Geometric morphisms

If X and Y are topoi, a geometric morphism u: X→Y is a pair of adjoint functors (u^∗,u_∗) such that u^∗ preserves finite limits. Note that u^∗ automatically preserves colimits by virtue of having a right adjoint.

By Freyd's adjoint functor theorem, to give a geometric morphism X → Y is to give a functor u^∗: Y → X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales.

If X and Y are topological spaces and u is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi.

Points of topoi

A point of a topos X is a geometric morphism from the topos of sets to X.

If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk F_x has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map x: 1 → X.

Essential geometric morphisms

A geometric morphism (u^∗,u_∗) is essential if u^∗ has a further left adjoint u_!, or equivalently (by the adjoint functor theorem) if u^∗ preserves not only finite but all small limits.

Ringed topoi

A ringed topos is a pair (X,R), where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through for ringed topoi. The category of R-module objects in X is an abelian category with enough injectives. A more useful abelian category is the subcategory of quasi-coherent R-modules: these are R-modules that admit a presentation.

Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks.

Homotopy theory of topoi

Michael Artin and Barry Mazur associated to any topos a pro-simplicial set. Using this inverse system of simplicial sets one may sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory.

The pro-simplicial set associated to the etale topos of a scheme is a pro-finite simplicial set. Its study is called étale homotopy theory.