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_infinity 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.