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.
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