categorification
homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
Roughly speaking, vertical categorification is a procedure in which structures are generalized from the context of set theory to category theory or from category theory to higher category theory.
What precisely that means may depend on circumstances and authors, to some extent. The following lists some common procedures that are known as categorification. They are in general different but may in cases lead to the same categorified notions, as discussed in the examples.
See also categorification in representation theory.
What has a more specific definition is the process of decategorification: this concretely quotients out k-morphisms in a n-category to produce a $k$-category for some $k \lt n$, in particular a (possibly large) set (the set of isomorphism classes or equivalence classes of objects) when $k = 0$.
One may understand vertical categorification as any operation that is a section of this decategorification operation, and many examples in the literature are of this kind.
The maybe archetypical example of categorification as something that is taken by decategorification to the identity operation is the categorification of the set $\mathbb{N}$ of natural numbers to the category FinSet of finite sets:
The set $\mathbb{N}$ is a 0-category, while $FinSet$ is a 1-category. The isomorphism classes of $FinSet$ however are in canonical bijection with the elements of $\mathbb{N}$:
So $\mathbb{N}$ is the decategorification of $FinSet$ and accordingly $FinSet$ is a (vertical) categorification of $\mathbb{N}$.
From the point of view of category theory, there is some justification to the converse of this statement: the study of the natural numbers is nothing but the study of the isomorphism classes in $FinSet$. In a way, the very notion of counting is about this. This point is made nicely in BaDo98.
But there are other categorifications of the natural numbers, for instance the category $FinVect$ of finite dimensional vector spaces (over any given field). This is not equivalent to $FinSet$, but still, its set of isomorphism classes is also canonically in bijection with the natural numbers:
Common structures such as groups, rings, etc. may be defined entirely diagrammatically as a collection of objects, morphisms and 2-morphisms in a category like Set (the 2-morphisms then are the identities required to hold between the given morphisms).
For instance a group is an object $G$ in Set equipped with a morphism $\mu : G \times G \to G$ and so on, and equipped with a 2-morphism from $\mu \circ Id_G \times \mu$ to $\mu \circ \mu \times Id_G$.
In such diagrammatic incarnation, these definitions may be internalized into other categories. For instance a group internal to Diff is a Lie group.
But similarly one can also internalize in categories of higher categories. Let Cat be the category of (small) categories, regarded as an ordinary category. Then a group internal to Cat is a strict 2-group. This is thought of as a notion of a categorified group.
And under the decategorification functor $Cat \to Set$ every categorified group $\mathbf{G}$ in this sense maps to an ordinary group $G$ and one could speak of $\mathbf{G}$ being “a categorification” of $G$. But notice again how highly non-unique such categorification is. In particular, every ordinary group may trivially be regarded as a strict 2-group. As such every ordinary group is at the same time one of its own categorifications, in this sense.
Not every 2-group is a strict 2-group that is a group object internal to the 1-category Cat. In general, a 2-group is a weak group object in Cat: where in Set all 2-morphisms in the diagrammatic description of the concept of group necessarily had to be identities, in Cat they could be taken to be non-trivial. But if they are, one will usually want 3-morphisms (which now are necessarily identities) to relate various combinations of these 2-morphisms.
One speaks of this process of categorification using weak internalization of diagrammatic descriptions as categorification of structures up to coherent higher equivalences or up to coherent higher homotopies . One way to make this systematic is discussed below.
In the sense of “weakly internalizing in a higher categorical context” we have the following examples of categorification:
the categorification of sheaf is stack, 2-stack, 3, stack, … ∞-stack.
accordingly, the categorification of (sheaf-)topos is (stack-)2-topos, … ∞-stack (∞,1)-topos
using these ∞-stack (∞,1)-topos notions one can then speak of ∞-spaces Lie ∞-groupoids and the like.
Since simplicial objects in any category usually model higher categorial structures, many simplicial objects may be regarded as secretly being objects in higher category theory. For instance simplicial groups, being Kan complexes, may be thought of as ∞-groupoids equipped with a strict group structure, i.e. as strict $\infty$-groups. By the Dold-Kan correspondence it follows then that many objects in homological algebra such as chain complexes are equivalent incarnations of these simplicial (abelian) groups. This way large parts of homological algebra may be regarded as being secretly about “categorified linear algebra” in some sense. For instance the dg-category of chain complexes in many contexts plays the role of a stable (∞,1)-category of ”$(\infty,1)$-vector spaces”. Notably L-∞-algebras may be thought of as Lie algebras weakly internalized in $(\infty,1)$-vector spaces understood in this sense.
When understanding higher linear algebra in this sense, there are important $\infty$-categorifications of notions like integral transforms, Fourier transforms etc. Examples of such categorified linear algebra are Fourier-Mukai transforms or more generally integral transforms. A general theory of this is described at geometric ∞-function theory. Also geometric Langlands duality fits into this context.
The above process of categorification by coherently weak internalization into higher categorical contexts can be made systematic at least in some cases.
If the structures being defined are algebras over an operad $T$ one may think of regarding $T$ as an (∞,1)-operads and then consider the structures of its algebras as algebras over an $(\infty,1)$-operad. These are infinity-categorified versions of the original structures.
