nLab
locally ringed topos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Higher geometry

Contents

Idea

A locally ringed topos is a locally algebra-ed topos for the theory of local rings.

Definition

Definition

A ringed topos (X,𝒪 X)(X, \mathcal{O}_X) with enough points (such as the sheaf topos over a topological space) is a locally ringed topos if all stalks 𝒪 X(x)\mathcal{O}_X(x) are local rings.

This is a special case of the following equivalent definitions:

Definition

A locally ringed topos is a topos equipped with a commutative ring object (see ringed topos) that in addition satisfies the axioms

  • (0=1)false(0 = 1) \vdash false
  • x+y=1 z(xz=1) z(yz=1)x + y = 1 \vdash \exists_z (x z = 1) \vee \exists_z (y z = 1)

(note these are axioms for a geometric theory, interpreted according to Kripke-Joyal semantics in a topos).

Definition

A ringed topos (X,𝒪 X)(X, \mathcal{O}_X) is a locally algebra-ed topos for the theory of local rings:

Properties

Proposition

Definition 1 is indeed a special case of def. 3.

This is for instance in ([Johnstone]) and in (Lurie, remark 2.5.11)

References

Section VIII.6 of

Section abc of

Section 2.5 of

Section 14.33 of

Revised on January 23, 2012 14:28:54 by Todd Trimble (74.88.146.52)