A topos view of the type-2 fuzzy truth value algebra

TL;DR

This paper explores the topos-theoretic structure of type-2 fuzzy truth value algebra, showing its isomorphism to complex algebras of subobjects in the topos of étalé spaces, generalizing the connection between fuzzy logic and topos theory.

Contribution

It establishes a topos-theoretic framework for type-2 fuzzy truth values, linking them to subobjects of constant relational étalé spaces, and generalizes the construction to arbitrary topological spaces.

Findings

Type-2 fuzzy truth value algebra is isomorphic to a complex algebra of subobjects.

The framework generalizes to any topological space with a lattice of open sets.

Provides a categorical perspective on fuzzy logic structures.

Abstract

It is known that fuzzy set theory can be viewed as taking place within a topos. There are several equivalent ways to construct this topos, one is as the topos of \'{e}tal\'{e} spaces over the topological space $Y=[0,1)$ with lower topology. In this topos, the fuzzy subsets of a set $X$ are the subobjects of the constant \'{e}tal\'{e} $X\times Y$ where $X$ has the discrete topology. Here we show that the type-2 fuzzy truth value algebra is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by the type-1 fuzzy truth value algebra $\mathfrak{I}=([0,1],\wedge,\vee,\neg,0,1)$. More generally, we show that if $L$ is the lattice of open sets of a topological space $Y$ and $\mathfrak{X}$ is a relational structure, then the convolution algebra $L^\mathfrak{X}$ is isomorphic to the complex algebra formed from the subobjects of the constant relational \'{e}tal\'{e} given by $\mathfrak{X}$ in the topos of \'{e}tal\'{e} spaces over $Y$.