Contents

topos theory

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

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

## Definition

###### Definition

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

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

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

###### Definition

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

## Properties

###### Proposition

Definition is indeed a special case of def. .

This is for instance in Johnstone (2002) and in Lurie (2009), remark 2.5.11

## References

Original references