Contents

topos theory

Contents

Idea

In point set topology, an open map is a continuous map that sends open sets to open sets. The notion of an open geometric morphism is a generalization of this notion from topology to topos theory.

From a logical perspective, a geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff it preserves the interpretation of first order logic. This contrasts with general geometric morphisms which are only bound to preserve geometric logic.

Definition

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

is called open if the following equivalent conditions hold

Properties

Proposition

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff the canonical map $\lambda:\Omega_\mathcal{E}\to f_\ast(\Omega_\mathcal{F})$ of poset objects in $\mathcal{E}$ has an internal left adjoint $\mu :f_\ast(\Omega_\mathcal{F})\to\Omega_\mathcal{E}$.

(cf. Mac Lane-Moerdijk (1994), p.502)

Proposition

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff the pullback of any bounded geometric morphism with codomain $\mathcal{E}$ is skeletal iff the pullback of any localic geometric morphism with codomain $\mathcal{E}$ is skeletal.

This result appears as corollary 4.9 in Johnstone (2006).