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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...

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

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

The algebraic characterization of tangles by Freyd, Turaev and Yetter has led to the discovery of new invariants for links. In this dissertation, we prove an analogous result one dimension higher: that the 2-category of unframed, unoriented 2-tangles is the free semistrict braided monoidal 2-category with duals on one unframed self dual object. We give appropriate definitions of the 2-category of 2-tangles, and of duality for monoidal and braided monoidal 2-categories. We use the movie moves of Carter, Rieger and Saito, to show that there is a 2-functor from this 2-category to any braided monoidal 2-category with duals containing an unframed self dual object. Knotted surfaces in 4-space are naturally included in this characterization, sinc...

Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘n-categories’, algebraic structures having objects, morphisms between objects, 2-morphisms between morphisms and so on up to n-morphisms. After a brief introduction to n-categories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle n-categories, cobordism n-categories, and the homotopy n-types of the loop spaces Ω k S k. We conclude by describing a definition of weak n-categories based on the theory of operads. 1

Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R 4 may be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. In this announcement we give a purely algebraic characterization of the 2-category of unframed unoriented 2-tangles in 4 dimensions as the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. A forthcoming paper will contain a proof of this result using the movie moves of Carter, Rieger and Saito. We comment on how one might use this result to construct invariants of 2-tangles. 1 Introduction Recent work on `quantum invariants' of knots, links, tangles, and 3-manifolds depends crucially on a purely algebraic characterization of tangles in 3-dimensional space. It follows from work of Freyd and Yetter, Turaev, and Shum [13, 18, 19, 20] that isotopy classes of framed oriented tangles in 3 dimensions are the morphisms of a cert...

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cal structures suggests, then sustained interaction is essential. Bringing people together will result in significant technical progress and significantly greater conceptual understanding of these structures. Despite their complexity, if developed coherently, these structures, like categories themselves, should eventually become part of the standard mathematical culture. PROJECT DESCRIPTION EXPLORATIONS OF HIGHER CATEGORICAL STRUCTURES AND THEIR APPLICATIONS PROPOSED RESEARCH 1. Historical background and higher homotopies Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their 1945 paper [47]. The language they introduced transformed modern mathematics. Their focus was not on categories and functors, but on natural transformations, which are maps between functors. Higher category theory concerns higher level notions of naturality, which can be viewed as maps between natural transformations, and maps between such maps, and so for

d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surfaces, and so on through n-manifolds that have boundaries with corners. The structure encodes cobordisms between cobordisms between cobordisms. This is an n-category with additional structure, and one needs analogously structured linear categories as targets for the appropriate "functors" that define the relevant TQFT's. One could equally well introduce the basic idea in terms of formulations of programming languages that describe processes between processes between processes. A closely analogous idea has long been used in the study of homotopies between homotopies between homotopies in algebraic topology. Analogous structures appear throughout mathematics. In contrast to the original Eilenb