On 7 Mar 2004, at 19:43, Tom Leinster wrote:
I'd interpret "nice" differently. (Apart from anything else, the trivial example in my previous paragraph would otherwise make the golden object problem uninteresting.) "Nice" as I understand it is not a precise term - at least, I don't know how to make it precise. Maybe I can explain my interpretation by analogy with the equation T = 1 + T^2. A nice solution T would be the set of finite, binary, planar trees together with the usual isomorphism T -~-> 1 + T^2; a nasty solution would be a random infinite set T with a random isomorphism to 1 + T^2. (Both these solutions are in the rig category Set with its standard + and x.) I regard the first solution as nice because I can see some combinatorial content to it (and maybe, at the back of my mind, because it has a constructive feel), and the second as nasty because I can't. I'm not certain what I think of the solution given by the set of not-necessarily-finite binary planar trees (nice?), or by the set of binary planar trees of cardinality at most aleph_5 (probably nasty).
From a computer science point of view, both the first "nice" solution (finite binary trees) and the second "nice?" solution (possibly non-finite binary trees) are canonical, in the sense that the first is the carrier of the initial algebra for the endofunctor 1+X^2 on Set, while the second is the carrier of its final coalgebra. All the best, Pawel.