Cosheaves and distributions on toposes
type: Item
instance of: scholarly article
title: Cosheaves and distributions on toposes
zbMATH Open document ID: 0863.18002
DOI: 10.1007/BF01181872
published in: Algebra Universalis
publication date: 1997-06-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18D30
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18A35
zbMATH DE Number: 836828
zbMATH Keywords: indexed categories
zbMATH Keywords: topos
zbMATH Keywords: symmetric topos functor
zbMATH Keywords: cosheaves
zbMATH Keywords: Joyal-Tierney representation theorem
zbMATH Keywords: Grothendieck toposes
zbMATH Keywords: localic groupoids
zbMATH Keywords: categories of distributions
author: Marta Bunge
reviewed by: Peter Johnstone
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01181872
OpenAlex ID: W1986048956
description: scientific article; zbMATH DE number 836828