Examples of categorification

Question

What is your favorite example of categorification?

Answer 1

There are a bunch; I don't know that I have a favorite. Here's one for now:

The free commutative monoid functor is a categorification of the exponential function.

Edit: I have been asked to explain this, so I will. We'll interpret "commutative monoid" in any cocomplete symmetric monoidal category $C$ where $\otimes$ distributes over colimits (each $X \otimes -$ preserves colimits); the simplest way of ensuring that is to assume the category is symmetric monoidal closed.

Then, at the level of formulas, the free commutative monoid is

$$\exp(X) = \sum_{n \geq 0} X^{\otimes n}/\mathbf{n!}$$

where $\mathbf{n!}$ is the categorifier's notation for the symmetric group $S_n$, and we divide out by the canonical action of the $S_n$ on $X^{\otimes n}$.

There is an awful lot more to say about the categorified analogy, but I'll just say one. Using the hypotheses on the symmetric monoidal category $C$, the object $\exp(X)$ carries a commutative monoid structure, and in fact it is the free commutative monoid on the object $X$ (think of the symmetric algebra for the category $C = Vect$, for instance). Like any free functor, the left adjoint $\exp$ preserves colimits, for example coproducts. What is the coproduct of two commutative monoids (in the category of commutative monoid objects)? Their tensor product in $C$! Thus, we arrive at the exponential law

$$\exp(X + Y) \cong \exp(X) \otimes \exp(Y)$$

and this has many applications.

Answer 2

The move from betti numbers to homology groups.

Although this might not fit super tightly with the usual modern examples of "categorification" (in the way that say a monoidal category is a categorification of a monoid), it is probably the first and most important example of a concept being categorified, allowing for notions such as functoriality, naturality, etc. to flourish. [No way for a continuous map between spaces to induce a map between betti numbers! The old days before functoriality!].

Answer 3

Here's another example: the functor which maps a group to its classifying space is a categorification of taking the reciprocal.

Edit: The idea is that the total space $EG$ of the classifying bundle of $G$ is contractible and a cofibrant replacement of the point $1$ on which $G$ acts freely. Thus, the construction $BG = EG/G$ is taking a stack-y quotient $BG = 1//G$.

There is a bit more to this idea than may first appear; let me take a related example (which may appear to have some Eulerian "wishful thinking" in it, but have a little faith here!). One way of taking the reciprocal is to pass to a geometric series, so that one suggestive notation for the free monoid construction

$$\sum_{n \geq 0} X^{\otimes n}$$

(in a suitable monoidal category; see my other comment on categorifying exponentiation) is a categorified reciprocal $1/(1 - X)$. We can apply this idea in group cohomology for a group $G$ as follows: think of $\mathbb{Z}$ as being an abelianized point, and consider a standard $G$-free resolution of $\mathbb{Z}$ such as the normalized homogeneous bar resolution, which we can think of as an abelianized $EG$. In one way of constructing this bar resolution (see e.g. Hilton-Stammbach p. 217), the degree $n$ component of $EG$ is

$$\mathbb{Z}G \otimes IG^{\otimes n}$$

where $IG$ is the augmentation ideal, i.e., the kernel of the augmentation map $\varepsilon: \mathbb{Z}G \to \mathbb{Z}$. As a bare module (or seen in degree 0), $IG$ can be seen as an abelianized "$G - 1$". However, in the differential-graded world, it is better to think of it as in degree 1, and this degree 1 shift $\Sigma IG$ can be seen as a categorified "$-IG = 1 - G$" (this may make more sense in the "super-world"; see for example my old notes on the Lie operad when I was doing some work with Saunders Mac Lane, or consider for example the occurrence of signs in the Euler characteristic). So now the total space of the bar resolution $EG$ is the sum of the degree $n$ components

$$\mathbb{Z}G \otimes \sum_{n \geq 0} (\Sigma IG)^{\otimes n}$$

which is an abelianized categorified form of $g \cdot \sum_{n \geq 0} (1 - g)^n$ which is formally $1$ by the geometric series. Very similar types of categorified geometric series constructions occur in Joyal's theory of species (see especially his article on virtual species in Springer LNM 1234), which constructs the Lie operad by categorified constructions [if you read between the lines!], and in the bar resolution for operads as discussed by Ginzburg-Kapranov; I tried to amplify this in my notes on the Lie operad.

Just to put one final gloss on this: consider the Schubert cell decomposition of projective space as a finite geometric series. For a field $k$ we have

$$\mathbb{P}^{n-1}(k) = \frac{k^n - 1}{k - 1} = 1 + k + k^2 + \ldots + k^{n-1}$$

(the '$1$' in the numerator is a zero vector, and the denominator is nonzero scalars $k^\ast$). We can pass to a limit and get infinite-dimensional projective space. Keeping in mind that degree shifts introduce some sign changes in the geometric series, the infinite-dimensional projective space $\mathbb{RP}^\infty$ would be a model of the homotopy quotient $1//\mathbb{R}^* \simeq 1//\mathbb{Z}_2$.

Answer 4

The classical BGG resolution as a categorification of the Weyl character formula.

Answer 5

Grassmannian varieties as categorifying (q-)binomial coefficients.

