MODEL STRUCTURES ON PRO-CATEGORIES
HALVARD FAUSK AND DANIEL C. ISAKSEN
Abstract.We introduce a notion of a filtered model structure and use this
notion to produce various model structures on pro-categories. This frame*
*work
generalizes the examples of [13], [15], and [16]. We give several examp*
*les,
including a homotopy theory for G-spaces, where G is a pro-finite group.*
* The
class of weak equivalences is an approximation to the class of underlyin*
*g weak
equivalences.
1.Introduction
The goal of this paper is to give a general framework for constructing model
structures on pro-categories. Given a proper model structure on C, there is a s*
*trict
model structure on pro-C [15]. However, for most purposes, the class of strict
weak equivalences on pro-categories is too small. For example, generalized coho-
mology theories in the strict model structure on pro-spaces do not typically ha*
*ve
good computational properties. A more useful class of maps are those that induce
pro-isomorphisms on pro-homotopy groups [13]. It turns out that this class of e*
*quiv-
alences is exactly the class of maps that are isomorphic to strict n-equivalenc*
*es for
all integers n. In this paper we axiomatize this situation so that it includes *
*many
other interesting model structures on pro-categories. Also, we desire to stream*
*line
the technical arguments that are usually required in establishing a model struc*
*ture
on a pro-category. In the example described above, the approach in this paper
avoids many technical issues from [13] involving basepoints.
The key idea is the notion of a filtered model structure. Our main theorem (s*
*ee
Section 5) states that every proper filtered model structure on C gives rise to*
* a
model structure on pro-C.
Because the list of axioms for a filtered model structure is complicated (see
Section 4), for now we will give an example that gives a feeling for filtered m*
*odel
structures. Let C be a category, and let A be a directed set such that (Wa, Ca,*
* Fa)
is a model structure on C for every a in A. Moreover, assume that Wa and Ca
are contained in Wb and Cb respectively for a b (i.e., the classes Wa and Ca *
*are
"decreasing", so the classes Fa are automatically "increasing"). This is a part*
*icular
example of a filtered model structure, so there results an associated model str*
*ucture
on pro-C. In pro-C, a pro-map f is a weak equivalence if for all a in A, f is
isomorphic to a pro-map that belongs to Wa levelwise. The cofibrations are defi*
*ned
analogously. The fibrations, as usual, can be defined via a lifting property, b*
*ut we
will give a more concrete description of them.
A filtered model structure is a generalization of the situation in the previo*
*us
paragraph. We still have a directed set A and classes Wa, Ca, and Fa of maps for
____________
Date: July 14, 2005.
1991 Mathematics Subject Classification. 55U35 Primary ; Secondary 55P91, 1*
*8G55 .
1
2 HALVARD FAUSK AND DANIEL C. ISAKSEN
each a in A. However, we do not assume that (Wa, Ca, Fa) is necessarily a model
structure. For example, instead of requiring the two-out-of-three property for *
*each
class Wa, we only require an "up-to-refinement" property. Namely, for every a in
A, there must exist a b in A such that if two out of the three maps f, g, and g*
*f are
in Wb, then the third is in Wa. One concrete example of this kind of phenomenon
occurs when A is the set of natural numbers and Wn is the class of n-equivalenc*
*es
of spaces. This may seem like an unnatural generalization of the more natural
situation from the previous paragraph, but it is important in our main examples.
The very nature of pro-categories suggests that up-to-refinement definitions ar*
*e a
sensible approach.
We suspect that this axiomatization is more complicated than it has to be, but
we do not know any way to simplify it so that it still includes all the example*
*s of
interest.
A t-model structure is a stable model structure C with a t-structure on its (*
*trian-
gulated) homotopy category, together with a lift of the t-structure to C. This *
*notion
is studied in detail in [8], where it is shown that a particularly well-behaved*
* filtered
model structure on C (and thus a model structure on pro-C) can be associated to
any t-model structure.
