### What is Categorification?

#### Posted by John Baez

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 $S$ is to find a category $C$ and a function

$p: Decat(C) \to S$

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

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

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

In these examples $p$ 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 $S$ is the set of natural numbers, $C$ is the category of finitely generated abelian groups, and $p$ is ‘rank’.

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

Khovanov categorified the set $S$ of Laurent polynomials with integer coefficients using the category $C$ of bounded chain complexes of $\mathbb{Z}/2$-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 $p$ is onto, but not one-to-one. If $p$ 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!

## Re: What is Categorification?

Maybe one can argue that there is a single use of the word which deserves to be addressed as the single correct one, but maybe also the word is commonly being used in different ways.

Apart from the

which you describe in the above entry there is also

E.g. 2-groups as “categorified groups”, $L_\infty$-algebras as “categorified Lie algebras” etc.

I recalled this point discussed on Jeffrey Morton’s blog at Two Kinds of Categorification.

Maybe we need more descriptive terms! Categorification in the first sense is maybe more descriptively described as “categorical resolution” (as in: stacky quotient), whereas the second is maybe better “homotopification”.

(Some people say “vertical categorification” for this second meaning and then “horizontal categorification” for “oidification”, i.e. for passing from single to many object versions (group tp groupoid, algebra to algebroid).)