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

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 u0027{e}talu0027{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 u0027{e}talu0027{e} $Xtimes Y$ 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 u0027{e}talu0027{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 u0027{e}talu0027{e} given by $mathfrak{X}$ in the topos of u0027{e}talu0027{e} spaces over $Y$.