A MODEL STRUCTURE ON THE CATEGORY OF
PRO-SIMPLICIAL SETS
DANIEL C. ISAKSEN
Abstract.We study the category pro-SSof pro-simplicial sets, which arises
in 'etale homotopy theory, shape theory, and pro-finite completion. We e*
*stab-
lish a model structure on pro-SSso that it is possible to do homotopy th*
*eory
in this category. This model structure is closely related to the strict *
*structure
of Edwards and Hastings. In order to understand the notion of homotopy
groups for pro-spaces we use local systems on pro-spaces. We also give s*
*ev-
eral alternative descriptions of weak equivalences, including a cohomolo*
*gical
characterization. We outline dual constructions for ind-spaces.
1.Introduction
If C is a category, then the pro-category pro-C [1, Expos'e 1, Section 8] is *
*the
category whose objects are small cofiltered systems in C (of arbitrary shape) a*
*nd
whose morphisms are given by the formula
Hom (X, Y ) = limscolimtHomC(Xt, Ys).
While investigating the 'etale homotopy functor, Artin and Mazur [2] studied the
category pro-Ho(SS), where Ho(SS) is the homotopy category of simplicial sets.
They also introduced a notion of weak equivalence of pointed connected pro-spac*
*es
that involved isomorphisms of pro-homotopy groups.
However, an Artin-Mazur weak equivalence is not the same as an isomorphism
in pro-Ho(SS). This suggests that pro-Ho(SS) is not quite the correct category *
*for
studying 'etale homotopy.
Around the same time, Quillen [14] developed the fundamental notions of homo-
topical algebra by realizing that model structures allow one to do homotopy the*
*ory
in many different categorical contexts. A model structure on a category is a ch*
*oice
of three classes of maps (weak equivalences, cofibrations, and fibrations) sati*
*sfying
certain axioms. The weak equivalences are inverted to obtain the associated hom*
*o-
topy category, while the cofibrations and fibrations serve an auxiliary role. Q*
*uillen
[14, II.0.2] observed that the homotopy theory of pro-spaces would be an intere*
*sting
application of model structures.
At least two model structures for pro-spaces are already known to exist. Ed-
wards and Hastings [6] established a "strict" model structure on pro-spaces for*
* the
purposes of shape theory and proper homotopy theory, but their weak equivalences
did not generalize those of Artin and Mazur.
____________
Date: June 18, 2001.
1991 Mathematics Subject Classification. Primary 18E35, 55Pxx, 55U35; Second*
*ary 14F35,
55P60.
Key words and phrases. Closed model structures, pro-spaces, 'etale homotopy.
The author was supported in part by an NSF Graduate Fellowship.
1
2 DANIEL C. ISAKSEN
Also, Grossman [9] described a different model structure for pro-spaces that *
*are
countable towers. The weak equivalences of this theory are appropriately related
to the Artin-Mazur equivalences. The category of towers is suitable in many app*
*li-
cations from proper homotopy theory because it is reasonable to assume that the
neighborhoods at infinity have a countable basis.
However, applications of pro-spaces to the algebro-geometric concept of 'etale
cohomology require more general pro-spaces. General cofiltered systems of spaces
are necessary for essentially the same reason that the sheaf theory of Grothend*
*ieck
topologies is necessary to define 'etale cohomology.
This paper gives a generalization of Grossman's model structure to the entire
category of pro-spaces. Weak equivalences between pointed connected pro-spaces
are precisely Artin-Mazur weak equivalences. The Edwards-Hastings strict struc-
ture is an intermediate stage to building our structure. Our homotopy theory is*
* the
P -localization of the strict homotopy theory, where P is the functor that repl*
*aces
a space with its Postnikov tower.
Our weak equivalences have several alternative characterizations, one of which
uses twisted cohomology. This is important because it is often difficult to ch*
*eck
that a map of pro-spaces induces an isomorphism on homotopy groups. Usually
it is much easier to verify a cohomology isomorphism and then conclude that the
map is a weak equivalence.
Another characterization of weak equivalences is in terms of "eventually n-
connected" maps (see Theorem 7.3 (d)), which is a very convenient property in
practice. The equivalence of this property with the definition of weak equivale*
*nce
is not obvious. One must use the full power of the model structure to prove the
equivalence.
We mention two applications of this homotopy theory of pro-spaces. First, the
model structure gives a more convenient category for studying 'etale homotopy b*
*e-
cause it allows the reinterpretation of the central ideas of the theory in term*
*s of
the established notions of model structures. It is also a start towards the def*
*inition
of generalized cohomology of pro-spaces. For example, define K0 to be represent*
*ed
by the constant pro-space BU. The realization of K-theory as a generalized co-
homology theory requires more understanding of the category of pro-spectra. The
'etale K-theory of a scheme [5] is probably most clearly expressed as a general*
*ized
cohomology theory applied to the 'etale homotopy type.
Second, pro-spaces arise in the study of pro-finite completion [15] [4] [13].*
* Again,
the new model structure provides a better category in which to study such compl*
*e-
tions. For example, Mandell [11] has used the model structure to compare his new
algebraic construction of p-adic homotopy theory to Sullivan's p-pro-finite com*
*ple-
tion.
We describe briefly the model structure; the formal definitions appear in Sec*
*tion
6. Given a pointed pro-space X (i.e., a map from the one-point constant pro-spa*
*ce
* to X or equivalently a pro-object in the category of pointed spaces), one can*
* apply
the functor ßn(-, *) to each space in the pro-system to get a pro-group ßn(X, **
*).
The most obvious notion of equivalence of pro-spaces X ! Y is the requirement
that the map induce an isomorphism of pro-homotopy groups ßn(X, *) ! ßn(Y, *)
for every n 0 and every point * of X.
However, points of a pro-space are rather awkward. In fact, some non-trivial *
*pro-
spaces have no points whatsoever. Local systems permit the discussion of homoto*
*py
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 3
groups without choosing basepoints. Grossman's work on towers [9] inspired this
trick.
Cofibrations of pro-spaces are maps that are isomorphic to levelwise cofibrat*
*ions
of systems of spaces of the same shape. The model category axioms then force
the definition of fibrations. These fibrations are similar to those of Edwards *
*and
Hastings [6, 3.3], but they satisfy an extra condition that compares the homoto*
*py
groups of the total pro-space to the homotopy groups of the base pro-space.
The model structure has a few interesting aspects. For example, it is not cof*
*i-
brantly generated. This means that the proof of the model axioms is quite diffe*
*rent
from the standard arguments. One result of this fact is that the factorizations*
* are
not functorial. Almost all naturally arising model structures are cofibrantly g*
*ener-
ated. Hence pro-spaces are an interesting example of a non-cofibrantly generated
model structure.
Also, we only know how to work with pro-simplicial sets, not pro-topological
spaces. We use in a very significant way the fact that finite dimensional skele*
*tons
are functorial for simplicial sets. Since relative cell complexes do not have f*
*unctorial
skeletons, the same proofs do not apply.
Finally, note that the category of ind-spaces has a similar model structure. *
*We
do not provide details in this paper because we have no application in mind. Se*
*veral
comments throughout the paper explain where the significant differences occur.
The paper is divided into three main parts. Sections 2-5 give some background
material and introduce language and tools for later use. Sections 6-10 describe
the model structure, state the main theorems, and make comparisons to other
homotopy theories. Sections 11-19 provide proofs of the main theorems.
We assume familiarity with model structures. See [10] or [14] for background
material.
Many of the important ideas in this paper come from Grossman [9]. I thank
Peter May, Bill Dwyer, Brooke Shipley, Greg Arone, Michael Mandell, and Charles
Rezk for many helpful conversations throughout the progress of this work.
2. Preliminaries on Pro-Categories
First we establish some terminology for pro-categories.
Definition 2.1.For a category C, the category pro-C has objects all small cofil-
tering systems in C, and
Hom pro-C(X, Y ) = limscolimtHomC(Xt, Ys).
Objects of pro-C can be thought of as functors from arbitrary small cofilteri*
*ng
categories to C. See [1] or [2] for more background on the definition of pro-ca*
*tegories.
We use both set theoretic and categorical language to discuss indexing categori*
*es;
hence "t sä nd "t ! s" mean the same thing.
