OPERADIC CATEGORIES, A∞-CATEGORIES AND n-CATEGORIES

When I offered to talk here, I asked myself if I had something suitable for a nice short 20 minute talk, and sent in an abstract about “cores of spaces and spectra”. However, since I have been given 50 minutes, I have decided to scrap that for something a little more substantial. It is brand new. I assume that I am among friends and that you won’t mind if some of what I say turns out to be nonsense. On April 1, a paper of mine on triangulated categories was posted on the algebraic K-theory web site. The next day, Maxim Kontsevich called me from Paris to talk about a possible use of “h-categories”, which are homotopical analogues of “A∞-categories”, as an alternative way of thinking about triangulated categories. At the time, I had no idea what an A∞-category was. Now I do, and the notion fits together with some ideas I have long had about parametrizing the composition in a category by use of operads. I wrote up some notes for Kontsevich on this and sent them off on April 15. Two days later, Carlos Simpson gave an Adrian Albert talk at Chicago on n-categories. It was immediately apparent to me that the ideas in my notes for Kontsevich would give a considerably simpler and probably equivalent alternative way of thinking about n-categories. This is based on a model category of enriched operadic A∞-categories. The model category point of view also gives an easy construction of the triangulated derived category of complexes of modules over an A∞-category. I shall try to explain the ideas in the simplest possible terms. There are no real applications yet, but the concepts just feel right to me. I will give the idea by presenting an intuitive example that is naturally related to topological conformal field theory. You have all seen the pair of pants picture of a composite of cobordisms, with say two inputs (around the ankles) and one output (around the waist). Now pants hide more structure, legs with knee joints. Instead of glueing along matching outputs and inputs, one might more naturally and flexibly sew in a cobordism with one input and one output at each joint. For symmetry, we don’t want different looking knees, so we sew in the same cobordism at joints at the same level. Formally, the idea looks like this. Consider the moduli space M (j, k) whose points are (possibly disconnected) Riemann surfaces Σ, say of arbitrary genus or more modestly just of genus zero, together with j + k biholomorphic maps from the disc D into Σ, with disjoint interiors. We think of the first j disks as inputs and the last k as outputs. Let C (j) be the Cartesian product of j − 1 copies of M (1, 1). We get maps θ : C (j)×M (ij−1, ij)× · · · ×M (i0, i1) −→ M (i0, ij) as follows. For (c1, . . . , cj−1) ∈ C (j), we sew the input disc of cr to each of the ir output discs of a point of M (ir−1, ir) and we sew the output disc of cr to each of the