nLab
open geometric morphism

Idea

The notion of open geometric morphism is the notion of open map for geometric morphisms between toposes.

Definition

A geometric morphism

$$f:=(f^{*}⊣f_{*}):ℰ\to ℱ$$f := (f^* \dashv f_*) : \mathcal{E} \to \mathcal{F}

is called open if the following equivalent conditions holds

• the localic reflection of $f$ is an open map of locales;

• the inverse image $f^{*}$ preserves first order logic, hence is a Heyting functor.

References

• Peter Johnstone, Open maps of toposes, Manuscripta Mathematica, Volume 31, Numbers 1-3

• André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309 (1984).