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.
A topos is called -scattered if the subtopos of double negation sheaves is an open subtopos.
A topos is called scattered if every closed subtopos of is -scattered.
Every Boolean topos is -scattered since it obviously coincides with and is open in itself (and closed as well). Accordingly, is scattered since subtoposes of Boolean toposes are Boolean.
A simple example of a non-Boolean scattered topos is the Sierpinski topos that consists of two copies of glued together such that one copy corresponding to is open and the other one is closed. Since and the closed copy of are both -scattered the claim follows. (For another simple example see at hypergraph.)
That the Sierpinski-topos is -scattered is an instance of the more general fact that the topos of sheaves on a -space is -scattered iff open points are dense in (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 is scattered iff every non degenerate subtopos has a point (cf. Esakia-Jibladze-Pataraia 2000, p.103).
The concept was introduced in
For the wider context see also
Last revised on March 16, 2018 at 13:27:00. See the history of this page for a list of all contributions to it.