Grothendieck topos

Remarks

Example: for a topological space $X$, the category $\mathrm{Sh}(X)$ of sheaves on the site of open sets of $X$ is a Grothendieck topos. Example: for a small category $\mathcal{C}$, the category of presheaves on $\mathcal{C}$ is a Grothendieck topos, for the topology in which only maximal sieves cover. Non-example: the category of finite sets is an elementary topos but not a Grothendieck topos, since it lacks infinite coproducts, which every Grothendieck topos has. Every Grothendieck topos is an elementary topos and is complete and cocomplete. Giraud's theorem gives an intrinsic characterization.