nLab locally ringed topos

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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 is indeed a special case of def. .

This is for instance in Johnstone (2002) and in Lurie (2009), remark 2.5.11

References

Original references

See also:

Last revised on April 16, 2023 at 08:55:53. See the history of this page for a list of all contributions to it.