Stone duality provides the dual equivalence between algebras
and spaces which is central in the relationship between modal algebras and
Kripke semantics as well as the central mechanism for the relationship
between specification of program logics and denotational semantics for
these logics as layed out in Abramsky's Domain Theory in Logical Form.
In seeking to extend the scope of this duality, one may ask for natural
dualities for various related structures or for extended Stone or
Priestley type dualities. The latter corresponds to the coalgebraic
picture and an algebraic tool for studying the scope and limits of
extended Stone and Priestley dualities is provided by canonical
extensions. This talk will highlight a few of the recent developments in
this area that I expect are pertinent to work in coalgebraic logic.