CLOSED CLASSES
WOJCIECH CHACHOLSKI
1. Introduction
A non empty class C of connected spaces is said to be a closed class if
it is closed under weak equivalences and pointed homotopy colimits. A closed
class can be characterized as a non empty class of connected spaces which is
closed under weak equivalences and is closed under certain simple operations:
arbitrary wedges, homotopy push-outs and homotopy sequential colimits. The
notion of a closed class was introduced by E. Dror Farjoun [6].
Two important constructions give rise to examples of closed classes. The
first one is the Bousfield-Dror periodization functor PA [2]. The class of tho*
*se
spaces X, such that PA X is weakly contractible, forms a closed class. By looki*
*ng
just at the properties of this class we can prove, for example, that PA X is
weakly equivalent to PA X (see [2], [4]). The second construction is E. Dror
Farjoun's colocalization functor CWA . The class of those spaces X, for which
there exists a space Y , such that X is weakly equivalent to CWA Y , forms a
closed class. This class is denoted by C(A) and is called the class of A-cellul*
*ar
spaces. By looking just at the properties of the class C(A) we can prove, for
example, that CWA X is weakly equivalent to CWA X (see [4], [6]).
We say that a closed class C is closed under extensions by fibrations, if
for every fibration sequence (Z ! E ! B), such that Z and B belong to C, E
belongs to C. A closed class C is closed under extensions by fibrations if and
only if for every diagram F : I ! C , such that the classifying space BI belongs
to C, the unpointed homotopy colimit hocolimIF belongs to C.
The purpose of this paper is to understand to what extent a closed class
is closed under extensions by fibrations and under taking unpointed homotopy
colimits. We start with proving a theorem that, in particular, implies:
o Let F : I ! Spaces? be a pointed diagram, such that the classifying
space BI belongs to C. If for every i 2 I, F (i) belongs to C, then so
does the unpointed homotopy colimit hocolimIF .
o Let (Z ! E ! B) be a fibration sequence with a section. If Z and B
belong to C, then so does E.
o Let F : I ! C and G : I ! C be diagrams and : F ! G be a natural
transformation. If hocolimIF belongs to C, then so does hocolimIG.
1
2 WOJCIECH CHACHOLSKI
Surprisingly these and many other results are the consequences of just one
statement, see theorem 5.1.
We continue with investigating the properties of a base space B (respec-
tively of the classifying space BI), which will guarantee that a closed class C*
* is
closed under extensions by fibrations with base B (respectively C is closed un-
der taking the unpointed homotopy colimit of diagrams F : I ! C ). We study
the following class:
D(C) = {BI | ifF : I ! C is a diagram, then hocolimIF 2 C}
The main result of this paper is:
Theorem. The class D(C) is a closed class and it is closed under extensions
by fibrations.
Using this theorem, we can characterize the class D(C) as follows:
D(C) = {B | ifZ ! E ! B is a fibration sequence and Z 2 C; then E 2 C}
Since D(C) is a closed class, it is closed under weak equivalences. This
is a very non trivial fact itself. It is obvious that ? belongs to D(C). What is
not clear at all is that for any diagram F : I ! C over a contractible category
I, the homotopy colimit hocolimIF belongs to C. As a result we get a new
characterization of a closed class:
A non empty class C of connected spaces is a closed class if and only if it is
closed under weak equivalences and for any unpointed diagram F : I ! C over
a contractible category I, the homotopy colimit hocolimIF belongs to C.
The techniques that are used to prove the main theorem involve studying
the homotopy fiber of a map f : X ! Y through inverse images of simplices in
Y (a simplicial analogues of point inverse images). We prove, roughly, that if
point inverse images belong to a closed class, then so does the homotopy fiber,
see corollary 7.9. One consequence of this is that if the point inverse images *
*of
f are acyclic with respect to some homology theory, then so is the homotopy
fiber of f .
Techniques, we have introduced, and properties of D(C) are applied to
prove a generalization of a theorem of E. Dror Farjoun (see 9.1 and [7, theorem
I]):
Theorem. Let : E ! B be a natural transformation between unpointed
diagrams E : I ! Spaces and B : I ! Spaces . If for every i 2 I, the homo-
topy fiber F ib(i : E(i) ! B(i)) belongs to a closed class C, then so does
F ib( : hocolimIE ! hocolimIB ).
CLOSED CLASSES 3
2. 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 mono-
tone maps of sets. The morphisms of are generated by codegeneracy maps
si : [n]! [n + 1] (i = 0; 1; : :;:n) and coface maps di : [n - 1]! [n], subject
to well-known cosimplicial identities (see [3]). A simplicial set is a functor
K : op ! Sets , where Sets denotes the category of sets [9, x2]. The set K([*
*n])
is usually denoted by Kn . A map between two simplicial sets is by definition a
natural transformation of functors. A simplicial set K can be interpreted as a
collection of sets Kn together with face maps di : Kn ! Kn-1 and degeneracy
maps si : Kn+1 ! Kn (i = 1; 2; : :;:n) which satisfy the duals of the cosim-
plicial identities (see [3]). For description of haw to do homotopy theory with
simplicial sets see [3] and [9].
If oe 2 Kn , then oe is called an n-dimensional simplex of K. The dimensi*
*on
of oe will be denoted by dim(oe). The object [n] is the standard n-simplex,
given by [n]k = mor ([k]; [n]) (see [3]). There is a distinguish n-dimensional
simplex o 2 [n]n , the one that comes from the identity map [n] ! [n]. It is
easy to check that for any simplicial set K, the assignment 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 .
A pointed simplicial set is a pair (K; k), where k is a chosen simplex
of dimension zero in K. We will refer to this 0-dimensional simplex as the
basepoint of (K; k). A map between pointed simplicial sets is a map of simplici*
*al
sets which preserves the basepoints. 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.
If I is a small category, the nerve of I, denoted by N (I), is the simpli*
*cial
set whose n-simplices are n-tuples (i0 ! i1 ! . .i.n) of composable morphisms
in I (see [3]).
If C is a category and K is an object in C, then by C=K we denote the
category of objects of C over K [8, 1x6]. This is the category whose objects
are morphisms f : X ! K in C and maps from f : X ! K to g : Y ! K are
those morphisms h : X ! Y in C such that gOh = f .
