12 February 2007

an apology, and some category theory

I apologize for being so absent from the blogosphere. My computer died — it moved from epileptic through comatose to completely blind — in January, and since then I've been spending maybe ten minutes a day on the internet. I'm not taking classes, so my math exposure consists mostly of reading (I'm currently enjoying Hartry Fields' Science Without Numbers); instead, my time is consumed by an almost complete dance project (our main performance is Friday), running the hiring committee for next year's staff in my dorm, and getting into (and soon visiting) graduate schools around the country.

I did find the time recently to catch up on This Week's Finds, and was particularly interested in the discussion in Week 244. In it, Baez discusses an article by Tom Leinster, called The Euler characteristic of a category, which I skimmed and need to go read more thoroughly.

I certainly haven't grokked most of the material, but I did want to make two comments. First, a small one: Baez wonders at the coincidence that the character 1 has Euler characteristic 1, whereas the character 0 has Euler characteristic 0. I suggest that this has, in fact, real meaning: 1 presumably comes from a has mark, a line, a dot; any natural character for 1 ought to be readily contractible to a point. But 0, on the other hand, should represent exactly an absence: it must be filled with a point (equivalently a disk) to become a disk (equivalently a point). And this exactly defined its Euler characteristic to be 0. Somehow, the ancients, or perhaps the forces of calligraphic natural selection, understand enough naive topology to build Euler characteristics into the system.

My second comment is a little more substantive. In defining the Euler characteristic of a (for now, finite) category, we're trying to generalize
1. Euler characteristic of topological space. If our category is a groupoid (all morphisms are iso), then equivalently our category is the homotopy groupoid of the space formed by including a point for each object in our category, an edge for each morphism, a disk for each commuting triangle, a 3-ball for each commuting tetrahedron, etc.
2. Euler characteristic, as defined by Rota, of a poset. This motivates the idea of defining on our category such objects as a ζ function, a μ function, convolution of functions, etc., with which the Euler characteristic is defined.

Which is all well and good. But then Leinster et al decide on what seems to me to be a bizarre idea: a category with two objects and one morphism (between them), i.e.
o---->----o

should also (along with o--<--->--o) become an interval, and so should have Euler characteristic 1.

See, in a poset, this was exactly the right thing to do. But I claim that we're then wrong to think of a poset as a category, at least in the normal way. In a category, unlike a poset, you can have morphisms both directions. An isomorphism is really two morphisms, glued together by a commuting disk. There's a difference in a category between that and a unidirectional morphism. On the other hand, in a poset, there's no chance at morphisms going both ways. Instead, a poset is really just a space, but one with a (transitive) sense of "up" and "down".

It seems really bizarre to throw away so much information from the category by ignoring the direction of morphisms when taking Euler characteristics. Instead, why not follow Baez's suggestion: "If we were willing to make up new kinds of numbers, we could make up a new number for the size of this category."

I'm willing. What properties should our new number have? I thought at first that an isomorphism was the sum of two inverse morphisms. If this were so, then we'd want to assign some value X to a "forward" morphism, and some value X^{-1} to a "backwards" one. (Of course, forward and backward are local properties, so these values might want to live locally, say as a field rather than a function, but we are also going to want to sum them.) Then, I thought, we should want X + X^{-1} to be an isomorphism, which we know has Euler characteristic -1 (either object has characteristic 1). We can, of course, solve this: X+X^{-1}=-1 means that X = -(1/2)\pm(i\sqrt{3}/2). And, I said to myself, this is cool: the "Op" operation that reverses categories is just like complex conjugation!

Alas, however, this method fails a horrible death. Because an isomorphism is not the sum of two morphisms, but the sum of two morphisms and a commuting disk. And if we decide that disks should have characteristic 1, then each morphism really should have characteristic -1: we're solving X+X^{-1}+1=-1. And a commuting disk, at least the kind in an isomorphism, really really ought to have characteristic 1, because it can go in any direction. (I am interested by those pairs of morphisms for which, e.g., fg=1 but gf is not 1. I've never totally understood how to treat them and their commuting diagrams.)

So, yes, what should our method be? Certainly I want to assign weights to morphisms that are sensitive to their direction. Perhaps I should even attend to composition and the like: in a one-object category, a nilpotent morphism should be different from an isomorphism? So I want to build some sort of field on my category, and then integrate it against some sort of measure? I definitely want Op to act like "complex conjugation". Perhaps we'll end up with choices — certainly we'll write down natural (local) conditions that the values of our field should satisfy, but their values will depend on what field algebraic space (I'm using "field" in the physicists' sense) we solve the equations in — but then what should happen is that we get some sort of "free" object that can map to, for instance, the reals. And so the Euler characteristic, as described by Leinster, is one (and possible the most natural) representation. But in the same way that knot theorists find more powerful tools when they look for representations that attend to crossings, we should look for representations that attend to the direction of our morphisms.