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 $\mathfrak{I}=([0,1],\wedge ,\vee ,\neg ,0,1)$. 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 étalé given by $\mathfrak{X}$ in the topos of étalé spaces over $Y$.