ON THE FUNCTORS CWA AND PA
WOJCIECH CHACHOLSKI
1. introduction
Let A be a pointed and connected space. A pair of spaces (Y; X) is called a r*
*elative
A-CW-complex if, roughly speaking, Y can be obtained from X by wedging with
suspensions of A and attaching cones on suspensions of A (see [5, corollary 3.7*
*]).
If A = S1, then a relative S1-CW-complex is essentially an ordinary relative CW-
p
complex. Any pointed map f : X ! Y can be factored as a composition (X ! Y 0!
Y ), where (Y 0; X) is a relative A-CW-complex and p induces a weak equivalence*
* of
mapping spaces p?: map?(A; Y 0)! map?(A; Y ).
Let X be a pointed space. By factoring ? ! X, we get a map CWAX ! X, where
(CWAX; ?) is a relative A-CW-complex and the induced map map?(A; CWAX) !
map?(A; X) is a weak equivalence. The assignment X 7! CWAX can be made
functorial, in such a way that the map CWAX ! X is natural.
By factoring X ! ?, we get a map X ! PAX, where (PAX; X) is a relative A-
CW-complex and the space map?(A; PAX) is weakly contractible. The assignment
X 7! PAX can be made functorial, in such a way that the map X ! PAX is natural.
The functors CWA and PA are crucial in studying spaces through "the eyes" of
A. The functor CWA assigns to a space X the largest sub-object CWAX ! X,
which is totally "visible" by A. While the functor PA associates with X the lar*
*gest
quotient X ! PAX, which is totally "invisible" by A. The space CWAX contains
all the information about X that can be detected by A, while PAX contains all t*
*he
information about X, that can not be detected by A at all.
The purpose of this paper is to study the relationship between the functors C*
*WA
and PA. We study these functors by looking at their images and kernels. The ima*
*ge
of CWA (respectively of PA) is the class of all spaces X, for which there exist*
*s Y ,
such that X is weakly equivalent to CWAY (respectively X is weakly equivalent to
PAY ). The kernel of CWA (respectively of PA) is the class of all spaces X, for*
* which
CWAX is weakly contractible (respectively PAX is weakly contractible).
We investigate to what extent the following sequence is "exact":
. .-.PA!cSpaces? CWA-!cSpaces? PA-!cSpaces? CWA-!cSpaces? PA-!. . .
where cSpaces? is the category of pointed and connected spaces.
1
2 WOJCIECH CHACHOLSKI
As the first result, we prove that the image of PA coincides with the kernel *
*of
CWA. It can be described as the class of all spaces X, such that map?(A; X) is
weakly contractible (see theorem 15.2).
A more difficult and subtle problem is the correlation between the image of C*
*WA
and the kernel of PA.
We characterize the image of CWA as the smallest closed class C(A), which con*
*tains
A (see example 4.6). That is the image of CWA consists of all those spaces that
can be built from A using certain simple operations. These operations are taki*
*ng
wedges, homotopy push-outs and homotopy sequential colimits (see theorem 8.2).
This constructive characterization was originally proven by E. Dror Farjoun [5].
The crucial property of the kernel of PA is that it is closed under extension*
*s by
fibrations. If (Z ! E ! B) is a fibration, where Z and B belong to the kernel o*
*f_PA,_
then so does E. We characterize the kernel of PA as the smallest closed class C*
*(A) ,
which contains A and is closed under extensions by fibrations (see example 4.9.*
*) That
is the kernel of PA consists of all those spaces that can be built from A using*
* certain
simple operations. In addition to taking wedges, homotopy push-outs and homotopy
sequential colimits, as it is in the case of C(A), we add one extra operation, *
*which is
taking extensions by fibrations. As a result we get:
Theorem. The kernel of PA is the closure of the image of CWA under taking
extensions by fibrations.
We actually_prove_more. We show that the homotopy fiber F ib(X ! PAX) belongs
to the class C(A) (see theorem 17.1). We prove this by constructing the functo*
*r PA
in terms of CWA. The process can be described by the following algorithm:
(1)take X0 = X,
(2)take the homotopy cofiber: X1 = Cof(CWAX0 ! X0),
(3)take the homotopy cofiber: X2 = Cof(CWAX1 ! X1),
(4)continue the process (possibly a transfinite number of times),
(5)take the colimit: colim(X0 ! X1 ! X2 ! . .)..
The space that we get is weakly equivalent to PAX (see the proof of theorem 17.*
*1).
Finally we describe the functor CWA in terms of PA. This process can be expre*
*ssed
by the following algorithm:
i W j
(1)take the homotopy cofiber: X0 = Cof h2[A;X]A ! X ,
(2)apply the functor PA to X0,
(3)take the homotopy fiber: F ib(X ! X0 ! PA X0).
The space that we get is weakly equivalent to CWAX (see theorem 20.5). It follo*
*ws
that we can built the functor CWA in such a way, so the map CWAX ! X is a
principal fibration (see corollary 20.7).
We use this construction of CWA, to show that the n-dimensional sphere Sn be-
longs to C(Sn+1) if and only if n = 1; 3; 7 (see corollary 20.10). The proof of*
* this
ON THE FUNCTORS CWA AND PA 3
corollary was suggested by W. Dwyer.
__________
We also show that for every n, Sn belongs to C(Sn+1) (see corollary 20.13).
This gives_a_non_trivial_example of a space, for which there is a proper inclus*
*ion
C(Sn+1) ( C(Sn+1) .
4 WOJCIECH CHACHOLSKI
Acknowledgment: I would like to thank William Dwyer for his encouragement,
support and many discussions. Being W. Dwyer's advisee has been mathematically
very fruitful and most enjoyable.
2. Organization of the paper
In section 3 we state the notation.
In section 4 we define and list crucial properties of closed classes.
Sections 5 to 7 contain detailed discussion regarding the CWA construction in*
* the
category of simplicial sets. Most of the results in these sections are due to E*
*. Dror
Farjoun [5]. The approach presented in this paper is different from the one of *
*E. Dror
Farjoun. Instead of giving a constructive description, we start with pointing o*
*ut the
universal property of A-cellular spaces. We also prove that the functor CWA can*
* not
be defined on the category of unpointed simplicial sets (see proposition 7.4).
In sections 8 to 10, we characterize the image and the kernel of CWA. We also*
* prove
various strong cellular inequalities, a notion that was introduced by E. Dror F*
*arjoun
[6]. We write X A, if and only if X belongs to the image of CWA. Although
most of these inequalities were already known (see [6]), the proofs presented i*
*n this
paper are original and use only basic properties of closed classes. In particul*
*ar, we
show:
o X A if and only if X is simply connected and X A (see theorem 10.8).
o Let f : X ! Y be a map. If Y is a connected space, then F ib(f : X ! Y )
F ib(f : X ! Y ) (see theorem 10.9).
In section 11, as a corollary, we get the result of E. Dror Farjoun regarding*
* CWA
and loop spaces: CWAX ' CWA X (see corollary 11.2 and [5, section 4.1]).
In sections 12 to 14 we define and list basic properties of the PA functor. *
*We
follow the approach of A. K. Bousfield [1]. In section 13, we prove that A-cel*
*lular
equivalences are preserved under unpoined and pointed homotopy colimits. Althou*
*gh
to show this one could use the universalIproperty of hocolim (this is A. K. Bou*
*sfield's
argument), we are taking advantage of F construction.
