nLab
open geometric morphism

Context

Topos Theory

Could not include topos theory - contents

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: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

A geometric morphism

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

is called open if the following equivalent conditions hold

Examples

Properties

Proposition

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

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

References

Revised on August 2, 2015 09:13:50 by Thomas Holder (89.204.138.58)