categories: Kripke equivalence frames as coalgebras?

["Prof. T.Porter" <t.porter@bangor.ac.uk>]

Dear All,

I note the large amount of activity in the study of coalgebras and am
looking for a way to introduce elements of this into seminar discussions
with a group of people working in   parts of computer science and
artificial intelligence that normally do not see much categorical light
(and who have little categorical background).  In a series of seminars I
have been discussing epistemic logic (that is various extensions of the
modal logic S5 and its multimodal versions) and have thus introduced
Kripke equivalence frames as sets of possible worlds with an equivalence
relation and have talked about their morphisms (bounded or p-morphisms
to the modal logicians).

Various coalgebraic sources make reference to Kripke frames as
coalgebras for the power set functor, P : Sets -> Sets, but I have so
far been unable to find a relatively elementary treatment of cofree
coalgebras for this context.  The `dual' category (for the S5 case) is
of monadic algebras and it  has free algebras but I have been trying to
avoid using duality too much in the seminars I have been giving and
certainly do not want to go into questions of `generalised frames'.
Explicitly I would like references for answers to the following:
(i) Is there a clear description in the literature (e.g. in coequational
form or categorically) of the category of Kripke \emph{equivalence}
frames as coalgebras?
(ii) Where can I find descriptions (as direct and simple as possible!)
of limits in the category of equivalence frames (or does the lack of a
complete duality muck things up as it does in the category of ALL Kripke
frames)
(iii) Given a Kripke equivalence frame, F, is there a nice cofree
coalgebra construction which is not `too horrendous for words' e.g.
avoiding going via canonical models and the like.

Thanking you all in advance,
 and slightly belated Happy New Year to everyone,

Tim Porter