3. The homotopy colimit
In this section we describe the notion of a diagram indexed by a simplici*
*al
set and define the homotopy colimit of such a diagram. The particular form
of the homotopy colimit which is going to be used was introduced by E. Dror
4 WOJCIECH CHACHOLSKI
Farjoun [7]. Various properties of the homotopy colimit are also listed in the
appendix. The reference for the proofs is [5].
The motivation for these constructions comes from the fact that any map
f : X ! K of simplicial sets, can be reconstructed (up to a weak equivalence)
from the homotopy colimit of a diagram indexed essentially by the range of K.
The constituents of this diagram are the analogues (in the simplicial category)
of the point inverse images of f . The ability to build a map in a homotopically
meaningful way from its range and its point inverse images is going to be
explored in the following sections.
3.1. Definition. Let K : op ! Sets be a simplicial set. The category asso-
ciated to K, sometimes called the transport category of K or Grothendieck
construction on K, is the category whose objects are pairs ([n]; oe) where [n]
is an object of op and oe 2 Kn . A morphism ([n]; oe) ! ([m]; o ) is a map
' : [m] ! [n] in (or ' : [n]! [m] in op) such that K(')(oe) = o .
To avoid introducing to many notation we will denote the category asso-
ciated to a simplicial set K by the same symbol K, and speak of "functors with
domain K". One can think about the category K as having as objects the sim-
plices of K and morphisms generated by the arrows di : oe ! dioesi: oe ! sioe,
subject to some relations that come from the simplicial structure of K. The
morphisms si : oe ! sioeis called the degeneracy morphisms and di : oe ! dioe
the boundary morphisms.
The notion of the category associated to a simplicial set can be used to
define the subdivision of K (see [11 ]):
3.2. Definition. If K is a simplicial set, the subdivision of K is the nerve
(see section 2) of the category associated to K. The subdivision of K will be
denoted by sdK.
3.3. Example. The category associated to [0] = ? is isomorphic to op.
Diagrams over ? are simplicial spaces.
If K is a simplicial set, a functor F : K ! Spaces over the category as*
*soci-
ated to K will be called a diagram with shape K. Diagrams are our main object
of interest. As example 3.3 suggests, the category associated to a simplicial s*
*et
can be quite complicated. Since we care about diagrams, in order to simplify
the situation, we will distinguish the class of bounded diagram which are tech-
nically more manageable. It turns out that all the examples of diagrams, we
are going to consider, are bounded.
3.4. Definition. F : K ! Spaces is a bounded diagram if for any degeneracy
morphism si : oe ! sioe, F (oe) = F (sioe) and F (si) = idF (oe)
3.5. Example. A bounded diagram with shape ? is determined by the value
on the only zero dimensional simplex of ?. The category of bounded diagrams
with shape ? is equivalent to the category of simplicial sets.
CLOSED CLASSES 5
3.6. Example. The category of bounded diagrams over [1] is equivalent
to the category of diagrams of the form A B ! C, so called push-out
diagrams. Out of a bounded diagram F : [1] ! Spaces , a push-out diagram
can be extracted in the following way:
F (0) F (0; 1) ! F (1)
3.7. Example. Let f : X ! K be a map. For every simplex oe 2 K, let Ff(oe)
be the space that fits into the following pull-back square:
Ff(oe) --- - ! X
?? ?
y ?yf
[dim(oe)] ---oe-! K
Roughly speaking Ff(oe) is the inverse image in X of the simplex oe of K. This
construction clearly defines a functor Ff : Kop ! Spaces . Out of Ff we can
built a bounded diagram df : sdK ! Spaces such that for an n-dimensional
simplex v = oe0 '0!oe1 '1!::: 'n-1!oenin sdK:
df (v) = Ff(oen )
(
df (di : v ! div) = id: Ff(oen )! Ff(oen ) ifi < n
Ff('n-1 ) : Ff(oen )! Ff(oen-1 ) ifi = n
The construction d is natural. If f : X ! K , g : Y ! K and h : X ! Y are
maps such that gOh = f , then there is a natural transformation df ! dg. In par-
ticular there is a natural transformation : df ! d(id) induced by f : X ! K
itself. It turns out that:
colimsdK : colimsdK df ! colimsdK d(id) = (f : X ! K )
We will see in example 3.12 that the diagram df has nice homotopic properties
with respect to the map f .
If f : X ! K is a fibration, then the functor Ff : Kop ! Spaces has *
*the
property that for every morphism ', Ff(') is a weak equivalence. In this case
Ff(oe) is weakly equivalent to the homotopy fiber of f . It is clear from the
definition that the diagram df : sdK ! Spaces inherits the same properties.
This motivates the following definition:
3.8. Definition. F : K ! Spaces is a good diagram if it is a bounded diagram
and for every morphism ' 2 K, F (') is a weak equivalence.
The construction d defines a functor:
d : Spaces=K ! {bounded diagrams over sdK}
(f : X ! K ) 7! (df : sdK ! Spaces )
such that fibrations are carried out to good diagrams.
6 WOJCIECH CHACHOLSKI
We will now introduce a construction which will allow us to recover, up to
homotopy, a map f from the diagram df . One can think about this construction
as sort of an "inverse" of d:
3.9. Definition. The homotopyIcolimit is the following functor:
: {diagrams over K} ! Spaces
I K
i F j .
F = oe2K [dim(oe)] x F (oe) ~
K
where ~ is an equivalence relation generated by:
let' 2 mor ([n]; [m]) ,o 2 Km ,x 2 F (o ) ,t 2 [n]
(['](t); x) ~ (t; F (')(x))
3.10. Definition.ZThe pointed homotopy colimit is the following functor:
: {pointed diagrams over K} ! Spaces?
Z K
i W j .
F = oe2K [dim(oe)] x F (oe) = [dim(oe)] x {?} ~
K
where ~ is an equivalence relation generated by:
let' 2 mor ([n]; [m]) ,o 2 Km ,x 2 F (o ) ,t 2 [n]
(['](t); x) ~ (t; F (')(x))
3.11. Example. If F : K ! Spaces? is a constant diagram F (oe) = X and
F (') = idX , then:
I Z
F = K x X ; F = (K x X)=(K x {?}) = K n X
K K
I
In case X = ?, we get that ? = K, where ? : K ! Spaces denotes the
K
constant diagram whose value is ?.
Let F : K ! Spaces be a diagram. There is a natural transformation
F ! ? between F and the constant diagram ?. This natural transformation
induces the following map:
I I
F ! ? = K
K K
Let F : K ! Spaces? be a pointed diagram. This means that there is a
natural transformation ? ! F betweenIthe constantIdiagram ? and F . This
transformation induces the map K = ? ! F , which is a section of the
I K K
map F ! K.
