Ben Webster: Perhaps something could be said about an extended TQFT ? My understanding was that the decategorification of was given by ; is this right?
Urs Schreiber: that process certainly makes an -dimensional QFT becomes an -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.
This is a functor
from the category (or even -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 and all suitable notions of higher categories one can consider decategorifications
For instance forming the homotopy category of an (∞,1)-category means decategorifying as
There may also be extra structure induced more directly on . 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.