This paper grew out of an attempt to find useful model structures on the cate*
*gory
of pro-G-spaces and pro-G-spectra when G is a pro-finite group. Section 8 conta*
*ins
a detailed description of a model structure on G-spaces as an illustration of o*
*ur
general theory. Analogous results for pro-G-spectra are presented in detail in *
*[8]
and [9].
We now summarize our interest in pro-G-spaces when G is a pro-finite group.
Let G be a finite group. There is an obvious generalization to pro-G-spaces of
the model structure for pro-non-equivariant spaces in the first paragraph. Now *
*the
weak equivalences are maps f such that for every n, f is equivalent to a level *
*map
g with the property that gH is a levelwise n-equivalence for every subgroup H *
*of
G. One can make similar model structures for arbitrary topological groups.
When G is a pro-finite group, the model structure on pro-G-spaces described
above is probably not the right construction. For a pro-finite group, it is the*
* continu-
ous cohomology colimUH*(G=U; M) rather than the group cohomology H*(Gffi; M)
that is of interest. Here U ranges over the finite-index normal subgroups of G,*
* M
is a discrete continous G-module, and Gffiis the group G considered as a discre*
*te
group. For finite groups, the group cohomology of G is equal to the Borel cohom*
*ol-
ogy of a point. In other words, the homotopy-orbit space *hG of a point is BG. *
*In
model-theoretic terms, what is happening is that one takes a cofibrant replacem*
*ent
EG for * and then takes actual G-orbits to obtain BG.
We desire a model structure on pro-G-spaces such that the system {E(G=U)}
plays the role of the free contractible space EG. In other words, {E(G=U)} shou*
*ld
be a cofibrant replacement for *. Then *hG is equal to {B(G=U)}, and the co-
homology of *hG is the continuous cohomology of G. Such a model structure is
obtained using our machinery by defining a weak equivalence of pro-G-spaces to
be a map such that for every integer n, the map is isomorphic to a levelwise map
f with fU an n-equivalence for some finite-index subgroup U of G. The details of
this particular example are given in Section 8.
In fact, the n-equivalences in the previous paragraph are irrelevant for the *
*pur-
poses of ordinary continuous cohomology. We could just as well consider a weak
MODEL STRUCTURES ON PRO-CATEGORIES 3
equivalence to be a map that is isomorphic to a levelwise map f with fU a weak
equivalence for some finite-index subgroup U of G. The resulting model structure
still behaves well with respect to continuous cohomology, but it does not behav*
*e well
with respect to generalized continuous cohomology theories. The non-equivariant
analogue of this phenomenon is explained in detail in [16].
In the context of equivariant model categories that behave well with respect *
*to
continuous cohomology, the paper [10] should also be mentioned.
1.1. Organization. We begin with a review of pro-categories, including a tech-
nical discussion of essentially levelwise properties. Afterwards, we define fi*
*ltered
model structures and prove our main result in Theorem 5.15, which establishes t*
*he
existence of model structures on pro-categories. The next section considers Qui*
*llen
functors in this context.
Then we proceed to examples. We first give a few examples of our general
theory. Then we focus on constructing G-equivariant homotopy theories when G is
a pro-finite group.
1.2. Background. We assume that the reader is familiar with model categories,
especially in the context of equivariant homotopy theory. The original referen*
*ce
is [18], but we will refer to more modern treatments [11] [12]. This paper is a
generalization of [15], and we use specific pro-model category techniques from *
*it.
2.Pro-categories
We begin with a review of the necessary background on pro-categories. This
material can be found in [1], [2], [5], [6], and [15].
2.1. Pro-Categories.
Definition 2.1. For a category C, the category pro-C has objects all cofiltering
diagrams in C, and
Hom pro-C(X, Y ) = limscolimtHomC(Xt, Ys).
Composition is defined in the natural way.
