J. F. Jardine - Fuzzy sets and presheaves

compositionality:13506 - Compositionality, December 20, 2019, Volume 1 (2019) - https://doi.org/10.32408/compositionality-1-3
Fuzzy sets and presheavesArticle

Authors: J. F. Jardine 1

  • 1 Department of Mathematics, University of Western Ontario, London, Ontario, Canada

This note presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. A presheaf-theoretic method is used to show that the category of fuzzy sets is complete and co-complete, and to present explicit descriptions of classical fuzzy sets that arise as limits and colimits. The Boolean localization construction for sheaves and presheaves on a locale L specializes to a theory of stalks if L approximates the structure of a closed interval in the real line. The system V(X) of Vietoris-Rips complexes for a data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed in this paper, in stages.


Volume: Volume 1 (2019)
Published on: December 20, 2019
Imported on: May 2, 2024
Keywords: Mathematics - Category Theory
Funding:
    Source : OpenAIRE Graph
  • Funder: Natural Sciences and Engineering Research Council of Canada

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