The word "systemä lways refers to an object of a pro-category, while the word
"diagram" refers to a diagram of pro-objects.
A subsystem of an object X : I ! C in pro-C is a restriction of X to a cofilt*
*ering
subcategory J of I. A subsystem is cofinal if for every s in I, there exists so*
*me t
in J and an arrow t ! s in I. A system is isomorphic in pro-C to any of its cof*
*inal
subsystems.
A directed set I is cofinite if for every t, the set of elements of I less th*
*an t
is finite. Except in Section 11, all systems are indexed by cofinite directed *
*sets
4 DANIEL C. ISAKSEN
rather than arbitrary cofiltering categories. This is no loss of generality [6,*
* 2.1.6]
(or [1, Expos'e 1, 8.1.6]). The cofiniteness is critical because many construct*
*ions
and proofs proceed inductively.
Whenever possible we avoid mentioning the structure maps of a pro-object X
explicitly. When necessary the notation (X, OE) indicates a system of objects {*
*Xs}
with structure maps OEts: Xt! Xs.
Definition 2.2.Let I be a cofinite directed set. For each s in I, the height h(*
*s)
of s is the value of n in the longest chain s > s1 > s2 > . .>.sn starting at s*
* in S.
In particular, h(s) = 0 if and only if there are no elements of I less than s.
We always assume that systems have no initial object or equivalently that any
system has objects of arbitrarily large height. This is no loss of generality s*
*ince it is
possible to add isomorphisms to a system so that it no longer has an initial ob*
*ject,
and this new system is isomorphic to the old one.
We frequently consider maps between two pro-objects with the same index cat-
egories. In this setting, a level map X ! Y between pro-objects indexed by I is
given by maps Xs ! Ys for all s in I. Up to isomorphism, every map is a level m*
*ap
[2, Appendix 3.2].
A map satisfies a certain property levelwise if it is a level map X ! Y such *
*that
each Xs ! Ys satisfies that property.
Lemma 2.3. A level map A ! B in pro-C is an isomorphism if and only if for all
s, there exists t s and a commutative diagram
At _____//_Bt
| ---- |
| -- |
fflffl|""fflffl|--
As _____//Bs.
Proof.The maps Bt! As induce an inverse. |___|
3. Preliminaries on Simplicial Sets
Now we review some definitions and results about simplicial sets. Let SS be t*
*he
category of simplicial sets. We use the expressions "spaceä nd "simplicial se*
*t"
interchangeably.
For simplification, we often use the same notation for a basepoint and its im*
*age
under various maps (e.g., ßn(X, *) ! ßn(Y, *)).
Definition 3.1.A map f : X ! Y of simplicial sets is an n-equivalence if for all
basepoints * in X, f induces an isomorphism ßi(X, *) ! ßi(Y, *) for 0 i < n
and a surjection ßn(X, *) ! ßn(Y, *). The map f is a co-n-equivalence if for a*
*ll
basepoints * in X, f induces an isomorphism ßi(X, *) ! ßi(Y, *) for i > n and an
injection ßn(X, *) ! ßn(Y, *).
Definition 3.2.Set
Jn = { mk! m |m 0} [ {@ m ! m |m > n}.
A map of simplicial sets is a co-n-fibration if it has the right lifting proper*
*ty with
respect to all maps in Jn. A map of simplicial sets is an n-cofibration if it h*
*as the
left lifting property with respect to all co-n-fibrations.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 5
In other words, a co-n-fibration is a Jn-injective, and an n-cofibration is a*
* Jn-
cofibration [10, 12.4.1]. Note that co-n-fibrations and n-cofibrations are char*
*acter-
ized by lifting properties with respect to each other. When n = -1 or n = 1, the
definitions reduce to the usual definitions of trivial cofibrations and fibrati*
*ons or to
the definitions of cofibrations and trivial fibrations.
We will see below that n-cofibrations are just maps that are both cofibrations
and n-equivalences. Also, co-n-fibrations are just maps that are both fibration*
*s and
co-n-equivalences. These facts motivate the terminology.
Proposition 3.3.For any n, each map f of simplicial sets factors as f = pi,
where i is an n-cofibration and p is a co-n-fibration.
Proof.Apply the small object argument [10, 12.4]. |___|
Lemma 3.4. A map f : E ! B is a co-n-fibration if and only if f is a fibration
and co-n-equivalence.
Proof.First suppose that f is a co-n-fibration. The generating trivial cofibrat*
*ions
are contained in Jn, so f is also a fibration. Now we must show that f is also a
co-n-equivalence.
Consider test diagrams of the form
@ m _____//E< n. Let g : m ! B be a constant map with image *, and let F be the
fiber of f over *. Since lifts exist in the test diagram, ßm F = 0 for m n. U*
*sing
the long exact sequence of homotopy groups of a fibration, it follows that f is*
* a
co-n-equivalence.
Now suppose that f is a fibration and co-n-equivalence. It follows from the l*
*ong
exact sequence of homotopy groups that ßm F = 0 for m n, where F is any
fiber of f. There are lifts in the test diagrams shown above for m > n because *
*the __
obstructions to such lifts belong to ßm-1 F . Hence f is a co-n-fibration. *
* |__|
Definition 3.5.If A ! X is a cofibration of simplicial sets, then the relative
n-skeleton X(n)is the union of A and the n-skeleton of X.
Lemma 3.6. A map f : A ! X is an n-cofibration if and only if f is a cofibrati*
*on
and n-equivalence.
Proof.Since trivial fibrations are co-n-fibrations, n-cofibrations are also cof*
*ibra-
tions.
Suppose that f : A ! X is a relative Jn-cell complex [10, 12.4.6]. Then A !
X(n)is a weak equivalence since X(n)is obtained from A by gluing along maps of
the form mk! m . Note that X(n)! X is an n-equivalence, so A ! X is also
an n-equivalence. Arbitrary n-cofibrations are retracts of relative Jn-cell com*
*plexes
[10, 13.2.9], so all n-cofibrations are n-equivalences.
Conversely, suppose that f is a cofibration and n-equivalence. We show that f*
* is
an n-cofibration by demonstrating that it has the left lifting property with re*
*spect
to all maps that are both fibrations and co-n-equivalences. By Lemma 3.4, this
means that f has the left lifting property with respect to all co-n-fibrations.
6 DANIEL C. ISAKSEN
Factor f as
A __j_//_Yq__//X,
where j is a trivial cofibration and q is a fibration. Note that q is also an *
*n-
equivalence. Let p : E ! B be a map that is both a fibration and co-n-equivalen*
*ce,
and consider a diagram
A _____________//E77nn
j|| n n n |p|
fflffl|nnn fflffl|
Y __q__//X____//_B.
The dashed arrow exists because j is a trivial cofibration and p is a fibration.
This gives the diagram
A _____//Y_____//_E
f || |p|
fflffl| fflffl|
X _____//X_____//B.
There is no obstruction to lifting over X(0)since ß0A ! ß0X is surjective.
Obstructions to finding a lift over the higher relative skeletons of X belong*
* to
ßm F , where F is some fiber of p. These obstructions lie in the image of ßm G,
where G is a fiber of q. For every m, either ßm G or ßm F is zero. Hence there_*
*are
no obstructions to lifting. |__|
Remark 3.7.For ind-spaces, consider the set
In = { mk! m |m 0} [ {@ m ! m |m < n}
to define n-fibrations and co-n-cofibrations. All n-fibrations are both fibrati*
*ons and
n-equivalences, but the converse is not true. If f : A ! X is a co-n-cofibratio*
*n,
then f is a cofibration and the induced map X(n-1)! X is a weak equivalence.
4. Local Systems
The language of local systems is necessary in order to state the idea of homo*
*topy
groups for pro-spaces. Recall that a local system on a space X is a functor X !
Ab, where X is the fundamental groupoid of X and Ab is the category of abelian
groups. Denote by LS(X) the category of local systems on X or equivalently the
category of locally constant sheaves on X.
For example, nX is a local system on X for n 2. It is defined by ( nX)x =
ßn(X, x) with isomorphisms ( nX)x ! ( nX)y given by the usual maps on ho-
motopy groups induced by paths.
Occasionally we refer to local systems with values in non-abelian groups. For
example, 1X is such a local system. We emphasize the notational distinction
between X (a groupoid) and 1X (a local system).
