Cosheaves and distributions on toposes
This paper is a contribution to the study of toposes bounded over a base topos \({\mathcal S}\) (which may be thought of as the topos of sets and functions) as cocomplete \({\mathcal S}\)-indexed categories. The key result is that the forgetful 2-functor from \({\mathcal S}\)-toposes to cocomplete \({\mathcal S}\)-indexed categories satisfying a `small presentation' condition has a left adjoint, the symmetric topos functor. Applying this functor to an \({\mathcal S}\)-topos \({\mathcal X}\), one obtains a topos \(\Sigma {\mathcal X}\) which classifies the theory of `distributions on \({\mathcal X}\)' in the sense of Lawvere; the latter may be identified with cosheaves on any (small) site of definition for \({\mathcal X}\). The author also exploits the Joyal-Tierney representation theorem for Grothendieck toposes in terms of localic groupoids, to give a similar representation for categories of distributions on Grothendieck toposes.
scientific article; zbMATH DE number 836828
scientific article