K
CLOSED CLASSES 7
As a consequence we get that the homotopy colimit can be seen as a
functor with values in Spaces=K:
I
: {diagrams over K} ! Spaces=K
K
I
(F : K ! Spaces ) 7! F ! K
K
which carries out pointed diagrams into maps with a section.
If F : K ! Spaces? is a pointed diagram, then the following is a cofib*
*ra-
tion sequence:
I Z
K ! F ! F
K K
As a consequence we get that if F : K ! Spaces? is a pointed diagram over
weakly contractible simplicial set, then the unpointed and the pointed homo-
topy colimits of F are weakly equivalent.
3.12. Example. Let f : X ! K be a map. Out of f we have constructed
a diagram df : sdK ! Spaces (see example 3.7). The main property of df is
that in the following commutative diagram, all the horizontal arrows are weak
equivalences:
I I
df -id df -! colimsdK df = X
sdK? sdK? ? ?
y y ?y ?yf
I
sdK - d(id) -! colimsdK d(id) = K
sdK
It impliesIthat every map f : X ! K is weakly equivalent to a map of the
form F ! L for some bounded diagram F : L ! Spaces . Since every map
L
is weakly equivalent to a fibration, we can assume that F : L ! Spaces is a
good diagram whose values are weakly equivalent to the homotopy fiber of f
(see example 3.7).
3.13. Example. Let _ [n] ! L be a map, where _ [n] is the boundary of [n].
We are going to consider diagrams over L [ _[n][n].
o Let F : L [ _[n][n] ! Spaces be a diagram. One can show that:
I I I I
F = colim F F ,! F
L[ _[n][n] L _[n] [n]
8 WOJCIECH CHACHOLSKI
o Let F : L [ _[n][n] ! Spaces be a bounded diagram. If o 2 ([n])n
is the only non degenerate simplex and F (o ) = X, one can conclude
that in this case:
I I
F = colim F _[n] x X ,! [n] x X
L[ _[n][n] L
3.14. Example. A functor F : I ! Spaces over a small category I defines a
bounded diagram Fsd : N (I) ! Spaces over the nerve of I. Let:
oe = (a0 '0!a1 '1!::: 'n-1!an) 2 N (I)n
Fsd is defined as follows:
Fsd(oe) = F (a0)
(
Fsd(di : oe ! dioe) = F ('0) : F (a0)! F (a1) ifi = 0
id: F (a0)! F (a0) ifi > 0
It can be shown that in this case:
I
Fsd = hocolimIF
N(I)
where hocolimIF denotes the homotopy colimit of F in the sense of Bousfield
and Kan [3].
In case of a pointed functor F : I ! Spaces? there is a similar equalit*
*y:
Z
Fsd = phocolimIF
N(I)
where phocolimIF denotes the pointed homotopy colimit of F in the sense of
Bousfield and Kan [3].
4. Closed Classes
In this section we state the definition and give some examples and basic
properties of closed classes. The notion of a closed class was introduced by E.
Dror Farjoun [6], [7]. The definition presented in this papers is slightly diff*
*erent
from the one given by E. Dror Farjoun [6, definition 2.1]. We think about a
closed class as a class of unpointed and connected simplicial sets. A good
example, to keep in mind, is the class of acyclic spaces with respect to some
homology theory.
4.1. Definition. (E. Dror Farjoun [6]). A non empty class C of connected sim-
plicial sets is said to be a closed class if it 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
Observe that a closed class is assumed to be non-empty. Notice also that
since the empty space is not connected, it does not belong to any closed class.
CLOSED CLASSES 9
4.2. Notation.
o Throughout this article C always denotes a closed class.
o By F : K ! C we denote a diagram such that for every simplex oe 2 K,
F (oe) belongs to C.
o Let f : X ! Y be a map. We say that the homotopy fiber F ib(X !f Y )
belongs to a closed class C, if the homotopy fibers of f over every
component of Y belong to C. In particular, the homotopy fibers of f ,
over various components, are connected and f induces an isomorphism
on ss0. If F ib(X !f Y ) belongs to C, then we will write F ib(X !f Y *
*) 2
C.
4.3. Remark. The definition of the pointed homotopy colimit 3.10 implies
that a class C is closed if and only if:
o C is non-empty.
o Let X and Y be weakly equivalent simplicial sets. If X 2 C, then
Y 2 C.
o Let (Xi)i2I be a familyWof simplicial sets. If Xi 2 C, then for any ch*
*oice
of basepoints in Xi, i2IXi 2 C.
o Let X? be a simplicial space. If for every n 0, Xn 2 C, then the
realization |X?| 2 C.
It follows that a closed class can be characterized as a class of connect*
*ed
simplicial sets, such that:
o C is non-empty.
o C is closed under weak equivalences.
o C is closed under taking arbitrary wedges.
o Let X1 X2 ! X3 be a diagram. If Xi 2 C, then the following
simplicial set belongs to C:
hocolim(X1 X2 ! X3)
o Let ( X0 ! X1 ! X2 ! . . .) be a diagram. If Xi 2 C, then the
following simplicial set belongs to C:
hocolim(X0 ! X1 ! X2 ! . .).
4.4. Examples. Here is a list of some examples of closed classes:
o Let A be a connected simplicial set. The smallest closed class C(A)
such that A 2 C(A). This class is called the class of A-cellular space*
*s.
This class was introduced by E. Dror Farjoun, see [4],[6] and [7].
o The class of acyclic spaces with respect to some homology theory.
o The class C(?) of weakly contractible spaces.
o The class C(Sn+1 ) of n-connected spaces.
o {X 2 cSpaces | H"i(X; G) is trivial fori n}.
10 WOJCIECH CHACHOLSKI
o Let A be a pointed and connected Kan simplicial set.
{X 2 cSpaces | for any choice of basepoints in X, map?(X; A) ' ?}.
o Let A be a connected simplicial set.
{X 2 cSpaces | ifY is Kan and the basepoint evaluation map
map(A; Y ) ! Y is a weak equivalence, then map(X; Y ) ! Y
is also a weak equivalence }.
This class is called the class A-acyclic spaces, see [4, definition 16*
*.1].
The following two propositions give some examples of elements of a closed
class.
4.5. Proposition. (E. Dror Farjoun [6, section 2.3]). If C is a closed class,
then ? 2 C.
