Subject: categories: From finite sets to Feynman diagrams

From: John Baez <[email-masked]>
Date: Wed, 26 Apr 2000 02:20:46 -0700 (PDT)

Here's a paper that might be of interest to some category theorists,
although it's actually aimed at a general audience, and secretly tries
to get them interested in categories.  Later we'll write a more technical
paper about Feynman diagrams and `stuff operators'.

>From Finite Sets to Feynman Diagrams
John C. Baez and James Dolan

To appear in "Mathematics Unlimited - 2001 and Beyond",
eds. Bjorn Engquist and Wilfried Schmid, Springer Verlag.

Abstract: `Categorification' is the process of replacing equations by
isomorphisms.  We describe some of the ways a thoroughgoing emphasis on
categorification can simplify and unify mathematics.  We begin with
elementary arithmetic, where the category of finite sets serves as a
categorified version of the set of natural numbers, with disjoint union
and Cartesian product playing the role of addition and multiplication.
We sketch how categorifying the integers leads naturally to the infinite
loop space Omega^infinity S^infinity, and how categorifying the positive
rationals leads naturally to a notion of the `homotopy cardinality' of a
tame space.  Then we show how categorifying formal power series leads to
Joyal's `especes des structures', or `structure types'.  We also
describe a useful generalization of structure types called `stuff
types'.  There is an inner product of stuff types that makes the
category of stuff types into a categorified version of the Hilbert space
of the quantized harmonic oscillator.  We conclude by sketching how this
idea gives a nice explanation of the combinatorics of Feynman diagrams.

Available at:

http://arXiv.org/abs/math.QA/0004133