We give characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (respectively connected and locally connected, atomic, compact, presheaf) topos, and exploit the existence of multiple sites of definition for a given topos to establish various results on quotients of theories of presheaf type.
Keywords: Grothendieck topos, site characterizations, geometric logic, presheaf topos
2000 MSC: 03G30, 18C10, 18B25
Theory and Applications of Categories, Vol. 26, 2012, No. 6, pp 176-193.