Quantum Gravity Seminar  Winter 2004
Quantization and Categorification
John Baez and Derek Wise
Here are some lecture notes on
quantization and categorification. They are a continuation of the
Fall 2003
notes. As before, these notes have been written by
Derek Wise, based on
lectures by John Baez.
In the fall session we talked about how to quantize and then categorify the
simplest physical system of all: the harmonic oscillator.
Lots of people know how to quantize the harmonic oscillator, but
fewer know that if you then categorify it you recover
Joyal's theory of "structure types", which is important
in combinatorics.
This gives a purely combinatorial explanation of the discreteness
arising in the quantum harmonic oscillator!
In this quarter we'll generalize
this discussion to include quantum field theory, delve deeper into
the category theory behind structure types, and introduce "stuff
types" 
which appear when you categorify the theory of Feynman diagrams.
Besides the lecture notes you can also get homework problems,
which contain miniessays on related topics. You can also get
answers to some of these problems. Different answers shed different
light on the problems, and some include cool extra stuff!
All of these are PDF files:

Week 1  Review of structure
types. This quarter's plan. Fibonacci numbers.

Week 2  The quantum harmonic
oscillator with many degrees of freedom: counting states of a
given energy.

Week 3  The quantum harmonic
oscillator with many degrees of freedom: a square of functors
commuting up to natural isomorphism.
(Only one lecture this week, since
Danny
Stevenson spoke about gerbes on Thursday.)

Week 4 
Violin string theory: quantizing the open string with Dirichlet
boundary conditions. Partition numbers and the partition function
of the quantum string.

Week 5  Two roles of generating functions
in quantum theory: as states of the harmonic oscillator, and as
partition functions. Partition functions in classical and quantum
statistical mechanics.

Week 6 
Generating functions from partition functions: the analogy between
thermodynamic systems and structure types.
Groupoids as "sets with fractional cardinality".
How to evaluate a structure type at a set and get a groupoid 
a categorified version of evaluating a power series at a natural
number and getting a real number.

Week 7 
Applying the structure type F to the set Z and
getting the groupoid F(Z) of "Fstructured, Zcolored
sets": examples. Categorified hyperbolic trigonometry.
Halfcolored sets, double factorials, hypercubes and the square
root of e.

Week 8 
Applying a structure type F to a groupoid Z and getting a groupoid F(Z) 
a categorified version of evaluating a power series at
a real number and getting a real number.
The necessary technology: taking the
weak quotient of a groupoid by a group acting on it.
How to define the weak quotient by a universal property.

Week 9 
How to construct the weak quotient.
F(Z) as the groupoid of "Fstructured finite
sets with elements labelled by objects of the groupoid Z".
Examples: the groupoid of "finite sets labelled by kelement
sets", which has cardinality e^{1/k!}. The groupoid
of "finite sets with elements labelled by finite sets", which
has cardinality e^{e}.

pp. 87 & 88 in color 
much better pedagogically,
but adding these color pages makes the files huge, so we
have included them as a separate file.

Week 10 
Composing structuring types  a categorified version of composing
functions given by power series. Examples: the structure type
"being a finite set with elements labelled by 2element sets",
which has generating function e^{z2/2!}.
Categorified hyperbolic trigonometry revisited.
Exponentiation and connected structures.
The need for stuff types.
Mike Stay
has kindly created an index to these notes:
For the continuation of this course, check out the
Spring 2004 notes!
You can also find links to more references there.
If you discover any errors in the course notes
please email me, and we'll try to correct them.
We're keeping a list of errors that
haven't been fixed yet.
Here's a terse outline
of the whole course:
Yet another view of the same material can
be found in this paper:

John Baez and James Dolan,
From finite sets to
Feynman diagrams, in
Mathematics Unlimited  2001 and Beyond, vol. 1,
edited by Björn Engquist and Wilfried Schmid, SpringerVerlag,
Berlin, 2001, pp. 2950.
If you're interested in structure types and generating functions,
you'll also enjoy these books:

Herbert Wilf, Generatingfunctionology,
Academic Press, Boston, 1994. Available for free
at http://www.cis.upenn.edu/~wilf/.

F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species
and TreeLike Structures, Cambridge, Cambridge U. Press, 1998.
You can also download
TeX or LaTeX files of the
homework problems and some solutions, if for some bizarre reason you want
them. However, the authors keep all rights to this work, except
when stated otherwise.
© 2004 John Baez and Derek Wise
baez@math.removethis.ucr.andthis.edu