nLab
localic groupoid

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.

References

• The Elephant, chapter C5.
• Andre Joyal, M. Tierney, An extension of the Galois theory of Grothendieck Mem. Amer. Math. Soc. no 309 (1984)