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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2505.07131 |
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| _version_ | 1866915283133792256 |
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| author | Menni, Matí as |
| author_facet | Menni, Matí as |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-singular maps in toposes with a local state classifier Menni, Matí as Category Theory 18F10, 18B25, 03G30, 14F06 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. |
| title | Non-singular maps in toposes with a local state classifier |
| topic | Category Theory 18F10, 18B25, 03G30, 14F06 |
| url | https://arxiv.org/abs/2505.07131 |