Proof. Let X 2 C and X !? X be the constant map. Notice that the following
space is contractible and belongs to C:
hocolim(X !? X !? X !? X . .).
It implies that ? 2 C. __|_ |
4.6. Proposition. (E. Dror Farjoun [6, theorem 2.8]). Let K and X be sim-
plicial sets. If X 2 C, then for any choice of basepoint in X, K n X 2 C (see
example 3.11).
Proof. Lets consider the constant diagramZX : K ! C , X(oe) = X and X(') =
idX . Since C is a closed class, K n X = X 2 C. __|_ |
K
5. Closed classes and unpointed homotopy colimits
Closed classes are not usually closed under unpointed homotopy colimits
(see [6, section 2.3]). This section contains the first approach to this questi*
*on,
to what extent a closed class is closed under unpointed homotopy colimits.
The motivation for the following theorem can be found in the next section.
This theorem is going to be applied to prove various properties of closed class*
*es
with respect to fibrations. Surprisingly all those properties are just particul*
*ar
cases of this one statement.
5.1. Theorem. Let F : K ! C , GI: K ! C be diagrams and : F ! G be
a natural transformation. If h : F ! Y is a map such that Y 2 C, then:
K
i I I h j
hocolim G F ! Y 2 C
K K
CLOSED CLASSES 11
I
5.2. Lemma. For every diagram F : [n] ! C , F belongs to C.
[n]
Proof. Let o 2 ([n])n be the non-degenerate simplex. Observe that by choos-
ing a vertex in F (o ), we can think about F : [n] ! C as a pointed diagram
(this chosen vertex determines a basepoint in F (oe), for all oe 2 [n]). Since *
*[n]
is contractible, pointed and unpointed homotopy colimits are weakly equiva-
lent. It proves the lemma. __|_ |
Proof of the theorem. the proof will be by the induction on the dimension of
K. It is obvious that the theorem is true for K such that dim(K) = 0. Lets
assume that the theorem is true for K such that dim(K) < n. Let dim(L) < n
and _[n] ! L be a map. We prove that the theorem holds for K = L[ _[n][n].
Lets consider the following commutative diagram:
Y - - id-- Y --id--! Y
x? x x
? h ??h ??h
I I I
F - - - - F --- - ! F
L? _[n]? [n]?
?y ?y ?y
I I I
G - - - - G --- - ! G
L _[n] [n]
By the inductive assumption:
i I I h j
hocolim G F ! Y 2 C
L L
i I I h j
hocolim G F ! Y 2 C
_[n] _[n]
I I
According to lemma 5.2, F and G belong to C. Because C is closed
[n] [n]
under homotopy push-outs we get:
i I I h j
hocolim G F ! Y 2 C
__ K K
|_ |
5.3. Corollary. Let F : K ! CI , G : K ! C Ibe diagrams and : F ! G be
a natural transformation. If F 2 C, then G 2 C.
K K
I
Proof. Apply theorem 5.1 to the case when Y = F and h = id. __|_ |
K
12 WOJCIECH CHACHOLSKI
5.4. ICorollary. Let K be a simplicial set. If F : K ! C is a diagram such
that F 2 C, then K 2 C.
K
Proof. Since there is a natural transformation F ! ? between F and the
constantIdiagram ? : K ! Spaces , whose value is ?, corollary 5.3 implies that
K = ? belongs to C. __|_ |
K
6. Closed classes and fibrations
The behavior of a closed class with respect to fibrations has been studied
by E. Dror Farjoun [6],[7]. This section contains generalizations of his result*
*s.
6.1. Definition. We say that a closed class C is closed under extensions by
fibrations if for every fibration sequence Z ! E ! B such that Z and B belong
to C, E belongs to C. 2mm
Closed classes are not usually closed under extensions by fibrations [7].
6.2. Examples. Here is a list of some examples of closed classes that are
closed under extensions by fibrations:
o The class of acyclic spaces with respect to some homology theory.
o The class C(?) of weakly contractible spaces.
o The class C(Sn+1 ) of n-connected spaces.
o Let A be a connected simplicial set.
{X 2 cSpaces | ifY is Kan and the basepoint evaluation map
map(A; Y ) ! Y is a weak equivalence, then map(X; Y ) ! Y
is also a weak equivalence }.
The next theorem is a geometric interpretation of theorem 5.1.
6.3. Theorem. Let p1 : E1 ! B , p2 : E2 ! B and s : E1 ! E2 be maps such
that p1 = p2Os and the homotopy fibers F ib(E1 p1!B), F ib(E2 p2!B) belong to
C. For any map h : E1 ! Y , where Y 2 C:
hocolim(E2 s E1 !h Y ) 2 C
Proof. Example 3.12 implies that p1 and p2 are weakly equivalent, respectively
to maps of the form:
I I
F ! sdB ; G ! sdB
sdB sdB
where F has values weakly equivalent to the homotopy fibers of p1 and G has
values weakly equivalent to the homotopy fibers of p2. Since the definitions we*
*re
CLOSED CLASSES 13
natural, s : E1 ! E2 induces a natural transformation : F ! G . Theorem 5.1
implies then:
I I
hocolim(E2 E1 ! Y ) ' hocolim( G F ! Y ) 2 C
sdB sdB
__|_ |
The following corollaries are particular cases of theorem 6.3.
6.4. Corollary. Let p1 : E1 ! B , p2 : E2 ! B and s : E1 ! E2 be maps such
that p1 = p2Os. If the homotopy fibers F ib(E1 p1!B), F ib(E2 p2!B) and E1
belong to C, then so does E2.
Proof. Apply theorem 6.3 to the case when Y = E1 and h = id. __|_ |
6.5. Corollary. (E. Dror Farjoun [7]). Let p : E ! B be a map such that
the homotopy fiber F ib(E !p B) belongs to C. If E 2 C, then B 2 C.
Proof. Apply corollary 6.4 to the case when p1 = p, p2 = idB and s = p. __|*
*_ |
6.6. Corollary. (E. Dror Farjoun [7]). Let F ! E !p B be a fibration se-
quence. If B and F belong to C, then so does E.