A category I is cofiltering if the following conditions hold: it is non-empty*
* and
small; for every pair of objects s and t in I, there exists an object u togethe*
*r with
maps u ! s and u ! t; and for every pair of morphisms f and g with the same
source and target, there exists a morphism h such that fh equals gh. Recall tha*
*t a
category is small if it has only a set of objects and a set of morphisms. A dia*
*gram
is said to be cofiltering if its indexing category is so. Beware that some pape*
*rs
on pro-categories, such as [2] and [17], consider cofiltering categories that a*
*re not
small. All of our pro-objects will be indexed by small categories.
Objects of pro-C are functors from cofiltering categories to C. We use both s*
*et-
theoretic and categorical language to discuss indexing categories; hence "t s*
*" and
"t ! s" mean the same thing when the indexing category is actually a cofiltering
partially ordered set.
The word pro-object refers to an object of a pro-category. A constant pro-
object is one indexed by the category with one object and one (identity) map. L*
*et
c : C ! pro-C be the functor taking an object X to the constant pro-object with
value X. Note that this functor makes C into a full subcategory of pro-C.
4 HALVARD FAUSK AND DANIEL C. ISAKSEN
2.2. Level Maps. 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 a category 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 pro-isomorphic, in the category of arrows in pro-C, to a levelwi*
*se
M-map.
We will return to level maps in more detail in Section 3.
2.3. Cofiniteness. A partially ordered set (I, ) is directed if for every s an*
*d t
in I, there exists u such that u s and u t. A directed set (I, ) is cofini*
*te if
for every t, the set of elements s of I such that s t is finite. A pro-object*
* or level
map is cofinite directed if it is indexed by a cofinite directed set.
Every pro-object is isomorphic to a cofinite directed pro-object [6, Th. 2.1.*
*6] (or
[1, Expos'e 1, 8.1.6]). Similarly, up to isomorphism, every map is a cofinite d*
*irected
level map. Cofiniteness is critical for inductive arguments.
Let f : X ! Y be a cofinite directed level map. For every index t, the relati*
*ve
matching map Mtf is the map
Xt! lims n. The results of [13, Sec. 3] imply that Wn is *
*the
class of n-equivalences, and Fn is the class of co-n-fibrations, i.e., the clas*
*s of maps
that are both fibrations and co-n-equivalences. The hypotheses of Proposition 4*
*.19
are satisfied because the class of n-equivalences is closed under retract and b*
*ecause
if any two of f, g, and gf are n-equivalences, then the third is an (n-1)-equiv*
*alence.
22 HALVARD FAUSK AND DANIEL C. ISAKSEN
Before we can use Theorem 5.15 to obtain a model structure on pro-simplicial
sets, we must also observe that the filtered model structure of the previous pa*
*ra-
graph is proper. This follows from the results of [13, Sec. 3]; see the end of*
* the
proof of Proposition 8.12 and Lemma 8.13 for the topological analogue.
The resulting model structure on the category of pro-simplicial sets is the s*
*ame
as the model structure of [13, Thm. 6.4]. The weak equivalences are the pro-maps
f such that for every n 0, f is isomorphic to a levelwise n-equivalence. This*
* fact
is not clearly stated in [13], but see [13, Prop. 6.8] for a "morally equivalen*
*t" claim.
Compared to the technical methods of [13] involving basepoints, this approach is
much simpler.
Example 7.3. Similarly to Example 7.2, there is a filtered model structure on
the category of topological spaces. The class Cn is the class of Serre cofibrat*
*ions,
i.e., retracts of relative cell complexes; the class Wn is the class of n-equiv*
*alences;
and the class Fn is the class of co-n-fibrations, i.e., Serre fibrations that a*
*re also
co-n-equivalences. In this context, In is the set of generating cofibrations Sk*
*-1 !
Dk with k > n. In the same way as the previous example, one can show that
the hypotheses of Proposition 4.19 are satisfied and that the associated filter*
*ed
model structure is proper. The resulting model structure on pro-spaces is Quill*
*en
equivalent to the model structure on pro-simplicial sets from the previous exam*
*ple.
