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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.07131 |
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Table of Contents:
- 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.