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