We sometimes want to know when two Abelian categories $\mathscr{A}$ and $\mathscr{B}$ are "similar", and the usual notion is the equivalence of categories. This may be too demanding and we sometimes ask for a (triangulated) equivalence between $D(\mathscr{A})$ and $D(\mathscr{B})$. This seems to be used to understand representations of groups/algebras/quivers. My questions is about the same problem, but applied to topological spaces.
For any topological space $X$, one may create a topos $\operatorname{Sh}(X)$. As is well-known, $X$ may be recovered from its category of sheaves if it is sober. Usually, we are interested only in the homological informations of a topological space, and so it seems natural to study $D(X) := D(\operatorname{Ab}(X))$, so my question is the following: if two sober spaces $X$ and $Y$ are derived equivalent (i.e. $D(X) \cong D(Y)$), what is the relation between $X$ and $Y$?