# 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 $f:\mathcal{F}\to\mathcal{E}$ 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)

## References

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