Proof. Since B ! F ! E is a fibration sequence such that B and F belong
to C, corollary 6.5 implies that E belongs to C. __|_ |
6.7. Corollary. Let F ! E !p B be a fibration sequence and B !s E be a
section of p. If h : B ! Y is a map such that Y 2 C, then:
colim(E s B !h Y ) 2 C
Proof. Apply theorem 6.3 to the case when p1 = idB , p2 = p. __|_ |
6.8. Corollary. Closed classes are closed under split extensions. Let F !
E !p B be a fibration sequence such that p has a section. If F 2 C and B 2 C,
then E 2 C.
Proof. Apply corollary 6.7 to the case when Y = B and h = idB and s is a
section of p. __|_ |
6.9. Corollary. (W. Dwyer [6]). Closed classes are closed under products. If
X 2 C and Y 2 C, then X x Y 2 C .
Proof. Notice that X ! X x Y ! Y is a fibration sequence with a section.
According to corollary 6.8, X x Y belongs to C. __|_ |
14 WOJCIECH CHACHOLSKI
7. Homotopy properties of shapes of diagrams
In this section the behavior of a closed class under unpointed homotopy
colimits is going to be investigated further (see section 5).
A diagram F : K ! Spaces consist of bunch of spaces which are related
to each other by various maps. Those relations are coming from the geometry
of K. We will try to understand how the geometry of K effects the homotopy
colimit functor of diagrams over K.
I
7.1. Definition. D(C) = {K | ifF : K ! C is a diagram, then F 2 C}
K
Class D(C) consists of those simplicial sets that carry enough information so
by gluing elements of class C, according to K, we get back a space in C.
7.2. Proposition. D(C) C
I
Proof. Let K 2 D(C). Since ? 2 C, according to the definition, K = ?
K
belongs to C.
Notice that corollary 5.4 is stronger thanIthis proposition. It says that
if there exist a diagram F : K ! C such that F 2 C, then automatically
__ K
K 2 C. |_ |
Observe that lemma 5.2 implies:
7.3. Proposition. For every n, [n] 2 D(C).
It turns out that D(C) has nice homotopic properties. The next theorem
suggests that to some extant, not geometry but the homotopy type of a simpli-
cial set K plays a crucial role toward the homotopic properties of the homotopy
colimit functor of diagrams over K.
7.4. Theorem. Class D(C) is closed under weak equivalences.
7.5. Lemma.
o Let K L ,! M be a push-out diagram such that L ,! M is a
cofibration. If K, L and M belong to D(C), then so does:
K [L M = colim(K L ,! M )
o Let be the category associated with an ordinal number (see [8,
page 11]). Let G : ! Spaces be a functor such that for every mor-
phism ' 2 , G(') is a cofibration. If for every 2 , G() belongs
to D(C), then so does colim G .
Proof. We will prove only the first part of the lemma. The second part can be
proven in the same way.
Let F : K [L M ! C be a diagram. According to example 3.13:
CLOSED CLASSES 15
I i I I I j
F = colim F F ,! F
K[LM K L M
I I I
By the assumption F , F and F belong to C. Since any closed class is
K L M
closed under taking homotopy push-outs:
i I I I j
hocolim F F ,! F 2 C
K L M
Notice that the cofibration assumption implies that the following map is a weak
equivalence:
i I I I j i I I I j
hocolim F F ,! F ! colim F F ,! F
K L M K L M
I
It implies that F belongs to C. __|_ |
K[LM
7.6. Lemma. Let 0 k n. [n; k] belongs to D(C), where if o 2 [n]n is
the non degenerate simplex, [n; k] is the simplicial subset of [n], generated
by simplices {dio }i6=k.
Proof. We are going to present [n; k] as a sequence of push-outs of standard
simplices. In order to do this we have to introduce some notation:
o Let i 2 {0; 1; . .;.n}. i denotes the simplicial subset of [n] generat*
*ed
by the simplex dio .
o {i} denotes the simplicial subset of [n] generated by the vertex {i}.
o Let {i; j} 2 {0; 1; . .;.n}. i;jdenotes the simplicial subset of [n]
generated by the simplex di-1djo if i > j, or by dj-1 dio if i < j.
There are obvious inclusions i;j! i ! [n]. Let X be the colimit of the
following diagram:
k+2x -id! k+2x id! nx 1x !id k-1x
? ? ? ? ?
k+1;k+2? k+2;k+3? . . . n;0? 0;1? . . . k-2;k-1?
y y y y y
id
k+1 k+3 id! 0 -id! 0 ! k-2
Out of the construction of X, we have two natural inclusions k-1 ! X,
k+1 ! X. Notice that there is a cofibration map k-1;k+1 _{k}k-1;k+1 ! X
which is the wedge of the following maps:
k-1;k+1 ! k-1 ! X
{k} ! k-1 ! X
k-1;k+1 ! k+1 ! X
By laborious but straightforward calculation one can show that:
16 WOJCIECH CHACHOLSKI
id_id
[n; k] = colim k-1;k+1 - k-1;k+1 _{k} k-1;k+1 ! X
Since [n; k] is built from standard simplices by push-out process, where the
maps involved are cofibrations, according to the lemma 7.5, [n; k] belongs to
D(C). __|_ |
Proof of the Theorem. The proof will be divided into several steps.
Step 1. Let E '! B be a fibration and a weak equivalence. If E 2 D(C), then
B 2 D(C).
I
Proof. Let F : B ! C be a diagram. We have to show that F 2 C. Accord-
B
ing to section A.1, the following is a pull-back square:
I I
F Op --- - ! F
E? B?
?y ?y
E ---p- ! B
I I
Since p is a fibration and a weak equivalence, F Op ! F is a weak equiva-
I E B I
lence as well. Because E 2 D(C), F Opbelongs to C. It implies that F 2 C.
E B
Step 2. Let f : X ! Y be a weak equivalence. If X 2 D(C), then Y 2 D(C).
Proof. We are going to construct by the induction a sequence of spaces and
inclusions:
(X0 ! X1 ! X2 ! . .).
together with a sequence of maps:
{il: X ! Xl }l0 ; {pl : Xl ! Y }l0
such that Xl 2 D(C), il is a cofibration and a weak equivalence, ilOpl= f and
the map colimit(pl) : colimit(Xl) ! Y is a fibration. We denote colimit(pl) by
p : X ! Y , in particular X = colimit(Xl).
Let X0 = X, i0 = idX and p0 = f . Lets assume that the construction has
been carried out for i < l. Let J be the set of all commutative diagrams of the
form:
[n; k] --- - ! Xl-1
?? ?
y ?ypl-1
[n] --- - ! Y
CLOSED CLASSES 17
where [n; k] ! [n] is the canonical inclusion. Xl is defined to be the simpli-
cial set that fits into the following push-out square:
F l-1
J[n;?k] -- - - ! X?
