nLab
locally closed subtopos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

A locally closed subtopos generalizes the concept of a locally closed subspace from topology to toposes.

Definition

A subtopos j\mathcal{E}_j\hookrightarrow\mathcal{E} is called locally closed if it is the meet j= o c\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c of an open subtopos o\mathcal{E}_o and a closed subtopos c\mathcal{E}_c in the lattice of subtoposes of \mathcal{E}.

Properties

  • Since open and closed subtoposes are complemented in the lattice of subtoposes, the join of the complements ¯ o¯ c\bar\mathcal{E}_{o}\cup\bar\mathcal{E}_{c} provides a complement for a locally closed subtopos j= o c\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c.

  • j\mathcal{E}_j is locally closed iff j\mathcal{E}_j\hookrightarrow\mathcal{E} can be factored into an open followed by a closed inclusion.

References

  • M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (Exposé IV, ex.9.4.9., pp.462-463)

  • Peter Johnstone, Conditions Related to de Morgan’s Law , pp.479-491 in LNM 753 Springer Heidelberg 1979.

  • A. Kock, T. Plewe, Glueing analysis for complemented subtoposes , TAC 2 (1996) pp.100-112. (pdf)

Revised on August 17, 2015 03:23:13 by Thomas Holder (89.204.130.205)