Function spaces and shape theories
by
Jerzy Dydak
University of Tennessee
Coauthors: Slawomir Nowak

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 iff the induced function f^*:[Y, P] --> [X, P] is a bijection for all P in ANR. A map f:X --> Y of k-spaces is a strong shape equivalence iff the induced map f^*:Map(Y, P) --> Map(X, P) is a weak homotopy equivalence for all P in 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 morphisms form a set). Due to pecularities of function spaces, extending the concept of a 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 in ANR, and one can extend it to morphisms of pro-Top. However, the authors do not know if the corresponding localization exists.

Here is an application of our methods: A map f:X --> Y of k-spaces is a strong shape equivalence iff f×id_Q:X×_k Q --> Y×_k Q is a shape equivalence for each CW complex Q.

Date received: June 1, 1999