Function spaces and shape theories
Jerzy Dydak, Sławomir Nowak
2002
Fundamenta Mathematicae
The purpose of this paper is to provide a geometric explanation of strong shape theory and to give a fairly simple way of introducing the strong shape category formally. Generally speaking, it is useful to introduce a shape theory as a localization at some class of "equivalences". We follow this principle and we extend the standard shape category Sh(HoTop) to Sh(pro-HoTop) by localizing pro-HoTop at shape equivalences. Similarly, we extend the strong shape category of Edwards-Hastings to
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... -Top) by localizing pro-Top at strong shape equivalences. A map f : X → Y is a shape equivalence if and only if the induced function f * : is a weak homotopy equivalence for all P ∈ ANR. One generalizes the concept of being a shape equivalence to morphisms of pro-HoTop without any problem and the only difficulty is to show that a localization of pro-HoTop at shape equivalences is a category (which amounts to showing that the morphisms form a set). Due to peculiarities of function spaces, extending the concept of strong shape equivalence to morphisms of pro-Top is more involved. However, it can be done and we show that the corresponding localization exists. One can introduce the concept of a super shape equivalence f : X → Y of topological spaces as a map such that the induced map f * : Map(Y, P ) → Map(X, P ) is a homotopy equivalence for all P ∈ ANR, and one can extend it to morphisms of pro-Top. However, the authors do not know if the corresponding localization exists. Here are applications of our methods: Theorem. Suppose f : X → Y is a map of topological spaces. (a) f is a shape equivalence if and only if the induced function f * : [Y, M ] → [X, M ] is a bijection for all M = Map(Q, P ), where P ∈ ANR and Q is a finite CW complex. (b) If f is a strong shape equivalence, then the induced function f * : [Y, M ] → [X, M ] is a bijection for all M = Map(Q, P ), where P ∈ ANR and Q is an arbitrary CW complex. (c) If X, Y are k-spaces and the induced function f * : [Y, M ] → [X, M ] is a bijection for all M = Map(Q, P ), where P ∈ ANR and Q is an arbitrary CW complex , then f is a strong shape equivalence. 2000 Mathematics Subject Classification: Primary 55P55, 55N20, 55N07.
doi:10.4064/fm171-2-2
fatcat:egvxgnexmrbljjyybzladpdo3q