K
In sections 15 to 18, we characterize the image and the kernel of PA. We also
prove various weak cellular inequalities, a notion that was introduced by E. Dr*
*or
Farjoun [6]. We write X > A, if and only if X belongs to the kernel of PA. Alth*
*ough
most of the inequalities presented in this section were already known (see [6])*
*, the
proofs given in this paper are original and use only basic properties of closed*
* classes
and the characterization of the kernel of PA. In particular, we show that X > A
if and only if X is simply connected and X > A. As a simple corollary, we get
the result of A. K. Bousfield and E. Dror Farjoun regarding PA and loop spaces:
PAX ' PA X (see corollary 19.2 and [1, theorem 3.1]).
Finally in section 20, we describe the functor CWA in terms of PA.
ON THE FUNCTORS CWA AND PA 5
3.Notation
The symbol denotes the simplicial category [9, x2], in which the objects are*
* the
ordered sets [n] = {0; 1; : :;:n}, and morphisms are weakly monotone maps of se*
*ts.
The morphisms of are generated by coface maps di: [n - 1]! [n]and codegeneracy
maps si: [n]! [n + 1](i = 0; 1; : :;:n), subject to well-known cosimplicial ide*
*ntities
[2]. A simplicial set is a functor K : op ! Sets, where Sets denotes the categ*
*ory of
sets [9, x2]. The set K([n]) is usually denoted by Kn. A map between two simpli*
*cial
sets is, by definition, a natural transformation of functors. The set of maps b*
*etween
K and L is denoted by hom(K; L).
A simplicial set K can be interpreted as a collection of sets (Kn)n0 togethe*
*r with
face maps di: Kn ! Kn-1 and degeneracy maps si: Kn+1 ! Kn (i = 1; 2; : :;:n),
which satisfy the duals of the cosimplicial identities [2, 8x2]. For descriptio*
*n of how
to do homotopy theory with simplicial sets see [2] and [9].
If oe 2 Kn, then oe is called a n-dimensional simplex of K. The object [n]
is the standard n-simplex given by [n]k = mor ([k]; [n]) [2, 10x2]. There is a
distinguished n-dimensional simplex o 2 [n]n, the one that comes from the ident*
*ity
map [n] ! [n]. It is easy to check that for any simplicial set K, the assignme*
*nt
f ! f(o) gives a bijection between the set of maps [n] ! K and Kn. If oe 2 Kn
we will denote the corresponding map by oe : [n] ! K.
Let o 2 ([n])n be the distinguished simplex. By _[n] we denote the simplici*
*al
subset of [n], which is generated by the set of simplices {di(o)}0in . By [n;*
* k]
we denote the simplicial subset of [n], which is generated by the set of simpli*
*ces
{di(o)}(0in ; i6=k).
The function complex between K and L is a simplicial set map(K; L), whose
n-dimensional simplices are maps [n] x K ! L. There is a natural inclusion
L ! map(K; L), such that a n-dimensional simplex oe : [n] ! Lis sent to the map
pr1 oe
([n] x K ! [n] ! L). If we choose a simplex k of dimension 0 in K, then we
can also define the basepoint evaluation map map(K; L) ! L. This map is defined
by sending a map f : [n] x K ! L to a simplex, which is represented by the map:
f
([n] = [n] x {k} ,! [n] x K ! L).
A pointed simplicial set is a pair (K; k), where k is a chosen simplex of dim*
*ension
zero in K. We will refer to this 0-dimensional simplex as the basepoint of K. *
* A
map between pointed simplicial sets is a map of simplicial sets which preserves*
* the
basepoints. The set of pointed maps between K and L is denoted by hom?(K; L).
If X is a pointed simplicial set andiY is a simplicial set, thenjY n X is a p*
*ointed
simplicial set defined as: Y n X := (Y x X)=(Y x {?}); < Y x {?}> . By choosin*
*g a
basepoint in Y , we get an inclusion X ! Y n X.
If X and Y are both pointed, then Y ^ X is a pointed simplicial set defined
as: Y ^ X := (Y n X)=X. We will regard _[n + 1] as a pointed simplicial set,
where {0} is the chosen basepoint. If X is pointed, then nX := _ [n + 1] ^ X and
6 WOJCIECH CHACHOLSKI
enX := ( _[n + 1] x X) [ ([n + 1] x {?}) [n + 1] x X.
There is an inclusion map X = {0} x X ,! enX. It is not difficult to notice, *
*that
this map is weakly equivalent to the map X ! _ [n + 1] n X. In particular, we g*
*et
that nX is weakly equivalent to the homotopy cofiber of the map X ! enX.
The pointed function complex between pointed simplicial sets K and L is a sim*
*pli-
cial set map?(K; L) whose n-dimensional simplices are pointed maps [n] n K ! L.
If K and L are pointed simplicial sets and L is Kan [9, definition 1.3], then w*
*e denote
the set of relative -basepoint homotopy classes of pointed maps between K and L
by [K; L]. If A is a simplicial subset of K, then we denote the set of relativ*
*e -A
homotopy classes of pointed maps between K and L by [K; L]A.
The homotopy cofiber of a map A ! X is denoted by Cof(A ! X). By Cof(A !
X ! Y ) we denote the homotopy cofiber of the composition (A ! X ! Y ).
Let us choose a basepoint in X. The homotopy fiber of a map A ! X at the
chosen basepoint is denoted by F ib(A ! X). By F ib(A ! X ! Y ) we denote
the homotopy fiber of the composition (A ! X ! Y ). If X is connected, then the
homotopy type of F ib(A ! X) does not depend on the choice of a basepoint in X.
By X we denote the homotopy fiber of the basepoint map ? ! X. We call X the
loop space of X.
We will use the following notation for some categories which frequently occur:
o Spaces denotes the category of simplicial sets.
o cSpaces denotes the category of connected simplicial sets.
o Spaces? denotes the category of pointed simplicial sets.
o cSpaces? denotes the category of pointed and connected simplicial sets.
In some proofs, we willIuse theZnotion of the homotopy colimits of diagrams o*
*ver
simplicial sets. By F and F we denote respectively the unpointed and the
K K
pointed homotopy colimit of F . The references are [3] and [4].
4. Closed classes
In this section we state the definition and properties of closed classes. The*
* refer-
ences for the proofs are [3] and [5]. The notion of a closed class was introduc*
*ed and
studied by E. Dror Farjoun [5]. The following definition is slightly different *
*from the
one given in [5]. We think of a closed class as a class of unpointed and conne*
*cted
simplicial sets.
Definition 4.1.A non empty class C of connected simplicial sets is closed if the
empty simplicial set does not belong to C and C is closed under weak equivalenc*
*es
and taking pointed homotopy colimits. If F : K ! Spaces?Z is a pointed diagram
such that for every simplex oe 2 K, F (oe) 2 C, then F 2 C.
K
Notation. Let C be a closed class.
