# Gradual and fuzzy subsets

## TL;DR

This paper explores the categorical structure of fuzzy sets and groups using strong alpha-levels, establishing a functorial correspondence that enhances the theoretical framework of fuzzy set theory.

## Contribution

It identifies fuzzy sets and groups with subcategories of functorial categories, providing a new categorical perspective on fuzzy theory.

## Key findings

- Establishes a one-to-one correspondence using strong alpha-levels.
- Categorifies fuzzy sets and groups as subcategories of functorial categories.
- Enhances the theoretical understanding of fuzzy structures through category theory.

## Abstract

In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong $\alpha$--levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively fuzzy groups, with subcategories of the functorial categories $\mathcal{S}\textit{et}^{(0,1]}$, resp. $\mathcal{G}\textit{r}^{(0,1]}$.