If $K$ and $L$ are Grothendieck n-toposes, an **$n$-geometric morphism** $f:K\to L$ consists of an adjoint pair $f^*\dashv f_*$, where $f_*:K\to L$ is the *direct image* and $f^*:L\to K$ 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 $f^*:L\to K$ 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 $Gr n Top$ of Grothendieck $n$-toposes and $n$-geometric morphisms.

Created on February 17, 2009 at 06:39:53. See the history of this page for a list of all contributions to it.