(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.
Category Theory – What Precisely is Categorification?
big-listbig-picturecategorificationct.category-theory
Related Solutions
I tried to discuss this geometric series example of categorification in one of my answers to another MO question by Jan Weidner, here. I can't tell whether this reply was considered unsatisfactory, but what one considers satisfactory would have to depend on what one is looking for (especially as "categorification" is a vague term -- intentionally so).
Qiaochu has already given one interpretation, rewriting the linear fractional transformation $L = \frac{1}{1-x}$ in the form $L = 1 + xL$ and categorifying that. There are general ways of "categorifying" fixed points of functions, replacing endofunctions by endofunctors and equations by isomorphisms, but one is generally interested in a canonical solution. To illustrate this in the present case, one may categorify the endofunction $f: s \mapsto 1 + xs$ (on $\mathbb{R}$, say) to an endofunctor $F: S \mapsto 1 + X \times S$ on the category of sets. Now, there will generally be many "fixpoint solutions" of endofunctors (meaning a set $L$ together with an isomorphism $F(L) \cong L$), but many people (for example, those who like to talk about datatypes from a categorical perspective) tend to favor a canonical fixpoint solution that arises by applying the following result of Joachim Lambek.
- If $F: C \to C$ is an endofunctor, define an $F$-algebra to be an object $c$ of $C$ together with a morphism $F(c) \to c$. Morphisms are defined in the obvious way (involving a commutative square). Theorem (Lambek): if $(c, \alpha: F(c) \to c)$ is initial in the category of $F$-algebras, then $\alpha$ is an isomorphism.
For $F(S) = 1 + X \times S$ on $Set$, the initial $F$-algebra turns out to be the free monoid on $X$ as already indicated by Qiaochu. Another canonical fixpoint is obtained by dualizing Lambek's theorem, referring instead to terminal coalgebras of endofunctors. The first type of solution is typically recursive and algebraic; the second solution co-recursive and coalgebraic.
But perhaps this interpretation is not considered fully satisfactory if one is after a direct categorification of division or reciprocation which does not fall back on rewriting an equation multiplicatively. For example, when a topologist writes, in categorification mode as it were,
$$"BG = 1//G"$$
for the classifying space (take '1' here to be $EG$ which is homotopy equivalent to a point, and divide out by the action of $G$ on $EG$), he clearly doesn't mean $G \times BG \cong EG$. People like Baez and Dolan have thought about what it means to categorify reciprocals; in the decategorification direction, they define the cardinality of a groupoid $G$, when $G$ is equivalent to a disjoint sum of finite groups $G_x$, where $x$ ranges over the set of connected components, to be
$$card(G) = \sum_{x \in \pi_0(G)} 1/|G_x|$$
so that for example, the cardinality of the groupoid of finite permutations is e. In particular, the cardinality of a finite group is the reciprocal of its order.
In general, as far as I understand things, categorified division doesn't involve dividing by a set, but by a suitable (usually free) group action. Hints of this can be seen in my first answer to the other categorification request linked to above, where the categorified term $X^n/\mathbf{n!}$ means dividing by the usual action of the symmetric group $\mathbf{n!}$ on a tensor power $X^n$. A thorough discussion of this point would lead to considerations in $(\infty, 1)$-category theory, but to give a taste, one may think of a "space" $BG = 1/G$ (taking $G$ for now to be discrete) as given by the topos
$$1/G = Set^G$$
where the '1' here is the one-point "space" given by the topos $Set$; here one can take advantage of an equivalence
$$EG = Set^G/G \simeq Set$$
where the middle term is a slice topos (note: the notation for a slice should not be interpreted as division!), and define a "bundle projection" between toposes:
$$Set^G/G \to Set^G$$
which is the geometric morphism right adjoint to pulling back along $G \to 1$ in $Set^G$; concretely, it takes a morphism $p: X \to G$ in $Set^G/G$ to the internal object of sections.
To return now to the question, where one is attempting to categorify $\frac1{1-x}$, one needs somehow to construe $1-x$ as a group $G$ with a suitable action on a contractible object playing the role of 1. The question is: what is $x$ here (what does it categorify to)? My best attempt at an answer (which I tried to give, maybe not very successfully, in one of my answers to the other MO question) is to write $x = 1 - G = -(G - 1)$, interpreting here $G$ actually as a group object $\mathbb{Z}G$ (more precisely, a cocommutative Hopf algebra, which is a group object in the cartesian category of cocommutative coalgebras), then interpreting $G - 1$ as the so-called augmentation ideal $IG$ fitting in the exact sequence
$$0 \to IG \to \mathbb{Z}G \stackrel{\pi}{\to} \mathbb{Z} \to 0$$
where $\pi$ is an augmentation map which sits at the right end of a bar construction for $EG$ (in an abelian sense, meaning a free resolution for computing cohomology of $G$), and finally interpreting the additive inverse $-(G-1)$, or rather additive inversion generally, as the odd degree shift functor on the category of $\mathbb{Z}_2$-graded abelian groups (I'll call it $\Sigma$, for suspension; it takes a a graded object $(V_0, V_1)$ to $(V_1, V_0)$; I tried to explain why this is sensible here). The point I had alluding to is that there are various models for the contractible simplicial object $EG$, but the relevant one for purposes of this categorification problem seems to be where
$$EG_n = \mathbb{Z}G \otimes (\Sigma IG)^{\otimes n}$$
(more precisely, the free resolution $EG$ is taken to be a normalized bar resolution; see the reference to Hilton-Stammbach in my earlier answer); here $EG_n$ lives in degree $n \pmod 2$. Dividing the total graded space $EG$ by the action of $\mathbb{Z}G$, one is left with the model
$$BG = \sum_{n \geq 0} (\Sigma IG)^{\otimes n}$$
and this ultimately is how I am interpreting the categorification of the equation
$$\frac1{1-x} = \sum_{n \geq 0} x^n$$
This was quite a long reply! I'm having trouble previewing; let's see how this looks...
It should definitely be mentioned here that one often-used categorification of the integers is the sphere spectrum, $\mathbb S$, i.e. the infinite loop space $\mathbb S = \varinjlim \Omega^n S^n$. I've heard this idea attributed to Waldhausen. There's a fun exposition in TWF 102 by John Baez.
The idea is to think of $\mathbb Z$ is the (additive) group completion of the rig $\mathbb N$, and to categorify the group completion process. A natural categorification of the rig $\mathbb N$ is the 2-rig $\mathsf{FinBij}$ of finite sets and bijections, with $\times$ distributing over $\amalg$. So we group complete $\mathsf{FinBij}$ the $\amalg$ part. The thing is, rather than creating a 1-groupoid with two monoidal structures and the appropriate universal property, it's more natural to regard $\mathsf{FinBij}$ as an $\infty$-groupoid with two monoidal structures, and to construct a group completion in the world of $\infty$-groupoids. But by the homotopy hypothesis, an $\infty$-groupoid is the same thing as a space. So we end up with a space that has two monoidal structures, one distributing over the other. The "additive" monoidal structure is appropriately commutative, so this structure can be regarded as the structure of an infinite loop space, otherwise known as a connective spectrum. And the multiplicative structure makes it a "multiplicative infinite loop space", otherwise known as a ring spectrum.
Note that $\pi_0(\mathbb S) = \mathbb Z$, so this is a categorification in the most basic sense.
Best Answer
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.