Zen from three conferences
Ross Street – 9 April 1997
This was a brief report on three conferences I have just returned from: Northwestern University Algebraic Topology Conference, 23-27 March 1997 (in honour of Prof M. Mahowald) Workshop on Higher Category Theory and Physics, 28-30 March 1997 (Northwestern University, Evanston, Illinois, Organizers: E. Getzler, M. Kapranov) Memorial Conference for Garrett Birkhoff, 1 April 1997 (Jefferson Building Room 250, Harvard University, Oxford St, Cambridge, MA)
In very brief summary, I found that: categories are well used in a wide range of mathematical activities; operads are experiencing an exciting resurgence; and the demand for the development of higher dimensional categories is coming from many fronts outside category theory.
In relation to (c), John Baez gives a case for higher categories in field theory. Other speakers at Conference #II gave other needs from physics. The needs of knot theory, homotopy theory, algebraic geometry, cohomology theory, etc were well articulated.
Good problems bring, with their solutions, more widely applicable techniques. The problem of defining weak n-categories has stimulated two magnificent ideas which I believe will be vastly more applicable than to the problem for which they were devised. These are: (Baez-Dolan) the "plus construction on typed operads"; (Batanin) the omega-category of trees and the globular set T* of a tree T.