From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2589 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: mystification and categorification Date: 07 Mar 2004 19:43:05 +0000 Message-ID: <1078688585.20775.171.camel@tl-linux.maths.gla.ac.uk> References: <002a01c401ab$cd50b370$1767eb44@grassmann> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018764 4876 80.91.229.2 (29 Apr 2009 15:26:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:04 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Mar 8 16:49:21 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:21 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0Rcb-0007a8-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:45:09 -0400 In-Reply-To: <002a01c401ab$cd50b370$1767eb44@grassmann> X-Mailer: Ximian Evolution 1.2.2 (1.2.2-5) Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 70 Xref: news.gmane.org gmane.science.mathematics.categories:2589 Archived-At: Steve Schanuel wrote: > a category with both plus and times as extra monoidal structures. > (Does anyone know an example of interest in which the plus is not > coproduct?) Here are two examples that I've come across previously of rig categories in which the plus is not coproduct: (i) the category of finite sets and bijections, with + and x inherited from the category of sets; (ii) discrete rig categories, which are of course the same thing as rigs. > This freedom is unnecessary; a little algebra plus Robbie > Gates' theorem provides a solution G to G^2=G+1 which satisfies no > additional equations, in an extensive category (with coproduct as plus, > cartesian product as times). If you *do* allow yourself the freedom to use any rig category then an even simpler solution exists, also satisfying no additional equations: just take the rig freely generated by an element G satisfying G^2 = G + 1 and regard it as a discrete rig category. > Since in the category of sets, any nasty old infinite set satisfies > the golden equation and many others, I have taken the liberty of > interpreting 'nice' to mean at least 'satisfying no unexpected > equations'. 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). Maybe the finding of a "nice" solution is similar in spirit to the finding of a "concrete interpretation" of a combinatorial identity. As an extremely simple example, consider the identity saying that each entry in Pascal's triangle is the sum of the two above it, (n+1 choose r) = (n choose r-1) + (n choose r). This is a doddle to prove, but all the same you'd be missing something if you didn't know the standard "concrete interpretation": choosing r objects out of n+1 objects amounts to EITHER choosing the first one and then choosing r-1 of the remaining n OR ... . Even if the challenge of finding a "nice solution" or "concrete interpretation" isn't made precise, I think there is a shared sense of what would count as an answer, and finding an answer is in general not straightforward. Best wishes, Tom