Let $\mathcal{E}$ be an elementary topos. I know that the subobject classifier $\Omega$ of the topos is always a Heyting algebra.
I’m interested in versions of the converse — given any (let's say finite) Heyting algebra $L$, can one construct a topos $\mathcal{E}_L$ for which it serves as a subobject classifier? What are the details of this construction, if it can exist?
Best Answer
So I haven't actually checked this but I believe it should be true that if $L$ has small joins, or equivalently is a frame, then it is the (Heyting algebra of points of the) subobject classifier of the topos of sheaves on the locale $\text{Spec } L$. This is even some kind of left adjoint, I think. So this is a pretty natural class of examples.