Topology and Higher-Dimensional Category Theory: the Rough Idea

Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the higher-dimensional diagrams one draws to represent these structures can be taken quite literally as pieces of topology. Examples of this are the braids in a braided monoidal category, and the pentagon which appears in the definitions of both monoidal category and A∞-space. I will try to give a Friday-afternoonish description of some of the dreams people have for higher-dimensional category theory and its interactions with topology. Grothendieck, for instance, suggested that tame topology should be the study of n-groupoids; others have hoped that an n-category of cobordisms between cobordisms between . . . will provide a clean setting for TQFT; and there is convincing evidence that the whole world of n-categories is a mirror of the world of homotopy groups of spheres. These are notes from talks given in London and Sussex in summer 2001. I thank Dicky Thomas and Roger Fenn for their invitations and the audiences for their comments, many of which are incorporated here. What I want to give you in the next hour is an informal description of what higher-dimensional category theory is and might be, and how it is relevant to topology. There will be no real theorems, proofs or definitions. But to whet your appetite, here’s a question which we’ll reach an answer to by the end: Question What is the close connection between the following two facts? A No-one ever got into trouble for leaving out the brackets in a tensor product of several objects (abelian groups, etc.). For instance, it’s safe to write A⊗B ⊗ C instead of (A⊗B)⊗ C or A⊗ (B ⊗ C). B There exist non-trivial knots (that is, knots which cannot be undone) in R.