Explicit description of a topos of sheaves on an internal boolean algebra

Question

I have a question on how to calculate a topos of sheaves on an internal site.

Let $F$ be the category of finite sets and functions so that the topos ${\widehat{F}}$ of presheaves on $F$ classifies Boolean algebras. Let $B$ be the generic boolean algebra. It is an internal Boolean algebra in $\widehat{F}$. So we may consider the topos $E$ of sheaves on the internal site ${(B, c)}$ where $c$ is the coherent coverage on $B$.

The topos $E$ is bounded over $\widehat{F}$, which is bounded over sets. So $E$ is a Grothendieck topos. Can you provide an external site for $E$?

Thanks.

Answer 1

The topos of sheaves over a boolean algebra $B$ is the classifying topos of the theory of points of $B$, that is of boolean algebra morphism $B \to \{0,1\}$.

So, this $E$ is the classifying topos of the theory "a boolean algebras B together with a point of B", that is boolean algebras over $\{0,1\}$.

This is an essentially algebraic theory so it is classified by $Fun(C,Set)$ where $C$ is the category of its finitely presented models. Explicitely, $C$ is the category of finite boolean algebras together with a boolean algebra morphism $B \to Set$. So, through Stone duality, this is the opposite of the category of finite pointed sets.

So $E$ is $\widehat{F_\bullet}$ where $F_\bullet$ is the category of finite pointed sets.

The natural geometric morphism $E \to \widehat{F}$ is the one induced by the forgetfull functor $F_\bullet \to F$ (in the sense that its action on points extend this).