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 TQFTs are to be described as n-category representations. First we describe a ‘suspension’ operation on ncategories, 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 ndimensional 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.