nLab
scattered topos

Contents

Context

Category theory

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 scattered topos is the topos-theoretic analogue of a scattered topological space and therefore provides a natural environment for an internal interpretation of provability logic.

Definition

A topos \mathcal{E} is called \bot\negthinspace-scattered if the subtopos Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) of double negation sheaves is an open subtopos.

A topos \mathcal{E} is called scattered if every closed subtopos of \mathcal{E} is \bot-scattered.

Examples

Every Boolean topos \mathcal{E} is \bot-scattered since it obviously coincides with Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) and is open in itself (and closed as well). Accordingly, \mathcal{E} is scattered since subtoposes of Boolean toposes are Boolean.

A simple example of a non-Boolean scattered topos is the Sierpinski topos Set Set^\rightarrow that consists of two copies of SetSet glued together such that one copy corresponding to Sh ¬¬(Set )Sh_{\neg\neg}(Set^\rightarrow) is open and the other one is closed. Since Set Set^\rightarrow and the closed copy of SetSet are both \bot-scattered the claim follows. (For another simple example see at hypergraph.)

Properties

  • That the Sierpinski-topos is \bot-scattered is an instance of the more general fact that the topos Sh(X)Sh(X) of sheaves on a T 0T_0-space XX is \bot-scattered iff open points are dense in XX (cf. Esakia-Jibladze-Pataraia 2000, p.101).

  • A topos is scattered iff the (internal) Heyting algebra of Lawvere-Tierney topologies is Boolean (cf. Esakia-Jibladze-Pataraia 2000, p.103). Since Boolean algebras are precisely the Heyting algebras where every element is complemented this says that a topos where all subtoposes are complemented is scattered.

  • A spatial topos Sh(X)Sh(X) is scattered iff every non degenerate subtopos has a point (cf. Esakia-Jibladze-Pataraia 2000, p.103).

References

The concept was introduced in

  • Leo Esakia, Mamuka Jibladze, Dito Pataraia, Scattered Toposes , APAL 103 (2000) pp.97-107.

For the wider context see also

  • Leo Esakia, Quantification with a provability smack , Bull. Sect. Log. 27 (1998) pp.26-28. (pdf)

Last revised on March 16, 2018 at 09:27:00. See the history of this page for a list of all contributions to it.