Paul Taylor wrote in part:
Toby Bartels asked me:
My intuition is that polynomials with overt discrete coefficients should have overt discrete initial algebras, while those with compact Hausdorff coefficients should have compact Hausdorff final coalgebras. Have you any thoughts about that question?
In fact, what I have to say about this (in the setting of the existing established theory for locally compact spaces) is little more than Toby's "intuition". The existence of these spaces would follow from the limit--colimit coincidence, which is sketched in Remark 10.16 of "Geometric and higher-order logic in ASD" www.PaulTaylor.EU/ASD/loccpct#geohol
Yes, there it is! That's what I get for not reading them in order. (^_^)
The symmetry between => T /\ = some overt discrete free algebra and <= F \/ != all compact Hausdorff cofree coalgebra is very strong in this, but not perfect, because N is overt discrete Hausdorff not compact 2^N is compact Hausdorff not discrete overt I have not managed to isolate convincingly the precise point where the symmetry breaks down.
I was about to say that this comparison is not really fair, since N is the initial algebra of X |-> X + 1, while 2^N is the final coalgebra of X |-> 2 x X, but I guess that the final coalgebra of X |-> X + 1 is also overt but not discrete. Perhaps the asymmetry is simply between initial algebras and final colagebras. One is a colimit and the other is a limit; there are already several asymmetries between these, such as that products distribute over sums but not vice versa. Indeed, if final coalgebras preserve "some"s, this might be more than just a bad pun. --Toby