scattered topos


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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 \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.


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.


  • 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 .1


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)

  1. cf. Esakia-Jibladze-Pataraia (2000, p.101).

Revised on May 14, 2016 13:12:02 by Thomas Holder (