I’m going to use this as an opportunity to talk out loud and verbalize some thoughts that I’m trying to develop. Some of the them are connected with the TQFTs I’ve been writing about. I started reading Atiyah’s 1988 article again to compare his axioms against Kock’s. They imply the same but they assume slightly different axioms. For example, Atiyah supposes that if an (n-1)-manifold is sent to a vector space V then with the opposite orientation, will be sent to the dual of V, V*. This is derived in Kock.
I have a sequel to “TQFT – elementary examples and consequences” which works through a couple toy problems Kock has and in the process the trace of a matrix comes up, though artificially. I think I can work through some details and make it more natural, but Atiyah brings it up casually. He talks about the cobordism , which he obtains by identifying the opposite ends of the cylinder . He states “…then our axioms imply that ” (where Z is the TQFT functor).
Why the trace? There are lots of linear functionals, why that one? This made me realize that I don’t know much about the trace beyond it being the sum of eigenvalues of an operator (in finite dimensions). I’ve started poking around and there’s another idea to pursue, that its related to the determinant. In particular in Lie theory. The determinant is a map between Lie groups like GL and R. The trace is a linear transformation between their Lie algebras and R. The impression I get, though have not worked through, is that the trace is or is related to the differential of the determinant at the identity. That’s certainly provocative.
All the TQFT formalism combined with reading ‘On space and Time’ has made me pick up Hartle’s book on gravity again and start working through it. I had a really unsatisfactory course in GR a couple of years ago and really want to work a lot of examples. Hartle’s book is a ‘physics first approach’ which I normally don’t go for. In GR though I like it, he throws metrics at you constantly and you get to develop your intuition, something incredibly valuable in GR where you have to forget a great deal of your former physics intuition. I’ll be blogging interesting bits and pieces as they arise.
On the quantum end of things I think I’ve settled on George Mackey’s book, “Mathematical Foundations of Quantum Mechanics” which has been reprinted by Dover. It’s a really nice text, far more accessible than something like Von Neumann’s classic, though same damned font. Mackey’s book starts with classical mechanics in the setting of manifolds and Lie groups, though gently. Look forward to posts from it.
I’m still plugging away on the tensor post, hung up on a couple of details there as well.
On another note, I fetched that paper of Lawvere’s which Majid claimed had something to do with geometry beginning in a co-Heyting algebra (posted in “On Space Time a book review (in part)“). It was in the proceedings of a category theory conference in 1990. It’s entitled “Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes.” I’ll give you the opening paragraph,
“Certain lattices, such as that of all closed subsets of a topological space or that of all subtoposes of a given topos, may be called “co-Heyting algebras” in that they enjoy a “subtraction” operator left adjoint to join, dually to the “implication” operator right adjoint to meet which Heyting algebras have. The cited examples illustrate that such latices may occur in practice directly, not only as formal opposites of Heyting algebras. In particular, there is for each element A of a co-Heyting algebra a smallest element non A whose join with A is the top element, and the meet A and non A is a further element which deserves to be called the boundary of A.”
The article is just barely two and a half pages long and ends with this ‘punchline,’
Corollary: In the topos of simplicial sets, if , then for the co-Heyting boundaries in respectively.
What’s pretty funny is that there is only one reference, a book he edited from a workshop on category theory and the foundations of continuum mechanics called “Categories in Continuum Physics” published by Springer in 1986 (Lawvere began in physics before becoming seduced by category theory).
I understand pitifully little of the cited article, but more than I would have a year ago. I have a lot of ideas out on the table and a bunch of reading to go through and not nearly enough time. However it’s coming, little by little.