categories: Re: Kripke equivalence frames as coalgebras?

[jim stasheff <stasheff@email.unc.edu>]

"Prof. T.Porter" wrote:
> 
> 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

I think both Michaelis and Block have written on cofree coalgebras
and Michaelis is nearing completion of a lengthy survey article
on coalgebras.

jim