Michael Shulman Grothendieck 2-topos

Definition

A Grothendieck 2-topos is a 2-category that is equivalent to the 2-category of 2-sheaves (aka stacks) on some 2-site. More generally, a Grothendieck $n$-topos is an $n$-category that is equivalent to the $n$-category of $n$-sheaves on some $n$-site (for $n\le 2$); see truncated 2-topos.

The 2-Giraud theorem due to StreetCBS characterizes Grothendieck $n$-toposes as the infinitary n-pretoposes having a small eso-generating set of objects.

Cores and opposites

The most familiar Grothendieck 2-toposes are those of 2-sheaves on a 1-site. Even 2-presheaves on a 2-category that is not a 1-category can exhibit unexpected behavior; for instance they do not in general have cores or opposites. (2-presheaves on a (2,1)-category do have cores and opposites, though.)

The 2-Giraud theorem implies that if $K$ is a Grothendieck 2-topos, so is $K^{co}$. In the case of 2-presheaves, we can identify this as:

$$[C,Cat]^{co} \simeq [C^{co},Cat^{co}] \simeq [C^{co},Cat]$$

since we have $(-)^{op} : Cat^{co} \simeq Cat$. In other words, the “2-topos of presheaves” functor from 2-categories to 2-toposes preserves the 2-cell-dual involution. Another way to say this is that only for 2-sheaves on a (2,1)-category is “oppositization” an endofunctor; in other cases “passage to opposites” means passage to a different 2-topos.