?y ?y
F l
J[n] -- - - ! X
il is defined to be the following composition:
X il-1!Xl-1 ! Xl
pl is defined to be the push-out of the following maps:
F F
Xl-1 pl-1!Y ; J [n; k] ! Y ; J [n] ! Y
By the inductive assumption Xl-1 2 D(C). Since Xl is built by gluing
lots of [n] along [n; k] to Xl-1 , according to lemma 7.5, Xl 2 D(C). Observe
that the natural map i : X = X0 ! X is a weak equivalence. Notice also that
pOi = f . By Quillen's small object argument (see [10 ]), p is a fibration. Sin*
*ce f
and i are weak equivalences, so is p.
Lemma 7.5 implies that X 2 D(C). Since p : X ! Y is a fibration and a
weak equivalence, according to step 1, Y 2 D(C).
Step 3. If X is contractible, then X 2 D(C).
Proof. Since ? 2 D(C) and ? ! X is a weak equivalence, step 2 implies that
X 2 D(C).
I
Step 4. Let F : K ! C be a diagram. The homotopy fiber F ib( F ! K)
K
belongs to C.
Proof. Lets choose a connected component of K and a fibration P K ! K
such that P K is contractible and the image of P KIis in the chosen component.
According to corollary A.2, the homotopy fiber of F ! K over the chosen
I K
component is weakly equivalent to F. Since P K is contractible, according
I P K
to step 3, F belongs to C.
P K
Step 5. Let Z ! E ! B be a fibration sequence. If B 2 D(C) and Z 2 C,
then E 2 C.
18 WOJCIECH CHACHOLSKI
Proof. We can assume thatIp is a fibration. Example 3.12 implies that E ! B
is weakly equivalent to dp ! sdB, where dp : sdB ! Spaces is a good
sdB
diagram whose values are weakly equivalent to Z. Proposition A.5 gives the
following weak equivalence:
I I I
(dp)Oloe! dp
B N(B=oe) sdB
I
Since N (B=oe) is contractible (dp)Oloe2 C. The assumption B 2 D(C)
N(B=oe)
implies:
I I
E ' (dp)Oloe2 C
B N(B=oe)
Step 6. Let f : X ! Y be a weak equivalence. If Y 2 D(C), then X 2 D(C).
Proof. Let FI: X ! C be a diagram. Notice that the homotopy fiber of the
composition F ! X f! Y is weakly equivalent to the homotopy fiber
I X
F ib( F ! X). According to step 4, it belongs to C. Since Y 2 D(C), Step 5
X I
implies that F 2 C. This proves that X 2 D(C).
__ X
|_ |
Theorem 7.4 implies an interesting characterization of a closed class (see
also[1]):
7.7. Corollary. Non empty class C of connected simplicial sets is closed if it*
* is
closed under weak equivalences and for every, not necessarilyIpointed diagram,
F : K ! C over a contractible simplicial set K, F 2 C.
K
The definition of a closed class says that it is closed under pointed ho-
motopy colimits. It means that for any pointed diagram F : K ! Spaces? the
homotopy cofiber: I Z
Cof K ! F = F
K K
belongs to C. The next corollary implies that the dual statement is also true
(see [6] and [7] for discussion of similar statements).
I
7.8. Corollary. Let F : K ! C be a diagram. F ib( F ! K) 2 C.
K
The following corollary says that if the the pre-images of simplices have
certain properties (belong to a closed class), then so does the homotopy fiber
(see also [7]).
CLOSED CLASSES 19
7.9. Corollary. Let f : X ! K be a map. If for every simplex oe 2 K:
f
pullback X ! K [dim(oe)] 2 C
then F ib(X !f Y ) 2 C.
Proof. AccordingIto example 3.12, f : X ! K is weakly equivalent to a map
of the form df ! sdK, where for v = (oe0 ! . . .! oen ) 2 sdK, df is a
sdK
diagram such that:
f
df (v) = pullback X ! K [dim(oen )]
I __
Corollary 7.8 implies F ib df ! sdK 2 C. |_ |
sdK
8. Class D(C)
In this section we present other characterizations of the class D(C). We
will restrict the class of diagrams on which a simplicial set should be tested *
*in
order to find out if it belongs to D(C). We will show also that the class D(C)
is a closed class and is closed under extensions fibrations.
8.1. Proposition.
I
D(C) = {K | ifF : K ! C is a bounded diagram, then F 2 C}
K
Proof. Let:
I
D0 = {K | ifF : K ! C is a bounded diagram, then F 2 C}
K
Inclusion D(C) D is obvious.
By the same arguments as in theorem 7.4, we can show that the class D0
is closed under weak equivalences.ILet K 2 D0 and F : KI! C be a diagram.
According to remark A.6, F is weakly equivalent to Fsd . Since sdK is
K sdK
weakly equivalentIto K, it belongs to D0.INotice that Fsd is a bounded diagram,
therefore Fsd 2 C. It implies that F 2 C and K 2 D(C). __|_ |
sdK K
8.2. Proposition.
D(C) = {B | ifZ ! E ! B is a fibration sequence and Z 2 C; then E 2 C}
Proof. Let:
D0 = {B | ifZ ! E ! B is a fibration sequence and Z 2 C; then E 2 C}
20 WOJCIECH CHACHOLSKI
I I
Let B 2 D and F : B ! C be a diagram. Since F ! F ! B is a
I P BI B
fibration sequence (see A.2) and F 2 C, we get F 2 C. It implies the
P B B
inclusion D0 D(C).
Let K 2 D(C) and Z ! E !p K be a fibration sequence. AccordingIto
example 3.12, E ! K is weakly equivalent to a map of the form F ! L,
L
where the values of F are weakly equivalent to Z. SinceIL is weakly equivalent
to K, it belongs to D(C). As a consequence we get E ' F 2 C. It proves
__ L
that K 2 D0 and D D0. |_ |
8.3. Corollary. A closed class C is closed under extensions by fibrations if
and only if C = D(C).
The next corollary says that class D(C) is usually quite big.
8.4. Corollary. If B is such that B 2 C, then B 2 D(C).
Proof. See corollary 6.6. __|_ |
8.5. Proposition. I
D(C) = {K | ifF : K ! C is a good diagram, then F 2 C}
K
Proof. Let:
I
D0 = {K | ifF : K ! C is a good diagram, then F 2 C}
K
Inclusion D(C) D0 is obvious.
