nLab localic groupoid



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Higher geometry



A localic groupoid is a internal groupoid in the category of locales. A special case is of localic groups.

Localic groupoids are important, among other reasons, because every Grothendieck topos can be presented as the topos of equivariant sheaves on some localic groupoid. This fact is due to Joyal and Tierney. For more see classifying topos of a localic groupoid.

See also


The Joyal–Tierney theorem appeared in

  • Andre Joyal, M. Tierney, An extension of the Galois theory of Grothendieck Mem. Amer. Math. Soc. no 309 (1984)

An expository account of the Joyal–Tierney theorem:

  • Graham Manuell, Joshua L. Wrigley, The representing localic groupoid for a geometric theory, arXiv:2305.15209v1.

Last revised on May 25, 2023 at 13:39:35. See the history of this page for a list of all contributions to it.