Categorification and Geometrisation from Representation Theory, University of Glasgow

April 15, 2009

The relation between n-categories and topology is clarified by a collection of hypotheses, some of which have already been made precise and proved. The "homotopy hypothesis" says that homotopy n-types are the same as n-groupoids. The "stabilization hypothesis" says that each column in the periodic table of n-categories stabilizes at a certain precise point. The "cobordism hypothesis" gives an n-categorical description of cobordisms, while the "tangle hypothesis" does the same for tangles and their higher-dimensional relatives. We shall sketch these ideas, describe recent work by Lurie and Hopkins on the cobordism and tangle hypotheses, and say a bit about how these ideas are related to other lines of work on categorification.

Click on this to see the transparencies of the talk:

- Categorification and Topology - in PDF and Postscript

- John Baez and James Dolan, Higher-Dimensional Algebra and Topological Quantum Field Theory.
- John Baez and James Dolan, Categorification.
- John Baez and Laurel Langford, Higher-Dimensional Algebra IV: 2-Tangles.
- John Baez and Aaron Lauda, A Prehistory of n-Categorical Physics.
- Julie Bergner, A Survey of (∞,1)-Categories.
- Jacob Lurie, On the Classification of Topological Field Theories.

- Jacob Lurie, TQFT and the cobordism hypothesis. Videos and lecture notes.

© 2009 John Baez

except for many figures drawn by Aaron Lauda and Mike Stay

baez@math.removethis.ucr.andthis.edu