I’d just like to post something about a conceptual clarification that came up recently. Last week I gave the first of a couple of talks in the Algebra seminar in our department, about the ideas of structure types and stuff types, more or less as outlined in this paper which I put out a couple of years ago. It summarizes and traipses a little way beyond the matter of the 2003/2004 quantum gravity seminar at UCR, whence on this paper by John Baez and Jim Dolan, and even further back on work by André Joyal, particularly in the paper “Foncteurs analytiques et espèces de structures“, which regrettably doesn’t seem to be available either online. (I gave a blackboard version of the talk, but it was an expanded form of this one hour version.)
(Semantic side note: these espèces de structures are often referred to as “combinatorial species” in English. This is the more common translation than “structure type”, but unfortunately, it doesn’t capture the modifier “de structures“, instead choosing the more generic “combinatorial”, which makes it hard to distinguish “structure types” from “stuff types” in the Baez-Dolan sense. Also, “species” is probably over-specific as a translation of “espèces” in a way that “type” isn’t. The generic sense of “species” as “a kind of” in English is a bit recherché.)
In any case, what I’m interested in this post is the sense in which stuff types give a “categorification” of a vector space. In a nutshell, a stuff type is a groupoid over (the groupoid whose objects are finite sets, and whose morphisms are bijections). That is, it’s really a functor , which we call the “underlying set” functor. For example, consider the groupoid of all binary trees, where the underlying set is the set of nodes (or, a different example, the set of leaves). Any isomorphism between two such trees gives a bijection between the underlying sets, so this actually is a functor. Or one could take the functor , where the “underlying set” of a pair of sets is just , and likewise for morphisms. (Notice that different bijections “up above” in the bundle may give the same bijection “below” – in cases where this doesn’t happen, we have one of Joyal’s “structure types”). In some ways, it’s better to think of it as a bundle of groupoids – one fibre over each object in
The thing is, that map gives an invariant for objects in the category of groupoids, but not a complete invariant. Unlike, say, finite sets and the natural numbers. Natural numbers correspond exactly to isomorphism classes of sets – not so with groupoid cardinalities. So there’s an equivalence relation, and reducing the object set modulo that equivalence relation gives a structure – but it’s not the minimal throwing-away of information about objects that taking isomorphism classes would be.
But in any case, it’s the whole category of groupoids (over ) which gets “decategorified” down to a vector space, in that world. There is a concept of groupoid cardinality, which is given by Baez and Dolan in the paper above, and which is also linked to Tom Leinster’s definition of the Euler characteristic of a category. This adds up, over all the isomorphism classes of objects, , the reciprocals of the sizes of automorphism groups. Reasons why this is the nicest concept of cardinality are described in some of those references, but all that really matters here is that groupoid cardinality gets along with disjoint unions of groupoids (corresponding to sums of cardinalitys), and products of groupoids (which get the product of the two cardinalities). That is, the categorical coproduct and product, respectively, define operations on the set of cardinalities!
In particular, taking stuff types – groupoids over , we can take the cardinalities of the fibres over sets of each size giving the coordinate in a vector. So then is, the slice category has this “cardinality” on objects into a set, and the structure of the category gives well-defined operations on this set, turning it into a vector space. In fact, there’s an operation (weak pullback) which makes it an inner product space. (To make this work in complex cardinalities takes some fudging with phases in , but it can be done.)
The details are interesting, and I’m coming back to looking at some of this again, but what I want to point out at the moment is a more fundamental point, which has to do with the offhanded use of the handy, but imprecise, term “categorify”. With the category of (-) stuff types, we have a category with a “decategorification” map that compresses it into a vector space. This sure sounds like a “categorified vector space”. In fact, this seems to be what people who hear the term “categorification” often want it to mean: I look for a categorification of mathematical object X by finding a category which, secretly, looks like X.
The problem is, there’s another concept attached to the phrase “categorified vector space”, namely that of 2-vector space in the sense of Kapranov and Voevodski, as discussed, say, here. There’s a different level of abstraction at work here. The specific category of stuff types provides a categorification (if that indeed is the right word to use) of a specific vector space. The concept of a KV 2-vector space categorifies the concept of a regular vector space in a particular way: putting “additive” structure on objects, and “C-linear” structure on morphisms. (The Baez-Crans version does the same job in a different way).
You don’t think of a specific KV 2-vector space “decategorifying to” a specific vector space. Indeed, just taking the “minimal” equivalence relation – isomorphism classes of objects – what we get from a KV 2-vector space is more like an -module (over a rig, not a ring). Basically, 2-vectors have components which are vector spaces, and therefore classified by their dimension. The relationship between THIS kind of 2-vector space and the non-categorified concept is that real vector spaces show up as the -sets in a KV 2-vector space.
Elucidating exactly what’s going on with these two forms of categorification would be nice – perhaps somebody’s done it, but if so, I don’t know who. I also don’t know any nice conditions that tell you when you have a “category that can be mistaken for a vector space”, like stuff types: a good characterization of these things would be nice. Or again: both versions of “categorification” of vector space have special relationships to groupoids – but of two very different natures (in one, the groupoids can be interpreted as 2-vectors – in the other, there are whole 2-vector spaces associated to groupoids). Just a coincidence?
Another possibility that comes to mind would be to form some kind of hybrid structure – where the “vector spaces” which show up in the -sets in a KV 2-v.s. are secretly this fake-vector space type of category. Since both types seem to have physics-y ambitions, such a setup that combines both approaches is appealing, rather than a muddled and confusing competition for the term “categorification”.
I don’t have a good ending to this story, which is why this is a blog, not a book.