categorification
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.
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.
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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