topos theory

# 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 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 May 31, 2016 03:56:46 by Urs Schreiber (131.220.184.222)