If f : X ! Y is a map of spaces, then f induces a map of local systems nX !
f* nY on X, where f* is the pullback functor LS(Y ) ! LS(X). Recall that the
functor f* is exact in the sense that it preserves finite limits and colimits, *
*and it is
also exact in the sense that it preserves exact sequences.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 7
Lemma 4.1. Let f : X ! Y be a map of spaces. Then f is a weak equivalence if
and only if ß0f is an isomorphism and nX ! f* nY is an isomorphism of local
systems on X for all n 1.
Proof.This is a restatement without reference to basepoints of the definition_of
weak equivalence of simplicial sets. |__|
We now extend the definitions to pro-spaces.
Definition 4.2.A local system on a pro-space X is an object of colimsLS(Xs). If
L is a local system on (X, OE) represented by a functor Ls : Xs ! Ab and M is
another local system on X represented by a functor Mt : Xt ! Ab, then a map
from L to M is an element of colimuHom LS(Xu)(OE*usLs, OE*utMt). Denote by LS(X)
the category of local systems on X.
A local system on X is represented by a local system on Xs for some s. For
example, for n 1, nXs is a local system on X for each s.
A map between two local systems on X is a map of representing local systems
pulled back to some Xu. For example, for n 1 and t s, nXt ! nXs is a
map of local systems on (X, OE) because nXt! OE*ts nXs is a map of local syste*
*ms
on Xt.
Let f : X ! Y be a map of pro-spaces, and let L : Ys ! Ab be a local system
on Ys. Choose any map fts: Xt ! Ys representing f and consider the functor
L O fts: Xt ! Ys ! Ab . This gives a well-defined functor colimsLS(Ys) !
colimtLS(Xt).
Definition 4.3.Let f : X ! Y be a map of pro-spaces. The pullback f* :
LS(Y ) ! LS(X) is the functor colimsLS(Ys) ! colimtLS(Xt).
Lemma 4.4. If f : X ! Y is a map of pro-spaces, then f* : LS(Y ) ! LS(X) is
an exact functor in the sense that it preserves finite limits and colimits.
Proof.Without loss of generality, we may assume that f is a level map. Given a
finite diagram of local systems L on Y , there is an s such that each Liis repr*
*esented
by a local system Lison Ys. Then for each i, f*Li is represented by f*sLis. Now
colimLi in LS(Y ) is represented by colimLisin LS(Ys), so f*(colimLi) is repre-
sented by f*s(colimLis). Also, colimf*Li in LS(X) is represented by colimf*sLis*
*in
LS(Xs). But f*scommutes with finite colimits, so colimf*Li and f*(colimLi) are
isomorphic since they are represented by the same local system on Xs. __
An identical argument shows that f* commutes with finite limits. |__|
It follows from the lemma that f* is exact in the sense that it preserves exa*
*ct
sequences.
5. Homotopy Groups
With the notions of local systems in place, we can define homotopy groups of
pro-spaces as pro-objects in a category of local systems. The local systems are
necessary to avoid mentioning basepoints.
Definition 5.1.If X is a pro-space and n 2, then nX is the pro-local system
on X given by { nXs}. Also, ß0X is the pro-set given by {ß0Xs}, and 1X is the
pro-local system on X with values in non-abelian groups given by { 1Xs}.
8 DANIEL C. ISAKSEN
Note that a map of pro-spaces f : X ! Y induces a map nX ! f* nY in
pro-LS(X).
Lemma 5.2. If f : X ! Y is a map of pro-spaces and ß0f is an epimorphism in
the category of pro-sets, then a map g of pro-local systems is an isomorphism in
pro-LS(Y ) if and only if f*(g) is an isomorphism in pro-LS(X).
Proof.Without loss of generality, assume that f is a level map. Note that pro-L*
*S(Y )
is an abelian category [2, Appendix 4.5]. Since f* is exact, it suffices to con*
*sider a
local system L on Y such that f*L = 0 and conclude that L = 0.
A pro-local system M is zero if and only if for every i, there exists j i s*
*uch
that the map Mj ! Mi is trivial. For any i, choose j i so that f*Lj ! f*Li is
trivial. Let Lisand Ljsbe local systems on Ys representing Li and Lj respective*
*ly.
Choose s large enough so that f*sLjs! f*sLisis a trivial map of local systems on
Xs.
Since ß0f is an epimorphism, there exists some t and a map ß0Yt! ß0Xs such
that the map ß0Yt ! ß0Ys factors through ß0Xs. Since f*sLjs! f*sLisis trivial,
the map Ljs! Lisis trivial when restricted to the components of Ys in the image
of Xs. Now the image of ß0Xs in ß0Ys contains the image of ß0Yt, so Ljs! Lis
becomes trivial when pulled back to Yt. Hence the map Lj ! Li is trivial. This_
means that the pro-local system L is zero. |__|
Remark 5.3.A similar statement is true for pro-local systems with values in non-
abelian groups.
Lemma 5.4. A map of pro-spaces f : X ! Y induces an isomorphism of pro-sets
ß0f : ß0X ! ß0Y if and only if f is isomorphic to a level map f0 : X0 ! Y 0such
that f0 induces a level isomorphism ß0f0 : ß0X0! ß0Y 0.
Proof.One direction is clear because level isomorphisms are pro-isomorphisms.
Assume that ß0f is an isomorphism. We may also assume that f is a level map.
Define X0= X and Y 0= Y xß0Yß0X. Here we identify pro-sets with pro-spaces
of dimension zero.
Then Y 0is isomorphic to Y since ß0X ! ß0Y is an isomorphism. Let f0 : X0!
Y 0be the map induced by f : X ! Y and the projection X ! ß0X. Pullbacks can
be constructed levelwise in pro-categories, so for all s, Ys0= Ys xß0Ysß0Xs and*
* f0s
is induced by fs : Xs ! Ys and Xs ! ß0Xs. Note that f0sinduces an isomorphism_
ß0X0s! ß0Ys0. Hence f0 is the desired map. |__|
If f : X ! Y is a map of spaces such that ß0f is an isomorphism and ß1f is
an isomorphism for every basepoint, then f* induces an equivalence of categories
LS(Y ) ! LS (X). The following lemma makes an analogous statement for pro-
spaces.
Lemma 5.5. If f : X ! Y is a map of pro-spaces such that ß0f and 1X !
f* 1Y are isomorphisms, then the functor f* : LS(Y ) ! LS(X) is an equivalence
of categories.
Proof.We only prove that f* is essentially surjective in the sense that every o*
*bject
L of LS(X) is isomorphic to f*M for some M in LS(Y ). We leave the rest of the
proof to the interested reader. We will use only the surjectivity in this work.
With no loss of generality, we may assume that f is a level map. By Lemma 5.4,
we may also assume that ß0f is a level isomorphism.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 9
Let L be a local system on (X, OE) represented by a local system Ls on Xs. Th*
*ere
exists t s and a commutative diagram
1Xt _______//_f*t 1Yt
| qqqqq |
| qqqq |
fflffl|xxqq fflffl|
OE*ts 1Xs___//OE*tsf*s 1Ys
of local systems on Xt.
Choose one point xi in each component of Xt. Let yi be the image of xi in Yt;
this is a choice of one point in each component of Yt. Let x0ibe the image of x*
*i in
Xs.
By evaluating the above diagram at xi, the map ß1(Xt, xi) ! ß1(Xs, x0i) facto*
*rs
as
ß1(Xt, xi)___//_ß1(Yt,_yi)gi//_ß1(Xs, x0i).
The maps gi : ß1(Yt, yi) ! ß1(Xs, x0i) and the local system Ls determine a local
system Mt on Yt by setting (Mt)yi= Lx0iwith the ß1(Yt, yi)-action induced by gi.
Let M be the local system on Y represented by Mt. Note that f*tMtis isomorphi*
*c_
to OE*tsLs. Hence f*M is isomorphic to L. |__|
Remark 5.6.A similar statement applies to local systems with values in non-abel*
*ian
groups.
6.Model Structure
Now we explicitly describe the model structure on the category of pro-spaces.
Definition 6.1.A map of pro-spaces f : X ! Y is a weak equivalence if ß0f is an
isomorphism of pro-sets and nX ! f* nY is an isomorphism in pro-LS(X) for
all n 1.
