Naughty Numbers
The Everything Seminar continues its series on infinite sums of ordinals using either analytic or zeta function regularisation. Meanwhile Matti Pitkanen had the cool idea of generalising the categorical interpretation (at p=1) to p-adic series.
So what are we actually doing here? The problem is that in n-logos theory we have to be really, really careful about defining numbers. A finite ordinal is the number of elements of a finite set. In other words, a finite ordinal is the equivalence class of all finite sets. But replacing an equivalence class by a single object is a very categorical operation. Moreover, the disjoint union of two finite sets becomes the sum of two positive integers, and the Cartesian product of two sets becomes multiplication, so this tells us something about what numbers are. Essentially, the category of finite sets has been decategorified, since we don't attach arrows $n \rightarrow m$ to the collection of numbers. That is, somehow the numbers $n \in \mathbb{N}$ are each (-1)-categories, and their collection is a 0-category, or set.
Most of the time the elements $n \in \mathbb{N}$ are taken to be objects in a 1-category, but that might just be because one has taken a functor selecting the set $\mathbb{N}$ as an identity arrow in the topos of sets. It's all very confusing. Basically, numbers need to be in many places at once, just like distance measurements.
As Baez points out in TWF 191, a structure type is a functor from the category of finite sets (and bijections) to the category of sets. Here the category of finite sets may be thought of as the category with objects $n$ and morphisms $n \rightarrow n$ the permutation group $S_{n}$. The category of structure types (and natural transformations) is then a categorified version of the polynomials $\mathbb{N} [x]$, which form a rig because they don't have negatives. That is, it is a kind of 2-rig. Commutativity of addition has been replaced by a symmetric monoidal structure. And moreover, as Baez points out, a monoid object in the category of structure types is an operad (and we like those).
Now we want to categorify everything again, because numbers are really dimensions of weak n-categories rather than counting numbers. This is analogous to categorifying Betti numbers to cohomology, or Jones polynomials to Khovanov homology. That would bring us to tricategorical rigs, whereupon the right tensor structure is now dimension raising, so somehow numbers come from all dimensions at once!
So what are we actually doing here? The problem is that in n-logos theory we have to be really, really careful about defining numbers. A finite ordinal is the number of elements of a finite set. In other words, a finite ordinal is the equivalence class of all finite sets. But replacing an equivalence class by a single object is a very categorical operation. Moreover, the disjoint union of two finite sets becomes the sum of two positive integers, and the Cartesian product of two sets becomes multiplication, so this tells us something about what numbers are. Essentially, the category of finite sets has been decategorified, since we don't attach arrows $n \rightarrow m$ to the collection of numbers. That is, somehow the numbers $n \in \mathbb{N}$ are each (-1)-categories, and their collection is a 0-category, or set.
Most of the time the elements $n \in \mathbb{N}$ are taken to be objects in a 1-category, but that might just be because one has taken a functor selecting the set $\mathbb{N}$ as an identity arrow in the topos of sets. It's all very confusing. Basically, numbers need to be in many places at once, just like distance measurements.
As Baez points out in TWF 191, a structure type is a functor from the category of finite sets (and bijections) to the category of sets. Here the category of finite sets may be thought of as the category with objects $n$ and morphisms $n \rightarrow n$ the permutation group $S_{n}$. The category of structure types (and natural transformations) is then a categorified version of the polynomials $\mathbb{N} [x]$, which form a rig because they don't have negatives. That is, it is a kind of 2-rig. Commutativity of addition has been replaced by a symmetric monoidal structure. And moreover, as Baez points out, a monoid object in the category of structure types is an operad (and we like those).
Now we want to categorify everything again, because numbers are really dimensions of weak n-categories rather than counting numbers. This is analogous to categorifying Betti numbers to cohomology, or Jones polynomials to Khovanov homology. That would bring us to tricategorical rigs, whereupon the right tensor structure is now dimension raising, so somehow numbers come from all dimensions at once!
posted by Kea | 12:21 PM
2 Comments:
Dear Kea,
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.
I noticed that the introduction of p-adic generating functions is completely analogous to the transition from ordinary to p-adic thermodynamics since x^n is completely analogous to Boltzman weight for a harmonic oscillator like system.
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;-).
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