Also partition function is also book-keeping device since coefficient of Boltzmann weight is the number of states with given energy.

The next question is of course whether also partition function could be interpreted as category-theoretic isomorphism;-).

]]>thanks for clarifying the intuitive thinking behind categorization, decategorization, and numbers. I am becoming more and more confused but in fruitful manner;-).

In nut shell the generalization of rig idea is the replacement of natural numbers with p-adic integers which can be regarded as “super-naturals” including both finite and infinite naturals the latter having finite size in p-adic topology and mappable to finite reals by canonical identification.

This allows set theoretic definition of negative rationals as infinite p-adic fractals: pinary expansion represented as a tree with n:th level containing x_n+1 branches at each node or its equivalent as wavelet expansion making sense in TGD inspired theory of cognitive representations.

Complex algebraic numbers which can be interpreted as “real” p-adic numbers for very many primes become “cardinalities” of category theoretical objects but p-adic cardinalities are of course more natural.

Moreover, one obtains ALL algebraic complex numbers rather than those associated with polynomials with natural numbers as coefficients and generating functions converge for x=0 without any need for analytic continuation as in real case. Hence p-adics are very natural in categorification business.

The connection between numbers and categories is really fuzzy: doesn’t these results mean that one can assign to objects algebraic complex numbers and realize them as isomorphisms but it does not mean that complex numbers can be defined categorically. Is de-categorification necessary and should it allow sets/linear spaces which can have super-natural number of elements/ dimension.

Note that the notion of super-naturals conforms also nicely with quantum TGD based on fusion of various number fields along common rationals (and algebraics).

There must be also also connection with infinite primes/ integers/ rationals, which are in one-one correspondence with states of repeatedly quantized arithmetic QFT: these can be mapped to polynomials which in turn are realized as isomorphisms in rig theory.

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