Function spaces and shape theories

Abstract

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 sSh(pro-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 : (Y;P )! (X;P ) is a bijection for all P 2 ANR. A map f : X ! Y of k-spaces is a strong shape equivalence if and only if the induced map f : Map(Y;P ) ! Map(X;P ) is a weak homotopy equivalence for all P 2 ANR. One generalizes the concept of being a shape equivalence to morphisms of pro-HoTop without any problem and the only dicult y 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 2 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. A map f : X! Y of k-spaces is a strong shape equivalence if and only if f idQ : X k Q! Y k Q is a shape equivalence for each CW complex Q. 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 2 ANR and Q is a nite 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 2 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 2 ANR and Q is an arbitrary CW complex , then f is a strong shape equivalence.