For instance this way
the notion of ordinary associative algebra is sent to that of $A_\infty$-algebra
the notion of ordinary Lie algebra to $L_\infty$-algebra.
Some people also speak of horizontal categorification as categorification. This is to be distinguished from vertical categorification.
Some people just say ‘oidification’ for horizontal categorification, in which case it is consistent to speak of vertical categorificaton as just categorification .
(Vertical) categorification can often be usefully decomposed into two operations.
In groupoidal categorification, which may also be called homotopification or groupoidification (although the latter term also has a different meaning, we allow objects to come with automorphisms, those automorphisms to come with automorphisms, and so on. In the limit, this involves replacing sets by ∞-groupoids or homotopy types (hence the name “homotopification”).
In directed categorification, which may also be called directification or laxification, we allow morphisms that were previously required to be invertible to instead be noninvertible (i.e. “directed”).
This all looks good, with one complaint: I don't think anyone would consider laxification (passage from sets to posets, groupoids to categories, categories to $2$-posets, etc) to be ‘categorification’. That is, the term ‘directed categorification’ works best for the combination of groupoidal categorification followed by laxification. Do you agree, Mike? —Toby
Mike Shulman: I actually think those are definitely a type of categorification. I can think of several places where people have explicitly categorified from categories to (1,2)-categories, either with some particular application in mind, or as a waypoint on the way to 2-categories. I can see that the term “directed categorification” might most easily be interpreted to refer to the combination, but I do think “pure directification” is also a form of categorification.
Toby: I think that examples of ‘categorifying’ form categories to $2$-posets would help me understand. (Clear examples of ‘categorifying’ from $(\infty,n)$-categories to $(\infty,n+1)$-categories would be maximally motivating, if you have them too.)
Of course, $2$-posets can be thought of as special $2$-categories, and that may come even more easily than thinking of posets as special categories. But moving from categories to $2$-posets or from sets to posets seems like such a different thing. It probably doesn't help that I don't like the term horizontal categorification either; categories may be a common generalisation of groupoids, posets, and monoids, but out of all of those three originals, only moving from posets (or something even simpler such as sets) to categories seems like ‘categorification’ to me; only that moves us higher in the hierarchy of $n$-categories.
Mike Shulman: I think moving from $(\infty,n)$-categories to $(\infty,n+1)$-categories is definitely vertical categorification. It’s exactly like moving from $n$-categories to $(n+1)$-categories in so many ways. In particular, the collection of all $(\infty,n)$-categories is an $(\infty,n+1)$-category. Or consider the notion of symmetric monoidal (∞,1)-category; I think this is clearly a categorification of the notion of symmetric monoidal ∞-groupoid, i.e. infinite loop space. Not only can a definition of symmetric monoidal (∞,1)-category obtained from a definition of symmetric monoidal ∞-groupoid in a straightforward way, but it also satisfies a version of the microcosm principle: you can define symmetric monoidal objects in any symmetric monoidal (∞,1)-category. The beauty of working in the ∞-context from the start is that you don’t have to invent new coherence isomorphisms when you categorify, but I don’t think that makes it any less categorification.
For the case of (2,1)-categories, here are a few papers that come to mind:
The point about “horizontal categorification” is, of course, that the intuition of “moving us higher in the hierarchy of $n$-categories” is only applicable, basically by definition, to vertical categorification. Horizontal categorification is supposed to be something different. I’d rather not call that “categorification” either, but it is certainly a process of “generalizing something to use categories.” However I do feel, as I’ve said, that “directification” does “move us higher in the hierarchy of $n$-categories”—it’s just that that “hierarchy” is not linear! Remember the periodic table of (n,r)-categories.
Toby: All right, yes, that helps. I'll sleep on it, but there's a good chance that you've convinced me.
If you like negative thinking, then instead of saying that categorification ‘replaces sets by categories’ (to quote Wikipedia), you can 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, and 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).
However, when we apply “replace truth values by sets” to the truth values of the equality relation of a set, we end up with a groupoid, since the equality of a set is symmetric. Thus, 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. Thus, this gives us groupoidal categorification, or homotopification.
To get from groupoids to categories, we need to also allow things which were previously invertible to be noninvertible, i.e. perform “directification.” We could also do this first starting from a set, obtaining a poset (in which the symmetric relation “is equal to” has been replaced by the non-symmetric one “is less than or equal to”). Then when we homotopify a poset, we get a category: 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 $y$.
This can also be understood naturally in the language of (n,r)-categories. Recall that an $(n,r)$-category can be defined as an ω-category in which all cells above dimension $r$ are invertible, and all cells above dimension $n$ are trivial. Thus, groupoidal categorification can be understood as increasing $n$ but keeping $r$ constant, while directification can be understood as increasing $r$ but keeping $n$ constant.
An early article where the concept of categorification is described is
A bit of $n$-Café discussion on this subject can be found here:
A general notion of categorification for structures defined by cartesian monads, which specializes to produce weak n-categories in the sense of Leinster, can be found here: