Contents

category theory

topos theory

# Contents

## Idea

The (right) Ore condition is a simple condition on the morphisms in a category $\mathcal{C}$ in order ensure that sieves generated by singletons $\{ f\}$ behave well under pullback. It can be viewed as weaker form of the existence of pullbacks in $\mathcal{C}$.

## Definition

###### Definition

A category $\mathcal{C}$ is said to satisfy the (right) Ore condition if for any diagram

$\array{ & & A \\ & & \downarrow\\ B & \to & C }$

there is an object $D$ and arrows $D \to A, B$ such that the following diagram commutes:

$\array{ D & \to & A \\ \downarrow & & \downarrow\\ B & \to & C }$

## Properties

• A category $\mathcal{C}$ obviously satisfies the Ore condition when it has pullbacks.

• When $S$ is a sieve generated by a singleton $\{ f\}$ then the pullback $h^\ast (S)$ is nonempty provided $\mathcal{C}$ satisfies the Ore condition. More generally, a category $\mathcal{C}$ satisfies the Ore condition precisely when the collection of nonempty sieves forms a Grothendieck topology on $\mathcal{C}$ (cf. atomic site).

• A presheaf topos $Set^{\mathcal{C}^{op}}$ is a De Morgan topos precisely if $\mathcal{C}$ satisfies the Ore condition (cf. De Morgan topos).

## Remark

A category $\mathcal{C}$ satisfies the amalgamation property precisely if ${\mathcal{C}^{op}}$ satifies the Ore condition. Since the former is an important property in model theory, the De Morgan property is via the Ore condition dually bound to play a similar role.

## Reference

Last revised on August 9, 2016 at 15:45:20. See the history of this page for a list of all contributions to it.