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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.

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.

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 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 May 23, 2017 18:53:21 by Richard Williamson (84.202.247.109)