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locally ringed topos

Contents

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)(X, \mathcal{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)\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)false(0 = 1) \vdash false
  • x+y=1 z(xz=1) z(yz=1)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,𝒪 X)(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 January 23, 2012 14:28:54 by Todd Trimble (74.88.146.52)