Toposes over which essential implies locally connected

TL;DR

This paper introduces and studies classes of toposes, EILC and CILC, characterized by the local connectedness of geometric morphisms, with examples including sheaf toposes on certain spaces and Boolean toposes.

Contribution

It defines the notions of EILC and CILC toposes, providing examples and establishing their properties and relationships with known classes of toposes.

Findings

Sheaves on Hausdorff or Jacobson spaces are EILC.

Boolean étendues and classifying toposes of compact groups are EILC.

Boolean elementary toposes are CILC.

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.