  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 $Y$.

Logic