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

• $\left(0=1\right)⊢\mathrm{false}$
• $x+y=1⊢{\exists }_{z}\left(xz=1\right)\vee {\exists }_{z}\left(yz=1\right)$

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

###### Definition

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

• a topos $X$

• equipped with a geometric morphism

$X\stackrel{\stackrel{{𝒪}_{X}}{←}}{\underset{}{\to }}\mathrm{Sh}\left({\mathrm{CRing}}_{\mathrm{fp}}^{\mathrm{op}},Z\right)$X \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{}{\to}} Sh(CRing^{op}_{fp}, Z )

into the Zariski topos, the classifying 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)