The n-Category Café

Freed on Chern-Simons

Posted by David Corfield

After all these years there’s a pleasing amount I can understand of Daniel Freed’s very interesting Remarks on Chern-Simons Theory, dedicated to MSRI on its $25^{th}$ birthday. Something I don’t remember ever being discussed is the need to “integrate the scale-dependent and scale-independent aspects” of quantum field theory, as Freed does from p. 26 onwards.

Before this, concerning categorification, he tells us

…an $n$ category has category number $n$, a set has category number $0$, and an element in a set has category number $−1$. It is a feature of many parts of geometry over the past 25 years that the category number of objects and theorems has increased. Whereas theorems about equivalence classes–sets–used to be sufficient, new questions demand that automorphisms be accounted for: whence categories. This trend has affected–some would say infected–parts of quantum field theory as well. (p. 17)

I must find a student to work on metaphors of diseases applied to pieces of mathematics –e.g., Gruppenpest, q-disease (any others?). We might imagine such metaphors being used to deplore, perhaps ironically, a turn taken by a field. There’s a hint of some negative sentiment in Freed’s paper.

I once joked that every mathematician also has a category number, defined as the largest integer $n$ such that (s)he can think hard about $n$-categories for a half-hour without contracting a migraine. When I first said that my own category number was one, and in the intervening years it has remained steadfastly constant whereas that of many around me has climbed precipitously, if not suspiciously.

Of course, the concern could be more that some people are climbing the ladder too rapidly for their own good, even while admitting that the ladder should be climbed.