Michael Shulman 2-geometric morphism

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.