In Corollary 7.5 we will see that for pointed connected pro-spaces, a level m*
*ap
X ! Y is a weak equivalence if and only if ßnX ! ßnY is a pro-isomorphism
for all n. This works because there is no need for arbitrary basepoints. Hence
Artin-Mazur weak equivalences [2, Section 4] are also weak equivalences.
Definition 6.2.A map of pro-spaces is a cofibration if it is isomorphic to a le*
*vel-
wise cofibration.
Definition 6.3.A map of pro-spaces is a fibration if it has the right lifting p*
*roperty
with respect to all trivial cofibrations.
Theorem 6.4. The above definitions give a proper simplicial model structure on
pro-SS(without functorial factorizations). This model structure is not cofibran*
*tly
generated.
Proof.The axioms for a proper simplicial model structure are verified in Sectio*
*ns
11 through 17.
Limits and colimits exist by Proposition 11.1. The two-out-of-three axiom is
Proposition 13.1. Retracts preserve weak equivalences because weak equivalences
are defined in terms of isomorphisms. Retracts preserve fibrations because retr*
*acts
preserve lifting properties. Corollary 12.2 is the retract axiom for cofibrati*
*ons.
Propositions 15.1 and 15.2 are the factoring axioms, while Proposition 15.4 is *
*the
10 DANIEL C. ISAKSEN
non-trivial lifting axiom. The axioms for a simplicial model structure are demo*
*n-
strated in Proposition 16.3. Proposition 17.1 shows that the model structure i*
*s __
proper, and Corollary 19.3 states that it is not cofibrantly generated. *
* |__|
The model structure can be considered in two stages. The "strict" structure of
Edwards and Hastings [6, 3.3], in which the weak equivalences are defined level*
*wise,
is an intermediate step; see Section 10 for details. The situation is not unlik*
*e the
Bousfield-Friedlander strict and stable model structures for spectra [3].
We assume Theorem 6.4 for the rest of this section and for the next three sec*
*tions.
Sections 11-19 contain the details of the proof of the theorem.
In practice we need a more concrete description of fibrations. The next defin*
*ition
and proposition provide such a description.
Definition 6.5.A map is a strong fibration if it is isomorphic to a level map of
pro-spaces X ! Y indexed by a cofinite directed set such that for all t,
Xt! Ytxlims s and a
commutative diagram
Yt _____//_Zt
| ____|
| __ |
fflffl|~~__fflffl|
Ys ____//_Zs.
By restricting to cofinal subsystems, we may assume that such a diagram exists *
*for
every t > s.
Let J be the directed set of indecomposable arrows of I. The domain and range
functors J ! I are both cofinal since I is cofinite. For each OE : t ! s in J, *
*factor
the map Zt! Ys as
iOE pOE
Zt ____//_AOE___//Ys,
where iOEis a cofibration and pOEis a fibration.
Let BOEbe the pullback XsxYsAOE, and let COEbe the pushout WtqZtAOE. These
objects fit into a commutative diagram
18 DANIEL C. ISAKSEN
Xt _____//Yt____________//_Zt___//_Wt
| | """"| |
| | "" | |
fflffl| || ~~"" || fflffl|
BOE____________//_AOE___________//COE
| " |
| | "" | |
| | "" | |
fflffl| fflffl|~~"" fflffl| fflffl|
Xs _____//Ys____________//Zs___//_Ws.
Note that BOE! AOEis a weak equivalence because it is a pullback of a weak
equivalence along a fibration. Also, AOE! COEis a weak equivalence because it
is a pushout of a weak equivalence along a cofibration. Hence the composition
B ! A ! C is a levelwise weak equivalence. This composition is isomorphic_to
ghf since B ~=X, Y ~=A ~=Z, and W ~=C. |__|
The above proof works for any pro-category pro-C provided that C is a proper
model category. In fact, a minor variation of the proof works when C is either *
*left
proper or right proper. It is possible to prove the other parts of the two-out-*
*of-three
axiom for strict weak equivalences with similar techniques.
Proposition 10.5.[6] A map of pro-spaces is a strict fibration if and only if i*
*t is
a retract of a strong strict fibration.
The relationship between the strict model structure and our model structure is
expressed in the following results.
Let Hostrict(pro-SS) be the homotopy category associated to the strict struct*
*ure.
Proposition 10.6.The category Ho(pro-SS) is a localization of Hostrict(pro-SS).
Proof.Every levelwise weak equivalence is a weak equivalence in the sense_of De*
*fi-
nition 6.1. |__|
Corollary 10.7.If f is a fibration, then f is also a strict fibration.
Proof.The class of trivial cofibrations contains the class of strictly trivial_*
*cofibra-
tions by Proposition 10.6. |__|
Corollary 10.8.A map of pro-spaces is a trivial fibration if and only if it is a
strictly trivial fibration.
Proof.Cofibrations are the same as strict cofibrations. |_*
*__|
Proposition 10.9.Let X be a pro-space, and let Y be a pro-space such that each
Ys has only finitely many non-zero homotopy groups. Then
[X, Y ]pro~=Hom Hostrict(pro-SS)(X, Y ).
Proof.The condition on Y ensures that its strictly fibrant replacement Y 0is al*
*so a
fibrant replacement. To calculate morphisms from X to Y in either homotopy cat-
egory, consider morphisms from X to Y 0modulo the simplicial homotopy relation._
Hence the morphisms are the same. |__|
For example, the proposition applies when Y is a system of Eilenberg-Maclane
spaces. Edwards and Hastings described a relationship between homological algeb*
*ra
and the strict homotopy theory of such pro-spaces [6, Section 4]. It follows t*
*hat
the relationship works just as well for our homotopy theory of pro-spaces.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 19
11. Limits and Colimits
The rest of the paper concentrates on the technical details of the main theor*
*ems
stated in Section 6.
We provide specific constructions of limits and colimits in pro-categories. T*
*he
existence of all colimits seems to be a little-known fact. In this section only*
*, consider
pro-objects indexed by arbitrary cofiltering categories, not just ones indexed *
*by
cofinite directed sets.
Proposition 11.1.If C is complete, then pro-C is also complete. If C is cocom-
plete, then pro-C is also cocomplete.
Proof.Artin and Mazur [2, Appendix 4.2] showed that pro-C has all equalizers
(resp. coequalizers) provided that C does. It suffices to construct arbitrary p*
*roducts
(resp. coproducts) when these exist in C.
Let A be a set and let Xffbe a pro-objectQfor each ff in A. Let Iffbe the cof*
*iltering
index categoryQfor Xff. Define X = ff2AXffto be the cofiltering system with
objects ff2BXffsffand index category I consistingQof pairs (B, (sff)) where B*
* is
a finite subset of A and (sff) is an element of ff2BIff. A morphism of I from
(B, (sff)) to (C, (tff)) corresponds to an inclusion C B and morphisms sff! t*
*ff
in Ifffor all ff in C. Use of finite subsets B of A is essential because we use*
* the
fact that finite products commute with filtered colimits.
Direct calculation shows that for any Y in pro-C,
Y
Hom pro-C(Y, X) ~= Hom pro-C(Y, Xff).
ff2A
Thus arbitrary products exist. `
To construct`the coproduct, define X = ff2AXffto beQthe cofiltering systemQ
with objects ff2AXffsffand index category {(sff) 2 ff2AIff}. Note that If*
*fis
cofiltering since each Iffis.
In order to show that X has the correct universal mapping property, it suffic*
*es
to see that
Y
Hom pro-C(X, Y ) ~= Hom pro-C(Xff, Y ).
ff2A
for Y any object of C (i.e., Y is a constant system in pro-C). This can be chec*
*ked
directly, using the fact that colimits indexed on product categories commute wi*
*th_
the relevant products. Thus arbitrary coproducts exist. |_*
*_|
12.Retract Axioms
The class of fibrations is obviously closed under retracts. The class of weak*
* equiv-
alences is closed under retracts because weak equivalences are defined in terms*
* of
isomorphisms of pro-local systems. We must show that the class of cofibrations *
*is
also closed under retracts. We prove a general result and then apply it to cofi*
*bra-
tions.
Proposition 12.1.Let C be a category, and let C be any class of maps in C. Defi*
*ne
the class D as those maps in pro-C that are isomorphic to a level map that belo*
*ngs
to C levelwise. Then D is closed under retracts.
