nLab Ore condition in a category

This page seems to follow Johnstone 2002 in considering the Ore condition with respect to the full class of all morphisms. More generally see at Ore set.

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Idea

In category theory, the (right) Ore condition is a simple condition on the morphisms in a category $\mathcal{C}$ which 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 cospan diagram

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

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

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

Properties

Proposition

When $S$ is a sieve generated by a singleton $\{ f\}$ then the pullback sieve $h^\ast (S)$ 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).

Examples

Example

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

Example

A presheaf topos $Set^{\mathcal{C}^{op}}$ is a De Morgan topos (see there for more) precisely if its site $\mathcal{C}$ satisfies the Ore condition.

Remark

A category $\mathcal{C}$ satisfies the amalgamation property precisely if its opposite category ${\mathcal{C}^{op}}$ 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.

Related entries

• atomic site

• De Morgan topos

• amalgamation property

Reference

• Peter Johnstone, pp. 79, 1001 in: Sketches of an Elephant vols. I,II, Oxford (2002)