Grothendieck topos

Summary

A Grothendieck topos is a category equivalent to the category of sheaves of sets on a small site. Grothendieck introduced these categories as generalized spaces. A topological space is studied through its category of sheaves, and many sites that are not spaces still give such categories. Every Grothendieck topos is an elementary topos, and Giraud's theorem characterizes Grothendieck toposes by intrinsic properties. Different sites can give equivalent toposes, so the topos is the invariant object.