Example 7.4. The following example is a stable version of Example 7.2. Let C
be the category of spectra. Let A be Z, and define In to be the set of generati*
*ng
cofibrations whose cofibers are spheres of dimension greater than n. The result*
*s of
[16, Sec. 4] imply that Wn is the class of n-equivalences, and Fn is again the *
*class of
co-n-fibrations. As before, the hypotheses of Proposition 4.19 are satisfied, a*
*nd the
associated filtered model structure is proper. The resulting model structure on*
* the
category of pro-spectra is the same as the model structure of [16]. The weak eq*
*uiv-
alences can be described in terms of pro-homotopy groups, but the reformulation
is not quite as obvious as one might expect. See [16] for details.
Example 7.5. Let C be the category of spaces, and let A equal N. Let h* be a
homology theory on C that satisfies the colimit axiom. Let Wn be the class of m*
*aps
f such that hi(f) is an isomorphism for i < n and hn(f) is a surjection. In ord*
*er
to obtain a proper filtered model structure, we must assume that Wn is preserved
by base changes along fibrations. If we let Cn be the class of cofibrations and*
* Fn
be the class inj-(Wn \ Cn), then we get a proper filtered model structure on sp*
*aces.
We outline the verification of the axioms for a filtered model structure in t*
*his
case. Axioms 4.2 and 4.3 are obvious, as is the first half of Axiom 4.4. We def*
*er
the second half of Axiom 4.4 until later.
The first factorization of Axiom 4.5 is given by factorizations into cofibrat*
*ions
followed by acyclic fibrations. For the second part of this axiom, an adaptati*
*on
of the small object argument in [3, Sec. 11] gives the desired factorization wh*
*en
the source is cofibrant. The basic idea is to replace all statements of the fo*
*rm
"h*(K, L) = 0" with statements of the form "hi(K, L) = 0 for i n". The
factorization in general follows from standard arguments with model categories,
together with properness for the category of spaces.
Now we return to the second half of Axiom 4.4. Using the factorization of the
previous paragraph, a retract argument shows that proj-Fn equals Wn \ Cn.
MODEL STRUCTURES ON PRO-CATEGORIES 23
Having identified proj-Fn, the factorizations required by Axiom 4.6 are provi*
*ded
by factorizations into cofibrations followed by acyclic fibrations.
Axiom 4.9 follows from consideration of the long exact sequence in homology
associated to a cofiber sequence. Axiom 4.10 is satisfied by our assumption on *
*Wn.
If h* is a periodic cohomology theory, then the model structure of this examp*
*le is
just the strict model structure associated to the h*-local model structure on s*
*paces
[3].
8.The underlying model structure for pro-finite groups
Recall our goal of finding a model structure for pro-G-spaces in which {E(G=U*
*)}
is a cofibrant replacement for *, where G is a pro-finite group and U ranges ov*
*er
all finite-index normal subgroups of G. We begin by describing the equivariant
analogue of Example 7.2, but this model structure turns out not to have the des*
*ired
property.
Let G be a (topological) group, and let C be the category of G-spaces. Let C *
*be
the class of retracts of relative G-cell complexes; this is the class of cofibr*
*ations in
one of the usual model structures on the category of G-spaces. Now let A be N, *
*and
let In be the set of generating cofibrations of the form Sk-1 x G=H ! Dk x G=H,
where k > n and H is a (closed) subgroup of G. The framework of Proposition
4.19 applies; the hypotheses of this result can be verified just as in [13, Sec*
*. 3].
The class Wn turns out to be the class of G-equivariant n-equivalences, i.e., t*
*he
maps f : X ! Y such that fH : XH ! Y H is an n-equivalence for all (closed)
subgroups H of G. The class Fn is the class of equivariant co-n-fibrations, i.e*
*., the
maps f : X ! Y such that fH : XH ! Y H is a co-n-fibration for all (closed)
subgroups H of G.
The resulting model structure is a G-equivariant analogue of the model struct*
*ure
of [13, Thm. 6.4]. It can be shown that a map f : X ! Y of pro-G-spaces is a we*
*ak
equivalence if and only if ss_nf : ss_nX ! ss_nY is a pro-isomorphism of pro-co*
*efficient
systems for all n 0.
