nLab
scattered topos

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Category theory

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topos theory

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In higher category theory

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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.