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 nn-topos is an nn-category that is equivalent to the nn-category of nn-sheaves on some nn-site (for n2n\le 2); see truncated 2-topos.

The 2-Giraud theorem due to StreetCBS characterizes Grothendieck nn-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 KK is a Grothendieck 2-topos, so is K coK^{co}. In the case of 2-presheaves, we can identify this as:

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

since we have () op:Cat coCat(-)^{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.

Revised on February 23, 2009 02:27:12 by Mike Shulman (75.3.140.11)