"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 18-Jan-2002 08:31:45 -0400,2810;000000000001-00000000