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. Aclosed
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 ofclosed classes. The
first one is the Bousfield-Dror periodization functor PA [2]. The class ofthose
spaces X, such that PA X is weakly contractible, forms a closed class. By looki*
*ng
just at the properties of this class wecan prove, for example, that PA X is
weakly equivalent toPA X (see [2], [4]). The second construction is E. Dror
Farjoun's colocalization functor C WA. The class of those spaces X, for which
there exists a space Y, such that X is weakly equivalent toC WAY , 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 ofthe class C (A) we can prove, for
example, that CWAX 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 andB 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 closedclass
is closed under extensions by fibrations and under taking unpointed homotopy
colimits. We start with proving a theorem that, in particular, implies:
fflLet F : I ! Spaces? be a pointed diagram, such that the classifying
space BI belongs to C. If for every i 2 I, F (i)b elongs to C , thenso
does the unpointed homotopy colimit hocolimIF.
fflLet (Z ! E ! B) be a fibration sequence with a section. If Z and B
belong to C, then so does E.
fflLet F : I ! C andG : I ! C be diagrams and : F ! G be a natural
transformation. If hocolimIF belongs to C, then so does hocolimIG.
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 ) =fB I jifF : I ! C isa diagram, then hocolimIF 2 Cg
The main result of this paper is:
Theorem. The class D(C) is a closed class and it is closedunder extensions
by fibrations.