Category theory is used to study structures in various branches of
mathematics, and higher-dimensional category theory is being developed to
study higher-dimensional versions of those structures. Examples include
higher homotopy theory, higher stacks and gerbes, extended TQFTs,
concurrency, type theory, and higher-dimensional representation theory. In
this talk we will present two general methods for "categorifying" things,
that is, for adding extra dimensions: enrichment and internalisation. We
will show how these have been applied to the definition and study of
2-vector spaces, with 2-representation theory in mind. This talk will be
introductory; in particular it should not be necessary to be familiar with
any category theory other than the basic idea of categories and functors.