Higher-dimensional algebra and topological quantum field theory

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 topological quantum field theories (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≥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.