2-geometric morphism

If $K$ and $L$ are Grothendieck n-toposes, an **$n$-geometric morphism** $f:K\to L$ consists of an adjoint pair ${f}^{*}\u22a3{f}_{*}$, 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 $\mathrm{Gr}n\mathrm{Top}$ of Grothendieck $n$-toposes and $n$-geometric morphisms.

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