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Fine shape of metrizable spaces as a left fraction localization

Vladislav Zemlyanoy

Abstract

The strong shape category of compact metrizable spaces (compacta) is very well-studied; extending it to noncompact spaces, however, introduces computational complexity that makes it hard to work with. The fine shape category, as defined by Melikhov, seems to hold promise in terms of both applicability and simplicity: it is a different extension of compact strong shape to a generalized homotopy theory of metrizable spaces that is compatible with both Čech cohomology and Steenrod-Sitnikov homology, and its definition lends itself to straightforward proofs. Further research seems to be in order. One goal to have in mind is to show the fine shape category to be a homotopy category in Quillen's sense, which implies representation as a localization. But the strong shape of compacta was shown to be a left fraction localization in several ways; we extend the representation given by Cathey to fine shape, introducing the notion of FDR-embeddings to extend Cathey's SSDR-maps. In the process, we also introduce what we call the mapping cylinder of an approaching map; such a construction has been defined by Ferry and elaborated on by Mrozik in the compact case, yet it seems the direct extension on noncompact spaces is not possible. Thus we resort to a somewhat different definition.

Fine shape of metrizable spaces as a left fraction localization

Abstract

The strong shape category of compact metrizable spaces (compacta) is very well-studied; extending it to noncompact spaces, however, introduces computational complexity that makes it hard to work with. The fine shape category, as defined by Melikhov, seems to hold promise in terms of both applicability and simplicity: it is a different extension of compact strong shape to a generalized homotopy theory of metrizable spaces that is compatible with both Čech cohomology and Steenrod-Sitnikov homology, and its definition lends itself to straightforward proofs. Further research seems to be in order. One goal to have in mind is to show the fine shape category to be a homotopy category in Quillen's sense, which implies representation as a localization. But the strong shape of compacta was shown to be a left fraction localization in several ways; we extend the representation given by Cathey to fine shape, introducing the notion of FDR-embeddings to extend Cathey's SSDR-maps. In the process, we also introduce what we call the mapping cylinder of an approaching map; such a construction has been defined by Ferry and elaborated on by Mrozik in the compact case, yet it seems the direct extension on noncompact spaces is not possible. Thus we resort to a somewhat different definition.
Paper Structure (13 sections, 38 theorems, 18 equations)

This paper contains 13 sections, 38 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces
  4. Metrizable joins and metrizable mapping cylinders
  5. Approaching maps
  6. Absolute retracts
  7. Fine shape
  8. Left fraction localization
  9. FDR-embeddings
  10. The left fraction localization
  11. The S functor
  12. Approaching and fine shape cylinders
  13. The T functor

Key Result

Theorem 1.3

(Corollary fSh_represent) For any fine shape class $[\phi] \in [X,Y]_{fSh}$, there exist a space $Z$ along with maps (which can be chosen to be closed embeddings) $u\colon X \to Z$ and $i\colon Y \to Z$ such that $i$ is a fine shape equivalence and $[i]_{fSh}^{-1}\circ[u]_{fSh} = [\phi]$.

Theorems & Definitions (97)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 87 more