From: Pawel Sobocinski <pawel@brics.dk>
To: categories@mta.ca
Subject: Re: mystification and categorification
Date: Tue, 9 Mar 2004 10:54:02 +0000 [thread overview]
Message-ID: <13EDA32B-71B8-11D8-BE9C-000A95A85E4A@brics.dk> (raw)
In-Reply-To: <1078688585.20775.171.camel@tl-linux.maths.gla.ac.uk>
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.
next prev parent reply other threads:[~2004-03-09 10:54 UTC|newest]
Thread overview: 12+ messages / expand[flat|nested] mbox.gz Atom feed top
[not found] <schanuel@adelphia.net>
2004-03-04 5:44 ` Stephen Schanuel
2004-03-05 16:55 ` David Yetter
2004-03-06 6:49 ` Vaughan Pratt
2004-03-07 21:04 ` Mike Oliver
2004-03-08 10:20 ` Steve Vickers
2004-03-07 19:43 ` Tom Leinster
2004-03-09 10:54 ` Pawel Sobocinski [this message]
2004-03-12 13:50 ` Quillen model structure of category of toposes/locales? Vidhyanath Rao
2003-02-20 0:16 More Topos questions ala "Conceptual Mathematics" Galchin Vasili
2003-02-20 18:48 ` Stephen Schanuel
2003-02-21 0:57 ` Vaughan Pratt
2003-06-10 21:23 ` Galchin Vasili
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