nLab locally ringed topos

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:

• a topos $X$

• equipped with a geometric morphism

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

  into the Zariski topos, the classifying topos for the theory of local rings.

Properties

Proposition

Definition is indeed a special case of def. .

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

Related concepts

• Cole's theory of spectrum

• ringed space, locally ringed space

• ringed site, locally ringed site

• ringed topos, locally ringed topos

• locally algebra-ed topos

• locally algebra-ed (∞,1)-topos

References

Section VIII.6 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

Section abc of

• Peter Johnstone, Sketches of an Elephant

Section 2.5 of

• Jacob Lurie, Structured Spaces

Section 14.33 of

• Aise Johan de Jong, The Stacks Project (pdf) (project website)