||[Mar. 21st, 2006|04:41 pm]
I tried reading this paper by John Baez about n-categories again, but page 25 exploded my brain again, just like last time I tried, maybe a year ago. Everything up to that point is either mundane category theory and therefore easy reasing, or else it's stuff I don't understand, but saved from hardness by being merely suggestive and fuzzy — these latter parts I can skim without feeling guilty. Section 4.2 actually revs up the formal machinery that gives the real definition of opetops, and it's fucking intense.
One way to say what's going on that makes the definition so awesome is this: think about a monoid. On the one hand, a monoid is an algebraic structure that satisfies a certain theory. You got the monoid axioms, you're required to cough up a set and some operations that satisfy them, no big deal. But on the other hand, the notion of monoid action on a set X means that every monoid can itself be thought of as a little theory. A theory of what? Functions from X to itself. The monoid consists of a bunch of elements, and the action is required to map each element of the monoid to an endomorphism on X in such a way that composition of these endomorphisms is compatible with the monoid's operation. So all of the equations that expressions in the monoid "happen" to satisfy need, by virtue of the action axioms, to hold of the functions that make up the action.
What operads and the "slice operad" construction are, as best as I can tell, are a way of pumping enough generality into the situation so that these two views of an algebraic structure,
- As an instance of a theory
- (via the notion of "algebras over X" or "actions on X") As a theory itself
are both available simultaneously, letting one construct an infinite chain of algebras of the same sort, such that each is the theory of the next.
And then he defines opetopes somehow after this, and weak ω-categories after that. I ♥ math.