Michael Shulman
2-geometric morphism

If K and L are Grothendieck n-toposes, an n-geometric morphism f:KL consists of an adjoint pair f *f *, where f *:KL is the direct image and f *:LK is the inverse image, such that f * preserves finite limits (in the n-categorical sense).

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

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

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

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