ON THE FUNCTORS CWA AND PA 7
o By F : K ! C we denote an unpointed diagram such that for every simplex oe 2*
* K,
F (oe) 2 C.
o Let f : X ! Y be a map. We say that the homotopy fiber of f belongs to C, if *
*the
homotopy fibers of f over every component of Y belongs to C. In particular, t*
*he
homotopy fibers of f over various components of Y are connected and f induces
an isomorphism on ss0. If the homotopy fiber of f belongs to C, we will writ*
*e:
F ib(f : X ! Y ) 2 C.
Proposition 4.2. A non empty class C of connected simplicial sets, which does n*
*ot
contain the empty simplicial set, is closed if and only if the following proper*
*ties are
satisfied:
o Let X and Y be weakly equivalent. If X 2 C, then Y 2 C.
o Let (Xi)i2Ibe aWfamily of simplicial sets. If Xi 2 C, then for any choice of *
*base-
points in Xi, i2IXi2 C. We say that C is closed under taking arbitrary wedge*
*s.
o Let X1 X2 ! X3 be a diagram. If Xi belongs to C, then so does the homo-
topy push-out: hocolim(X1 X2 ! X3). We say that C is closed under taking
homotopy push-outs.
o Let ( X0 ! X1 ! X2 ! . .).be a diagram. If Xi belongs to C, then so does the
telescope: hocolim(X0 ! X1 ! X2 ! . .).. We say that C is closed under taking
homotopy sequential colimits or telescopes.
The following proposition gives some basic examples of elements of closed cla*
*sses.
Proposition 4.3. Let C be a closed class.
o If X is weakly contractible, then X 2 C.
o Let K and A be simplicial sets. If A 2 C, then for any choice of a basepoint *
*in A,
K n A 2 C. In particular, "n A 2 C (see section 3).
o If A 2 C, then nA 2 C.
o If X 2 C and Y is a retract of X, then Y 2 C.
The following theorem is the main tool, we are going to use, to study closed *
*classes.
It is a generalization of E. Dror Farjoun's theorem [3, theorem 8.1].
Theorem 4.4. Let E : K ! Spaces , B : K ! Spaces beidiagrams and :jE ! B
be a natural transformation. If for oe 2 K, F ib oe : E(oe)! B(oe) 2 C, then
i I I j
F ib : E ! B 2 C.
K K
Particular cases of this theorem are listed in the following proposition:
Proposition 4.5.
8 WOJCIECH CHACHOLSKI
o Let E : K ! Spaces , B : K ! Spaces be pointed diagrams andi : E ! B be aj
natural transformation. If for every simplex oe 2 K, F ib oe : E(oe)! B(oe) *
*be-
i Z Z j
longs to C, then so does F ib : E ! B .
K K
o Let F : K ! Spaces be a diagram.I If K is contractible and for every simplex
oe 2 K, F (oe) 2 C, then F 2 C.
K
o Let the following be a homotopy push-out square:
f
A - --! B
?? ?
?y ??y
g
X - --! Y
If the homotopy fiber of f belongs to C, theniso does the homotopyjfiber of g.
o Let f : A ! X be a map. If A 2 C, then F ib X ! Cof(f : A ! X ) 2 C.
Example 4.6. Let A be a connected simplicial set. By C(A) we denote the smallest
closed class such that A 2 C(A). One can think of C(A) as the class of simplic*
*ial
sets that are built from A using certain simple operations: taking arbitrary we*
*dges,
homotopy push-outs and homotopy sequential colimits. It turns out that there is*
* a
universal property that classifies the elements of this class (see section 10).
If A = M(Z=pk; n) is Z=pk-Moore space, then X 2 C(A) if X is (n-1)-connected,
ssnX is generated by by elements of order pk and for i > n, ssiX is a p-group.
The following theorem lists some properties of closed classes with respect to*
* fibra-
tions.
Theorem 4.7. Let ( Z ! E ! B) be a fibration sequence over a connected simplic*
*ial
set B.
o If Z 2 C, B 2 C and this fibration has a section, then E 2 C.
o If X 2 C and Y 2 C, then X x Y 2 C.
o If B 2 C and Z 2 C, then E 2 C.
Definition 4.8.A closed class is closed under extensions by fibrations if for e*
*very
fibration sequence ( Z ! E ! B), where Z and B belong to C, E belongs to C.
_____
Example 4.9. Let A be a connected simplicial set. By C(A) we denote the smallest
closed class which contains A and is closed under extensions by_fibrations._ As*
* in
the case of the class C(A) (see example 4.6), one can think of C(A) as the cla*
*ss of
simplicial sets that are built from A using certain operations. In addition to *
*taking
arbitrary wedges, homotopy push-outs and homotopy sequential colimits, as it is*
* in
the case of C(A), we add one extra operation which is taking extensions by fibr*
*ations.
ON THE FUNCTORS CWA AND PA 9
It turns_out_that there is a universal property which characterizes the element*
*s of the
class C(A) (see section 18). _____
If A = M(Z=p; n) is Z=p-Moore space, then X 2 C(A) if X is (n - 1)-connected
and for i n, ssiX is a p-group. As a consequence we get that C(A) is not closed
under extensions by fibration (see example 4.6). Another example of a spaces A *
*for
which C(A) is not closed under extensions by fibrations is given in corollary 2*
*0.13.
To measure to what extent a closed class C is closed under extensions by fibr*
*ations,
a new class D(C) is introduced [3, section 6].
Definition 4.10. Let C be a closed class.
I
D(C) := {K 2 cSpaces | ifF : K ! C is a diagram, then F 2 C}
K
In case of C(A), D(C(A)) will be denoted simply by D(A).
Theorem 4.11.
o D(C) is a closed class and it is closed under extensions by fibrations.
o D(C) = {B 2 cSpaces | ifZ ! E ! B is a is a fibration sequence for which
Z 2 C; then E 2 C}
o D(C) C.
o C is closed under extensions by fibrations if and only if D(C) = C.
o Let B be a connected simplicial set. If B 2 C, then B 2 D(C) (see theorem 4.7*
*).
Throughout sections 5 to 9, A is assumed to be a pointed and connected simpli*
*cial
set. From section 10 on the assumption that A is pointed will be dropped.
5. A-cellular simplicial sets
Let A be a pointed and connected simplicial set. In this section we will defi*
*ne the
class of A-cellular simplicial sets, a notion introduced by E. Dror Farjoun [5]*
*. The
approach presented in this paper is different from E. Dror Farjoun's one [5, ex*
*ample
2.2]. We define an A-cellular simplicial set by giving its universal property f*
*irst.
Definition 5.1.Let X be a connected simplicial set. A simplicial set X is call*
*ed
A-cellular if for any choice of a basepoint x 2 X and for any map of pointed Kan
simplicial sets f : Y ! Zi, for whichjf?:imap?(A; Yj)! map?(A; Z) is a weak eq*
*uiv-
alence, the map f?: map? (X; x); Y ! map? (X; x); Z is also a weak equivalenc*
*e.
Remarks.
o Although A is pointed, the class of A-cellular simplicial sets consists of co*
*nnected
but not pointed simplicial sets.
10 WOJCIECH CHACHOLSKI
o Let X be a connected simplicial set. Notice that if for some vertex x0 2 X, a
map f : Y ! Z between pointediand connectedjKanisimplicialjsets induces a weak
equivalence: f?: map? (X; x0); Y ! map? (X; x0); Z , then for any choice of a
i j i j
basepoint x 2 X, f?: map? (X; x); Y ! map? (X; x); Z is a weak equivalence.
