Contents

Idea

A geometric morphism is locally connected if it behaves as though its fibers are locally connected spaces. In particular, a Grothendieck topos $E$ is locally connected iff the unique morphism to $\mathrm{Set}$ (the terminal Grothendieck topos, i.e. the point in the category of topoi) is locally connected.

Definition

A geometric morphism $F\underset{f_{*}}{\overset{f^{*}}{\leftrightarrows }}E$ is locally connected if it satisfies the following equivalent conditions:

1. It is essential, i.e. $f^{*}$ has a left adjoint $f_{!}$, and moreover $f_{!}$ can be made into an $E$-indexed functor?.

2. For every $A\in E$, the functor $f^{*}:E/A\to F/f^{*}A$ is cartesian closed?.

3. For any morphism $h:A\to B$ in $E$, the canonically defined transformation $f^{*}\circ {\Pi }_h\to {\Pi }_{f^{*}h}\circ f^{*}$ is an isomorphism.

Properties

If $f$ is locally connected, then it makes sense to think of the left adjoint $f_{!}$ as assigning to an object of $F$ its “set of connected components” in $E$. In particular, if $f$ is locally connected, then it is moreover connected if and only if $f_{!}$ preserves the terminal object. However, not every connected geometric morphism is locally connected.

Examples

If the terminal global section geometric morphism is locally connected, one speaks of a locally connected topos.

References

• Section C3.3 of the Elephant