Answer 6

The Monster Vertex Algebra (aka the Moonshine Module) categorifies Klein's $j$-invariant, in the sense that it is a graded vector space whose graded dimension is the $q$-expansion of $j-744$. More generally, vertex operator algebras often categorify modular functions and (quasi-)modular forms. This has something to do with invariance properties of torus partition functions.

The Monster Lie Algebra categorifies the Koike-Norton-Zagier $j$-function product identity, in the sense that the Weyl-Kac-Borcherds denominator formula of the Lie algebra is precisely this identity. More generally, physicists seem to use constructions with words like "BPS states" and "D-branes" in a way that categorifies automorphic forms on higher rank orthogonal groups (but I don't how it works).

Answer 7

I like one of the simplest and most well known examples: the category of finite sets and bijections functions (see below for comments) categorifies the natural numbers. Or rather it un-de-categorifies the de-categorification that led to much of mathematics in the first place. That makes it pretty special, even if it is rather basic compared with other examples.

Answer 8

A small example, but I think it's nice. The generating function $C(t) = \sum_{n \ge 0} \frac{1}{n+1} {2n \choose n} t^{2n}$ of the Catalan numbers is defined by the identity $C(t) = 1 + t^2 C(t)^2$. So one might try to find a "Catalan object" in some category satisfying an isomorphism generalizing this identity. One can take the corresponding combinatorial species in the sense of Joyal, but another choice is to take the invariant subalgebra of the tensor algebra of the defining representation of $\text{SU}(2)$!

Answer 9

Here's an example I learned from Todd Trimble. Recall that the degree $k$ part of the exterior algebra on a vector space $V$ of dimension $n$ has dimension ${n \choose k}$, and similarly the degree $k$ part of the symmetric algebra has dimension ${n+k-1 \choose k} = \left( {n \choose k} \right)$. So one can think of these constructions as categorifying binomial coefficients. More precisely, the exterior algebra categorifies its Hilbert series $(1 + t)^n$, and the symmetric algebra categorifies its Hilbert series $\frac{1}{(1 - t)^n}$.

But there's more! The duality between the Hilbert series above manifests itself in the identity $\left( {-n \choose k} \right) = (-1)^k {n \choose k}$, which categorifies to the following statement: "the exterior algebra is the symmetric algebra of a purely odd supervector space." So isomorphisms in the category of supervector spaces categorify identities involving negative binomial coefficients.

Answer 10

The plethystic monoidal product, or the substitution product of Joyal species, as a categorification of functional composition.

Answer 11

The empty category is a categorification of the empty set :-))

Answer 12

The sphere spectrum as categorification of the integers, as remarked in a comment of Thomas Kragh below his answer here, and which I believe is due to Joyal.

Hm, that's a lot of answers from me. Should I stop now, or keep going?

Answer 13

I don't know if I have a favourite either, but here's another one:

crossed modules in $Grp$ are strict 2-groups aka group objects in $Cat$ aka category objects in $Grp$.

Answer 14

Of course, my favorite example is the $2$-category of $2$-tangles (defined below) is a categorification of the category of tangles. The category of tangles is a monoidal category with objects that correspond to the non-negative integers, morphisms are generated by $|$, $\cup$, $\cap$, $X$ and $\bar{X}$. In this $1$-category, the Reidemeister moves (and zig-zag and $\psi$-move) are identities.

In the $2$-category of $2$-tangles, the $2$-morphisms are generated by $\{ \} \leftrightarrow O$ (birth or death), $| \ |\leftrightarrow \stackrel{\cup}{\cap}$ (saddle), and the aforementioned five Reidemeister moves (I, II, III, zig-zag, and $\psi$). These are subject to the full set of (35 or so) movie moves. The $2$-category of $2$-tangles is a braided monoidal $2$-category with duals. In fact, it is the free braided monoidal $2$-category with duals on one self-dual object generator (Baez and Langford).

Answer 15

Okay then, here's another. 2-Hilbert spaces as a categorification of Hilbert spaces, and the categorified Gram-Schmidt process (which I first learned from James Dolan).

This may be used to derive a $\mathbb{Z}$-linear basis for the representation ring of $S_n$ that consists of permutation representations, hence a combinatorial alternative to the basis consisting of irreducible representations. The reference above sketches how this works in the case $Rep(S_4)$.

Answer 16

The canonical example in my mind is:

Sets ~> vector spaces ~> linear categories

This is not so trivial -- it is relevant to the topic of extended TQFTs.

Answer 17

The category of groupoids as a categorification of the ring of rational numbers. See this MO question and this n-category cafe post.

Answer 18

Is it perverse to just quote the original inception by Crane?

An obvious nice collection would be the paper with Yetter on examples of Categorification.

However, I actually like another Paper of Yetter's better in this direction; categorical linear algebra.

Also, Rosenberg's Noncommutative spectrum is a categorification: pdf-link. Not in the strict sense, but "morally". That would be undoubtedly my favorite.

Answer 19

In my limited experience of categories, I liked Quillen's notion of homotopy fibre (his paper Higher Algebraic K-Theory I) for a functor between categories modelling the homotopy fibre of any map.