nLab
Ore condition

Contents

Context

Category theory

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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}.

Definition

Definition

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 }

Properties

  • 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).

Remark

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.

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.