20 DANIEL C. ISAKSEN
Proof.Suppose that f : W ! Z is a retract of g : X ! Y , where g is a level map
that belongs to C levelwise. Hence there is a commutative diagram
W _____//X_____//W
f || g || |f|
fflffl| fflffl|fflffl|
Z _____//_Y____//_Z
where the horizontal compositions are identity maps. We must show that f also
belongs to D. Choose a level representative for f.
By adding isomorphisms to the systems for f or g, make the cardinalities of t*
*he
index sets equal. Choose an arbitrary isomorphism ff from the index set of f to
the index set of g.
Define a function t(s) inductively on height satisfying several conditions. F*
*irst,
choose t(s) large enough so that Xt(s)! Ws and Yt(s)! Zs represent respectively
the maps X ! W and Y ! Z. Also, choose t(s) large enough so that t(s) ff(s).
Finally, choose t(s) large enough so that for all u < s, t(u) < t(s) with a com*
*muting
diagram
Xt(s)__________//Ws
vvv ____||
--vvv || ""__ |
Xt(u)____________//Wu |
| | || ||
| fflffl| | fflffl|
| Yt(s)____ |____//Zs
| v | _
| vvv | ___
fflffl|--vv fflffl|""__
Yt(u)___________//Zu.
__ __ __
Now_the function t defines cofinal subsystems X and Y of X and Y where X
and Y have the same index sets_as_W_and Z.
Repeat this_process_on _g: X ! Y to obtain another function u inducing cofinal
subsystems W and Z of W and Z. The result is a level diagram
_______//______//
W X W
_f| _ | |
| g | |f
_fflffl|_fflffl|fflffl|_
Z _____//_Y____//_Z
representing (up to isomorphism) f as a retract of g.
Since W ! X ! W and Z ! Y_!_Z are_identity maps, u_can_be_chosen so
that the composites Wu(s)= W s ! X s! Ws and Zu(s)= Zs ! Y s! Zs are
structure maps of W and Z for all s.
Since g belongs to C levelwise, the same is true for _g. Define a pro-space ^*
*W by
starting with the system_W and replacing the single map Wu(s)! Ws with the
pair of maps Wu(s)! X s! Ws. Define ^Zsimilarly.
Note that W and Z are cofinal subsystems of W^ and ^Zrespectively, so W is
isomorphic to ^W and Z is isomorphic to ^Z. Thus it suffices to show that the l*
*evel
map ^f: ^W! ^Zbelongs to D.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 21
__
The subsystem of ^W on objects {X s}_is_also a cofinal subsystem, and the same
is_true for the subsystem on_objects {Y s} in ^Z. Beware that the subsystem of *
*^Won
{X s} is not isomorphic_to X because_the structure maps are different. The same
warning applies to {Y s} and Y.__ __
Restrict ^fto the subsystem {X s} ! {Y s}. This last map belongs to C lev-
elwise, so it belongs to D. Hence f also belongs to D because D is closed_under
isomorphisms. |__|
Corollary 12.2.The class of cofibrations of pro-spaces is closed under retracts.
Proof.Apply Proposition 12.1 to the class of all cofibrations in SS. *
* |___|
13. Weak Equivalences
We begin with the two-out-of-three axiom.
Proposition 13.1.Let f : X ! Y and g : Y ! Z be maps of pro-spaces. If any
two of the maps f, g, and gf are weak equivalences, then so is the third.
Proof.For n 1, the map nX ! f*g* nZ factors as
nX ! f* nY ! f*g* nZ.
Also, the map ß0X ! ß0Z factors through ß0Y . This immediately proves two of
the three cases.
For the third case, suppose that f and gf are weak equivalences. Then f* nY !
f*g* nZ is an isomorphism for all n 1. By Lemma 5.2, nY ! g* nZ is also_
an isomorphism for all n 1. |__|
The following lemma is a surprising generalization to pro-groups of an obvious
fact about groups. Bousfield and Kan [4, III.2.2] stated without proof a speci*
*al
case.
Lemma 13.2. Let U be the forgetful functor from pro-groups to pro-sets. Then a
map f of pro-groups is an isomorphism if and only if U(f) is an isomorphism of
pro-sets.
Proof.For simplicity write the group operations additively, even though the gro*
*ups
are not necessarily abelian.
We may assume that f : X ! Y is a level map. If f is an isomorphism, then
Uf is also an isomorphism by Lemma 2.3.
Now suppose that Uf is an isomorphism. By Lemma 2.3 applied twice, for every
s, there exist u t s and a commutative diagram
Xu _____//Yu
| g ____|
| __ j|
|fflffl~~fflffl|__ft
Xt _____//Yt
__
OE||h____ ||
|fflffl~~fflffl|__
Xs _____//Ys
where the diagonal maps are not necessarily group homomorphisms.
22 DANIEL C. ISAKSEN
However, the composite map Yu ! Xs is in fact a group homomorphism. For
every x and y in Yu,
hj(x + y) = h(jx + jy) = h(ftgx + ftgy) = hft(gx + gy)
because of commutativity in the top square. Also,
hft(gx + gy) = OE(gx + gy) = OEgx + OEgy
because of commutativity in the bottom square. Finally, OEg = hj.
Therefore, there is a commutative diagram of groups
Xu _____//Yu
| ----|
| -- |
fflffl|""fflffl|--
Xs _____//Ys.
By Lemma 2.3, f is an isomorphism. |___|
The formal nature of Definition 6.1 is often too abstract for comfort in tech*
*nical
situations. The following proposition shows that conditions (a) and (b) of Theo*
*rem
7.3 are equivalent, thus giving a less natural but more concrete equivalent def*
*inition
of weak equivalence.
Proposition 13.3.A level map of pro-spaces f : (X, OE) ! (Y, j) is a weak equiv-
alence if and only if for all n 0 and for all s, there exists some t s such*
* that
for all basepoints * in Xt, there is a commutative diagram
ßn(Xt, *)____//_ßn(Yt, *)
| qqqqq |
| qqq |
fflffl|xxqq fflffl|
ßn(Xs, *)____//ßn(Ys, *).
Remark 13.4.By the argument in the proof of Lemma 13.2, it is not important
whether we assume that the diagonal map is a group homomorphism or just a map
of sets. For convenience, we assume that it is a group homomorphism. Note that
the diagonal map is not geometrically induced; it just exists abstractly. The c*
*hoice
of t may depend on n and s, but it must work for every basepoint.
Proof.First suppose that f is a weak equivalence. Since ß0f is an isomorphism,
Lemma 2.3 gives the conclusion for n = 0. For n 1, Lemma 2.3 implies that, for
every s, there exists a t s and a commutative diagram
nXt _______//_f*t nYt
| |
| |
fflffl| fflffl|
OE*ts nXs____//OE*tsf*s nYs
of local systems on Xt.
In particular, for every basepoint * in Xt, there is a commutative diagram
ßn(Xt, *)____//_ßn(Yt, *)
| qqqqq |
| qqq |
fflffl|xxqq fflffl|
ßn(Xs, *)____//ßn(Ys, *).
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 23
This proves the ö nly if" part of the claim.
Now suppose that the diagrams in the statement of the proposition exist. By
Lemma 2.3, ß0f is an isomorphism. For n 1, we use the fact that a local syste*
*m L
on a space Z is determined up to isomorphism by its value Lx as a ß1(Z, x)-modu*
*le
for one point x in each component of Z. For every s, there exist u t s such
that for every basepoint * in Xu there are commutative diagrams
ßn(Xu, *)____//_ßn(Yu, *) ß1(Xu, *)_____//ß1(Yu, *)
| qqqqq | | qqqqq |
| qqq | | qqq |
fflffl|xxqq fflffl| fflffl|xxqq fflffl|
ßn(Xt, *)_____//ßn(Yt, *) ß1(Xt, *)____//_ß1(Yt, *)
| qqqqq | | qqqqq |
| qqq | | qqq |
fflffl|xxqq fflffl| fflffl|xxqq fflffl|
ßn(Xs, *)_____//ßn(Ys, *) ß1(Xs, *)____//ß1(Ys, *).
