Non-singular maps in toposes with a local state classifier

TL;DR

The paper develops a framework connecting local state classifiers to non-singular maps in toposes, showing that for any topos ${\cal E}$ with a local state classifier ${\Xi}$, the domain ${\mathcal E/ X}$ of non-singular maps over ${X}$ is itself a topos, with the inclusion into ${\cal E/ X}$ being the inverse image of a hyperconnected morphism. It introduces shells and probes to study coherent families of monomorphisms and establishes a bijection between shells on a presheaf topos ${\widehat{\mathbf C}}$ and saturated probes on ${\mathbf C}$, providing a concrete mechanism to measure singularities via the object ${\mathit{Eq}}$ and the map ${\sigma_X}$. The paper then defines non-singular maps with respect to a lax cocone, constructs the petit toposes ${\mathcal N_\mu(Y)}$, and proves they are hyperconnected-images, under mild hypotheses, while clarifying limitations (e.g., non-calibration in certain toposes like ${\widehat{\Delta_1}}$). By analyzing reflexive graphs and pre-cohesive toposes, it highlights when local state classifiers yield genuine topos-theoretic petit toposes and when they reflect broader space-vs-generalized-space distinctions, offering potential geometric applications in topos theory. Overall, the work ties historical ideas of singularities to modern topos-theoretic machinery, providing structural tools to bound morphism classes and to isolate 'petit' spaces within larger toposes.

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.

Theorems & Definitions (63)

• Definition 1.1
• Definition 1.2
• Proposition 1.3: Hora2024
• proof
• Definition 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof