nLab decategorification

Idea

In category theory, by “decategorification” one means (see below) the process which turns a category into a set, namely into its set of isomorphism classes.

Typically one is interested in the case where the category is equipped with extra higher structure (see further below), whence its set of isomorphism classes will carry the corresponding ordinary structure. For example: the decategorification of a monoidal category is canonically a monoid, the decategorification of a rig category is canonically a rig, the decategorification of a 2-group is canonically a plain group, etc.

(Combined with group completion, decategorification of monoidal categories is often known as a form of K-theory in degree 0, see for instance at K-theory of a permutative category.)

In this sense decategorification is a “left inverse” to (vertical) “categorification” (see there for more), namely to the process of asking for category theoretic higher structures analogous to given set theoretic structures.

Crucially, though, decategorification is a systematic process (in fact a 2-functor, see below) while categorification, being a (local) section of this functor involves making choices: There are in general several categorical structures which have the same decategorification. For instance, the rig monoidal categories FinSet (with its cartesian product) and FinDimVect (with its tensor product of vector spaces) both decategorify to the rig monoid of natural numbers (see further examples below).

More generally, in higher category theory there are higher sequences of “higher decategorification” functors which incrementally discard non-invertible higher morphisms and quotient by remaining invertible higher morphisms up to some degree (see below).

In particular, in the homotopy theory of groupoids, higher groupoids and $\infty$-groupoids, namely in $(\infty,0)$-category-theory, decategorification is nothing but truncation and the tower of decategorifications/$n$-truncations is known as the Postnikov tower.

Definitions

For categories

Given an (essentially small) category $\mathcal{C}$, its decategorification is the set $K(C)$ of isomorphism classes of objects of $\mathcal{C}$.

This construction extends to a 2-functor

(1)$$\begin{array}{rcl} \mathllap{K \;\colon\; } Cat &\longrightarrow& Set \\ \mathcal{C} &\mapsto& Obj(\mathcal{C})_{/iso} \end{array}$$

from the (2-)category Cat of (essentially small) categories to the category (or locally discrete 2-category) Set of sets.

Notice that we may think of sets as 0-categories, so that (1) may equivalently be thought of as being of the form

$$K \colon 1Cat \longrightarrow 0Cat \,,$$

which makes manifest that and how decategorification indeed decreases categorical degree.

For generalization of decategorification to higher category theory (below) it is useful to make explicit that the decategorification 2-functor (1) factors as

(2)$$K \,\colon\, Cat \overset{Core}{\longrightarrow} Grpd \overset{\tau_0}{\longrightarrow} Set \,,$$

where

1. $Core$ assigns the “core” of a category, namely the maximal groupoid inside it, hence $Core$ “discards” all non-invertible morphisms;

2. $\tau_0$ is the 0-truncation functor which turns a 1-groupoid into its 0-groupoid of connected components.

It may be interesting to notice here that:

• Core is right adjoint to the embedding $Grpd \hookrightarrow Cat$, hence is a co-reflection,

• $\tau_0$ is left adjoint to the embedding $Set \hookrightarrow Grp$, hence is a reflection.

For higher categories

For any sensible notion of higher categories one will have corresponding analogs of the core- and the truncation-operations used in (2), which allows to define decategorification of higher categories.

For example, for 2-categories there are the evident notions of core in a 2-category.

More generally, for any of the models of $(\infty,n)$-categories we have a coreflection

$$(\infty,n) Cat \underoverset {\underset{Core_n}{\longleftarrow}} {\hookrightarrow} {\;\; \bot \;\;} (\infty,n+1)Cat$$

which together with the $m$-truncation-operation to homotopy $m$-types yields towers of higher decategorification functors

$$(\infty,n+1)Cat \overset{Core_n}{\longrightarrow} (\infty,n)Cat \to \cdots \to (\infty,0)Cat \;\simeq\; \infty Grpd \overset{ \tau_m }{\longrightarrow} m Grpd \to \cdots \to 0 Grpd \;\simeq\; Set \,,$$

any stage of which may reasonably be addressed as an intermediate stage of higher decategorification.

Extra structure

If the category in question has extra higher structure, then this is usually inherited in some decategorified form by its decategorification. For instance if $C$ is a monoidal category then $K(C)$ is a monoid.

A famous example are fusion categories whose decategorifications are called Verlinde rings.

There may also be extra structure induced more directly on $K(C)$. For instance the K-group of an abelian category is the decategorification of its category of bounded chain complexes and this inherits a group structure from the fact that this is a triangulated category (a stable (∞,1)-category) in which there is a notion of homotopy exact sequences.

Examples

• The decategorifications of finite sets and finite dimensional vector spaces (over any ground field) are natural numbers

  $$FinSet_{/\sim} \simeq \mathbb{N}$$
  $$FinDimVect_{/\sim} \simeq \mathbb{N}$$

  For instance the rank-nullity theorem is the decategorification of the splitting lemma in the category FinDimVect.

• The decategorification of the 2-category Grpd of (small) groupoids is equivalent to the homotopy category of homotopy 1-types.

• The decategorification in the same sense of the 2-category of (small) categories is equivalent to the full homotopy category. (explain…)

Parable

From John Baez, https://math.ucr.edu/home/baez/week121.html

If one studies categorification one soon discovers an amazing fact: many deep-sounding results in mathematics are just categorifications of facts we learned in high school! There is a good reason for this. All along, we have been unwittingly “decategorifying” mathematics by pretending that categories are just sets. We “decategorify” a category by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a mere set: the set of isomorphism classes of objects.

To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and “count” it, setting up an isomorphism between it and some set of “numbers”, which were nonsense words like “one, two, three,…” specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented.

According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification. While the historical reality is far more complicated, categorification really has led to tremendous progress in mathematics during the 20th century. For example, Noether revolutionized algebraic topology by emphasizing the importance of homology groups. Previous work had focused on Betti numbers, which are just the dimensions of the rational homology groups. As with taking the cardinality of a set, taking the dimension of a vector space is a process of decategorification, since two vector spaces are isomorphic if and only if they have the same dimension. Noether noted that if we work with homology groups rather than Betti numbers, we can solve more problems, because we obtain invariants not only of spaces, but also of maps.