The map {E(G=U)} ! c(*) induces a pro-isomorphism after applying ssVnfor
any fixed V . However, it does not induce a pro-isomorphism after applying ss_*
*n.
The problem is that in choosing refinements, one must make different choices for
different values of V . There is no one choice that works for all V .
The point of the previous paragraphs is that we must work harder to obtain our
desired model structure. The rest of this section provides the details.
Let G be a pro-finite group. This means that G is a topological group such
that G ! limUG=U is an isomorphism, where U ranges over all normal subgroups
of G such that G=U is discrete and finite. Usually, a pro-finite group is view*
*ed
as a topological group, where the topology is totally disconnected, compact, and
Hausdorff. It is also possible to think of G as a pro-object in the category of*
* finite
groups. We will use both viewpoints.
Let C be the category of compactly generated weak Hausdorff spaces equipped
with a continuous G-action. First, recall that C is a simplicial category. If X*
* is a
G-space and K is a simplicial set, then X K is defined to be X x |K|, where t*
*he
realization |K| has a trivial G-action. Also XK is the topological mapping spa*
*ce
F (|K|, X) of non-equivariant continuous maps |K| ! X. If X and Y are both
G-spaces, then Map (X, Y ) is the simplicial set whose n-simplices are equivari*
*ant
maps X x | n| ! Y .
24 HALVARD FAUSK AND DANIEL C. ISAKSEN
Definition 8.1. Let U be a finite-index subgroup of G. A G-equivariant map
f : X ! Y is a U-weak equivalence if the map fV on V -fixed points is a weak
equivalence for all finite-index subgroups V of U, and it is a U-fibration if f*
*V is a
fibration for all finite-index subgroups V of U. It is a U-cofibration if it is*
* a retract
of a relative cell complex built from cells of the form G=V x Sn ! G=V x Dn+1,
where V is a finite-index subgroup of U.
Recall that an n-cofibration [13, Defn. 3.2] is a cofibration that is also an*
* n-
equivalence (i.e., an isomorphism on homotopy groups up to dimension n - 1 and a
surjection on nth homotopy groups). Dually, a co-n-fibration is a fibration tha*
*t is
also a co-n-equivalence (i.e., an isomorphism on homotopy groups above dimension
n and an injection on nth homotopy groups). We will now make similar definitions
in the equivariant situation.
Definition 8.2. A map f is a U-n-equivalence (resp., U-co-n-equivalence if
the map fV is an n-equivalence (resp., co-n-equivalence) for all finite-index *
*sub-
groups V of U. A map f is a U-co-n-fibration if fV is a co-n-fibration for all
finite-index subgroups V of U. Finally, a map is a U-n-cofibration if it is a
U-cofibration such that fV is an n-equivalence for all finite-index subgroups V*
* of
U.
The following two lemmas are proved in exactly the same way as their non-
equivariant analogues [13, Sec. 3].
Lemma 8.3. Any G-equivariant map can be factored into a U-n-cofibration fol-
lowed by a U-co-n-fibration.
Proof.Use the small object argument applied to the set of maps of the form G=V x
Sk ! G=V x Dk+1, where V is a finite-index subgroup of U and k n, together
with all maps of the form G=V x Im ! G=V x Im+1 , where V is a finite-index
subgroup of U and m is arbitrary. Here Im is the m-cube, and the map Im ! Im+1
is the inclusion of a face.
Lemma 8.4. The classes of U-n-cofibrations and U-co-n-fibrations are determined
by lifting properties with respect to each other.
Proof.This can be proved with an obstruction theory argument. See [13, Lem. 3.4]
and [13, Lem. 3.6] for more details.
Let A be the set consisting of all pairs (U, n), where U is a finite-index su*
*bgroup
of G and n is a non-negative integer. We write (U, n) (V, m) if U is containe*
*d in
V and if n m. This makes A into a directed set. In other words, given (U1, n1)
and (U2, n2), there exists (V, m) such that (V, m) (U1, n1) and (V, m) (U2,*
* n2).
