nLab
decategorification

Idea

Decategorification is the reverse of vertical categorification and turns an $n$-category into an $(n-1)$-category.

It corresponds in homotopy theory to truncation.

Ben Webster: Perhaps something could be said about an extended TQFT $F$? My understanding was that the decategorification of $F(X)$ was given by $F(X\times S^1)$; is this right?

Urs Schreiber: that process certainly makes an $n$-dimensional QFT becomes an $(n-1)$-dimensional one. It is pretty much exactly the mechanism of fiber integration.

So this certainly does have a flavor of decategorification. But the latter also has a precise sense in terms of taking equivalence classes in a category. So at face value fiber integration is something different. But perhaps there is some change of perspective that allows to regard it as decategorification in the systematic sense.

Definitions

Given a (small or essentially small) category $C$, the set of isomorphism classes $K(C)$ of objects of $C$ is called the decategorification of $C$.

This is a functor

$$K:\mathrm{Cat}\to \mathrm{Set}$$K : Cat \to Set

from the category (or even $2$-category) Cat of (small) categories to the category (or locally discrete 2-category) Set of sets. Notice that we may think of a set as 0-category, so that this can be thought of as

$$K:1\mathrm{Cat}\to 0\mathrm{Cat}.$$K : 1Cat \to 0Cat \,.

Decategorification decreases categorical degree by forming equivalence classes. Accordingly for all $n>m$ and all suitable notions of higher categories one can consider decategorifications

$$n\mathrm{Cat}\to m\mathrm{Cat}.$$n Cat \to m Cat \,.

For instance forming the homotopy category of an (∞,1)-category means decategorifying as

$$(\mathrm{\infty },1)\mathrm{Cat}\to 1\mathrm{Cat}.$$(\infty,1)Cat \to 1 Cat \,.

Therefore one way to think of vertical categorification is as a right inverse to decategorification.

Extra structure

If the category in question has extra 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.

Further examples

• The decategorifications of finite sets and finite dimensional vector spaces are natural numbers

  $$K(\mathrm{FinSet})\simeq \mathbb{N}$$K(FinSet) \simeq \mathbb{N}
  $$K(\mathrm{FinVect})\simeq \mathbb{N}$$K(FinVect) \simeq \mathbb{N}

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