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Topology and Higher-Dimensional Category Theory: the Rough Idea | | Tom Leinster
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28 Jun 2001 | | Subject: | Category Theory; Algebraic Topology; Quantum Algebra | math.CT math.AT math.QA | | Abstract: | 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. | | Source: | arXiv, math.CT/0106240 | | Services: | Forum | Review | PDF | Favorites | |
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