Topology and Higher-Dimensional Category Theory: the Rough Idea

TL;DR

This paper explores the topological nature of higher-dimensional category theory, discussing its connections with topology, algebraic structures, and potential applications in areas like TQFT and homotopy theory.

Contribution

It provides a conceptual overview of the dreams and goals for higher-dimensional category theory and its interplay with topology.

Findings

Higher-dimensional diagrams can be interpreted as topological objects.

Connections between n-categories and homotopy groups of spheres are suggested.

Potential applications include TQFT and cobordism categories.

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.