Tom Leinster (University of Cambridge)
T.Leinster@dpmms.cam.ac.uk
There has been a growing feeling recently that many objects naturally arising in topology and geometry can usefully be described using exotic or "higher" categorical structures. I will describe some of the motivating examples, including higher fundamental groupoids and structures relevant to topological quantum field theory. I will also give rough descriptions of some of the categorical structures people have been looking at: n-categories, A∞-categories, and perhaps the quasi-categories of Joyal - and try to give some idea of how actual progress so far measures up to people's dreams.
A link: My preprint "Topology and higher-dimensional category theory: the rough idea": http://uk.arxiv.org/abs/math.CT/0106240