topos theory

# Contents

## Idea

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

## Definition

A subtopos $\mathcal{E}_j\hookrightarrow\mathcal{E}$ is called locally closed if it is the meet $\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c$ of an open subtopos $\mathcal{E}_o$ and a closed subtopos $\mathcal{E}_c$ 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 $\bar\mathcal{E}_{o}\cup\bar\mathcal{E}_{c}$ provides a complement for a locally closed subtopos $\mathcal{E}_j=\mathcal{E}_o\cap\mathcal{E}_c$.

• $\mathcal{E}_j$ is locally closed iff $\mathcal{E}_j\hookrightarrow\mathcal{E}$ 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)