The n-Category Café

This is exactly why I’m running my seminar on “Applications of Category Theory” the way I am. These first few weeks I throw out just enough theory of categories qua categories to get our hands dirty, and then move directly to applications to semantics, logic, foundations, physics, universal algebra, and (of course) knot theory.

If we get as many of these nice “concrete” (as opposed to concrete) categories in students’ hands as soon as possible, maybe we can convince the next generation that these things are really useful, and not just abstract nonsense.

I think it’s a perfectly appropriate time and place to ask, Simon, especially since I myself brought up this business of rearrangement and cancellation of critical points! In fact, I owe you an apology: you did ask me before about this. Some other people have asked, too.

The last working title for the paper was “The Geometry of Gray Categories”, and had as its main aim the development of surface diagrams, with emphasis on progressive surface diagrams in the 3-cube, as a means to constructing the free Gray category generated from a Gray computad. The generating 3-cells of the computad are represented by “coning together” pairs of string diagrams (assumed to be in generic position in order to properly account for Gray interchange), and the 3-cells of the free Gray-category are isotopy rel boundary classes of surface diagrams appropriately labeled in the Gray computad. As you can see, this was supposed to be analogous to the progressive string diagrams of Joyal and Street, but raised by one categorical dimension.

The paper (such as it is) is in a somewhat scattered and incomplete state: at the first round of writing it (1996-1997!), there were technical difficulties that had not been properly addressed. It’s clear that surface diagrams in dimension 3 and beyond should be certain stratified spaces, but the critical questions about which geometric category to work in (e.g., topological, PL, or smooth, etc.), and how to circumvent noisome pathologies in a satisfactory way, sort of bogged down the project for a long time.

I guess around 1999 or 2000 I began to take seriously “tame geometry”, as developed by model theorists who work on so-called “o-minimal structures”, as the proper way to take care of issues of pathology. To deal properly with isotopy rel boundary, one of the things I needed was a version of Thom’s first isotopy lemma for stratified spaces that could be proven in the o-minimal framework. It was only in the last few years (after reading around and consulting some experts) that I was convinced that all this could indeed be carried out. But by then, the project had languished for a long time, and so far I haven’t gotten back into it to bring it to completion. Too many distractions…

Thanks for asking. If enough people hold my feet to the fire, I might finish the damned thing once and for all!