A diagram chase like that in the proof of Lemma 13.2 shows that the map
ßn(Yu, *) ! ßn(Xs, *) is actually a map of ß1(Xu, *)-modules, even though the
diagonal maps in the left diagram above are not maps of ß1(Xu, *)-modules. This
defines a commutative diagram
nXu _______//_f*u nYu
| ppppp |
| pppp |
fflffl|wwpp fflffl|
OE*us nXs____//OE*usf*s nYs
of local systems on Xu. __
Hence nX ! f* nY is an isomorphism by Lemma 2.3. |__|
Corollary 13.5.Suppose that f : X ! Y is a level map of pro-spaces indexed by
a cofinite directed set I for which there is a strictly increasing function n :*
* I ! N
such that fs : Xs ! Ys is an n(s)-equivalence. Then f is a weak equivalence.
Proof.For any s in I and any n 0, choose t s such that n(t) > n. For every
point * in Xt, there is a commutative diagram of solid arrows
~=
ßn(Xt, *)____//_ßn(Yt, *)
| q q |
| q q |
fflffl|xxq fflffl|
ßn(Xs, *)____//ßn(Ys, *).
Since the top horizontal map is an isomorphism, this diagram can be extended to
include the dashed arrow. Thus f satisfies the condition of Proposition 13.3,_s*
*o it
is a weak equivalence. |__|
14.Fibrations
The following lemma states some useful properties of strong fibrations that f*
*ollow
directly from the definition.
Lemma 14.1. If f : X ! Y is a strong fibration, then for all t the maps ft: Xt!
Yt and gt: lims a(s) for all s < t. Now select t2N-1, t2N-2, . .,.t1*
*, t0
so that ti> ti+1and there exist commutative diagrams
ßn(Ati, *)____//_ßn(Ati+1,7*)7
pp
| ppp |
| pppp |
fflffl|pp fflffl|
ßn(Xti, *)____//ßn(Xti+1, *).
for all 0 n N and all basepoints * in Ati. This is possible since j is a we*
*ak
equivalence and there are only finitely many conditions on the choice of each t*
*i.
Finally, choose a(t) so that a(t) > t0 and there exists a commutative diagram
ß0Aa(t)_____//_ß0At0::
| tttt |
| ttt |
fflffl|tt fflffl|
ß0Xa(t)____//_ß0Xt0.
Functorially factor each map Ati! Xtias Ati__ai_//Ytibi//_Xti, where ai is a
trivial cofibration and bi is a fibration.
Choose a basepoint * in Yti. Since ai is a weak equivalence, there exists a
basepoint ] in Atisuch that there is a path in Ytifrom * to the image of ]. For
0 n N, there is a diagram of solid arrows
28 DANIEL C. ISAKSEN
ßn(Ati, ])___________//_ßn(Ati+1,2])2eee
sss | eeeeeeeeeeeeppppp |
yysssse|eeeeeeee xxpp |
ßn(Xti, ])_________//_ßn(Xti+1, ]) |
| | |
| | | |
| fflffl| | fflffl|
| ßn(Yti, *)_____|_____//ßn(Yti+1, *)
| s |e e e2p2
| ssss e e e e | pppp
fflffl|yysseeee fflffl|xxpp
ßn(Xti, *)_________//_ßn(Xti+1, *)
where the vertical arrows are induced by the choice of path. Since the vertical
arrows are all isomorphisms, the dashed arrow also exists.
For similar reasons, there is also a commutative diagram
ß0Aa(t)_____//_ß0Yt0::
| tttt |
| ttt |
fflffl|tt fflffl|
ß0Xa(t)____//_ß0Xt0.
Now we have
Aa(t)____//At0____//At1____//._._._//At2N____//W
| -=|=
| a0| a1| a2N| - |
| | | | - |
| fflffl| fflffl| fflffl|- |
| Yt0_____//_Yt1___//._._._//Yt2N |q
| |
| | | | |
| b0| b1| b2N| |
fflffl| fflffl| fflffl| fflffl| fflffl|
Xa(t)_____//Xt0____//Xt1____//._._._//Xt2N____//Z
where the dashed arrow exists because a2N is a trivial cofibration and q is a f*
*ibration.
So we need only find a lift for the diagram
Aa(t)_____//Yt0___//_Yt2___//._._._//Yt2N-2____//_Yt2N___//W
| ||
| |
fflffl| fflffl|
Xa(t)____//_Xt0___//_Xt2___//._._._//Xt2N-2____//Xt2N____//Z.
Lemma 14.4 tells us that this diagram satisfies the hypotheses of Lemma 14.3._
Hence the desired lift exists. |__|
15. Lifting and Factorization Axioms
We now prove the lifting and factorizations axioms.
Proposition 15.1.If f : X ! Y is a map of pro-spaces, then f factors (not
functorially) as
p
X __i_//_Z___//_Y,
where i is a trivial cofibration and p is a strong fibration.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 29
Proof.We may assume that f is a level map. We construct the factorization in-
ductively.
Assume for sake of induction that the factorization is already constructed on*
* all
indices less than t. Recall the height function h(t) from Definition 2.2. Factor
Xt! YtxlimsE>__=__//E">>
"" ""
""" i||" |p|
"" fflffl|fflffl|"
A _____//_Y____//B>>">>~
~~
j|| "" q||~=~
fflffl|"fflffl|~~
X _____//B,
the lift in the lower left square exists because of the strict model structure,*
* and the
lift in the upper right square exists by the definition of fibrations. The_comp*
*osition
X ! Y ! E is the desired lift. |__|
16.Simplicial Model Structure
Recall that a model structure on a category C is simplicial if for every X in*
* C
and every simplicial set K, there are functorial constructions X K ("tensor") *
*and
XK (öc tensor") in C satisfying certain associativity and unit conditions. Also*
*, for
every X and Y in C, there is a simplicial function complex Map (X, Y ). These t*
*hree
constructions are related by the adjunctions
Map (X K, Y ) ~=Map (K, Map(X, Y )) ~=Map (X, Y K).
Finally, Map (-, -) must interact appropriately with the model structure as fol*
*lows.
If i : A ! X is a cofibration in C and p : E ! B is a fibration in C, then
Map (X, E) ! Map (A, E) xMap(A,B)Map (X, B)
is a fibration that is a weak equivalence if either i or p is a weak equivalenc*
*e.
We begin with a general proposition showing that tensors and cotensors defined
for finite simplicial sets automatically extend to all simplicial sets.
Proposition 16.1.Suppose that C is a model category with a simplicial function
complex Map (-, -). Also suppose that the tensor X K and the cotensor XK are
defined for all objects X of C and all finite simplicial sets K so that the axi*
*oms for a
simplicial model structure are satisfied when they make sense. Then the definit*
*ions
of tensor and cotensor can be extended to provide a simplicial model structure *
*for
C.
Proof.For any simplicial set K, let Kfinbe the filtering system of finite subsp*
*aces
of K. For an object X of C, define X K to be colims(X Kfins) and XK to
be limsXKfins. Using the fact that the system Kfinx Lfinis cofinal in the system
(K x L)fin, the required isomorphisms
X (K x L) ~=(X K) L
and
Map (X K, Y ) ~=Map (K, Map(X, Y )) ~=Map (X, Y K)
can be verified directly. |___|
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 31
Definition 16.2.If X and Y are pro-spaces and K is a simplicial set, define
Map (X, Y ) = Hom pro-SS(X x o, Y ) = limscolimtMap(Xt, Ys),
X K = colims(X x Kfins),
and fin
Y K = lims(Y Ks ).
For an arbitrary pro-space X and a simplicialfsetiK,nX x K can be constructed
as theflevelwiseiproductnwith K. Also, limsY Ks can be constructed as the system
{YtKs }, indexed by all pairs (s, t) in the product of the index categories.
Note that X K is not in general isomorphic to X x K because finite limits do
not always commute with filtered colimits in the category of pro-spaces. Howeve*
*r,
if K is finite, then X K is isomorphic to X x K since K itself is the terminal
object of Kfin. Also, when K is finite, Y K is the system {YsK} with the same i*
*ndex
category as that of Y .
Proposition 16.3.The above definitions make pro-SSinto a simplicial model cat-
egory.
Proof.By Proposition 16.1, it suffices to check the axioms only for finite simp*
*licial
sets. Most of the axioms are obvious; we verify only the non-trivial ones here.
