Toposes over which essential implies locally connected

Abstract: We introduce the notion of an EILC topos: a topos $\mathcal{E}$ such that every essential geometric morphism with codomain $\mathcal{E}$ is locally connected. We then show that the topos of sheaves on a topological space $X$ is EILC if $X$ is Hausdorff (or more generally, if $X$ is Jacobson). Further examples of Grothendieck toposes that are EILC are Boolean \'etendues and classifying toposes of compact groups. Next, we introduce the weaker notion of CILC topos: a topos $\mathcal{E}$ such that any geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is locally connected, as soon as $f^*$ is cartesian closed. We give some examples of topological spaces $X$ and small categories $\mathcal{C}$ such that $\mathbf{Sh}(X)$ resp. $\mathbf{PSh}(\mathcal{C})$ are CILC. Finally, we show that any Boolean elementary topos is CILC.