To see why this is true, just observe that U1 \ U2 is a finite-index subgroup o*
*f G
whenever U1 and U2 are finite-index subgroups.
For each (U, n) in A, we will define three classes CU,n, WU,n, and FU,n of G-
equivariant maps.
Definition 8.5. The class CU,n is the class of all U-cofibrations. The class WU*
*,n
is the class of maps that are V -n-equivalences for some V . The class FU,n is *
*the
class of all U-co-n-fibrations.
Note that CU,n does not actually depend on n, and WU,n does not actually
depend on U. The point of this seemingly confusing notation is that there is ju*
*st
one indexing set A for all three families of classes.
MODEL STRUCTURES ON PRO-CATEGORIES 25
As in Definition 4.7, we write F for the union of the classes FU,n.
Example 8.6. The following example emphasizes a subtlety in the definition of
WU,n, Consider the map
qU E(G=U) ! qU *,
where U ranges over the finite-index normal subgroups of G. This map is an
underlying weak equivalence. However, it is not a V -equivalence for any finit*
*e-
index subgroup V of G and thus does not belong to WV,nfor any (V, n).
Lemma 8.7. If (V, m) (U, n), then CV,m is contained in CU,n, WV,m is contained
in WU,n, and FU,nis contained in FV,m.
Proof.All three claims follow immediately from the definitions. If V is contain*
*ed
in U, then the set of generating V -cofibrations is a subset of the set generat*
*ing
U-cofibrations. This shows that CV,m is contained in CU,n.
If m n, then an m-equivalence is automatically an n-equivalence; this shows
that WV,m is contained in WU,n.
If m n, then a co-n-fibration is automatically a co-m-fibration. If V is co*
*n-
tained in U, then a U-fibration is automatically a V -fibration. This shows th*
*at
FU,nis contained in FV,m.
Lemma 8.8. The class inj-CU,n(i.e., the class of maps that have the right lifti*
*ng
property with respect to all U-cofibrations equals the class of U-acyclic fibra*
*tions
(i.e., maps that are both U-weak equivalences and U-fibrations). The class proj*
*-FU,n
(i.e., the class of maps that have the left lifting property with respect to al*
*l U-co-n-
fibrations) equals the class of U-n-cofibrations.
Proof.The first claim follows from standard equivariant homotopy theory. The
second claim is immediate from Lemma 8.4.
Recall that C is the category of compactly generated weak Hausdorff spaces wi*
*th
continuous G-actions and G-equivariant maps. The following definition is a spec*
*ial
case of Definitions 5.1, 5.2, and 5.3.
Definition 8.9. A map in pro-C is a cofibration if it is an essentially levelwi*
*se
CU,n-map for every (U, n) in A. A map in pro-C is a weak equivalence if it is an
essentially levelwise WU,n-map for every (U, n) in A. A map in pro-C is a fibra*
*tion
if it is a retract of a special F -map.
Theorem 8.10. Definition 8.9 is a proper simplicial model structure on the cate-
gory pro-C.
Proof.Using Theorems 5.15 and 5.16, we just need to verify that Definition 8.5 *
*is
a proper simplicial filtered model structure. This is provided below in Proposi*
*tions
8.11, 8.12, and 8.15.
Proposition 8.11. Definition 8.5 is a filtered model structure.
Proof.We have to verify Axioms 4.2 through 4.6.
We have already observed that A is a directed set. Lemma 8.7 says that the
containments given in Definition 4.1 are satisfied. Also, the category of G-spa*
*ces
is complete and cocomplete; limits and colimits are constructed in the underlyi*
*ng
category of topological spaces.