By the same arguments as in the theorem 7.4, we can show that the class
D0 is closed under weak equivalences. Let B 2 D0 and p : E ! B be a fibration
such that the fiber of p belongs to C. AccordingIto example 3.12, p : E ! B
is weakly equivalent to a map of the form dp ! sdB, where dp is a good
sdB
diagramIwhose values are weakly equivalent to the fiber of p. It implies that
E ' dp 2 C. This proves the proposition. __|_ |
sdB
8.6. Theorem.ILet G : K ! D(C) be a diagram. If K belongs to D(C), then
so does G .
K
I I
Proof. According to remark A.6, G is weakly equivalent to Gsd . Theo-
I K I sdK
rem 7.4 implies that G belongs to D(C) if and only if Gsd does. Since
K sdK
CLOSED CLASSES 21
Gsd is a bounded diagram, without loss of generality, it is enough to prove the
theorem for a bounded diagram G : K ! D(C) . I
Let G : D(C) ! b e a bounded diagram and F : G ! C be a diagram.
I K I I
Theorem A.9 implies that H F is weakly equivalent to F . Since
K G I sdK G(v)
G has values in D(C), then so does G, therefore F belongs to C.
IG(v)I
Because K 2 D(C), sdK 2 D(C) and it follows that F belongs to
I sdK G(v)
C. This proves H F 2 C. __|_ |
K G
8.7. Corollary. D(C) is a closed class and D(D(C)) = D(C), therefore D(C)
is closed under extensions by fibrations.
9. Theorem of E. Dror Farjoun
9.1. Theorem. Let : E ! B be a natural transformation between diagrams
E : K ! Spaces and B : K ! Spaces . If for every simplex oe 2 K the homo-
topy fiber F ib(E(oe) -oe!B(oe)) belongs to C, then:
I I
F ib E -! B 2 C
K K
9.2. Lemma. Lets consider the following commutative diagram:
E1 - - - - E2 --- - ! E3
?? ? ?
y p1 ?yp2 ?yp3
B1 - - f-- B2 ---g- ! B3
where the maps E2 ! E3, B2 !g B3 are cofibrations. If F ib(p1), F ib(p2) and
F ib(p3) belong to C, then:
F ib(E1 [E2 E3 ! B1 [B2 B3) 2 C
Proof. Without loss of generality we can assume that p1, p2 and p3 are fibra-
tions. Let p = colim(p1 p2 ! p3). According to corollary 7.9, it is enough
to prove that for every simplex, oe 2 B1 [B2 B3:
F (oe) = pullback(E1 [E2 E3 !p B1 [B2 B3 [dim(oe)]) 2 C
Let oe 2 B1 [B2 B3. Either oe lies in the image of B1 or B2. Lets assume that it
belongs to the image of B1. Let K = pullback([dim(oe)] ! B1 f B2). There
22 WOJCIECH CHACHOLSKI
is a natural map K ! B2. Let X1, X2 and X3 be simplicial sets that fit into
the following pull-back squares:
X1 --- - ! E1 X2 --- - ! E2 X3 --- - ! E3
?? ? ? ? ? ?
y ?yp1 ?y ?yp2 ?y ?yp3
[dim(oe)] --- - ! B1 K --- - ! B2 K --- - ! B3
Observe that the definition gives natural maps X2 ! X1 and X2 ! X3. By
straightforward combinatorial calculation one can show:
F (oe) = colim(X3 X2 ! X1)
Notice that the maps X3 ! K, X2 ! K and X2 ! X1 satisfy the assump-
tions of theorem 6.3, therefore hocolim(X3 X2 ! X1) 2 C. Cofibration
assumption of the lemma implies:
hocolim(X3 X2 ! X1) ' colim(X3 X2 ! X1)
It proves the lemma. __|_ |
Proof of the theorem. Instead of E : K ! Spaces , B : K ! Spaces we can co*
*n-
sider bounded diagramsIEsd : sdKI ! Spaces ,IBsd : sdK I! Spaces . Since the
homotopy fibers F ib E ! B and F ib Esd ! Bsd are weakly
K K sdK sdK
equivalent, it is enough to prove the theorem for bounded diagrams.
The proof will be by the induction on the dimension of K. If dim(K) = 0,
the theorem is obvious. Lets assume that the theorem is true for K such that
dim(K) < n. Let L be a simplicial of dimension less than n and _[n] ! L be
a map. We prove that the theorem holds for K = L [ _[n][n].
Let o 2 ([n])n be the only non degenerate simplex. Lets consider the
following diagram:
I I
E = colim E _ [n] x E(o ) ,! [n] x E(o )
K? L? ? ?
y y ?yidxo ?yidxo
I I
B = colim B _ [n] x B(o ) ,! [n] x B(o )
K L
I I
By the inductive assumption F ib E ! B belongs to C. Since the homo-
L L
topy fiber F ib(E(o ) o! B(o ))also belongs to C, according to lemma 9.2:
I I
F ib E ! B 2 C
__ K K
|_ |
As a corollary we get the theorem of E. Dror Farjoun
CLOSED CLASSES 23
9.3. Corollary. (E. Dror Farjoun [7, theorem I]). Let E : K ! Spaces? and
B : K ! Spaces? be pointed diagrams and : E ! B be a natural transfor-
mation. If for every simplex oe 2 K, F ib(E(oe) -oe!B(oe)) 2 C , then:
Z Z
F ib E -! B 2 C
K K
Proof. Lets consider the following diagram:
Z I
E ' hocolim ? K ! E
K? ? ? K?
y ?y ?yid y
Z I
B ' hocolim ? K ! B
K K
I I
According to theorem 9.1, F ib E ! B belongs to C. Since the ho-
K K
motopy fiber F ib(K id!K) belongs to C, applying once again theorem 9.1 we
get:
Z Z
F ib E ! B 2 C
__ K K
|_ |
9.4. Theorem. Let the following be a homotopy push-out square:
A ---f- ! B
?? ?
yi ?y
X ---g- ! Y
If the homotopy fiber F ib(A !f B) belongs to C, then so does the homotopy
fiber F ib(X !g Y ).
Proof. Lets consider the following diagram:
i id
X? ' hocolim X? A? ! A?
