March 18th, 2008

beartato phd

(no subject)

A very frustrating day, fighting off recurrence of PLAGUE and generalized shit-feeling. I know I need to work on this unification implementation and CSL paper before the end of the month, but I am finding myself unrestrainedly obsessed with a totally unrelated problem lately, having to do with a possibly quite cute connection between geometry and weak n-category theory. Since cute connections between the two abound, it's not terrifically likely that what I'm working on is novel, but it provides an addicting sort of (possibly false) satisfaction that I am slowly increasing the depth of my understanding of really foundational things.

To put it as a single question: why is it that composition of arrows in a category should be associative? Where do these sorts of axioms come from, really?

I think the answer is basically this diagram:

Even if composition of arrows itself is assumed weakly enough that all you get is a 2-cell mediating the connection between the path that you are composing and the direct path between source and target, you still are able to compose these 2-cells to get from one composition order to another, and this feels like an essentially geometric fact somehow. Of course you can get back, too because you know that that 2-cell is invertible. (indeed I am already using both directions of it in the above diagram at different points) And the identity laws are essentially just nullary versions of associativity.

The annoying thing is how difficult it is to formalize n-dimensional cell complexes in the appropriate way. This is also the part I'm most certain has been done a million times already, but it seems like good practice to bash my brain against it awhile.