26 HALVARD FAUSK AND DANIEL C. ISAKSEN
For Axiom 4.2, first observe that if any two of the maps f, g, and gf are V -
(n + 1)-equivalences, then a simple diagram chase shows that the third is a V -
n-equivalence. If any two of f, g, and gf belong to WU,n+1, then there exists a
finite-index subgroup V of G such that the two maps are V -(n + 1)-equivalences.
The third map is a V -n-equivalence, which means that it belongs to WU,n.
We now consider Axiom 4.3. The V -n-equivalences are closed under retract sin*
*ce
non-equivariant n-equivalences are closed under retract, so WU,n is closed under
retract. The class CU,nof U-cofibrations is closed under retract, finite compos*
*itions,
and arbitrary cobase changes because it is defined in terms of retracts of rela*
*tive
cell complexes. The class FU,n is defined by a right lifting property (see Lem*
*ma
8.4), so it is closed under retract, finite compositions, and arbitrary base ch*
*anges.
Axiom 4.4 is immediate from Lemma 8.8.
The first half of Axiom 4.5 is given by factorizations into U-cofibrations fo*
*llowed
by U-acyclic fibrations; these factorizations are supplied by standard equivari*
*ant
homotopy theory. The second half of Axiom 4.5 is given by Lemma 8.3.
For Axiom 4.6, let f belong to WU,n. This means that f is a V -n-equivalence *
*for
some V . Now factor f into a V -cofibration i followed by a V -acyclic fibratio*
*n p. By
Lemma 8.8, this means that p belongs to inj-CV,n. Because f is a V -n-equivalen*
*ce,
i is also a V -n-equivalence and hence a V -n-cofibration. By Lemma 8.8 again, *
*this
means that i belongs to proj-FV,n.
Proposition 8.12. Definition 8.5 is a proper filtered model structure.
Proof.We just need to verify Axioms 4.9 and 4.10.
For Axiom 4.9, suppose that f : A ! B is a U-cofibration and that g : A ! C is
a V -n-equivalence for some finite-index subgroups U and V of G. We may replace
V by V \ U to assume that V is a finite-index subgroup of U; this is allowed
because g is a still a V -n-equivalence. We will show that the map h : B ! B qA*
* C
is also a V -n-equivalence. If W is a finite-index subgroup of V , then (B qA C*
*)W
is equal to BW qAW CW ; this uses the fact that f is injective. Now W is a
finite-index subgroup of U, so fW is a non-equivariant cofibration. Since gW *
*is a
non-equivariant n-equivalence, we need only show that cobase changes along non-
equivariant cofibrations preserve non-equivariant n-equivalences. This is prove*
*d in
Lemma 8.13 below.
For Axiom 4.10, suppose that f : X ! Y is a U-fibration and that g : Z ! Y
is a V -n-equivalence for some finite-index subgroups U and V of G. Choose a
finite-index subgroup W contained in both V and U. Then f is a W -fibration
and g is a W -n-equivalence. Taking fixed points commutes with fiber products.
Therefore, in order to show that X xY Z ! X is a W -n-equivalence, we only need
prove that base changes of non-equivariant n-equivalences along non-equivariant
fibrations are n-equivalences. This last fact follows from the five lemma and *
*the
long exact sequence of homotopy groups for a fibration.
Lemma 8.13. Let f : A ! B be a cofibration of topological spaces, and let g : A*
* !
C be an n-equivalence. Then the map h : B ! B qA C is also an n-equivalence.
Proof.Since the usual model category on topological spaces is left proper, the
pushout B qA C is in fact a homotopy pushout. Therefore, we may replace g by a
weakly equivalent cofibration; this will not change the weak homotopy type of h.
MODEL STRUCTURES ON PRO-CATEGORIES 27
Now we have that g is an n-cofibration (i.e., a cofibration and an n-equivale*
*nce).
The class of n-cofibrations is determined by a left lifting property; this is t*
*he non-
equivariant version of Lemma 8.4. Therefore, the class of n-cofibrations is clo*
*sed
under arbitrary cobase changes, so h is also an n-cofibration.
