Some folks are starting to talk more and more about “categorification”. Others are getting more and more puzzled by what this word means.

Let me tell you what it means.

Actually, I’ll just tell you what it means to categorify a given set. Then you can guess for yourself what it means to categorify a function between sets — and I’ll tell you if your answer is right, if you want.

Since a huge amount of math is about sets and functions, this will give you some idea of what it means to categorify a given mathematical idea, or a given branch of mathematics. For example: to categorify the Jones polynomial, an invariant of knots.

Okay:

To categorify a set S S is to find a category C C and a function

p : Decat ( C ) → S p: Decat(C) \to S

where Decat ( C ) Decat(C) is the set of isomorphism classes of objects of C C .

The classic example: S S is the natural numbers, C C is the category of finite sets, p p is ‘cardinality’.

The other classic example: S S is the natural numbers, C C is the category of finite-dimensional vector spaces, p p is ‘dimension’.

In these examples p p is one-to-one and onto, which is very nice.

Can you guess what I mean by saying “if we categorify the natural numbers to the category of finite sets, addition gets categorified to disjoint union?”

Emmy Noether categorified the concept of ‘Betti number’. Here S S is the set of natural numbers, C C is the category of finitely generated abelian groups, and p p is ‘rank’.

Extending this idea a bit further, we can categorify the set S S of Laurent polynomials with natural number coefficients using category C C of bounded chain complexes of finitely generated abelian groups. Here p p is ‘Poincaré polynomial’.

Khovanov categorified the set S S of Laurent polynomials with integer coefficients using the category C C of bounded chain complexes of ℤ / 2 \mathbb{Z}/2 -graded finitely generated abelian groups. This was the first easy step towards his real accomplishment: categorifying a bunch of polynomial invariants of knots.

In these examples p p is onto, but not one-to-one. If p p isn’t onto, categorification becomes too easy to be interesting, unless we use other tricks, like the ‘Grothendieck group’ trick.

There are many other important examples… but if you know some math, you can probably find your own!