nLab
decategorification

Contents

Idea

Decategorification is the reverse of vertical categorification and turns an nn-category into an (n1)(n-1)-category.

It corresponds in homotopy theory to truncation.

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

Urs Schreiber: that process certainly makes an nn-dimensional QFT becomes an (n1)(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 CC, the set of isomorphism classes K(C)K(C) of objects of CC is called the decategorification of CC.

This is a functor

K:CatSet K : Cat \to Set

from the category (or even 22-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:1Cat0Cat. K : 1Cat \to 0Cat \,.

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

nCatmCat. n Cat \to m Cat \,.

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

(,1)Cat1Cat. (\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 CC is a monoidal category then K(C)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)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(FinSet) K(FinSet) \simeq \mathbb{N}
    K(FinVect) K(FinVect) \simeq \mathbb{N}

Revised on March 13, 2014 06:33:05 by Urs Schreiber (88.128.80.11)