(And what's it good for.)
Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
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Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
The Wikipedia answer is one answer that is commonly used: replace sets with categories, replace functions with functors, and replace identities among functions with natural transformations (or isomorphisms) among functors. One hopes for newer deeper results along the way.
In the case of work of Lauda and Khovanov, they often start with an algebra (for example ${\bf C}[x]$ with operators $d (x^n)= n x^{n-1}$ and $x \cdot x^n = x^{n+1}$ subject to the relation $d \circ x = x \circ d +1$) and replace this with a category of projective $R$-modules and functors defined thereupon in such a way that the associated Grothendieck group is isomorphic to the original algebra.
Khovanov's categorification of the Jones polynomial can be thought of in a different way even though, from his point of view, there is a central motivating idea between this paragraph and the preceding one. The Khovanov homology of a knot constructs from the set of $2^n$ Kauffman bracket smoothing of the diagram ($n$ is the crossing number) a homology theory whose graded Euler characteristic is the Jones polynomial. In this case, we can think of taking a polynomial formula and replacing it with a formula that inter-relates certain homology groups.
Crane's original motivation was to define a Hopf category (which he did) as a generalization of a Hopf algebra in order to use this to define invariants of $4$-dimensional manifolds. The story gets a little complicated here, but goes roughly like this. Frobenius algebras give invariants of surfaces via TQFTs. More precisely, a TQFT on the $(1+1)$ cobordism category (e.g. three circles connected by a pair of pants) gives a Frobenius algebra. Hopf algebras give invariants of 3-manifolds. What algebraic structure gives rise to a $4$-dimensional manifold invariant, or a $4$-dimensional TQFT? Crane showed that a Hopf category was the underlying structure.
So a goal from Crane's point of view, would be to construct interesting examples of Hopf categories. Similarly, in my question below, a goal is to give interesting examples of braided monoidal 2-categories with duals.
In the last sense of categorification, we start from a category in which certain equalities hold. For example, a braided monoidal category has a set of axioms that mimic the braid relations. Then we replace those equalities by $2$-morphisms that are isomorphisms and that satisfy certain coherence conditions. The resulting $2$-category may be structurally similar to another known entity. In this case, $2$-functors (objects to objects, morphisms to morphisms, and $2$-morphisms to $2$-morphisms in which equalities are preserved) can be shown to give invariants.
The most important categorifications in terms of applications to date are (in my own opinion) the Khovanov homology, Oszvath-Szabo's invariants of knots, and Crane's original insight. The former two items are important since they are giving new and interesting results.
One way to think of categorification is that it's a generalization of enumerative combinatorics. When a combinatorialist sees a complicated formula that turns out to be positive they think "aha! this must be counting the size of some set!" and when they see an equality of two different positive formulas they think "aha! there must be a bijection explaining this equality!" This is a special case of categorification, because when you decategorify a set you just get a number and when you decategorify a bijection you just get an equality. As a combinatorialist I'm sure you can come up with some examples that nicely illustrate how this sort of categorification is not totally well-defined. ("What exactly do Catalan numbers count?" has many answers rather than a single right answer.)
A more sophisticated kind of categorification in combinatorics is "Combinatorial Species" which categorify power series with positive coefficients.
When people talk about categorification they usually mean something less combinatorial than the above two examples because they're almost always thinking of a different categorification of the natural numbers: Vector spaces. Just like Sets vector spaces have a single invariant which is a nonnegative integer. So when a combinatorialist sees positive numbers they think "aha! the size of a set" the typical categorifier (there are exceptions) thinks "aha! dimensions of vector spaces!"
Furthermore categorification is often dealing with things with more structure. For example, if you're given a ring with a basis such that the product in that basis has positive structure constants (e.g. the Hecke algebra in the Kazhdan-Lusztig basis) you should think "this is Grothendieck group of some tensor category and the basis is the basis of irreducibles." Similarly possibly negative integers can be thought of as dimensions of graded vector spaces.
I think one important point that has been missed here is that there is not (currently) a precise answer to this question.
There is a loose answer along the lines of that which Pete Clark gave, but I think there may be a typo in that response. And, of course, there are specific instances which shed light (and provide new mathematics) as Scott has pointed out.
"As you can see from this article, one aspect of categorification is the systematic negation of "decategorification", which is the process of taking a category (e.g. the category of sets) and replacing it by the class of isomorphism classes of its objects (in that case, the class of cardinal numbers)."
Categorification is NOT the systematic negation of decategorification. Decategorification can be defined in various ways as a systematic process and categorification can be understood as the non-systematic (i.e. creative) process of undoing decategorification.
As a proponent of negative thinking, instead of saying that categorification ‘replaces sets by categories’ (to quote Wikipedia), I would say that we replace truth values by sets, especially the truth values of equations. That is, we acknowledge that there may be many different ways in which something may be true, in particular many different ways in which two things may be the same. And then it is meaningful to ask whether two ways in which these things are the same are the same way (and if so, whether two ways that they are the same are the same way, etc).
In particular, while two elements of a set simply may (or may not) be equal, two objects of a groupoid may be isomorphic in many different ways. And while two parallel isomorphisms in a groupoid may be equal, two parallel equivalences in a $2$-groupoid may be isomorphic in many different ways. Or while one element $x$ of a poset may precede an element $y$, there may be many different morphisms from one object $x$ of a category to an object $y$.