Remark 8.14. The reader may feel that it is not possible to prove Lemma 8.13
without using van Kampen's theorem. Van Kampen's theorem is necessary to prove
that the model category of topological spaces is left proper, so we are in fact*
* using
it in a disguised way.
Proposition 8.15. Definition 8.5 is a simplicial filtered model structure.
Proof.We have already observed that C is a simplicial category, so we just need*
* to
prove that Axiom 4.12 holds.
Let j : K ! L be a cofibration of finite simplicial sets, and let i : A ! B b*
*e a
U-cofibration. Standard equivariant homotopy theory implies that the map
f : A L qA K B K ! B L
is also a U-cofibration.
Similarly, if j is an acyclic cofibration, then standard equivariant homotopy
theory implies that f is a U-acyclic cofibration. This implies that f is a U-n-
cofibration, which means that it belongs to proj-FU,nby Lemma 8.8.
Next, suppose that j is a cofibration and that i belongs to proj-FU,n. By Lem*
*ma
8.8, this means that i is a U-n-cofibration. We want to conclude that f is also*
* a
U-n-cofibration. We have already shown that f is a U-cofibration, so we just ne*
*ed
to show that fV is an n-equivalence for every finite-index subgroup of V of U.
The map fV is equal to the map
AV L qAV K BV K ! BV L.
This follows from the fact that the G-actions on K and L are trivial and that t*
*aking
V -fixed points commutes with the pushout because A K ! A L is injective.
Now the map iV : AV ! BV is an n-equivalence because i is a U-n-cofibration, so
the desired conclusion follows from Lemma 8.16 below.
Lemma 8.16. Suppose that j : K ! L is a cofibration of simplicial sets, and
suppose that i : A ! B is an n-cofibration of non-equivariant topological space*
*s.
Then the map
f : A L qA K B K ! B L
is an n-cofibration.
Proof.Consider the diagram
A K __________//_B K
| |
| |
fflffl| fflffl|
A L ____//_XXXXXA LQqA K B K
XXXXX QQQQf
XXXXXXX QQQQ
XXXXXXX QQQQ
XXXXX,,X((
B L.
We will show that A K ! B K and A L ! B L are n-cofibrations.
Then, since n-cofibrations are preserved by cobase changes, it follows that the*
* map
28 HALVARD FAUSK AND DANIEL C. ISAKSEN
A L ! A L qA K B K is an n-cofibration. A small diagram chase proves
that f is an n-equivalence.
It remains only to prove that A K ! B K is an n-cofibration; the proof
for A L ! B L is identical. First, standard homotopy theory of topological
spaces says that Ax|K| ! B x|K| is a cofibration. Second, since homotopy groups
commute with products, the map A x |K| ! B x |K| is an n-equivalence.
Remark 8.17. The basic ideas of this section can be implemented in exactly the
same way for naive G-spectra. One small difference is that the indexing set A
consists of pairs (U, n) where U is a finite-index subgroup of G as before but *
*n is an
arbitrary integer, possibly negative. See [16] for details concerning n-cofibra*
*tions
and co-n-fibrations of spectra.
Finally, we can establish our main motivation for producing the model structu*
*re
of Theorem 8.10.
Proposition 8.18. The pro-G-space {E(G=U)} is a cofibrant replacement for the
constant trivial pro-space c(*) in the model structure of Theorem 8.10.
Proof.Note first that for each finite-index subgroup U, E(G=U) can be built from
cells of the form Sk-1 x G=U ! Dk x G=U. This means that the map from the
empty set to E(G=V ) is a U-cofibration whenever V is contained in U. It follows
that OE ! {E(G=U)} is a cofibration of pro-G-spaces.
Now we will show that the map E(G=U) ! * is a U-weak equivalence for each
U; this will imply that the map {E(G=U)} ! c(*) is a weak equivalence of pro-
G-spaces. If V is a finite-index subgroup of U then the V -fixed points of E(G=*
*U)
equals E(G=U). Since E(G=U) is contractible, it follows that the map
E(G=U)V ! *V
is a weak equivalence.
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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