arXiv subjects

Matí as Menni

Publications and source records attributed to Matí as Menni.

4 recordsLinked to original sources

arXiv2025

Non-singular maps in toposes with a local state classifier

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.

math.CT↗

arXiv2024

The étendue of a combinatorial space and its dimension

To each simplicial set $X$ we naturally assign an étendue ${É X}$ whose internal logic captures information about the geometry of $X$. In particular, we show that, for 'non-singular' objects $X$ and $Y$, the étendues ${É X}$ and ${É Y}$ are equivalent if, and only if, $X$ and $Y$ have the same dimension. Many of the results apply to presheaf toposes over 'well-founded' sites.

math.CT↗

arXiv2024

The least subtopos containing the discrete skeleton of $Ω$

Let $p: \mathcal{E} \to \mathcal{S}$ be a pre-cohesive geometric morphism. We show that the least subtopos of $\mathcal{E}$ containing both the subcategories $p^*: \mathcal{S} \to \mathcal{E}$ and $p^!: \mathcal{S} \to \mathcal{E}$ exists, and that it coincides with the least subtopos containing $p^*2$, where 2 denotes the subobject classifier of $\mathcal{S}$.

math.CT↗

arXiv2023

The successive dimension, without elegance

Experience shows that the poset of levels (or dimensions) of the topos of presheaves on some elegant Reedy categories may be equipped with a monotone increasing `successor' function which, as the case of simplicial sets shows, is different from Lawvere's Aufhebung in general. We prove that a similar result holds for the topos of presheaves on a small category with split-epi/mono factorizations; a typical feature of categories that are Reedy elegant, or skeletal, or graphic (von Neumann-)regular, but more general. In fact, we show that the more general `successor' may be described as a function on the poset of full subcategories of the site that are closed under subobjects.

math.CT↗