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.

June 15, 2008 at 1:23 pm

Urs discusses a distinction here, but I think it applies within the second of your categorifications.

June 16, 2008 at 2:43 am

Hi, David. I think you’re right. The second sense I suggest for the term “categorification” seems to be the most standard. It is a way of creating a categorical concept which mimics, but doesn’t necessarily reduce to, a specific set-based concept. In the discussion you’re linking to, the original concept is “quantum group”, which is a bit removed from what I’m looking at here. I suspect similar basic issues probably come up, though.

On the topic of categorifying vector spaces, though: of the two views in the linked discussion, where does the Kapranov-Voevodsky 2-vector space fit? Option (b), a “category object in Vect” is a Baez-Crans 2-vector space. So that suggests it follows (a) a a “vector space realized in Cat”. But what does “realized” mean?

In the case of KV, it doesn’t mean “internal to” – a vector space internal to Cat would presumably have a -linear set of objects as well as of morphisms. There’s some sense in which a KV 2-v.s. is “realized in” Cat – for example, instead of being an “abelian group”, it’s an “abelian category”, but to get the definition for the latter from the former takes a pretty big creative step. Various “generalized 2-vector space” ideas by Elgueta have the same feature.

In fact, of all the various ways I know of to “categorify” a vector space, only the Baez-Crans approach fits either of the two methods mentioned in that discussion. Probably there are others that do, also but clearly the meaning of “categorify” still hasn’t been entirely nailed down. Which is part of what makes it an interesting concept.

June 17, 2008 at 3:42 pm

I agree that prying apart the different notions of ‘categorification’, and explaining their relations, is a very important task.

However, it’s tricky. Here’s just one little symptom of how tricky it is:

I see your point… but actually I often

dothink of it working this way. Any Kapranov-Voevodsky 2-vector space has a ‘basis’ of objects, where we take one representative of each isomorphism class of simple objects. We can also form an ordinary vector space with the same basis. And, I often think of this vector space assome sort of decategorificationof the original 2-vector space.Of course, I’m being sloppy here! Every object in the 2-vector space determines a vector in the corresponding vector space, in an obvious way. Isomorphic objects give the same vector. Two objects that give the same vector must be isomorphic. But, not every vector comes from an object: only those vectors that are linear combinations of basis vectors with

natural numbercoefficients!So, as you note, the more restrictive concept of ‘decategorification’ – namely, the process that takes a category and returns the set of isomorphism classes of objects – would take a 2-vector space and return a free module over the natural numbers. Then, to get the vector space described above, we tensor this N-module with our field.

So, maybe we should call this process ‘decategorification followed by complexification’ (when our field is the complex numbers).

But this two-step construction is very important.

For example: suppose G is a finite group. Then the category Rep(G) consisting of finite-dimensional representations is a 2-vector space with one ‘basis object’ per isomorphism class of irreducible representation of G. The corresponding vector space is the vector space R(G) consisting of ‘class functions’ on G – that is, complex functions on G that are constant on conjugacy classes. The map sending objects of Rep(G) to elements of R(G) is called ‘taking the character of a representation’.

There are lots of other examples like this.

Yes: I’m willing to let the word ‘decategorification’ be a bit vague for a while, to avoid locking ourselves into some notion that limits our creativity. But, eventually we’ll want to be more precise….

June 18, 2008 at 7:59 pm

John:

According to the restrictive version of the term, neither (a) the way of getting a vector space from a KV 2-vectior space, nor (b) the way of getting a vector space from the category of stuff types is actually “decategorification”.

The first involves “decategorification” (taking isomorphism classes) and then “complexification”. Noting that the initial set is not linearly independent, one might ask that the set “remember” information about which objects were the basis objects, or failing that it should remember the linear dependencies that already exist.

The second involves decategorification in the same sense, followed by a further quotient by a new equivalence relation, which identifies all stuff types (or -stuff types) with the same cardinality. (It’s probably equivalent to say it’s an equivalence relation that gets along with sums and products, and identifies all the groupoids with cardinality zero). This is where interference happens.

Also, I’d point out the following. Consider:

(1) the procedure you described above for turning a KV 2-vector space into a vector space

(2) the “free KV 2-vector space on a groupoid” construction

(3) the “degroupoidification” functor described in your draft of HDA VII (taking the zeroth homology)

These do not get along nicely. Take the case is a finite group, seen as a 1-object groupoid. Then process (3) gives just a one-dimensional vector space whose basis vector is the lone object of . OTOH, if we do first (1), getting the category , and then (2), getting the vector space of class functions, this will generally be much bigger than what’s given by (3). (This was what I initially expected HDA VII to do).

I still don’t quite see the relation, though it seems pretty clear that given a way to pick a canonical map from the bigger into , one could realize the result of (3) as a quotient of the result of (1) and (2). But it does raise the question of in what sense the construction is a “categorification” of …

June 19, 2008 at 6:35 am

I agree that the restrictive sense of “decategorification” is the fundamental one: the process of turning a small category into a set, namely its set of isomorphism classes of objects. It’s often good to then compose this with other processes.

Degroupoidification is indeed very different in flavor than the hom(-,Vect) construction you mention, which might best be called ‘forming the category of representations of a groupoid’.

I know an answer to your last question, but it’s not thrilling. Take a finite set S and regard it as a discrete groupoid. Form the 2-vector space hom(S,Vect). Then apply the procedure I described to turn this into a vector space: namely, decategorify it and tensor the resulting N-module with C. We get hom(S,C).

Btw, this 2-group conference in Barcelona is pretty interesting. I’ve been talking to Derek a bit about your plans to straighten out the theory of infinite-dimensional 2-Hilbert space. Too bad you’re not here!

January 3, 2009 at 4:04 pm

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