Let X and Y be arbitrary pro-spaces, and let K be a finite simplicial set. We*
* use
the fact that Hom SS(K, colimsZs) is equal to colimsHom SS(K, Zs) for any filte*
*red
system Z of simplicial sets because K is finite. It follows by direct calculati*
*on that
Map (X K, Y ) ~=Map (X, Y K) ~=Map (K, Map(X, Y )).
We now show that the map
f : Map (B, X) ! Map (A, X) xMap(A,Y )Map(B, Y )
is a fibration whenever i : A ! B is a cofibration and p : X ! Y is a fibration
and that this map is a trivial fibration if either i or p is trivial. We proce*
*ed
by showing that f has the relevant right lifting property. Let j : K ! L be a
generating cofibration or a generating trivial cofibration. Note that K and L a*
*re
finite simplicial sets.
By adjointness, it suffices to show that the map
g : A L qA K B K ! B L
is a cofibration that is trivial if either i or j is trivial.
We may assume that i is a levelwise cofibration. For every s, As ! Bs is a
cofibration. Therefore, the map
As L qAs K Bs K ! Bs L
is also a cofibration. This is a standard fact about simplicial sets. Thus g *
*is a
levelwise cofibration.
In order to show that g is trivial whenever i or j is, it suffices to show th*
*at the
map A K ! B K is trivial if i is trivial and that the map A K ! A L
is trivial if j is trivial. This reduction follows from the two-out-of-three a*
*xiom,
the fact that trivial cofibrations are preserved by pushouts, and the commutati*
*ve
diagram
32 DANIEL C. ISAKSEN
A K __________//_A JLJ
| | JJJJ
| | JJ
|fflffl // fflffl| JJJJJ
_____YA L qA K B K
B K YYYYYYYYY UUU JJJ
YYYYYYYYY UUUUUU JJJ
YYYYYYYYUUUUUJJJ
YYYYYYYJJ%%,,Y**UUU
B L.
First suppose that j is trivial. The map A K ! A L is a levelwise weak
equivalence, so it is a weak equivalence of pro-spaces.
Now suppose that i is trivial. Since A K ! B K is constructed by levelwis*
*e__
product with K, condition (b) of Theorem 7.3 is easily verified. *
*|__|
17. Properness
We now show that the model structure of Theorem 6.4 is proper. Recall that
a model structure is left proper if weak equivalences are preserved under pusho*
*ut
along cofibrations. Dually, a model structure is right proper if weak equivalen*
*ces
are preserved under pullback along fibrations.
Proposition 17.1.The simplicial model structure of Theorem 6.4 is left and right
proper.
Proof.Left properness follows immediately from the fact that all pro-spaces are
cofibrant. We must show that the model structure is right proper.
Let p : E ! B be a fibration and let f : X ! B be a weak equivalence. Use
Theorem 7.3 to suppose that p and f are level maps with the same cofinite direc*
*ted
index set I for which there is a strictly increasing function n : I ! N such th*
*at fs is
a n(s)-equivalence. Let P be the pullback X xB E, which is constructed levelwis*
*e.
We must show that the projection P ! E is a weak equivalence.
We start with a special case. First suppose that p is a levelwise fibration. *
*Let *
be a basepoint in Ps. This yields a diagram
F _____//Ps___//_Xs
=|| || |fs|
fflffl| fflffl| fflffl|
F ____//_Esps__//Bs
in which the rows are fiber sequences. From the 5-lemma applied to the long exa*
*ct
sequences of homotopy groups of the fibrations, Ps ! Es is also an n(s)-equival*
*ence.
By Theorem 7.3, P ! E is a weak equivalence.
Now let p be an arbitrary fibration. By Proposition 6.6, there exists a stro*
*ng
fibration q : E0! B such that p is a retract of q. Note that q is a levelwise f*
*ibration
by Lemma 14.1.
Consider the commutative diagram
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 33
P __________//P_0_______//_P
~~ ___ """ |
~~~~~||__//0""__||___//_~~"" ||
E E E |
| | | | | |
|| fflffl|||__/fflffl|||/__/fflffl|/_
| ~X | _X | "X
| ~~ | __ | """
fflffl|""~~ fflffl|~~__ fflffl|~~"
B _________//_B________//_B,
where P 0= X xB E0. This diagram is a retract of squares in the sense that all
of the horizontal compositions are identity maps. The map P 0! E0 is a weak
equivalence by the special case. Since weak equivalences are closed under retra*
*cts,_
the map P ! E is also a weak equivalence. |__|
18.Alternative Characterizations of Weak Equivalences
We finish here the proof of Theorem 7.3 describing weak equivalences in other
terms. For expository clarity, we split the theorem into several parts. The equ*
*iva-
lence of (a) and (b) was shown in Proposition 13.3.
Proposition 18.1.A map of pro-spaces is a weak equivalence if and only if it is
isomorphic to a level map g : Z ! W indexed by a cofinite directed set I for wh*
*ich
there is a strictly increasing function n : I ! N such that gs : Zs ! Ws is an
n(s)-equivalence.
Proof.Corollary 13.5 showed that a map g satisfying the conditions of the propo-
sition is a weak equivalence.
Now suppose that f is a weak equivalence. We may assume that f is a level
map. Use Proposition 15.1 to factor f as
p
X __i_//_Z___//_Y,
where i is a trivial cofibration and p is a trivial fibration. By Corollary 10.*
*8, p is
also a strictly trivial fibration. In particular, p is isomorphic to a levelwis*
*e weak
equivalence. The proof of Proposition 15.1 indicates that i satisfies the condi*
*tions
of the proposition. By an argument similar to the proof of Proposition 10.4, f_*
*also_
satisfies the conditions of the proposition. |_*
*_|
Recall the Moore-Postnikov functor P from Definition 7.2.
Lemma 18.2. The canonical map X ! P X is a weak equivalence for any pro-
space X.
Proof.Condition (b) of Theorem 7.3 is easily verified. |_*
*__|
Proposition 18.3.A map of pro-spaces f : X ! Y is a weak equivalence if and
only if P f is a strict weak equivalence.
Proof.Suppose that P f is a strict weak equivalence. Then it is also a weak equ*
*iv-
alence. The maps X ! P X and Y ! P Y are weak equivalences by Lemma 18.2,
so f is also.
Now suppose that f is a weak equivalence. By Proposition 18.1, we may assume
that f is a level map indexed by a cofinite directed set I for which there is a*
* strictly
increasing function n : I ! N such that fs : Xs ! Ys is an n(s)-equivalence.
34 DANIEL C. ISAKSEN
Consider the subsystem X0 = {Pn(s)Xs|s 2 I} of X and the subsystem Y 0=
{Pn(s)Ys|s 2 I} of Y . Note that X0 and Y 0are cofinal in X and Y . Let f0 be t*
*he
level map X0 ! Y 0induced by f, so f0 is isomorphic to f. Since Xs ! Ys is an
n(s)-equivalence, the map Pn(s)Xs ! Pn(s)Ys is a weak equivalence. Hence f0 is *
*a__
levelwise weak equivalence, so f is a strict weak equivalence. *
*|__|
Proposition 18.4.A map of pro-spaces f : X ! Y is a weak equivalence if and
only if ß0f is an isomorphism of pro-sets, 1X ! f* 1Y is an isomorphism of
pro-local systems on X, and for all m and all local systems L on Y , the map
Hm (Y ; L) ! Hm (X; f*L) is an isomorphism.
Proof.Let f be a weak equivalence. By Proposition 18.1, we may assume that f
is a level map indexed by a cofinite directed set I for which there is an incre*
*asing
function n : I ! N such that fs is an n(s)-equivalence.
By the Whitehead theorem, fs induces a cohomology isomorphism in dimen-
sions less than n(s) for any local system on Ys. Hence f induces an isomorphism
Hm (Y ; L) ! Hm (X; f*L) in the colimit for every m.
Now suppose that f satisfies the conditions of the proposition. Factor f as
p
X __i__//Y_0___//Y,
where i is a cofibration and p is a strictly trivial fibration. Since p induce*
*s co-
homology isomorphisms by the first part of the proof, the map i still satisfies*
* the
hypotheses of the proposition. Therefore, we may assume that f is a level map t*
*hat
is a level cofibration.
