Are there any non-trivial finite elementary topoi? Title basically says it all: are there any finite topoi (that is, finite set of objects, finite hom-objects) other than $\textbf{1}$ (the terminal category) and $\textbf{2}$ (the category $\ast \rightarrow \ast$)? If so, have they been classified?
 A: Let $C$ be a category with finite products. If $C$ has objects $c, d$ such that there are at least $2$ morphisms $c \to d$, then there are at least $2^k$ morphisms $c \to d^k$, and it follows that $C$ has infinitely many morphisms. We conclude that if $C$ is a finite category with finite products then there is at most one morphism between any pair of objects, and hence $C$ is a preorder.
Now let $C$ be a preorder with a terminal object and a subobject classifier $\Omega$. Since there is at most one morphism $1 \to \Omega$ from the terminal object to $\Omega$, there is at most one equivalence class of monomorphisms to the terminal object. But in a preorder, the poset of equivalence classes of monomorphisms to the terminal object is precisely the poset of isomorphism classes of objects, and it follows that $C$ has at most one isomorphism class of object. (Edit, 6/24/21: Also, $C$ cannot be empty because it has a terminal object, so it has at least one isomorphism class of object.)
In particular, $2$ is not a finite topos. The only finite topos, up to equivalence, is $1$.