?yg ?yid ?yid ?yf
i f
Y ' hocolim X A ! B
Since F ib(A !f B) belongs to C, theorem 9.1 implies that F ib(X !g Y ) 2 C . *
* __|_ |
24 WOJCIECH CHACHOLSKI
9.5. Corollary. Let f : X ! Y be a map. If X belongs to C, then so does
the homotopy fiber F ib Y ! Cof (X !f Y ) .
Proof. Apply theorem 9.4 to the following homotopy push-out square:
X? __________! ??
?y ?y
Y - ! Cof (X !f Y )
__|_ |
Appendix A. The homotopy colimit
The reference for the proofs of the statements, listed in the appendix,
is [5].
A.1. Pulling-back of diagrams. Let F : K ! Spaces be a diagram over K
and f : L ! K be a map. We can pull-back F into a diagram F Of over L (F Of
will be often denoted simply by F ). Let o and oe be simplices in L and ' : o !*
* oe
be a morphism in the category associated with L. F Of : L ! Spaces is defined
as follows:
F Of(oe) = F (f (oe))
F Of (') = F (')
The basic property of the pull-back diagram F Of is that the following is
a pull-back square:
I I
F Of --- - ! F
L? K?
?y ?y
L ---f- ! K
As corollary of the this property we get:
A.2. Corollary. Let F : K ! Spaces be a diagram and P K ! K be a fibra-
tion such that P K is contractible. The following is a fibration sequence:
I I
F ! F ! K
P K K
CLOSED CLASSES 25
A.3. Diagrams over colimIG. Let G : I ! Space , F : colimIG ! Spaces
be diagrams. There is a family of maps {li: G(i) ! colimIG }i2I which satisfies
the universal property of the colimit of the diagram G : I ! Space (see [8, 3x*
*3]).
Out of this data we can construct a functor:
I - ! SpacesI
i 7- ! F Oli
G(i)
I H F OlbI
(a '! b) 7- ! ( F Ola G(')-! F Olb)
G(a) G(b)
This functor has the followingIproperty: I
F = colimG F Oli
colimIG G(i)
A.4. Diagrams over sdK. Let oe be a simplex in K. By K=oe we denote the
over category of K (see section 2). There is a functor:
K=oe ! K
(o ! oe) 7! o ; (o0 ! o1 ! oe) 7! (o0 ! o1)
This functor induces a map between simplicial sets:
loe: N (K=oe) ! N (K) = sdK
(o0 ! . .!.on ! oe) loe7!(o0 ! . .!.on )
One can verify that the family of maps {loe: N (K=oe) ! sdK }oe2K satisfies the
universal property of the colimit of the functor:
K ! Spaces ; oe 7! N (K=oe)
It implies:
sdK = colimK N (K=oe)
Let F : sdKI ! SpacesI be a diagram. AccordingIto subsection A.3:
F = F = colimK F Oloe
sdK colimK N(K=oe) N(K=oe)
A.5. Proposition.ITheInatural map: I I
F Oloe! colimK F Oloe= F
K N(K=oe) N(K=oe) sdK
is a weak equivalence.
A.6. Remark. Let F : K ! Spaces be a diagram. This means that F is a
functor over the category associated to K. According to example 3.14,Iit de-
fines a bounded diagram Fsd : sdK ! Spaces . It turns out that F is weakly
I I K
equivalent to Fsd and Fsd = hocolimK F , where hocolimK F is the
sdK sdK
homotopy colimit of F : K ! Spaces in the sense of Bousfield-Kan.
26 WOJCIECH CHACHOLSKI
I
A.7. Diagrams over G . Let G : K ! Spaces be a diagram. Out of G we
K
can construct a new diagram G : sdK ! Spaces . Let:
u = (o0 !0 o1 !1 ::: m-1!om) 2 (sdK)m
v = (oe0 '0!oe1 '1!::: 'n-1!oen) 2 (sdK)n
j : u ! v be a morphism in sdK
By definition 3.1, j is a morphism in such that j : [n]! [m] and sdK(j)(u) =
v. G : sdK ! Spaces is a diagram defined as follows:
G(v) = [dim(oen )] x G(oe0)
8
>>>id x id ifj(0) = 0 ; j(n) = m
< id x G( O . .O. ) ifj(0) > 0 ; j(n) = m
G(j) = > j(0)-1 0
>>:[ m-1 O . .O. j(n)] x id ifj(0) = 0 ; j(n) < m
[ m-1 O . .O. j(n)] x G( j(0)-1O . .O. 0) ifj(0) > 0 ; j(n) < m
Observe that G has the values weakly equivalent to the values of G.
A.8. Proposition.
I
G = colimsdK G
K
I
A.9. Theorem. Let G : K ! Spaces and F : G ! Spaces be diagrams. If
K
G is a bounded diagram,IthenIthe following natural map is a weak equivalence:
I I
F ! colimsdK F = H F
sdK G(v) G(v) K G
References
1. A.Amit, Direct limits over diagrams with contractible nerve, Master thesis,*
* Hebrrew
Univ. (1993).
2. A.K.Bousfield, Localization and periodicity in unstable homotopy theory, pr*
*eprint.
3. A.K.Bousfield and D.M.Kan, Homotopy Limits, Completions and Localizations, *
*Lect.
Notes in Math. 304, Springer (1972)
4. W. Chacholski, Functors CWA and PA , Ph.D. thesis, Univ. of Notre Dame (199*
*5).
5. W.Chacholski, Homotopy properties of shapes of diagrams, report No. 6, 1993*
*/94, Insti-
tut Mittag-Leffler.
6. E.Dror Farjoun, Cellular spaces, preprint.
7. E.Dror Farjoun, Cellular inequalities, Proc. to the conf. in Alg. Top. Nort*
*heastern Univ.
June 1993, Springer Verlag.
8. S. MacLane, Categories for working mathematician, Grad. Texts in Math. 5, S*
*pringer
(1971).
9. J.Peter May, Simplicial objects in algebraic topology, Van Nostrand Math. S*
*tudies 11,
(1987).
10. D. Quillen, Homotopical algebra, Lect. Notes in Math. 43, Springer (1967)
CLOSED CLASSES 27
11. G.Segal, Classifying spaces and spectral sequences, Inst. haut. Etul. sci.,*
* Publ. math. 34,
105-112 (1968).
Wojciech Chacholski, Department of Mathematics, University of Notre Dame,
Mail Distribution Center, Notre Dame, Indiana 46556-5683
E-mail address: wchacho1@kenna.math.nd.edu