Equivalent Grothendieck toposes

Question

Let $(\mathbb{C}, J)$ and $(\mathbb{D}, K)$ be small sites whose associated Grothendieck toposes are equivalent: $\mathsf{Sh}(\mathbb{C}, J) \simeq \mathsf{Sh}(\mathbb{D}, K)$. I am aware that a given Grothendieck topos can have many "inequivalent" sites of definition, but is there any relation between the sites $(\mathbb{C}, J)$ and $(\mathbb{D}, K)$ that can be deduced from the associated Grothendieck toposes being equivalent?

Answer 1

In general that is a very difficult question, and as far as I know there isn't yet a site characterization for the resulting topoi to be equivalent.

However, restricting to certain classes of topoi and sites there do exist characterizations. Just to give a flavour of this:

1. If $X$, $Y$ are sober spaces, they can be recovered from their topoi $Sh(X)$ and $Sh(Y)$ and so $Sh(X) \simeq Sh(Y)$ iff $X \cong Y$. More generally, if $L$, $K$ are distributive lattices then $Sh(L,J_{can}) \simeq Sh(K, J_{can})$ iff $K$ and $L$ have isomorphic co-completions.
2. If the topoi are presheaf topoi, then $Sets^{\mathcal{C}^{op}} \simeq Sets^{\mathcal{D}^{op}}$ iff $Ind(\mathcal{C}) \cong Ind(\mathcal{D})$ (the Ind-completion).
3. Theorem 3 from this paper by Moerdijk gives sufficient conditions for the topoi of sheaves on two topological groupoids to be equivalent.
4. Corollary 6.6 in this paper characterizes equivalence between presheaves on monoids algebraically.