The study of topological quantum field theories increasingly relies upon concepts from higherdimensional 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 ≥ 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.

We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or``S-operads,'' and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O + whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I 0+ =I, I(i+1)+ =(I i+ ) +

Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category of finite sets serves as a categorified version of the set of natural numbers, with disjoint union and Cartesian product playing the role of addition and multiplication. We sketch how categorifying the integers leads naturally to the infinite loop space Ω ∞ S ∞ , and how categorifying the positive rationals leads naturally to a notion of the 'homotopy cardinality' of a tame space. Then we show how categorifying formal power series leads to Joyal's espèces des structures, or 'structure types'. We also describe a useful generalization of structure types called 'stuff types'. There is an inner product of stuff types that makes the category of stuff types into a categorified version of the Hilbert space of the quantized harmonic oscillator. We conclude by sketching how this idea gives a nice explanation of the combinatorics of Feynman diagrams.

In a pilot study, a multicriteria-based patient decision aid for colorectal cancer screening improved patients' decision-making processes but had no effect on the implementation of screening plans.