Contents

categorification

# Contents

## 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.

## 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

$K : Cat \to Set$

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

$K : 1Cat \to 0Cat \,.$

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

$n Cat \to m Cat \,.$

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

$(\infty,1)Cat \to 1 Cat \,.$

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

## Decategorification of a 2-category

A precise way to define the decategorification of a 2-category in the above sense is to identify all 1-arrows which are 2-isomorphic (note that this defines an equivalence relation on 1-arrows and respects composition), and to discard the 2-arrows.

## Example

The decategorification, in the above sense, of the 2-category 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.

## 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

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

Last revised on May 20, 2022 at 04:35:06. See the history of this page for a list of all contributions to it.