maps can
be computed levelwise because cofibrations are defined levelwise. It follows t*
*hat
hocofibf belongs to M n-1 levelwise. Lemma 9.2 says that hocofibf belongs to
(pro-M) n-1. By definition, f is an n-equivalence.
A similar argument proves the second claim.
Proposition 9.8. The H*-model structure on pro-M and the t-structure on P of
Definition 9.1 are a non-functorial t-model structure on pro-M in the sense that
all the axioms of a t-model structure are satisfied except that the factorizati*
*ons
in the model structure and the factorizations into n-equivalences followed by c*
*o-n-
equivalences might not necessarily be functorial.
Proof.We showed in Theorems 6.3 and 6.13 that the H*-model structure is sim-
plicial, proper, and stable. We showed in Proposition 9.4 that Definition 9.1 i*
*s a
t-structure on P.
It remains only to produce (non-functorial) factorizations into n-equivalences
followed by co-n-equivalences. Let f : X ! Y be any map in pro-M, which
we may assume is a levelwise map. Using that the t-model structure on M has
functorial factorizations, we may factor f into a levelwise n-equivalence g : X*
* ! Z
followed by a levelwise co-n-equivalence h : Z ! Y . Finally, Lemma 9.7 implies
that g is an n-equivalence in pro-M, and h is a co-n-equivalence in pro-M.
Lemma 9.9. A map in pro-M is an n-equivalence if and only if it is an essential*
*ly
levelwise n-equivalence. A map in pro-M is a co-n-equivalence if and only if it*
* is
an essentially levelwise co-n-equivalence.
Proof.We prove the first claim. The proof of the second claim is dual.
One direction was already proved in Lemma 9.7. For the other direction, suppo*
*se
that f : X ! Y is an n-equivalence in pro-M. We may assume that f is a directed
cofinite level map. Factor f into a levelwise cofibration i : X ! Z followed by*
* a
special acyclic fibration p : Z ! Y . The map p is an H*-weak equivalence, so i*
* is
also an n-equivalence in pro-M. Moreover, the map p is a levelwise weak equiva-
lence. The class of essentially levelwise n-equivalences is closed under compos*
*ition
(the proof of [18, Lem. 3.5] applies), so it suffices to show that i is an esse*
*ntially
levelwise n-equivalence.
We know that i is an n-cofibration in pro-M, so it has the left lifting prope*
*rty
with respect to all co-n-fibrations. Factor the map i into a levelwise n-cofibr*
*ation
j : X ! W followed by a special co-n-fibration q : W ! Z. The map q is a co-n-
fibration in pro-M, so i has the left lifting property with respect to q by Lem*
*ma
4.6. Thus, the retract argument shows that i is a retract of j. But essential*
*ly
levelwise n-cofibrations are closed under retract [17, Cor. 5.6], so i is an es*
*sentially
levelwise n-cofibration and thus an essentially levelwise n-equivalence.
T-MODEL STRUCTURES 27
Lemma 9.10. The strict homotopy category of pro-(M 0 \ M 0) is the same as
the H*-homotopy category of pro-(M 0 \ M 0).
Proof.In order to compute strict weak homotopy classes from X to Y , one needs
to take a strict cofibrant replacement "Xof X and a strict fibrant replacement *
*^Yof
Y . But "Xis also an H*-cofibrant replacement for X, and Corollary 6.10 says th*
*at
^Yis an H*-fibrant replacement for Y .
We would like a description of the heart of the t-structure on P. We have
not been able to identify the heart in complete generality. However, in the pri*
*mary
applications to cochain complexes or to spectra, we can identify it using Propo*
*sition
9.11
Proposition 9.11. Suppose that there is a "rigidification" functor K : H(D) !
M 0 \ M 0 such that the composition H(D) ! M 0 \ M 0 ! H(D) is the
identity. Then the heart H(P) of the H*-homotopy category on pro-M is equivalent
to the category pro-H(D).