Note that M = (X x 1) qX Y is weakly equivalent to Y since M is constructed
levelwise and M is levelwise weakly equivalent to Y .
We prove the proposition by showing that for every strongly fibrant pro-space
Z, the map Map (M, Z) ! Map (X, Z) is a weak equivalence. A retract argument
then shows that the map Map (M, Z) ! Map (X, Z) is a weak equivalence for all
fibrant pro-spaces Z.
Assume that Z is an arbitrary strongly fibrant pro-space. Note that each Zs is
a fibrant simplicial set with only finitely many non-zero homotopy groups.
Recall that Map (X, Z) = limsMap(X, Zs). Also recall that for every t, the map
Zt! lims tZs is a fibration since Z is fibrant. Therefore, the map
Map (X, Zt) ! limsMtap(X, Zs) = Map (X, limsZts)
is a fibration. It follows that Map (X, Z) is weakly equivalent to the homotopy*
* limit
holimsMap(X, Zs). Similarly, Map (M, Z) is weakly equivalent to the homotopy
limit holimsMap (M, Zs).
Since homotopy limits are invariant under levelwise weak equivalence, we only
need show that Map (M, Zs) ! Map (X, Zs) is a weak equivalence of simplicial se*
*ts
for each s. Therefore, we may assume that Z is a fibrant simplicial set with on*
*ly
finitely many non-zero homotopy groups.
By adjointness, to show that Map (M, Z) ! Map (X, Z) is a weak equivalence, it
suffices to find lifts in the diagrams of pro-spaces
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 35
X _____//_Z N. Hence, the desired lifting exists when k 1.
Now consider k = 0. The argument given for k 1 does not work. The trouble
is that we cannot lift over p1 with obstruction theory because the first homoto*
*py
groups of the fiber are not necessarily abelian.
When k = 0, the map p is just the map Z ! *, so we need to find an s and a
factorization of Xs ! Z through Ms. Note that such factorizations are the same
as factorizations up to homotopy of Xs ! Z through Ys.
Artin and Mazur [2, Section 4] constructed such factorizations when Z is con-
nected. Their argument works even when Z is not connected provided that ß0X ~=
ß0Y . Here we use the fact that [X, Z]pro= Hom pro-Ho(SS)(X, Z) by Lemma 8.1._
This proves the result. |__|
19. Non-Cofibrantly Generated Model Structures
We prove in this section that the model structure of Section 6 is not cofibra*
*ntly
generated. The same argument shows that the strict model structure [6] is also *
*not
cofibrantly generated. See Section 10 for a description of the strict structure*
*. We
start with a general lemma about cofibrantly generated model structures.
A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 37
Lemma 19.1. Suppose that a model structure on a category C is cofibrantly gen-
erated with a set of generating cofibrations I. Let T be the set of targets of *
*maps in
I, and let X be any cofibrant object of C not isomorphic to the initial object.*
* Then
there exists some Y in T not isomorphic to the initial object with a map Y ! X
in C.
Proof.Let X be a cofibrant object of C. Then X is a retract of another object X*
*0,
where X0 is a transfinite composition of pushouts of maps in I [10, 14.2.12]. S*
*ince
there is a map from X0 to X, it suffices to find a map from some object of T to*
* X0.
Since X is not the initial object, X0 is also not the initial object. Hence X0 *
*is a
non-trivial transfinite composition of pushouts of maps in I. Let Z ! Y be a ma*
*p__
in I occurring in the construction of X0. Then there is a map from Y to X0. *
*|__|
The next proposition gives a construction of specific pro-sets with special p*
*rop-
erties.
Proposition 19.2.Let F be a small family of pro-sets (i.e., a set of pro-sets) *
*not
containing the empty pro-set. Then there exists a pro-set X such that for every*
* Y
in F , there are no maps Y ! X of pro-sets.
Proof.Choose an infinite cardinal ~ larger than the size of any of the sets occ*
*urring
in any of the objects of F . Let S be a set of size ~.
Define a pro-set X as follows. Consider the collection of all subsets U of S *
*whose
complements Uc are strictly smaller than S. Note that this implies that the size
of U is ~, but the converse is not true. These subsets form a pro-set, where t*
*he
structure maps are inclusions. This system is cofiltered because (U \V )c = Uc[*
*V c
is strictly smaller than S when Uc and V care.
Let Y be an object of F . Suppose that there is a map f : Y ! X of pro-sets.
Then there exists a t and a map ft,S: Yt! S representing f. Let A be the image *
*of
ft,S, so A is strictly smaller than S since Ytis strictly smaller than S. Consi*
*der the
set S - A, which occurs as an object in the system X. Since f is a map of pro-s*
*ets,
there exists a u t such that the composition Yu ! Yt! S factors through S - A.
Since Ytand S -A have disjoint images in S, this is only possible if Yu is the *
*empty
set. However, Yu cannot be the empty set because Y is not the empty pro-set._By
contradiction, the map f cannot exist. |__|
Corollary 19.3.There are no cofibrantly generated model structures on pro-spaces
for which every object is cofibrant.
Proof.We argue by contradiction. Suppose that there exists a cofibrantly genera*
*ted
model structure for which every object is cofibrant. Let I be the set of genera*
*ting
cofibrations, and let T be the set of targets of maps in I. Apply ß0 to T to ob*
*tain
a small family of pro-sets F .
Let X be the pro-set constructed in Proposition 19.2. We can think of X as a
pro-space by identifying a set with a simplical set of dimension zero. By Lemma
19.1, there exists a non-empty Y in T and a map Y ! X. This induces a map
ß0Y ! ß0X = X. However, such a map cannot exist by Proposition 19.2 because_
ß0Y belongs to F . |__|
This corollary applies in particular to the model structure of Section 6 and *
*to
the strict model structure.
38 DANIEL C. ISAKSEN
References
[1]M. Artin, A. Grothendieck, and J. L. Verdier, Theorie des topos et cohomolo*
*gie 'etale des
schemas, Lecture Notes in Mathematics, vol. 269, Springer Verlag, 1972.
[2]M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 1*
*00, Springer
Verlag, 1969.
[3]A. K. Bousfield and E. M. Friedlander, Homotopy theory of -spaces, spectra*
*, and bisimplicial
sets, Geometric Applications of Homotopy Theory, vol. II (Proc. Conf., Evans*
*ton, IL, 1977),
Lecture Notes in Mathematics, vol. 658, Springer Verlag, 1978, pp. 80-130.
[4]A. K. Bousfield and D. Kan, Homotopy limits, completions, and localizations*
*, Lecture Notes
in Mathematics, vol. 304, Springer Verlag, 1972.
[5]W. G. Dwyer and E. M. Friedlander, Algebraic and 'etale K-theory, Trans. Am*
*er. Math. Soc.
292 (1985), 247-280.
[6]D. A. Edwards and H. M. Hastings, Cech and Steenrod homotopy theories with *
*applications
to geometric topology, Lecture Notes in Mathematics, vol. 542, Springer Verl*
*ag, 1976.
[7]E. M. Friedlander, Etale K-theory I: Connections with 'etale cohomology and*
* algebraic vector
bundles, Invent. Math. 60 (1980), 105-134.
[8]____, Etale homotopy of simplicial schemes, Annals of Mathematics Studies, *
*vol. 104,
Princeton University Press, 1982.
[9]J. Grossman, A homotopy theory of pro-spaces, Trans. Amer. Math. Soc. 201 (*
*1975), 161-
176.
[10]P. Hirschhorn, Localization of Model Categories, preprint.
[11]M. A. Mandell, E1 algebras and p-adic homotopy theory, to appear in Topolog*
*y.
[12]J. P. May, Simplicial objects in algebraic topology, Van Nostrand Mathemati*
*cal Studies, vol.
11, Van Nostrand, 1967.
[13]F. Morel, Ensembles profinis simpliciaux et interpr'etation g'eom'etrique d*
*u foncteur T, Bull.
Soc. Math. France 124 (1996), 347-373.
[14]D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, *
*Springer Verlag,
1967.
[15]D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of *
*Math. 100
(1974), 1-79.
[16]G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics*
*, vol. 61,
Springer Verlag, 1978.
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
E-mail address: isaksen@mathematik.uni-bielefeld.de