The outcome was the realization that the notion of rig used to categorify the subset of algebraic numbers obtained as roots of polynomials with natural number valued coefficients generalizes trivially by replacing natural numbers by p-adic integers. As a consequence one obtains beautiful p-adicization of the generating function F(x) of structure as a function which converges p-adically for any rational x=q for which it has prime p as a positive power divisor.
Effectively this generalization means the replacement of natural numbers as coefficients of the polynomial defining the rig with all rationals, also negative, and all complex algebraic numbers find a category theoretical representation as "cardinalities". These cardinalities have a dual interpretation as p-adic integers which in general correspond to infinite real numbers but are mappable to real numbers by canonical identification and have a geometric representation as fractals as discussed in the previous posting.
1. Mapping of objects to complex numbers and the notion of rig
The idea of rig approach is to categorify the notion of cardinality in such a manner that one obtains a subset of algebraic complex numbers as cardinalities in the category-theoretical sense. One can assign to an object a polynomial with coefficients, which are natural numbers and the condition Z=P(Z) says that P(Z) acts as an isomorphism of the object. One can interpret the equation also in terms of complex numbers. Hence the object is mapped to a complex number Z defining a root of the polynomial interpreted as an ordinary polynomial: it does not matter which root is chosen. The complex number Z is interpreted as the "cardinality" of the object but I do not really understand the motivation for this. The deep further result is that also more general polynomial equations R(Z)= Q(Z) satisfied by the generalized cardinality Z imply R(Z)= Q(Z) as isomorphism. This means that algebra is mapped to isomorphisms.
I try to reproduce what looks the most essential in the explanation of John Baez and relate it to my own ideas but take this as my talk to myself and visit This Week's Finds to learn of this fascinating idea.
- Baez considers first the ways of putting a given structure to n-element set. The set of these structures is denoted by Fn and the number of them by Fn. The generating function F(x) = ∑nFnxn packs all this information to a single function.
For instance, if the structure is binary tree, this function is given by T(x)= ∑nCn-1xn, where Cn-1 are Catalan numbers and n>0 holds true. One can show that T satisfies the formula
since any binary tree is either trivial or decomposes to a product of binary trees, where two trees emanate from the root. One can solve this second order polynomial equation and the power expansion gives the generating function.
- The great insight is that one can also work directly with structures. For instance, by starting from the isomorphism T=1+T2 applying to an object with cardinality 1 and substituting T2 with (1+T2)2 repeatedly, one can deduce the amazing formula T7(1)=T(1) mentioned by Kea, and this identity can be interpreted as an isomorphism of binary trees.
- This result can be generalized using the notion of rig category (Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, available as math.CT/0212377). In rig category one can add and multiply but negatives are not defined as in the case of ring. The lack of subtraction and division is still the problem and as I suggested in previous posting p-adic integers might resolve the problem.
Whenever Z is object of a rig category, one can equip it with an isomorphism Z=P(Z) where P(Z) is polynomial with natural numbers as coefficients and one can assign to object "cardinality" as any root of the equation Z=P(Z). Note that set with n elements corresponds to P(Z)= n. Thus subset of algebraic complex numbers receive formal identification as cardinalities of sets. Furthermore, if the cardinality satisfies another equation Q(Z)= R(Z) such that neither polynomial is constant, then one can construct an isomorphism Q(Z)= R(Z). Isomorphisms correspond to equations which is nice!
- This is indeed nice that there is something which is not so beautiful as it could be: why should we restrict ourselves to natural numbers as coefficients of P(Z)? Could it be possible to replace them with integers to obtain all complex algebraic numbers as cardinalities? Could it be possible to replace natural numbers by p-adic integers? Oops! I told it! All tension of drama is now lost! Sorry!
The notions of generating function and rig generalize to the p-adic context.
- The generating function F(x) defining isomorphism Z in the rig formulation converges p-adically for any p-adic number x containing p as a factor so that the idea that all structures have p-adic counterparts is natural. In the real context the generating function typically diverges and must be defined by analytic continuation. Hence one might even argue that p-adic numbers are more natural in the description of structures assignable to finite sets than reals.
- For rig one considers only polynomials P(Z) (Z corresponds to the generating function F) with coefficients which are natural numbers. Any p-adic integer can be however interpreted as a non-negative integer: natural number if it is finite and "super-natural" number if it is infinite. Hence can generalize the notion of rig by replacing natural numbers by p-adic integers. The rig formalism would thus generalize to arbitrary polynomials with integer valued coefficients so that all complex algebraic numbers could appear as cardinalities of category theoretical objects. Even rational coefficients are allowed. This is highly natural number theoretically.
- For instance, in the case of binary trees the solutions to the isomorphism condition T=p+T2 giving T= [1+/- (1-4p)1/2]/2 and T would be complex number [p+/-(1-4p)1/2]/2. T(p) can be interpreted also as a p-adic number by performing power expansion of square root: this super-natural number can be mapped to a real number by the canonical identification and one obtains also the set theoretic representations of the category theoretical object T(p) as a p-adic fractal. This interpretation of cardinality is much more natural than the purely formal interpretation as a complex number. This argument applies completely generally. The case x=1 discussed by Baez gives T= [1+/-(-3)1/2]/2 allows p-adic representation if -3==p-3 is square mod p. This is the case for p=7 for instance.
- John Baez poses also the question about the category theoretic realization of Golden Object, his big dream. In this case one would have Z= G= -1+G2=P(Z). The polynomial on the right hand side does not conform with the notion of rig since -1 is not a natural number. If one allows p-adic rigs, x=-1 can be interpreted as a p-adic integer (p-1)(1+p+...), positive and infinite and "super-natural", actually largest possible p-adic integer in a well defined sense. A further condition is that Golden Mean converges as a p-adic number: this requires that sqrt(5) must exist as a p-adic number: (5=1+4)1/2 certainly converges as power series for p=2 so that Golden Object exists 2-adically. By using quadratic resiprocity theorem of Euler, one finds that 5 is square mod p only if p is square mod 5. To decide whether given p is Golden it is enough to look whether p mod 5 is 1 or 4. For instance, p=11, 19, 29, 31 (M5) are Golden. Mersennes Mk,k=3,7,127 and Fermat primes are not Golden. One representation of Golden Object as p-adic fractal is the p-adic series expansion of [1/2+/-51/2]/2 representable geometrically as a binary tree such that there are 0< xn+1≤p branches at each node at height n if n:th p-adic coefficient is xn. The "cognitive" p-adic representation in terms of wavelet spectrum of classical fields is discussed in the previous posting.
- It would be interesting to know how quantum dimensions of quantum groups assignable to Jones inclusions relate to the generalized cardinalities. The root of unity property of quantum phase (qn+1=1) suggests Q=Qn+1=P(Q) as the relevant isomorphism. For Jones inclusions the cardinality q =exp(i2π/n) would not be however equal to quantum dimension d(n)= 4cos2(π/n).