For spectra, the functor K takes an abelian group A to a functorial model for*
* the
Eilenberg-Mac Lane spectrum HA. For cochain complexes, K takes an R-module
A to the cochain complex with value A concentrated in degree 0.
Proof.The functor K extends to a levelwise functor pro-H(D) ! pro-(M 0 \
M 0). This gives us a functor F : pro-H(D) ! H(P) after composition with the
usual quotient functor (because the quotient functor takes pro-(M 0 \ M 0) into
both P 0 and P 0).
On the other hand, the quotient functor M 0 \ M 0 ! H(D) extends to a
levelwise functor pro-(M 0\M 0) ! pro-H(D). This functor takes levelwise weak
equivalences to (levelwise) isomorphisms, so the functor factors through the st*
*rict
homotopy category of pro-(M 0 \ M 0). Lemma 9.10 implies that the functor
also factors through the H*-homotopy category of pro-(M 0 \ M 0), which is the
same as H(P) by Lemma 9.6. Thus we obtain a functor G : H(P) ! pro-H(D).
It remains to show that F and G are inverse equivalences. The composition
F G is the identity because of the original assumption on K. On the other hand,
for every pro-object X, GF X is levelwise weakly equivalent to X. Thus GF is
isomorphic to the identity.
Without a rigidification functor, the most we can say is stated in the follow*
*ing
lemma.
Lemma 9.12. The functor G : H(P) ! pro-H(D) from the proof of Proposition
9.11 is fully faithful.
Proof.See Lemma 8.5.
As promised in Section 6, we now recharacterize H*-weak equivalences in terms
of the pro-cohomology functors Hn.
Theorem 9.13. A map f : X ! Y in pro-M is an H*-weak equivalence if and
only if it is an essentially levelwise m-equivalence for some m and Hn(f) is an
isomorphism in pro-H(D) for all n.
28 HALVARD FAUSK AND DANIEL C. ISAKSEN
Proof.This is an application of Theorem 3.8 to the H*-model structure. The
hypothesis of that theorem is proved at the end of Proposition 9.4. We also need
Lemma 9.9 to identify the m-equivalences in pro-M. Finally, we need Lemma 9.12
to recognize that a map g is an isomorphism in H(P) if and only if G(g) is an
isomorphism in pro-H(D) (where G is the functor of Lemma 9.12).
10.The Atiyah-Hirzebruch spectral sequence
In this section we construct a spectral sequence for computing in the homotopy
category of a t-model structure. We will also specialize this construction to *
*the
case of H*-model structures on pro-categories. Applied to the homotopy category
of spectra with the Postnikov t-structure we recover the Atiyah-Hirzebruch spec*
*tral
sequence for spectra.
Recall from Definition 2.11 that o q o q Y is isomorphic to -qHq(Y ), where
Hq(Y ) is the qth cohomology of Y with values in the heart of the t-structure.
Also recall from Definition 2.13 that Hp(X; E) is the pth cohomology of X with
E-coefficients, where E belongs to the heart H(D).
For brevity we write [X, Y ]n instead of D(X, nY ).
Theorem 10.1. For any X and Y in a t-model category M, there is a spectral
sequence with
Ep,q2= Hp(X; Hq(Y )).
The spectral sequence conditionally converges to [X, Y ]p+q if holimq!-1 T qY *
*is
contractible and if X is bounded above.
The construction of the spectral sequence is standard (for example, see [4, S*
*ec.
12] or [12, App. B] ). Conditional convergence of spectral sequences is defined*
* in
[4, Defn. 5.10].
To set up the spectral sequence, we only use the t-structure on the homotopy
category, but we use homotopy theory to state the convergence criterion. We are
using the functors T qrather than the functors o q so that we obtain an actual
tower
. .!.T q-1Y ! T qY ! T q+1Y ! . . .
whose homotopy limit we can consider. Recall that there are not necessarily maps
o q-1Y ! o q Y .
