Stone Coalgebras
Clemens Kupke, Alexander Kurz, Yde Venema
Abstract:
In this paper we argue that the category of Stone spaces forms an interesting
base category for coalgebras, in particular, if one considers the Vietoris
functor as an analogue to the power set functor on the category of sets.
We prove that the so-called descriptive general frames, which play a
fundamental role in the semantics of modal logics, can be seen as Stone
coalgebras in a natural way. This yields a duality between the category
of modal algebras and that of coalgebras over the Vietoris functor.
Building on this idea, we introduce the notion of a Vietoris polynomial
functor over the category of Stone spaces. For each such functor T we
provide an adjunction between the category of T-sorted Boolean algebras
with operators and the category of Stone coalgebras over T.
Since the unit of this adjunction is an isomorphism, this shows that
Coalg(T)^op is a full reflective subcategory of BAO_T.
Applications include a general theorem providing final coalgebras in
the category of T-coalgebras.