ABSTRACT
This paper 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. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval.
The system $V_{\ast}(X)$ of Vietoris-Rips complexes for a data set $X$ is both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.
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► References
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Cited by
[1] King-Yin Yan, Lecture Notes in Computer Science 13154, 327 (2022) ISBN:978-3-030-93757-7.
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