mystification and categorification

["Stephen Schanuel" <schanuel@adelphia.net>]

    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