M Theory Lesson 87

In a 2002 paper, Kennison proves the following result. First, let Stone be the category of Stone spaces and Bool be the category of Boolean algebras. There is a categorical equivalence $P$ from Bool to Stone which takes the space of all points in an object $B$. An arrow $t: X \rightarrow X$ in Stone is called Boolean cyclic if the corresponding arrow in Bool is cyclic, which means that the supremum over all equalisers of $t^{n}$ and $1_{X}$ is just $X$. Intuitively, these represent dynamical processes that eventually cycle. Now let $\mathbb{N}$ be the ordinals. An action is a map $\mathbb{N} \times X \rightarrow X$ given by $(n,x) \mapsto (t^{n}(x))$. Kennison showed that the property of being Boolean cyclic was equivalent to actually having an action by Z, the profinite integers, namely the product over ordinal primes $\prod \mathbb{Z}_{p}$ of the p-adic integers.

This is interesting because J. Borger has been looking at finite sets with Z actions in order to characterise $\Lambda$-rings in terms of finite sets with actions of $G \times \mathbb{N}^{+}$ for $G$ the absolute Galois group for the rationals. Recall that this group acts on Grothendieck’s ribbon graphs. A $\Lambda$-ring structure is a series of arrows $f_{p}: R \rightarrow R$ for a ring $R$. For example, one may take

$R = \frac{\mathbb{Z} [x]}{x^r – 1}$

along with the Frobenius maps $\psi_{p}: x \mapsto x^p$. Borger et al show that certain nice $\Lambda$-rings can only be a field if they are in fact the rationals. It turns out that all nice $\Lambda$-rings are subrings of products of the cyclotomic example above, for some $r$. This seems to be important somehow…

2 Responses so far »

  1. 1

    Doug said,

    Hi Kea,

    I am completely unfamiliar with category Stone or Stone Spaces.

    I have recently read Ash and Gross ‘Fearless Symmetry’ that was on the Vacation reading thread at MoonshineMath.

    I really like this book.

    In other reading Paul J Nahin, Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills, asserts these academic accomplishments by Paul AM Dirac. His Nobel biography seems to confirm this.

    BS – 1921 – Engineering (electrical)

    PhD – 1926 – Mathematics

    Employment – 1927 on – Mathematics

    Focus – 1928 on – Quantum mechanics

    Award – 1933 – Nobel in Physics “for the discovery of new productive forms of atomic theory” with Erwin Schrödinger.

    Maybe more physicists need to study the mathematics of electrical engineering?

    “Although Dirac made vastly important contributions to physics, it is important to realise that he was always motivated by principles of mathematical beauty.”

    http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Dirac.html

  2. 2

    Kea said,

    Hi Doug. It is certainly true, and a great tragedy, that in my generation physicists got away with almost no education in mathematics. I have had to teach myself almost everything I know (which is therefore frightfully little – sigh). Fortunately, the modern syllabus is adapting slowly to the times.


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