nLab
open geometric morphism

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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)

Proposition

A geometric morphism f: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).

References

Revised on August 7, 2015 04:59:52 by Thomas Holder (89.204.138.240)