nLab Ore condition



Category theory

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




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



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

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

there is an object DD and arrows DA,BD \to A, B such that the following diagram commutes:

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


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

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

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


A category 𝒞\mathcal{C} satisfies the amalgamation property precisely if 𝒞 op{\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.


Last revised on August 9, 2016 at 19:45:20. See the history of this page for a list of all contributions to it.