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… (More)

The theory of Lie algebras can be categorified starting from a new notion of ‘2-vector space’, which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector… (More)

Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating… (More)

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics.… (More)

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… (More)

We describe an interesting relation between Lie 2-algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where… (More)

While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the… (More)

'Categorification' is the process of replacing equations by isomor-phisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with… (More)

Let P → M be a principal G-bundle. We construct well-defined substitutes for “Lebesgue measure” on the space A of connections on P and for “Haar measure” on the group G of gauge transformations. More… (More)

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a… (More)