categorification
The geometric Langlands program
Decategorification is the reverse of vertical categorification and turns an $n$-category into an $(n-1)$-category.
It corresponds in homotopy theory to truncation.
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
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
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
For instance forming the homotopy category of an (∞,1)-category means decategorifying as
Therefore one way to think of vertical categorification is as a right inverse to decategorification.
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.
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.
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.
The decategorifications of finite sets and finite dimensional vector spaces are natural numbers
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Last revised on May 23, 2017 at 18:53:21. See the history of this page for a list of all contributions to it.