  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.
