The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a `suspension' operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k greater than or equal to n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n-dimensional unitary extended TQFTs as weak n-functors from the `free stable weak n-category with duals on one object' to the n-category of `n-Hilbert spaces'. We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction.
This paper is now available by anonymous ftp as the file. It is in LaTeX, but to LaTeX it you also need the files auxdefs.sty and diagram.sty, which are also in the directory baez at math.ucr.edu.