Giraud's theorem

Remarks

Coproducts are disjoint when the pullback of two different coproduct inclusions is an initial object. A construction is stable under pullback when pulling it back along any morphism gives a construction of the same kind. One direction is a check in the category of sheaves, where limits are computed objectwise and colimits are sheafifications of objectwise colimits. For the converse, take a small full subcategory $\mathcal{C}$ of generators. Give it the topology $J$ whose covering sieves are the families that are jointly epimorphic in $\mathcal{E}$. Then one shows that $E \mapsto \Hom(-, E)|_{\mathcal{C}}$ is an equivalence of $\mathcal{E}$ with $\mathrm{Sh}(\mathcal{C}, J)$. Example: $\mathbf{Set}$ has all six properties, with the one-point set as generator. Example: the category of finite sets satisfies conditions 1, 3, 4, 5 and 6. It fails condition 2, since it has no infinite coproducts, and it is not a Grothendieck topos.

Going deeper

For a student

The theorem is used to show that categories built in other ways are toposes, for example categories of sheaves glued from local pieces or categories of equivariant sheaves. It also shows that a Grothendieck topos determines a site only up to choice: any small generating full subcategory with its canonical topology is a site for it. Compared with the elementary definition of a topos, the extra conditions are the small coproducts and the small set of generators.