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Function spaces and shape theories

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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 ∈ 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 ∈ 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. A map f:X → Y of k-spaces is a strong shape equivalence if and only if $f × id_Q: X ×_k Q → Y ×_k Q$ 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 ∈ 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.
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  • Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A.
  • Institute of Mathematics, University of Warsaw, Banacha 2 02-097 Warszawa, Poland
  • Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-950 Warszawa, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-2-2
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