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

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