From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2588 Path: news.gmane.org!not-for-mail From: "Stephen Schanuel" Newsgroups: gmane.science.mathematics.categories Subject: mystification and categorification Date: Thu, 4 Mar 2004 00:44:46 -0500 Message-ID: <002a01c401ab$cd50b370$1767eb44@grassmann> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-Trace: ger.gmane.org 1241018764 4873 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: Original-X-From: rrosebru@mta.ca Fri Mar 5 10:21:43 2004 -0400 X-Keywords: X-UID: 9 Original-Lines: 55 Xref: news.gmane.org gmane.science.mathematics.categories:2588 Archived-At: I was unable to understand John Baez' golden object problem, nor his description of the solutions. He refuses to tell us what 'nice' means, but let me at least propose that to be 'tolerable' a solution must be an object in a category, and John doesn't tell us what category is involved in either of the solutions; at least I couldn't find a specification of the objects, nor the maps, so I found the descriptions 'intolerable', in the technical sense defined above. He is very generous, allowing one to use 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?) 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). Briefly, here it is. A primitive fifth root of unity z is a root of the polynomial 1+X+X^2+X^3+X^4, hence satisfies 1+z+z^2+z^3+z^4+z=z, which is of the 'fixed point' form p(z)=z with p in N[X] and p(0) not 0. Gates' theorem then says that the free distributive category containing an object Z and an isomorphism from p(Z) to Z is extensive, and its Burnside rig B (of isomorphism classes of objects) is, as one would hope, N[X]/(p(X)=X); that is, Z satisfies no unexpected equations. Since the degree of p is greater than 1, an easy general theorem tells us (from the joint injectivity of the Euler and dimension homomorphisms) that two polynomials agree at the object Z if and only if either they are the same polynomial or both are non-constant and they agree at the number z.Now the 'algebra': the golden number is 1+z+z^4. So G=1+Z+Z^4 satisfies G^2=G+1, as desired. It satisfies no unexpected equations, because the relation X^2=X+1 reduces any polynomial in N[X] to a linear polynomial, and these reduced forms have distinct Euler characteristics, i.e. differ at z. Thus the homomorphism from N[X]/(X^2=X+1) to B (sending X to G) is injective, and that's all I wanted. 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'. One could ask for more; the construction above has produced a distributive, but not extensive, category whose Burnside rig is N[X]/(X^2=X+1), the full subcategory with objects polynomials in G. (If it were extensive, it would be closed under taking summands, but every object in the larger category is a summand of G.) I don't know whether there is an extensive category with N[X]/(X^2=X+1) as its full Burnside rig; perhaps one or both of the examples John mentioned would do, if I knew what they were. While I'm airing my confusions, can anyone tell me what 'categorification' means? I don't know any such process; the simplest exanple, 'categorifying' natural numbers to get finite sets, seems to me rather 'remembering the finite sets and maps which gave rise to natural numbers by the abstraction of passing to isomorphism classes'. Finally, a note to John: While you're trying to give your audience some feeling for the virtues of n-categories, couldn't you give them a little help with n=1, by being a little more precise about objects and maps? Greetings to all, and thanks for your patience while I got this stuff off my chest, Steve Schanuel