Toposes of connectivity spaces. Morita equivalences with topological spaces and partially ordered sets in the finite case

TL;DR

This paper explores the relationship between connectivity spaces and topological spaces by showing that in the finite case, they are Morita-equivalent, thus establishing a deep categorical connection.

Contribution

It proves that every finite connectivity space is Morita-equivalent to a finite topological space, bridging the two concepts in the finite setting.

Findings

Finite connectivity spaces are Morita-equivalent to finite topological spaces

The topos of a connectivity space can be described as a sheaf topos

Establishes a categorical equivalence in the finite case

Abstract

This paper has two parts. First, we recall and detail the definition of the Grothendieck topos of a connectivity space, that is the topos of sheaves on such a space. In the second part, we prove that every finite connectivity space is Morita-equivalent to a finite topological space, and vice versa (we have given this proof in several, but we haven't yet shared this in writing).