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.
