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 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.