Closed Classes
نویسنده
چکیده
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 those spacesX, such that PAX is weakly contractible, forms a closed class. By looking just at the properties of this class we can prove, for example, that PAΩX is weakly equivalent to ΩPΣAX (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 CWAY , forms a closed class. This class is denoted by C(A) and is called the class of A-cellular spaces. By looking just at the properties of the class C(A) we can prove, for example, that CWAΩX is weakly equivalent to ΩCWΣAX (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: • Let F : I → Spaces? be a pointed diagram, such that the classifying space BI belongs to C. If for every i ∈ I, F (i) belongs to C, then so does the unpointed homotopy colimit hocolimIF . • Let (Z → E → B) be a fibration sequence with a section. If Z and B belong to C, then so does E. • Let F : I → C and G : I → C be diagrams and Ψ : F → G be a natural transformation. If hocolimIF belongs to C, then so does hocolimIG.
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تاریخ انتشار 1997