Re: mystification and categorification

[David Yetter <dyetter@math.ksu.edu>]

Categorification is a bit like quantization:  it isn't a construction so much
as a desideratum for a relationship between one thing and another (in the
case of categorification an (n+1)-categorical structure and an n-categorical
structure; in the case of quantization a quantum mechanical system and
a classical mechanical system).

Categorification wants to find a higher-dimensional categorical structure
corresponding to a lower-dimensional one by weakening equations to
natural isomorphisms and imposing new, sensible, coherence conditions.
In general, for the original purpose for which it was proposed--constructions
of TQFT's and models of quantum gravity--one wants the highest categorical
level to have a linear structure (hence Baez wanting tensor product
and a sum it distributes over, rather than cartesian product and coproduct).
Specific lower-dimensional categories with structure are 'categorified' by
finding a higher-dimensional category with the new structure which 'lies over'
the lower dimensional one in the way an additive monoidal category lies
over its Grothendieck rig.

For instance any (k-linear) monoidal category with monoid of isomorphism
classes M is a categorification of M, and more generally (k-linear) monoidal
categories are a categorification of monoids.

A simple example shows why it is not a construction:  commutative monoids
(as rather special categories with one object) admit two different
categorifications:  symmetric monoidal categories and braided monoidal
categories (each regarded as a kind of bicategory with one object).
There is a good argument for regarding braided monoidal categories
as the 'correct' categorification:  the Eckmann-Hilton theorem ('a group
in GROUPS is an abelian group'  or, really as the proof shows, 'a monoid
in MONOIDS is a commutative monoid') 'categorifies' to: A monoidal category
in MONCAT is a braided monoidal category.