Toposes over which essential implies locally connected
Jens Hemelaer

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 such that every essential geometric morphism with codomain is locally connected. We then show that the topos of sheaves on a topological space is EILC if is Hausdorff (or more generally, if 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 such that any geometric morphism is locally connected, as soon as is cartesian closed. We give some examples of topological spaces and small categories such that resp. are CILC. Finally, we show that any Boolean elementary topos is CILC.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topology and Set Theory · Rings, Modules, and Algebras