As you can see from these examples, I would distinguish categorification proper from the possibility of adding noninvertible arrows (which I would call ‘laxification’). Often one categorifies and then laxifies, but often one only categorifies.
There are already many good answers to this question given, but I would like to emphasize one aspect of (what it is good for) that hasn't been fully discussed yet.
It's all about the morphisms.
For example, knowing the knowing the Betti numbers of a topological space is really enough to identify cohomology spaces as vector spaces, but this is uninteresting. What is exciting is that suddenly the theory becomes functorial. There is no notion of a "morphism" from the betti numbers of one topological space to those of another, but having morphisms in cohomology effectively gives rise to all the interesting features one could desire - cup products, etc. In addition, one can now take invariants of morphisms (like traces on homology) instead of just invariants of the spaces themselves.
In similar fashion, if one has an additive category with the Krull-Schmidt property, then each element of the additive Grothendieck group uniquely identifies its corresponding object up to isomorphism. It is not in the objects of a categorification where any interesting new information lies, but in the morphisms. Quantum groups had a geometric categorification for some time now, but recent exciting work of Rouquier and Khovanov-Lauda have redescribed this same categorification (see results of Vasserot-Varagnolo). What makes the recent results exciting is that they give an explicit presentation of the morphisms in the category, which was previously not well understood. This has led to a number of new results, but the full implications are still being explored.
Categorification is not just a way to find new invariants, it is a way to add new layers of structure.
A longer answer is certainly called for (but I teach a class at 8 am). The article
http://en.wikipedia.org/wiki/Categorification
gives a good initial explanation. As you can see from this article, one aspect of categorification is the systematic negation of "decategorification", which is the process of taking a category (e.g. the category of sets) and replacing it by the class of isomorphism classes of its objects (in that case, the class of cardinal numbers).
A really beautiful instance of categorification (of the natural numbers) is given in Barry Mazur's article "When is one thing equal to some other thing?":
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf
Highly recommended!
There is now also an expanded nLab entry on this issue: vertical categorification.
Hopefully this will eventually be further expanded to do justice to this important (albeit vague) notion, but it does already mention some crucial aspects and examples not listed at Wikipedia.
-- There is a short paper by Khovanov, Mazorchuk and Stroppel, see "A brief review of abelian categorifications." Theory and Applications of Categories, Vol. 22, No. 19, 2009, pp. 479-508.
The definition suggested in this paper extends: Suppose that $(A,\cdot)$ is a $k$-algebra and $(C,\otimes)$ is a monoidal triangulated category then $C$ categorifies $A$ if the Grothendieck group of $C$ is isomorphic to $A$ (as $k$ algebras). $$(A,\cdot) \cong K_0(C, \otimes)\otimes_{\mathbb{Z}} k$$
The point is that elements $x,y\in A$ now correspond to objects $X,Y\in C$ and the $X\otimes Y$ transforms like $x\cdot y$. Now it makes sense to study, $Hom_C(X,Y)$. If $C$ is the derived category of an abelian category then these maps correspond to extensions which is what is destroyed by the Grothendieck group $K_0$.
If $C$ is a category of chain complexes over vector spaces (or some appropriate category) then for $X\in C$, $[X]\in K_0(C)$ is the Euler characteristic.
-- A different definition that can sometimes be found in the literature involves taking traces (Hochschild homology) rather than using the Grothendieck group.
Given a monoidal category $C$ and an algebra $D$. $C$ categorifies $D$ when $$End(C) \cong D.$$
For instance, this definition can be found in Toen and Vezzosi's paper on elliptic cohomology. One advantage of this definition is that it is easy to extend to higher categories. This definition can be related to the first definition by the Chern character when $End$ can be identified with Hochschild homology. In many examples, $$HH_0(C) \cong K_0(C)$$
-- Sometimes the word categorification of an object $X$ means thinking of $X$ as an object up to homotopy. This can be made precise when $X$ is an algebra over an operad. This usage isn't easy to relate to the other two definitions.
Categorification is a manner of synthesis.
The work of the early logicians (i.e. those people that helped set up the formal systems that modern symbolic logicians are using now), say Boole, Whitehead and Russell, Cantor, was very much an attempt to break down all mathematical objects into one primitive object - the set. This practice, taking place from 1800 - 1900 or so, was a practise of analysis. This led to a duality of true/false, set membership/ or non membership that characterises classical logic today.
Categorification can be understood as the opposite process, a process towards synthesis. Instead of a duality of equality/inequality, a spectrum (dare I say, a continuum) of emergent properties that can be observed as allowed as possibilities. Instead of a single isomorhpism, 2-morphisms also one to speak of a spectrum of properties that can be identified with the object.
What then is the purpose of having this synthesis, this spectrum? Perhaps it is the acknolwedgement that the symbolic duality today does not capture our full experience of space, time and other Platonic objects. A decategorified notion may suffice in the time of the early logicians, but as a whole range of experiences and possibilities, we seek greater differentiation and the possiblity of future mathematicians adding new experiences.
Let us use this answer to bring additional relevant links.
Here is a blog post Categorification step I by Peter Cameron based on a lecture by Igor Frenkel.
I myself heard an amazing lecture by David Kazhdan last year were the primary example was the step of moving from characters to representation.