The representing localic groupoid for a geometric theory

Graham Manuell and Joshua L. Wrigley

We give an expository, and hopefully approachable, account of the Joyal-Tierney result that every topos can be represented as a topos of sheaves on a localic groupoid. We give an explicit presentation of a representing localic groupoid for the classifying topos of a given geometric theory and discuss links with the topological groupoids of Forssell.

Keywords: Grothendieck topos, classifying topos, frame, semantic groupoid

2020 MSC: 18F10, 22A22, 03G30, 06D22, 18B25

Expositions in Theory and Applications of Categories, 2024, No. 2, pp 1-41.

Published 2024-04-26.

http://www.tac.mta.ca/tac/expositions/articles/2/te2.pdf

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