nLab locally ringed topos

Contents

Idea

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:

A locally ringed topos is a topos equipped with a commutative ring object (see ringed topos) that in addition satisfies the axioms

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

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.

Definition is indeed a special case of def. .

Original references

Alexander Grothendieck, Jean-Louis Verdier, Topos annelés, localisation dans les topos annelés, Section 11 in Exposé iv in: M. Artin et al., Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics 269, Springer (1972) [doi:10.1007/BFb0081551, pdf]
Monique Hakim, Chapitre III in: Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete 64, Springer (1972) [doi:10.1007/978-3-662-59155-0]