Michael Shulman
2-geometric morphism

If KK and LL are Grothendieck n-toposes, an nn-geometric morphism f:KLf:K\to L consists of an adjoint pair f *f *f^*\dashv f_*, where f *:KLf_*:K\to L is the direct image and f *:LKf^*:L\to K is the inverse image, such that f *f^* preserves finite limits (in the nn-categorical sense).

When n=1n=1 this is, of course, the usual definition. When n=(0,1)n=(0,1) this reduces to a continuous map between locales.

In general, we expect that Grothendieck nn-toposes satisfy an nn-categorical version of the adjoint functor theorem, so that any functor f *:LKf^*:L\to K that preserves finite limits and small colimits is the inverse image of some nn-geometric morphism.

We define transformations and modifications between nn-geometric morphisms to be transformations and modifications between their inverse image functors. We then have an (n+1)(n+1)-category GrnTopGr n Top of Grothendieck nn-toposes and nn-geometric morphisms.

Created on February 17, 2009 06:39:53 by Mike Shulman (75.3.140.11)