As an easy consequence of definition 5.1, we get:
Proposition 5.2.
(1)A is A-cellular.
(2)Let X and Y be weakly equivalent. If X is A-cellular, then so is Y .
(3)Let (Xi)i2Ibe a familyWof simplicial sets. If Xiare A-cellular, then for any*
* choice
of basepoints in Xi, I Xi is A-cellular.
(4)Let X1 X2 ! X3 be a diagram. If Xi are A-cellular, then so is the homotopy
push-out: hocolim(X1 X2 ! X3).
(5)Let X0 ! X1 ! X2 ! . . .be a diagram. If Xi are A-cellular, then so is the
telescope: hocolim(X0 ! X1 ! X2 ! . .)..
The properties (2) to (5) of proposition 5.2 say that the class of A-cellular*
* simplicial
sets is closed (see proposition 4.2). The property (1) implies that the smalles*
*t closed
class C(A), which contains A (see example 4.6), is included in the class of A-c*
*ellular
simplicial sets. We will see that in fact these properties determine the class*
* of A-
cellular simplicial sets. Thus the class of A-cellular simplicial sets is equal*
* to C(A)
(see theorem 8.2). Since to define C(A) we do not need A to be pointed, it sugg*
*ests
that to define an A-cellular simplicial set we do not have to choose a basepoin*
*t in A.
We will see that in order to perform some constructions, we need A to be pointe*
*d.
Example 5.3. Let A = _ [n + 1]. In this case the class of A-cellular simplicial*
* sets
is equal to the class of (n - 1)-connected simplicial sets.
The following two weakly equivalent simplicial sets belong to C(A), therefore*
* they
must be A-cellular (see proposition 4.3):
.
_ [n] n A = ( _[n] x A) ( _[n] x {?})
en-1A = ( _[n] x A) [ ([n] x {?}) [n] x A
These simplicial sets will play an important role in the following sections.
6. A-cellular equivalences
Definition 6.1 (E. Dror Farjoun [5]). A map f : X ! Y of pointed Kan simpli-
cial sets is called an A-cellular equivalence if f? : map?(A; X) ! map?(A; Y ) *
*is a
weak equivalence.
ON THE FUNCTORS CWA AND PA 11
Remark. An A-cellular equivalence is assumed to be a map of pointed Kan simplic*
*ial
sets.
As an immediate consequence of the definition we get:
Proposition 6.2.
o Let f : X ! Y be a map of pointed Kan simplicial sets and B be A-cellular. I*
*f f
is an A-cellular equivalence, then f is a B-cellular equivalence.
o Let f : X ! Y and g : Y ! Z be maps of pointed Kan simplicial sets. If two o*
*ut
of (f, g, gOf) are A-cellular equivalences, then so is the third.
o If f : X ! Y and f0: X0 ! Y 0are A-cellular equivalences, then so is the pro*
*duct
f xf0: X xX0 ! Y xY 0 .
Proposition 6.3 (E. Dror Farjoun [5]). Let f : X ! Y be a map of pointed and
connected A-cellular Kan simplicial sets. Then f is an A-cellular equivalence i*
*f and
only if f is a weak equivalence.
Proof.If f is a weak equivalence, then of course it is an A-cellular equivalenc*
*e.
Let f be an A-cellular equivalence. Since Y is A-cellular, f induces a bije*
*ction
f?: [Y; X]! [Y; Y.] It implies that there is a map g : Y ! X , such that fOg is
homotopic to idY . As f and fOg are A-cellular equivalences, then so is g. Th*
*us
g : Y ! X induces a bijection g? : [X; Y ]! [X; X]. It implies that there is a*
* map
h : X ! Y, such that gOh is homotopic to idX . It follows that h is homotopic t*
*o f
and g is the homotopy inverse of f. __|_ |
Corollary 6.4. Let X be a pointed and connected A-cellular Kan simplicial set. *
*If
map?(A; X) is weakly contractible, then so is X.
Let f : X ! Y be a map between pointed Kan simplicial sets. By definition th*
*is
map induces a weak equivalence f?: map?(A;iX) ! map?(A;jY ), ififor every point*
*edj
map h : A ! X and for every n 0, f?: ssn map?(A; X); h ! ssn map?(A; Y ); fOh
is an isomorphism. The set ssn(map?(A; X); h) consists of relative -A homotopy
classes of those maps g : _ [n + 1] n A! X, such that g|A = h. Since the pair
( _[n+1]nA; A) is weakly equivalent to (ne A; A), we can think of ssn(map?(A; X*
*); h)
as the set of relative -A homotopy classes of maps g : enA ! X, such that g|A *
*= h.
It is not difficult to see that for n > 0, g : enA ! X defines a zero element *
*in the
homotopy group ssn(map?(A; X); h), if and only if it can be extended along the *
*map
enA ,! [n+1]xA. It is also clear that pointed maps f : A ! X and g : A ! X are
f_g
homotopic relative the basepoint if and only if the composition (0e A '! A_A -!*
* X)
can be extended along e0A ,! [1] x A. We have showed:
Corollary 6.5. Let f : X ! Y be a map between pointed Kan simplicial sets.
12 WOJCIECH CHACHOLSKI
o The map f is an A-cellular equivalence if and only if for every n 0, it indu*
*ces
an isomorphism: f? : [ _[n] nA; X]A! [ _[n] nA; Y,]Aor equivalently, if and o*
*nly
if it induces an isomorphism: f? : [ne A; X]A! [ne A; Y ]A, where [ _[n] nA; *
*X]A
and [ne A; Y ]A are sets of relative -A homotopy classes of pointed maps.
o Assume: f has the property that if for a pointed map g : enA ! X, the compos*
*ition
fOg: enA ! Y can be extended along en A ,! [n + 1] x A, then g itself can be
extended along en A ,! [n + 1] x A. Under this assumption, f : X ! Y induces
a monomorphism f?: [ne A; X]A! [ne A; Y ]A.
7. The functor CWA
The purpose of this section is to construct a functor CWA : Spaces? ! Space*
*s?
and a natural transformation cwAX : CWAX ! X , such that CWAX is A-cellular
and if X is a pointed and connected Kan simplicial set, then cwAX : CWAX ! X*
* is
an A-cellular equivalence.
Construction. The construction of CWA and cwA, presented in this paper, is mod-
eled very closely on a construction of E. Dror Farjoun [5, section 3.4].
Let be a limit ordinal whose cofinality [7] is bigger then the cardinality o*
*f the
set of simplices of A. By we also denote the category whose objects are all or*
*dinal
numbers smaller than and for any two ordinal numbers j i, there is only one
morphism j ! i.
Let X be a pointed simplicial set. We are going to construct by induction a f*
*unctor
F (X): ! cSpaces? and a pointed map p : F (X) ! X, such that:
(1)If i = j + 1, then F (X)j ! F (X)i is a cofibration.
(2)For every ordinal i < , F (X)i belongs to C(A), so it is A-cellular.
(3)For every n 0, the map p0: F (X)0! X induces an epimorphism:
i j
(p0)?: hom? en A; F (X)0 ! hom?(ne A; X)
where hom?(Z; Y ) is the set of pointed maps between Z and Y .
