# Fuzzy sets and presheaves

## TL;DR

This paper introduces a presheaf-based framework for fuzzy sets, demonstrating their categorical properties and linking to classical and data-driven structures like Vietoris-Rips complexes.

## Contribution

It develops a presheaf-theoretic approach to fuzzy sets, showing their completeness, cocompleteness, and explicit limit and colimit descriptions, connecting to classical and data analysis contexts.

## Key findings

- Fuzzy sets form a complete and cocomplete category.
- Explicit descriptions of fuzzy sets as limits and colimits.
- Vietoris-Rips complexes can be viewed as fuzzy sheaves.

## Abstract

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.