Toposes of connectivity spaces. Morita equivalences with topological spaces and partially ordered sets 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).