(4)If i = j + 1 and g : enA ! F (X)jis a pointed map, for which pjOg:ienA ! X
g
can be extended along en A ,! [n + 1] x A, then the composition en A !
j
F (X)j ! F (X)i can also be extended along enA ,! [n + 1] x A.
Step 0. If i = 0, then:
_
o F (X)0 is defined as: F (X)0 = ne A,
n0
h2hom?(enA;X)_
o p0: F (X)0! X is defined as: p0 = h.
n0
h2hom?(enA;X)
ON THE FUNCTORS CWA AND PA 13
It is obvious from the definition that the conditions (1) through (4) are satis*
*fied.
Let us assume that the construction has been carried out for all ordinal numb*
*ers
smaller than i.
Step i. If i is not of the form j + 1, then:
o F (X)i is defined as: F (X)i = colimj n, ssiX ! ssiX, (g 7! gp) is an isomorphism.
13. A-periodic equivalences
By standard manipulations on mapping spaces one can show:
Proposition 13.1. Let f : X ! Y be a map and Z be a Kan simplicial set. The
following statements are equivalent:
o f?: map(Y; Z) ! map(X; Z) is a weak equivalence.
o For any choice of basepoints in X and Z, f? : map?(Y; Z) ! map?(X; Z) is a we*
*ak
equivalence.
ON THE FUNCTORS CWA AND PA 23
Definition 13.2 (A. K. Bousfield [1]).A map f : X ! Y is an A-periodic equiv-
alence if for any A-null Kan simplicial set Z, one of the statements of proposi-
tion 13.1 is true for f and Z.
As an immediate consequence of the definition we get:
Proposition 13.3.
o Let f : X ! Y and g : Y ! Z be maps. If two out of (f, g, gOf) are A-periodic
equivalences, then so is the third.
o Let f : X ! Y be a map between A-null Kan simplicial sets. Then f is an A-
periodic equivalence if and only if f is a weak equivalence.
The following proposition gives some obvious examples of A-periodic equivalen*
*ces.
Proposition 13.4.
o If X and Y are A-cellular, then any map X ! Y is an A-periodic equivalence. In
particular:
- A ! ? and ? ! A are A-periodic equivalences
- For any choice of a basepoint a 2 A, the map en (A; a) ,! [n + 1] x A is an
A-periodic equivalence (see proposition 4.3).
o Let (fi:`Xi ! Yi)i2Ibe`a`family of maps. If fi are A-periodic equivalences, t*
*hen
so is: Ifi: IXi! IYi.
o If f : X ! Y and g : Z ! W are A-periodic equivalences, then so is the prod*
*uct
map f x g : X x Z ! Y x W .
o Let the following be a commutative diagram:
X1 --- X2 - --! X3
?? ? ?
?yf1 ??yf2 ??yf3
Y1 --- Y2 - --! Y3
If fi are A-periodic equivalences, then so is hocolim(f1 f2 ! f3).
o Let G : ! Spaces be a diagram over the category associated to an ordinal nu*
*mber
. If for every ordinal number i < , G(i) ! G(i+1) is an A-periodic equivalenc*
*e,
then so is the natural map G(0) ! hocolim G.
Corollary 13.5. Let the following be a homotopy push-out square:
X ---! Y
?? ?
?yf ??yg
Z ---! W
If f is an A-periodic equivalence, then so is g.
Proof.Let us consider the following commutative diagram:
24 WOJCIECH CHACHOLSKI
i id j
Y? ' hocolim Y X ! X
?? ?? ?? ??
yg ?yid ?yid ?yf
i f j
W ' hocolim Y X ! Z
Since f : X ! Z , idX and idY are A-periodic equivalences, then so is g : Y ! W*
* . __|_ |
Theorem 13.6 (A. K. Bousfield [1]). Let F : K ! Spaces , G : K ! Spaces be
diagrams and : F ! G be a natural transformation. If for everyIsimplexIoe 2 *
*K,
oe : F (oe)! G(oe)is an A-periodic equivalence, then so is : F ! F .
K K
Proof.We can assume that F and G are bounded diagrams [4, proposition A.5].
We are going to prove the theorem by induction on the dimension of K. Since
the disjoint union preserves A-periodic equivalences, the theorem is true for a*
*ny zero
dimensional simplicial set K. Let us assume that the theorem is true for any K,
whose dimension is less that n. Let L be a simplicial set, whose dimension is *
*less
then n and _[n] ! L be a map. We are going to prove that the theorem is true f*
*or
K = L [ _[n][n]. Let o 2 ([n])n be the distinguished simplex (see section 3). L*
*et
us consider the following commutative diagram:
I i I j
F = colim F _[n] x F (o) ,! [n] x F (o)
K? L? ? ?
?y ?y ?? ??
I yidxo yidxo
i I j
G = colim G _[n] x G(o) ,! [n] x G(o)
K L
I I
By the inductive assumption, : F ! G is an A-periodic equivalence. Propo-
L L
sition 13.4 implies that _ [n] x F (o) ! _ [n] x G(o) and [n] x F (o) ! [n] x G*
*(o)
areIalso A-periodicIequivalences. Using once again proposition 13.4, we get th*
*at
: F ! G is an A-periodic equivalence. __|_ |
K K
Corollary 13.7. Let F : K ! Spaces? , G : K ! Spaces? be pointed diagrams and
: F ! G be a natural transformation.ZIf forZoe 2 K, oe : F (oe)! G(oe)is an A-
periodic equivalence, then so is : F ! F .
K K
Proof.Let us consider the following commutative diagram:
Z i I j
F ' hocolim ? K ! F
K? ? ? K?
?y ?? ?? ?
Z y yid y
i I j
G ' hocolim ? K ! G
K K
ON THE FUNCTORS CWA AND PA 25
I I
Theorem 13.6 implies that : F ! G is an A-periodic equivalence. It foll*
*ows
KZ KZ
from proposition 13.4, that : F ! G is also an A-periodic equivalence. *
* __|_ |
K K
14. The functor PA
In this section we state the theorem of the existence of the periodization fu*
*nctor
PA and list some of its basic properties.
Theorem 14.1 (A. K. Bousfield [1]). On the category of simplicial sets there e*
*x-
ist a functor PA : Spaces ! Spaces and a natural transformation pAX : X ! PAX *
* ,
such that PAX is A-null and the map pAX : X ! PAX is an A-periodic equivalenc*
*e.
Proposition 14.2.
o If f : X ! Y is an A-periodic equivalence, then up to homotopy, there exist a
unique map g : Y ! PAX such that gOf : X ! PAX is homotopic to the natural
map pAX : X ! PAX .
o Let f : X ! Y be a map. If Y is A-null, then up to homotopy, there exist a
pAX g
unique map g : PAX ! Y, such that f and the composition (X -! PAX ! Y ) are
homotopic.
o Let f : X ! Y be an A-periodic equivalence. If Y is A-null, then Y is wea*
*kly
equivalent to PAX and f : X ! Y is weakly equivalent to pAX : X ! PAX .
o A map f : X ! Y is an A-periodic equivalence if and only if PAf : PAX ! PAY
is a weak equivalence.
o If X is Kan, then X is A-null, if and only if pAX : X ! PAX is a weak equiv*
*a-
lence.
o If f : X ! Y is a weak equivalence, then so is PAf : PAX ! PAY . In partic*
*ular,
if X is weakly contractible, then so is PAX.
o PAX is weakly contractible if and only if X ! ? is an A-periodic equivalence.
