A Topos View of the Type-2 Fuzzy Truth Value Algebra

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 étalé 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 étalé $X\times Y$ 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 étalé given by the type-1 fuzzy truth value algebra $\mathfrak {I}=([0,1],\wedge ,\vee ,\lnot ,0,1)$. 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 $L^\mathfrak {X}$ Harding et al. (Algebra Universalis 17:33, 2018 [5]) is isomorphic to the complex algebra formed from the subobjects of the constant relational étalé given by $\mathfrak {X}$ in the topos of étalé spaces over Y.

Dedicated to Elbert Walker