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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 $f:X\to Y$ is a shape equivalence if and only if the induced function $f^*:[Y,P]\to [X,P]$ is a bijection for all $P\in {\rm ANR}$. A map $f:X\to Y$ of $k$-spaces is a strong shape equivalence if and only if the induced map $f^*:{\rm Map}(Y,P)\to {\rm Map}(X,P)$ is a weak homotopy equivalence for all $P\in {\rm ANR}$. 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 $f:X\to Y$ of topological spaces as a map such that the induced map $f^*:{\rm Map}(Y,P)\to {\rm Map}(X,P)$ is a homotopy equivalence for all $P\in {\rm ANR}$, 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 $f:X\to Y$ of $k$-spaces is a strong shape equivalence if and only if $f\times \mathop{\rm id}_Q:X\times_k Q\to Y\times_k Q$ is a shape equivalence for each CW complex $Q$.

Theorem. Suppose $f:X\to Y$ is a map of topological spaces.

(a) $f$ is a shape equivalence if and only if the induced function $f^\ast:[Y,M]\to [X,M]$ is a bijection for all $M={\rm Map}(Q,P)$, where $P\in {\rm ANR}$ and $Q$ is a finite CW complex.

(b) If $f$ is a strong shape equivalence, then the induced function $f^\ast:[Y,M]\to [X,M]$ is a bijection for all $M={\rm Map}(Q,P)$, where $P\in {\rm ANR}$ and $Q$ is an arbitrary CW complex.

(c) If $X$, $Y$ are $k$-spaces and the induced function $f^\ast:[Y,M]\to [X,M]$ is a bijection for all $M={\rm Map}(Q,P)$, where $P\in {\rm ANR}$ and $Q$ is an arbitrary CW complex, then $f$ is a strong shape equivalence.