Non-singular maps in toposes with a local state classifier

TL;DR

This paper explores non-singular maps in toposes with local state classifiers, showing their structural properties and potential applications in geometric contexts, especially in toposes of spaces.

Contribution

It introduces the notion of non-singular maps in toposes with local state classifiers and demonstrates their properties and relevance to geometric applications.

Findings

The domain of non-singular maps forms a topos.

In certain toposes, the local state classifier can be characterized as a limit.

The inclusion of non-singular maps corresponds to a hyperconnected geometric morphism.

Abstract

Recent progress on the question of the size of the class of connected and hyperconnected geometric morphisms from a given topos has led to the definition of {\em local state classifier}. We discuss a historical precedent which leads to the notion of {\em non-singular map} and we show that, for a topos ${\cal E}$ with a local state classifier, and each object $X$ therein, the domain of the full subcategory of ${{\cal E}/X}$ consisting of non-singular maps over $X$ is a topos, and that the inclusion is the inverse image functor of a hyperconnected geometric morphism. The prospective geometric applications direct our attention to local state classifiers in toposes `of spaces'. We show that, at least in the pre-cohesive topos of reflexive graphs, the local state classifier, which is a colimit by definition, may be characterized as a limit; more specifically, as a variant of a subobject classifier.