We present and characterize the classes of Grothendieck toposes having enough supercompact objects or enough compact objects. In the process, we examine the subcategories of supercompact objects and compact objects within such toposes and classes of geometric morphism which interact well with these objects. We also present canonical classes of sites generating such toposes.
Keywords: Grothendieck topos, supercompact object, morphism of sites, presheaf topos, regular topos, coherent topos
2020 MSC: 18F10, 18B25
Theory and Applications of Categories, Vol. 37, 2021, No. 32, pp 1017-1079.