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

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

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

Definition

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

• a topos $X$

• equipped with a geometric morphism

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

Related concepts

• 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)