What is decategorification?

Question

A decategorification is, roughly, some procedure $\Phi$ which inputs some sort of $n$-categorical data and outputs some sort of $(n-1)$-categorical data. Whatever this means, categorification is understood to be the art of "finding preimages under $\Phi$".

Usually one only talks about decategorification to set up a discussion of categorification. But I'd like to stop and think a bit more about decategorification. First of all, there are different kinds of decategorification.

Examples of decategorification:

1. Given a groupoid, we can extract the set of isomorphism classes of the groupoid.

   • More generally, given a space, we can extract the set of connected components of the space, or perhaps the homotopy groups of the space.

2. Given a category, we can extract the core (a groupoid).

   • More generally, given an $n$-category, we can extract the core $(n-1)$-category.

These two procedures are discussed in the nlab article linked above. But they are not exhaustive -- there are other ways to decategorify.

3. Given a symmetric monoidal category $\mathcal C$ and a dualizable objet $X \in \mathcal C$, we can extract the trace of $X$, which is an element of $End_\mathcal{C}(I)$ where $I \in \mathcal C$ is the unit.

   • This includes the example of taking the Euler characteristic of a finite chain complex or spectrum, or the Hochschild homology of various sorts of input.

   • A shadow on $\mathcal C$ is an example of a decategorification $\Phi : \mathcal C \to \mathcal D$. It turns out to be the same as a functor $THH(\mathcal C) \to \mathcal D$. This is an instance of the microcosm principle -- in order to decategorify the objects of $\mathcal C$, you first decategorify $\mathcal C$ itself.

4. Often we want to tweak the output of one of these procedures.

   • For instance, $K$-theory is a way to decategorify a symmetric monoidal category $\mathcal C$ to get an abelian group or a spectrum. You start by taking the core (as in (2)), but then you additionally group complete (or perhaps also split exact sequences) before arriving at your final decategorification $K(\mathcal C)$. Note that this "tweak" is not simply the composition of a "core" operation with another independent operation -- you need to remember the symmetric monoidal structure (or maybe even which objects are related by exact sequences) from the original input $\mathcal C$ to comptue this.

   • Algebraic $K$-theory (a decategorification procedure decreasing category number from 1 to 0) can be categorified the "noncommutative motives" construction (a decategorfication procedure decreasing category number from 2 to 1). See Blumberg-Gepner-Tabuada.

So I guess my questions are:

Question 1: Are there other kinds of "decategorification"?

Question 2: Is it possible to say more precisely what a "decategorification" is? Can we even give a general "theory of decategorification"?

And for good measure, if "decategorification" is "any procedure which reduces category number", then:

Question 3: What is the "category number" of a mathematical object?

From the above discussion, it already seems that we should maybe at least be talking about a "category binumber" -- given an $(m,n)$-category, a "decategorification" can decrease $m$ or $n$ or both it seems.

And perhaps it would be more appropriate to ask "when is the category number of one mathematical object greater than category number of another categorical object?" -- it seems wrong to think that there's really an "absolute" notion of category number. But rather, given two objects and some way of comparing them, we might be able to say when the comparison procedure is "reducing category number" or "increasing category number".

Answer 1

Taking loops (or, in categorical language, endomorphisms of the monoidal unit) is commonly seen as a type of decategorification.

For example, decategorifying von Neumann algebras produces Hilbert spaces and decategorifying Hilbert spaces produces complex numbers.

This observation is not a mere curiosity, but rather serves as a guiding principle for conjectural geometric models for spectra of increasing chromatic height such as topological K-theory, TMF, etc.