Proof.Consider the filtration
. .!.T q-1Y ! T qY ! T q+1Y ! . . .
of Y . We have a distinguished triangle
T q-1Y ! T qY ! -qHq(Y ) ! T q-1Y
in D by Lemma 2.17. If we apply the functor [X, -]p+q and set Dp,q2= [X, T qY *
*]p+q
and Ep,q2= [X, -qHq(Y )]p+q, we get an exact couple
(-1,1)
D2 _____________//D2
aaBBB _
BBB ___
(2,-1)BB ""_(0,0)___
E2
T-MODEL STRUCTURES 29
with the bidegrees of the maps indicated. This gives a spectral sequence where *
*dr
has bidegree (r, -r + 1). This follows from the definition of the differentials*
* given
after [4, 0.6].
Now we consider conditional convergence. Recall from [4, 5.10] that we need to
show that the limit limp!1 Dp,n-p2and the derived limit lim1p!1Dp,n-p2are both
zero, while the map colimp!-1 Dp,n-p2! [X, Y ]n is an isomorphism.
For the limit and the derived limit, Lemma 8.8 gives us a short exact sequence
lim1p!1[X, T n-pY ]n-1 ! [X, holimp!1T n-pY ]n ! limp!1[X, T n-pY ]n.
The middle group is zero by our assumption, so the first and last groups are al*
*so
zero.
For the colimit, we claim that the map colimp!-1 Dp,n-p2! [X, Y ]n is an
isomorphism for all n if and only if colimq!-1[X, T n-qY ]n is zero for all n.*
* This
follows from the distinguished triangle T n-pY ! Y ! T n-p+1Y ! T n-pY
and the fact that directed colimits of abelian groups respect exact sequences. *
*Under
our assumption, X is weakly equivalent to o m X for some m, so [X, T n-qY ]n is
zero whenever q < -m. Thus colimq!-1 [X, T n-qY ]n is zero.
We now specialize to the homotopy category P of the H*-model structure on
pro-M. We first give a lemma which shows that one of the conditions in Theorem
10.1 is always satisfied.
Lemma 10.2. For any Y in pro-M, holimq!-1 T qY is contractible in the H*-
homotopy category P.
Proof.Each map T q-1Y ! T qY is a fibration, so the homotopy limit is the same
as the ordinary limit limq!-1 T qY . If I is the indexing category for Y , the*
*n one
model for this limit is the pro-object Z indexed by I x N such that Zs,q= T qYs
[17, 4.1].
The map Zs,q! * is an n-equivalence whenever q n. This shows that Z ! * is
an essentially levelwise n-equivalence for all n, so it is an H*-weak equivalen*
*ce.
Theorem 10.3. Let M be a t-model category. Let X and Y be objects in pro-M.
There is a spectral sequence with
Ep,q2= Hp(X; Hq(Y ))
The spectral sequence converges conditionally to P(X, p+qY ) if:
(1) X is uniformly essentially levelwise bounded above (i.e., each Xs belong*
*s to
M n for some fixed n), or
(2) if Y is a constant pro-object and X is essentially levelwise bounded abo*
*ve
(i.e., each Xs belongs to M n for some n depending on s).
Recall that the object Hq(Y ) by definition belongs to the heart H(P) of the *
*H*-
homotopy category. However, when the conditions of Proposition 9.11 are satisfi*
*ed,
we can also view Hq(Y ) as the object of pro-H(D) obtained by applying Hq to Y
levelwise. When Y is a constant pro-object (i.e., belongs to M, not pro-M), then
Hq(Y ) belongs to H(D).
Proof.The spectral sequence and conditional convergence under the first hypothe*
*sis
follows from Theorem 10.1 and Lemma 10.2. Observe that an object X in pro-M
is is bounded above (in the sense that it belongs to P n for some n if and only*
* if
X is uniformly essentially levelwise bounded above; this follows from Lemma 9.2.
30 HALVARD FAUSK AND DANIEL C. ISAKSEN
As in the last paragraph of the proof of Theorem 10.1, it remains to show that
under the second hypothesis, colimq!-1 P(X, nT n-qY ) vanishes for all n. Be-
cause Y is a constant pro-object, Corollary 8.7 implies that the colimit is iso*
*morphic
to
colimq!-1colimsD(Xs, nT n-qY ).
Now exchange the colimits. By hypothesis, Xs belongs to M m for some m. Then
D(Xs, nT n-qY ) = 0 for q < -m, so colimq!-1 D(Xs, nT n-qY ) vanishes for
each s.
11. Tensor structures on pro-categories
This section contains some results on tensor structures on pro-categories.
Let C be a tensor category. There is a levelwise tensor structure on pro-C
given by letting {Xa} {Yb} be the pro-object {Xa Yb}. The unit object of pro*
*-C
is the constant pro-object with value the unit object in C. We only consider te*
*nsor
structures on pro-C that are levelwise tensor structures inherited from a tensor
structure on C.
If C is a cocomplete category, then pro-C is a cocomplete category [16, 11.1]*
*. We
recall the description of arbitrary direct sums and of coequalizers in pro-C.
Let A be an indexing set and let Xff2 pro-C for ff 2 A be a set of pro-object*
*s in
C.`Let Iffbe the cofiltered indexing category of the pro-object Xff. The coprod*
*uct
ff2AXffin pro-M is the pro-object`
{ ff2AXffiff}
Q
indexed on the cofiltered category ffIff.
Up to isomorphism we can assume that a coequalizer diagram is given by lev-
elwise maps {Xa} ' {Ya}. The coequalizer is the pro-object {coeq(Xa ' Ya)}
obtained by forming the coequalizer levelwise in C.
We now consider how direct sums and tensor`products interact. Let Y be a
pro-object indexed on J. We have that ff2A(Xff Y ) is the pro-object
`
{ ff(Xffiff Yjff)}
Q `
indexed on ff(Iffx J). On the other hand we have that ( ffXff) Y is the
pro-object `
{( ffXffiff) Yj}
Q ` `
indexed on ( ffIff)xJ. There is a canonical map from ff(Xff Y ) to ( ffXff)*
* Y .
Lemma 11.1. Let C be a cocomplete tensor category. If the tensor product in C
commutes with finite direct sums (coequalizers), then the tensor product in pro*
*-C
also commutes with finite direct sums (coequalizers).
Proof.This follows by cofinality arguments of the indexing categories.
The tensor product on pro-C might not commute with arbitrary direct sums
even if C is a closed tensor category. In particular, the tensor structure on p*
*ro-C is
typically not closed.
Example 11.2. Let C be a tensor category with arbitrary direct sums. Assume
that the tensor product on C commutes with arbitrary direct sums. Then the tens*
*or
product with a constant pro-object in pro-C respects arbitrary direct sums. In
general, however, tensor product with a pro-object does not commute with arbitr*
*ary
T-MODEL STRUCTURES 31
direct`sums. Let X be`a pro-object indexed on natural numbers. The canonical
map ( 10c(I)) X ! 10(c(I) X) is the map
` 1 ` 1
{ i=0Xni}{ni}2NN! { i=0Xn}n2N.
This map is typically not a pro-isomorphism. For example, it is not an isomorph*
*ism
if for each n we have that Xn is not a retract of Xm in C for any m < n. In par*
*ticular,
the tensor product on the category of pro abelian groups does not respect arbit*
*rary
sums.
In a closed tensor category C the tensor product with any object in C respects
epic maps. The same is true for a tensor product on pro-C, even though the tens*
*or
product is not closed.
Lemma 11.3. Let C be a closed tensor category. Then the tensor product on pro-C
respects epic maps.
Proof.A pro-map f : X ! Y is epic if and only if for any b in the indexing cate*
*gory
B of Y and any two maps Yb ' Z, for some Z 2 C, which equalize f composed
with the projection to Yb, there is a map Yb0! Yb which also equalizes the two
maps to Z.
Let f : X ! Y be an epic map. Assume that f 1W equalizes the two maps
Y W ' Z where Z is an object in C. Given two maps Yb Wk ' Z. Assume that
f 1W composed with the projection to Yb Wk equalize the two maps. Then there
is an a and a k0 so that Xa Wk0! Yb Wk equalizes these two maps. We have
that Xa ! Yb equalizes the two adjoint maps Yb ' F (Wk0, Z), where F denote the
inner hom functor. Hence by the assumption that f is epic there is a map Yb0! Yb
which also equalize the two maps. Now use the adjunction one more time to get
the conclusion that Yb0 Wk0! Yb Wk equalize the two maps Yb Wk ' Z.
One might consider monoids in pro-C. This is a more flexible notion than pro-
objects in the category of pro-(C-monoids). We have that`the category of monoids
in pro-C is the category of algebras for the monad TX = n 0X n . The category
of`commutative monoids in pro-C is the category of algebras for the monad PX =
n 0X n = n.
Lemma 11.4. Let C be a complete and cocomplet closed tensor category. Then the
category of (commutative) monoids in pro-C is complete and cocomplete.
Proof.We can follow [8, II.7]. Note that in the proof of Proposition II.7.2 in *
*[8] it
is not needed that the category is closed. It is only used that the tensor prod*
*uct
commutes with finite colimits and respects epimorphisms. This holds by Lemmas
11.1 and 11.4. Hence the result follows from [8, II.7.4].
12. Tensor model categories
We give conditions that guarantee that a tensor product on a model category
M induces a tensor product on the homotopy category of M. We also give more
specific conditions for a t-model category which guarantee that the induced ten*
*sor
product respects the triangulated structure and the t-structure on its homotopy
category.
The pushout product axiom for cofibrations says that if f : X ! Y and
f0 : X0! Y 0are cofibrations, then the pushout product map
(X Y 0) q(X X0)(Y X0) ! (Y Y 0)
32 HALVARD FAUSK AND DANIEL C. ISAKSEN
is a cofibration, and if in addition f or g is a weak equivalence, then the pus*
*hout
product map is also a weak equivalence.
Definition 12.1. A tensor model category M is a model category with a tensor
product such that
(1) M satisfies the pushout product axiom for cofibrations
(2) the functors - C and C - take weak equivalences to weak equivalences
for all cofibrant objects C in M.
See Hovey [14, 4.2.6] for more details on tensor model categories. Our defini*
*tion
is slightly stronger than his. If M is a tensor model category, then there is a*
* tensor
product on the homotopy category D of M [14, 4.3.2]. The homotopically correct
tensor product is given by first making a cofibrant replacement of at least one*
* of
the two objects and then form the tensor product.
Let M be a pointed simplicial model category and a symmetric tensor category.
Let ae be the functor from simplicial sets to M obtained by applying the simpli*
*cial
tensorial structure on M to the unit object in M.
Definition 12.2. We say that the tensor structure and the simplicial structure *
*on
M are compatible if there is a natural isomorphism between the simplicial tenso*
*rial
structure and the functor Id ae, restricted to finite simplicial complexes.
Lemma 12.3. Let M be a simplicial t-model category and a tensor category. If
the tensor product and the simplicial structure are compatible, then the simpli*
*cial
structure and the levelwise tensor structure on pro-M with the strict or H*-mod*
*el
structures are also compatible.
Proof.For a finite simplicial complexes the simplicial tensorial structure on p*
*ro-M
is given by applying the simplicial structure on M levelwise [16, 16]. Hence f*
*or
a finite simplicial set K we have that X ae(K) is naturally isomorphic to the
simplicial tensor of X with K.
We only consider the most na"ive compatibility of the tensor structure and the
triangulated structure on the homotopy category of a stable model category.
