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+) +, we call the operations of I (n−1)+ the ‘n-dimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ω-categories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘n-coherent O-algebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal n-categories’, ‘stable n-categories’, ‘virtual n-functors ’ and ‘representable n-prestacks’. We also describe how n-coherent O-algebra objects may be defined in any (n + 1)-coherent O-algebra. 1

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2-categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `Higher-Dimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...

‘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. 1

In this paper we establish a one-to-one correspondence between S 1-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with presently working as a postdoc at the University of Nottingham, UK 1 group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 1, satisfying a smoothness condition, where a homotopy between maps from [0,1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids. 1

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 ∗-structure, conjugate-linear on the hom-sets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H*-algebras, braided 2-H*-algebras, and symmetric 2-H*-algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized Doplicher-Roberts theorem stating that every symmetric 2-H*-algebra is equivalent to the category Rep(G) of continuous unitary finite-dimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2-H*-algebra on one even object of dimension n. 1

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 concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin ...

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an n-operad A in the author’s sense there exists a symmetric operad S n (A) called the n-fold suspension of A such that the

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one unframed self-dual object’. By this universal property, any unframed self-dual object in a braided monoidal 2-category with duals determines an invariant of 2-tangles in 4 dimensions. 1

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

Recent developments in higher-dimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2-categories, and a center construction for monoidal 2-categories which gives a braided monoidal 2-category. I give generalized center constructions for braided and sylleptic monoidal 2-categories which give sylleptic and symmetric monoidal 2-categories respectively, and I correct some errors in the original center construction for monoidal 2-categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3-types of arc-connected, simply connected spaces, and from higher-dimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...