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A Vietoris functor for d-frames

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If you have a question about this talk, please contact Neel Krishnaswami.

Coalgebraic logic behaves well whenever we have an adjunction between a category of spaces and a category of algebras. One example of such situation is when we take the category of Stone spaces and the category of Stone frames. To allow for non-determinism of coalgebras over Stone spaces one can use Vietoris hyperspace functor and the corresponding endofunctor on the algebraic (frame) side is then Johnstone’s powerlocale functor.

To admit four-valued reasoning, we need to change the base categories. We can take the adjunction between the category of Stone bitopological spaces for the category of spaces and the category of Stone d-frames as the category of algebras. To model non-determinism we need a hyperspace-like functor for bitopological spaces and powerlocale-like functor for d-frames.

We will show that this can be done. Following Johnstone’s step we first give a description of a free d-frame construction from a set of generators and relations and then show how we can use this to define a Vietoris functor for d-frames.

This talk is part of the Lab Lunch series.

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