nLab vertical categorification

Idea

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.

Variants

As a section of decategorification

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.

Examples

• 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 $\mathbb{N}$:

  $$\mathbb{N} \simeq FinSet_\sim .$$

  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 FinDimVect 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:

  $$\mathbb{N} \;\simeq\; FinDimVect_{/\sim} \,.$$

As internalization in $n Cat$

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.

Examples

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.

• categorified Gram-Schmidt process

• (∞,1)-categorification of differential calculus is Goodwillie calculus

  (∞,1)-categorification of derivative is Goodwillie derivative

  (∞,1)-categorification of chain rule is Goodwillie chain rule

  etc.

As homotopy coherent resolution

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.

Examples

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.

Contrast to horizontal categorification

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 .

Homotopification versus laxification

(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”).

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 ∞-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.

Related entries

• (∞,1)-categorification

• higher structures

• categorification in representation theory

• categorified Dold-Kan correspondence

References

The terminology goes back to

• Louis Crane, David Yetter, Examples of categorification [arXiv:q-alg/9607028]

and was further amplified in

• John Baez, James Dolan, Categorification, in: Higher Category Theory, Contemp. Math. 230, American Mathematical Society (1998) 1-36 [arXiv:math.QA/9802029]

Exposition:

• John Baez, What is categorification?

Lecture notes:

• Alistair Savage, Introduction to categorification [arXiv:1401.6037, slides:pdf]

• Dylan Madden, Introductory Categorification (2015) [pdf]

  (elementary introduction to category theory leading up to the example of the categorified Weyl algebra)

• Mikhail Khovanov (notes by You Qi), Introduction to categorification, lecture notes, Columbia University (2010, 2020) [web, web, full:pdf]

  (focus on link homology)

• Yuri Ximenes Martins, part II of Categorical and Geometrical Methods in Physics, Master’s thesis, UFMG, 2018. (pdf, slides)

• Yuri Ximenes Martins, Chapter 4 of Introdução à Teoria da Homotopia Abstrata, NEA, 2018. (arXiv:2008.05302, hal-02908896v1)

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:

• D. Bourn, J. Penon, Catégorification de structures définies par monade cartésienne, Cahiers de Topologie et Géométrie Différentielle Catégoriques 46, no. 1 (2005), p. 2-52, numdam.