15. The image of PA
In this section we are going to prove that essentially the image of the funct*
*or PA
is equal to the kernel of the functor CWA.
Definition 15.1. The image of PA is the class of all those simplicial sets X, f*
*or
which there exists a simplicial set Y , such that PAY is weakly equivalent to X.
Theorem 15.2. The following classes of simplicial sets are equal:
(1)The class of those simplicial sets X, such that every connected component of*
* X
is in the kernel of CWA.
(2)The image of PA.
(3)The class of those simplicial sets X, for which there exist a Kan simplicial*
* set
Y , which is weakly equivalent to X and Y is A-null.
26 WOJCIECH CHACHOLSKI
Proof.Definition 15.1 and proposition 14.2 imply immediately that classes (2) a*
*nd
(3) are equal.
Let X be a simplicial set such that every connected component of X is in the
kernel of CWA. Let Y be a Kan simplicial set, which is weakly equivalent to X. *
*Let
Y 0be any connected component of Y . According to proposition 9.2, for any choi*
*ce
of basepoints in A and Y 0, map?(A; Y 0) ' ?, thus Y 0is A-null. By proposition*
* 12.3,
Y is also A-null, It shows that classes (1) and (2) are equal. __|_ |
16. The kernel of PA
In this section we are going to show that the kernel of PA is closed under ex*
*tensions
by fibrations.
Definition 16.1. The kernel of PA is the class of all those simplicial sets X, *
*such
that PAX is weakly contractible. The elements of the kernel of PA will be call*
*ed
A-acyclic simplicial sets.
Remark. The kernel of PA consists of connected simplicial sets.
Proposition 14.2 implies:
Corollary 16.2.
o A simplicial set X is A-acyclic if and only if X ! ? is an A-periodic equival*
*ence.
o A is A-acyclic.
Proposition 16.3. Let F : K ! Spaces? beZa pointed diagram. If for every simpl*
*ex
oe 2 K, F (oe) is A-acyclic, then so is F .
K
Proof.Since F (oe) isZA-acyclic,ZF (oe) ! ? is an A-periodic equivalence. Accor*
*dingZ
to corollary 13.7, F ! ? = ? is also an A-periodic equivalence and so F *
* is
__ K K K
A-acyclic. |_ |
Corollary 16.4. The class of A-acyclic simplicial sets is a closed class and A-
cellular simplicial sets are A-acyclic.
Proposition 16.5. Let F : KI! Spaces be a diagram. If for every simplex oe 2 K,
F (oe) is A-acyclic, then F ! K is an A-periodic equivalence.
K
Proof.Since for every simplexIoe 2IK, F (oe) ! ? is an A-periodic equivalence,
theorem 13.6 implies that F ! ? = K is also an A-periodic equivalence. __*
*|_ |
K K
Corollary 16.6. Let F : K ! Spaces be a diagram.IIf K is A-acyclic and for eve*
*ry
simplex oe 2 K, F (oe) is A-acyclic, then F is A-acyclic.
K
ON THE FUNCTORS CWA AND PA 27
I
Proof.According to proposition 16.5, F ! K is an A-periodic equivalence. Sin*
*ce
K
K is A-acyclic, K ! ? is an A-periodic equivalence.IComposition of two A-period*
*ic
equivalences is an A-periodic equivalence, so F ! ? is an A-periodic equival*
*ence
I K
and F is A-acyclic. __|_ |
K
Corollary 16.7. Let (Z ! E ! B) be a fibration sequence, such that B is connect*
*ed.
If Z is A-acyclic, then E ! B is an A-periodic equivalence.
I
Proof.The map E ! B is weakly equivalent to a map of the form F ! K, where
K
F : K ! Spaces is a diagram such that for every simplex oe 2 K, F (oe) is weak*
*ly
equivalentIto Z, so it is A-acyclic [3, example 3.12]. Proposition 16.5 implie*
*s that
F ! K is an A-periodic equivalence. It shows that E ! B is also an A-periodic
K __
equivalence. |_ |
Using the same argument as in corollary 16.6, we get:
Corollary 16.8. Let (Z ! E ! B) be a fibration sequence. If F and B are A-
acyclic, then so is E. Thus the class of A-acyclic simplicial sets is closed u*
*nder
extensions by fibrations.
17. A-acyclic simplicial sets
Theorem 17.1.___If X is connected,_then the homotopy fiber F ib(pAX : X !_PAX_*
* )
belongs_to C(A) , where C(A) is the smallest closed class such that A 2 C(A) *
*and
C(A) is closed under extensions by fibrations (see example 4.9).
Proof.Let Y be a Kan simplicial set, which is weakly_equivalent to X. We are go*
*ing
to prove that F ib(pAY : Y ! PAY ) belongs to C(A) .
Let be a limit ordinal number, whose cofinality [7] is bigger then the cardi*
*nality
of the set of simplices of A. By we also denote the category associated to th*
*is
ordinal number (see section 7-Construction). Let us choose basepoints in A and *
*Y .
The idea of the proof is to define a functor G : ! Spaces? , such that:
(1)G0 = Y
(2)If i is not of the form j + 1, then Gi= hocolimj A and say that X is killed by A or A kills X.
It follows immediately from the definition that if X > B and B > A, then X > *
*A.
It is also straightforward to see that X A implies X > A. This_means_that if X
can be built by A, then it can be killed by A. Since the class C(A) is usually *
*bigger
than the class C(A) (for a non trivial example see corollary 20.13), the conver*
*se to
this statement is not true. If X can be killed by A, then in order to built X f*
*rom A,
we have to add an extra tool: taking extensions by fibrations. So the converse *
*would
be true if the class C(A) were closed under extensions by fibrations. This is t*
*he case,
if for example A = _ [n + 1].
According to the introduced definitions, notation and proven theorems and pro*
*po-
sitions the following statements are equivalent:
_____
o X belongs to C(A) .
o X is killed by A.
o X > A.
o X is A-acyclic.
o X is in the kernel of PA.
o For any choice of basepoints in A and X, if Y is a pointed Kan simplicial set*
* for
which map?(A; Y ) ' ?, then map?(X; Y ) ' ?. We will refer to this property *
*of
A-acyclic simplicial sets as their universal property.
Proposition 18.1. Let W be connected. The class C = {X 2 cSpaces | X > W }
is a closed class and it is closed under extensions by fibrations.
Proof.Let F : K ! Spaces? be a pointedZdiagramZsuch that for everyZsimplex oe 2*
* K,
______
F (oe) 2 C, so F (oe) 2 C(W ). Since F ' F , we get that F belongs to
______ Z K K K
C(W ). It implies: F 2 C and thus C is a closed class.
K
Let ( Z ! E ! B)_be_a fibration sequence for which_Z_2 C and B 2 C, so Z
and B belongito C(W ). We havejto show that E 2 C(W ). Consider the fibration
sequence F ib(E ! B) ! E ! B . Theorem 10.9 implies: F ib(E !
______
B) F ib(E_!_B) ' Z > W . As a result, we get: F ib(E ! B)_2_C(W_).
Since C(W ) is closed under extensions by fibrations, E 2 C(W ). __|_ |
Corollary 18.2. If X > A, then X > A.
ON THE FUNCTORS CWA AND PA 31
Proof.According to proposition 18.1, the class C = {X 2 cSpaces | X > A} is a
closed class_and it is closed under extensions by fibrations. Since obviously A*
* 2 C,
we get C(A) C. __|_ |
Proposition 18.3. Let W be connected. The following class is a closed class and*
* it
is closed under extensions by fibrations:
C = {X | X is simply connected andX > W }
Proof.Let F : K ! Spaces? be a pointed diagram such that for every simplex oe 2*
* K,
F (oe) 2 C. Since F is a pointed diagram, there is a natural transformation ? !*
* F ,
where ? : K ! Spaces? is the constant diagram whoseivalue is thejtrivial simpli*
*cial
set. By assumption, for every simplex oe 2 K, F ib ? ! F (oe) ' F (oe) belongs*
* to
______ iZ Z j Z ______
C(W ). According to proposition 4.5, F ib ? ! F ' F is also in C(W )*
* ,
K K K
and so C is a closed class.
Let (Z ! E ! B) be a fibration_sequence such that Z and B belong to C, thus
Z and_B_are_elements of C(W ). Since (Z ! E ! B) is a fibration_sequence_
and C(W ) is closed under extensions by fibrations, we get E 2 C(W ) . It follo*
*ws
that C is closed under extensions by fibrations. __|_ |
Corollary 18.4. Let X and A be simply connected. If X > A, then X > A.
Proof.By proposition 18.3, C = {X | X is simply connected andX > A} is a
closed class_and it is closed under extensions by fibrations. Since obviously A*
* 2 C,
we get C(A) C. __|_ |
Theorem 18.5. X > A if and only if X is simply connected and X > A.
Proof.Let X > A. Notice that C = {X | Xis simply connected andX > A} is a
closed class and_it_is_closed under extensions by fibrations. Since obviously A*
* 2 C,
it follows that C(A) C. This implies that X 2 C and thus X is simply connect*
*ed.
By corollary 18.4, X > A implies: X > A. According to proposition 10.7,
A A. It follows that X > A.
Let X be simply connected such that X > A. By corollary 18.2, we have X >
A. According to corollary 10.6, X X, thus X > A. __|_ |
19. The functor PA and loop spaces
Theorem 19.1 (A.K. Bousfield, E. Dror Farjoun). Let X be simply connected.
The loop of the natural map (pA X) : X ! PA X is an A-periodic equivalen*
*ce
and PA X is A-null.
Proof.Since PA X is A-null, map?(A; PA X) is weakly contractible. It follows
that map?(A; PA X) is also weakly contractible_and_thus_PA X is A-null.
By theorem 17.1, F ib(pA X : X_!_PA_ X ) 2 C(A) . Using theorem 18.5 we
have F ib(pA X : X ! PA X ) 2 C(A) and so F ib((pA X) : X ! PA X ) is
32 WOJCIECH CHACHOLSKI
A-acyclic. Corollary 17.2 implies that since X is simply connected, then PA X
is also simply connected and PA X is connected. Corollary 16.7 indicates that
X ! PA X is an A-periodic equivalence. __|_ |
Corollary 19.2. If X is simply connected, then PA X is weakly equivalent to
PAX and the loop of the natural map (pA X) : X ! PA X is weakly equivalent
to (pAX) : X ! PAX
20. Construction of CWA "from" PA
_______
Proposition 20.1. C(A) D(A) (see definition 4.10).
Proof.According to proposition 10.7, A A. By theorem 4.11, A A
implies: A 2 D(A). Since A 2 D(A) and D(A)_is_a closed class, which is closed
under extensions by fibrations, we get C(A) D(A). __|_ |
Corollary 20.2 (E. Dror Farjoun [6]).
o If X > A, then X A.
o Let f : X ! Y be a map. If Y > A and F ib(f : X ! Y ) A, then X A.
Theorem 20.3. Let A be pointed and connected and let X ! X0 be a map of
pointed and connected Kan simplicial sets. Assume that:
o F ib(X ! X0) A.
o The induced map [A; X] ! [A; X0] is the trivial map.
pA X0 0
Under these assumptions F ib(X ! X0 -! PA X ) is A-cellular and the map
pA X0 0
F ib(X ! X0 -! PA X ) ! X is an A-cellular equivalence.
Lemma 20.4. Let the following be a pull-back square of pointed simplicial set*
*s:
K - --! L
?? ?
?y ??y
p
E - --! B
f
where p : E ! B is a fibration. Let f : A ! L be a map. If (A ! L ! B) is homo-
topic to the constant map, then there exists a lifting A ! K, such that f : A !*
* L is
equal to the composition (A ! K ! L).
f
Proof.Using the fact that p : E ! B is a fibration, we can lift (A ! L ! B) to
A ! E. By the universal property of the pull-back, we can construct A ! K, such
that (A ! K ! L) is equal to f : A ! L. __|_ |
Proof of the theorem.We have the following fibration sequence:
pA X0 0 0 pA X0 0
F ib(X ! X0) ! F ib(X ! X0 -! PA X ) ! F ib(X -! PA X )
ON THE FUNCTORS CWA AND PA 33
pA X0 0 0
Since F ib(X0 - ! PA X ) > A and by assumption F ib(X ! X ) is A-cellular,
pA X0 0
corollary 20.2 implies: F ib(X ! X0 -! PA X ) A.
Let us consider the following pull-back square:
Z ---! X
?? ?
?y ??yp
P ---! PA X0
where P ! PA X0 is a fibration, such that P is contractible and p : X ! PA X0*
* is
pA X0 0
equal to the composition (X ! X0 - ! PA X ). We are going to show that the
map Z ! X is an A-cellular equivalence. Since Z ! X is homotopic to the map
F ib(p : X ! PA X0) ! X, the theorem will be proven.
Step 1. [A; Z] ! [A; X] is an epimorphism.
We are going to show that in fact hom?(A; Z) ! hom?(A; X) is an epimorphism.
Let f : A ! X be a pointed map. We want to show that there is A ! Z, such
f
that f : A ! X is equal to (A ! Z ! X). By assumption the composition (A !
X ! X0) is homotopic to the constant map. This implies that so is the following
f 0 pA X0 0
composition (A ! X ! X - ! PA X ). Using lemma 20.4, we can construct
A ! Z such that f : A ! X is equal to (A ! Z ! X).
Step 2. For n > 0, [ne A; Z]A ! [ne A; X]A is an epimorphism.
We are going to show that in fact hom?(ne A; Z) ! hom?(ne A; X) is an epimorphi*
*sm.
f 0
Let f : enA ! X be a pointed map. Consider the composition (A!ne A! X !X ).
f 0
By assumption this map is homotopic to the constant map and so (ne A ! X ! X )
factors through some map Cof(A ! en A) ! X0. Notice that Cof(A ! en A) '
nAi(see section 3), so it isjA-cellular. Since PA X0 is A-null, the composition
Cof(A ! en A) ! X0 ! PA X0 is homotopic to the constant map. As a result
f 0 0
we get that (ne A ! X ! X ! PA X ) is also homotopic to the constant map. By
lemma 20.4, we can find enA ! Z for which f : enA ! X is equal to (ne A ! Z !
X).
Step 3. For every n 0, [ne A; Z]A ! [ne A; X]A is a monomorphism.
We are going to use corollary 6.5 to prove step 3. Let us assume that g : enA *
*! Zis a
g en
pointed map such that (ne A ! Z ! X) can be extended along A ,! [n + 1] x A
by h : [n + 1] x A ! X . We are going to show that g itself can be extended alo*
*ng
enA ,! [n + 1] x A. Since [n + 1] x A is weakly equivalent to A, we get that
([n + 1] x A !h X ! X0) is homotopic to the constant map. It implies that
34 WOJCIECH CHACHOLSKI
p 0
([n + 1] x A !h X ! P X ) is also homotopic to the constant map. According to
lemma 20.4, we can find [n + 1] x A ! Z such that ([n + 1] x A ! Z ! X) is
equal to h : [n + 1] x A ! X . By the universal property of the pull-back, the *
*map
[n + 1] x A ! Z is an extension of g : enA ! Zand thus step 3 is proven. __|_*
* |
Theorem 20.5. Let A be a pointed simplicial set and X be a pointed Kan sim-
plicialiset. If X0 is a Kan simplicial set for which there is a weak equivalen*
*ce:
W j 0 0 pA X0 0
Cof h2[A;X]A ! X ! X , then F ib(X ! X - ! PA X ) is A-cellular and
pA X0 0
the map F ib(X ! X0 -! PA X ) ! X is an A-cellular equivalence.
i i W j j
Proof.Let X ! X0 be the composition X ! Cof h2[A;X]A ! X ! X0 . We are
going to prove that this map satisfies the assumptionsiof theorem 20.3.
W j
By proposition 4.5 the homotopy fiber F ib X ! Cof( h2[A;X]A ! X) is A-
i W j
cellular. Since the map Cof h2[A;X]A ! X ! X0 is a weak equivalence, thus
F ib(X ! X0) A.
It is clear that X ! X0 induces the trivial map [A; X] ! [A; X0]. __|_ |
Corollary 20.6 (E. Dror Farjoun [5]). Let A be a pointed simplicial set and X *
*be
a pointed Kan simplicial set. If [A; X] = ?, then F ib(pA X : X ! PA X ) is w*
*eakly
equivalent to CWAX and the map F ib(pA X : X ! PA X ) ! X is weakly equivalent
to cwAX : CWAX ! X .
Corollary 20.7. If X is pointed and connected Kan simplicial set, then the fibr*
*ation
cwAX : CWAX ! X is a principal fibration.
Corollary 20.8. Let A be a pointed simplicial set. The following classes are eq*
*ual:
o {X 2 cSpaces? | X is Kan, [A; X] = ? and X 2 C(A)}__
o {X 2 cSpaces? | X is Kan, [A; X] = ? and X 2 C(A) }
o {X 2 cSpaces? | X is Kan, [A; X] = ? and X 2 D(A)}
Theorem 20.9. If X is a pointediand connected Kan simplicial set, then X is A-
W j
cellular if and only if Cof h2[A;X]A ! X is A-acyclic.
i W j '
Proof.Let Cof h2[A;X]A ! X ! X0 be a weak equivalence such that X0 is Kan.
i W j
It follows that Cof h2[A;X]A ! X is A-acyclic if and only if X0 is.
If X0 is A-acyclic, then PA X0 is a weakly contractible simplicial set and t*
*he
pA X0 0
map F ib(X ! X0 - ! PA X ) ! X is a weak equivalence. Since the homotopy
pA X0 0
fiber F ib(X ! X0 -! PA X ) is weakly equivalent to CWAX (see theorem 20.5),
we get that X is A-cellular.
ON THE FUNCTORS CWA AND PA 35
Let us assume that X is A-cellular. Notice that X is connected and so is PA *
*X0.
Since X is an A-cellular Kan simplicial set, cwAX : CWAX ! X is a weak equi*
*va-
lence. By theorem 20.5, F ib(X ! PA X0) ! X is also a weak equivalence. Since
PA X0 is connected, we get that PA X0 is weakly contractible and thus X0 is A-
acyclic. __|_ |
Corollary 20.10 (W. Dwyer). Let Sn be a Kan simplicial set which is weakly
equivalent to _[n + 1]. Then Sn Sn+1 if and only if n = 1; 3; 7.
Lemma 20.11. Sn+1 Sn+1
Proof.It is obvious that Sn+1 Sn+1 ' Sn. By theorem 10.8, Sn+1 Sn. It
follows from corollary 10.2, that Sn+1 Sn ' Sn+1. __|_ |
Proof of the corollary.If n = 1; 3; 7, then Sn is an H-space and thus Sn is a r*
*etract
of Sn+1. According to proposition 4.3, Sn is Sn+1-cellular.i
W n+1
Let us assume: Sn Sn+1. By theorem 20.9, X0 = Cof h2[Sn+1;Sn]S !
j
Sn is Sn+1-acyclic. According the lemma, X0 is also Sn+1-acyclic. It follows
thatWX0 is n-connected (see example 17.5). As a consequence, we get that the map
h2[Sn+1;Sn]Sn+1 ! Sn induces an epimorphism on n-dimensional integral homol-
ogy. This means that there is h : Sn+1 ! Sn which also induces an epimorphism
on n-dimensional integral homology. By Hurewicz theorem, h has to be an epimor-
phism on ssn. This implies that there is Sn ! Sn+1 for which the composition
(Sn ! Sn+1 !h Sn) is a weak equivalence. It shows that Sn is a retract of an
H-space, so it is also an H-space. This can happen only for n = 1; 3; 7. __|_ |
Proposition 20.12. Let Sn be a pointed Kan simplicial set which is weakly equi*
*va-
lent to _[n + 1]. For every n 1, Sn > Sn+1.
Proof.For n = 1, since S1 is an H-space, S1 is a retract of S1 and so S1 S2.
Let n > 1. Let us consider the canonical map Sn ! Sn. By Freudenthal
suspension theorem the homotopy fiber F ib(Sn ! Sn) is (2n - 2)-connected and
so it is n-connected. This implies: F ib(Sn ! Sn) Sn+1 (see example 5.3).
According to corollary 10.6, Sn+1 Sn+1 and so Sn+1 Sn+1. It follows that
F ib(Sn ! Sn) Sn+1. Since F ib(Sn ! Sn) ! Sn ! Sn is a fibration
sequence, where the base and the homotopy fiber are Sn+1-cellular, the total sp*
*ace
Sn is Sn+1-acyclic. __|_ |
Corollary 20.13. If Sn is a pointed Kan simplicial set which is weakly equivale*
*nt to
_[n + 1], then:
______ __________
o C(Sn) = C(Sn) = C(Sn+1) for n 1,
o C(Sn) = C(Sn+1) if and only if n = 1; 3; 7.
36 WOJCIECH CHACHOLSKI
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Department of Mathematics, University of Notre Dame, Mail Distribution Cen-
ter, Notre Dame, Indiana 46556-5683, USA
E-mail address: wchacho1@math.nd.edu
The Fields Institute, 222 Colage St., University of Toronto, Toronto, Ontario
M5S 1A1, Canada