Lemma 12.4. Let M be a symmetric tensor model category with a compatible
based simplicial structure. Assume the tensor product respects pushouts. Then
there is a tensor triangulated structure on the homotopy category D in the sens*
*e of
a nonclosed version of [15, A.2].
We make use of the following property of a tensor triangulated category in
Proposition 12.11: There are natural isomorphisms ( X) Y ! (X Y ) and
X Y ! (X Y ) so the following holds: If X ! Y ! Z ! X is a dis-
tinguished triangle, then X W ! Y W ! Z W ! (X W ) is again a
distinguished triangle, where the last map is Z W ! ( X) W ! (X W ),
and similarly when W is tensored from the right.
Proof.The unit and associativity conditions stated in [15, A.2] follows from the
corresponding results for the tensor product. The results stated above follows *
*since
tensor products respects homotopy cofibers by our assumption.
Next we give a compatibility of the tensor structure with respect to the t-mo*
*del
structure. The conditions are used Proposition 12.11 to get a multiplicative st*
*ruc-
ture on the Atiyah-Hirzebruch spectral sequence.
T-MODEL STRUCTURES 33
Definition 12.5. Let D be a triangulated category. A t-structure D 0 and D 0
and a tensor structure on D are compatible if D 0 is closed under the tensor
product.
Thus for all integers i and j we have that if X 2 D i and Y 2 D j, then
X Y 2 D i+j.
Remark 12.6. If the t-structure on D is not constant, then the unit object of a
tensor structure compatible with the t-structure on D must be an object in D 0.
Proposition 12.7. Let M be a tensor model category. Then pro-M with the strict
model structure is also a tensor model category. In particular there is an indu*
*ced
tensor structure on its homotopy category. If in addition the simplicial struc*
*ture
is compatible with the tensor product, then the homotopy category is a triangul*
*ated
tensor category.
Proof.The pushout product axiom holds since the pushout product map can be
defined levelwise.
Let f be a weak equivalence in pro-M. We can assume that f is a levelwise weak
equivalence {fs : Xs ! Ys}. We can furthermore assume that the cofibrant object
is a levelwise cofibrant pro-object {Zt} indexed on a directed set T . We get t*
*hat
{fs} {Zt} and {Zt} {fs} are levelwise weak equivalences. The last statement
follows from Lemmas 12.3 and 12.4.
We do not get an induces tensor structure on the homotopy category P of pro-M
with the H*-model structure when M is a t-model category and a tensor model
category. This does not even hold when the t-structure and the tensor structure
on D respect each other. But in this case we do get a tensor product on the full
subcategory of P consisting of objects that are essentially levelwisebounded be*
*low.
Definition 12.8. Let M be a t-model category. Let M<1 be the full subcategory
of M with objects X so that X 2 M n for some n.
The category pro-M<1 is the strictly full subcategory of pro-M consisting of
objects that are essentially levelwisebounded below. It is larger than the cate*
*gory
(pro-M)<1 .
Lemma 12.9. Let M be a t-model category. Then the category M<1 inherits a
t-model structure from M.
The model structure M<1 has only finite colimits and limits. The classes of *
*cofi-
brations, weak equivalences, and fibrations are all inherited from the full inc*
*lusion
functor M<1 ! M.
Proof.If f : X ! Y is a map in M<1 , and X !g Z ! Y is a factorization
of f as an n-equivalence followed by a co-n-equivalence in M, then Z is also in
M<1 : Assume that X 2 M m for some m. In the homotopy category of M we
have a triangle hofib(g) ! X !g Z. Hence by Corollary 2.6 we have that Z is
in D max(m,n)so Z 2 M<1 . A similar argument shows that we have functorial
factorizations of any map in M<1 as an acyclic cofibration followed by a fibra*
*tion
and as a cofibration followed by an acyclic fibration. The rest of the t-model
category axioms are inherited from M.
