(10/25/2021, 08:40 PM)JmsNxn Wrote: I'm a little wary of the term Nixon-algebra; as I have no idea what it is and it has my name on it, LOL!
The term Hilbert space comes with a funny anecdote on its origin:
Saunders Mac Lane, one of the founding fathers of category theory, in [1989] writes
![[Image: image.png]](https://i.ibb.co/KbnMCyj/image.png)
Translation: "Dr. von Neumann, I would like to know what is a Hilbert space?" (tr. by \(n\)Lab)
[1989] S. Mac Lane, 'Concepts and categories in perspective: Addendum', pp. 323–365 in A century of mathematics in America, part I. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 1. American Mathematical Society (Providence, RI), 1988. An addendum to this article was published in History of Mathematics, part III (1989).
ps: A \(\diamond\)-algebra/Nixon Algebra is something much bolder than what you are thinking then, yet, originated from you. Let \((J,{}^+)\) be a set equipped with an function \({}^+:J\to J\), i.e. a discrete dynamical system maaping \(\sigma \mapsto \sigma^+\) that we call the system of ranks, define a Nixon algebra over \(J\) - with ranks in \(J\) - to be a pair \((M,\diamond)\) where
- \(M\) is a monoid \((M,\cdot, 1_M)\);
- equipped with an operation \(\diamond:J\times M\to M\) such that for every \(\sigma\in J\) and \(f\in M\) we have
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)