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From: Pawel Sobocinski <pawel@brics.dk>
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Subject: Re: mystification and categorification
Date: Tue, 9 Mar 2004 10:54:02 +0000
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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.