34 HALVARD FAUSK AND DANIEL C. ISAKSEN
Proposition 12.10. Let M be a t-model category with a tensor model structure so
that the tensor product on D is compatible with the t-structure. Then the model*
* cat-
egory pro-M<1 is a tensor model category and the tensor product on its homotopy
category is compatible with the t-structure. If in addition the simplicial stru*
*cture
and the tensor structure on M are compatible, then P is a tensor triangulated
category.
Proof.Let f be a weak equivalence in pro-M. We can assume that f is a levelwise
map {fs : Xs ! Ys} indexed on a directed set S such that for all n there is an *
*sn
such that fs is n-connected for all s sn [9, 3.2]. We can assume that the cof*
*ibrant
object is a pro-object {Zt} indexed on a directed set T so that Zt 2 D nt and
each Zt is cofibrant. We use that tensoring with a cofibrant object has the cor*
*rect
homotopy type. The indexing set {s, t 2 S xT | conn(fs)+conn(Xt) n} is cofinal
in S x T . Hence we have that {fs} {Zt} is an essentially levelwise n-equival*
*ence
for all n.
The first part of the pushout axiom follows by considering two levelwise cofi*
*bra-
tions. When one of the maps is a levelwise acyclic cofibration we use the previ*
*ous
paragraph and Lemma 3.7 to show that the pushout map is also a weak equivalence.
The last statement follows from Lemmas 12.3 and 12.4.
12.1. Multiplicative structures on the Atiyah-Hirzebruch spectral sequence.
We show that if Y is a monoid in a tensor triangulated category with a t-struct*
*ure
that is compatible with the tensor structure, then the Atiyah-Hirzebruch spectr*
*al
sequence is multiplicative.
Proposition 12.11. Let D be a symmetric tensor triangulated category with a
compatible t-structure. Let Y be a monoid in D. Then the spectral sequence in 1*
*0.1
is multiplicative.
Proof.For convenience let hn denote o n o n ~= -n Hn. It suffices to prove that
we have unique dotted maps
f
Y i Y j______//_____________Y i+j
| |
| |
fflffl|g fflffl|
hi(Y ) hj(Y_)___//__________hi+j(Y )
where f is compatible with the multiplication on Y . Consider the square
f
Y i Y j____//__________Y i+j
| |
| |
fflffl| fflffl|
Y Y ________//_Y_____//_Y i+j+1
Since Y i Y j2 D i+j we get that the map from Y i Y jto Y i+j+1vanish.
Hence there is a lift to Y i+j. This lift is unique since the difference of tw*
*o lifts
factor through -1Y i+j+12 D i+j+2. Hence there is a unique map f. We now
T-MODEL STRUCTURES 35
prove that there is a unique map g between the cohomology. Consider the square
Y i-1 Y j______//Y i Y _j____//_Y i+j
| |
| |
fflffl| fflffl|
hi(Y ) Y j___//___________hi+j(Y )
Since Y i-1 Y j2 D i+j-1 and Y i-1 Y j2 D i+j-2, we get that there
is a unique map making the diagram commute. A similar argument with the
distinguished triangle Y j-1 ! Y j ! hj(Y ) ! Y j-1 tensored from the
right by hi(Y ) gives a unique map hi(Y ) hj(Y ) ! hi+j(Y ) compatible with
Y i Y j! Y i+j.
The following remark says that it suffices to consider monoids in the homotopy
category of pro-M with the strict model structure to get a multiplicative struc*
*ture
on the Atiyah-Hirzebruch spectral sequence of Theorem 10.3.
Remark 12.12. We can apply Theorem 10.1 to the homotopy category of pro-M
with the strict model structure and with the levelwise t-structure. We get a sp*
*ectral
sequence that is isomorphic to the spectral sequence obtained from P with the
levelwise t-structure. This is seen by inspecting the definition of D2, E2 and *
*the
differentials using Propositions 5.3 and 8.4.
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Department of Mathematics, University of Oslo, 1053 Blindern, 0316 Oslo, Norw*
*ay
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
E-mail address: fausk@math.uio.no
E-mail address: isaksen@math.wayne.edu