A 2-group is a 'categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call 'weak' and 'coherent' 2-groups. A weak… (More)
To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the Kac-Moody Lie algebra associated with the graph.
We categorify one-half of the quantum group associated to an arbitrary Cartan datum.
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs). These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of… (More)
We categorify Lusztig's ˙ U – a version of the quantized enveloping algebra U q (sl 2). Using a graphical calculus a 2-category ˙ U is constructed whose split Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilin-ear… (More)
A 2-category was introduced in math.QA/0803.3652 that categorifies Lusztig's integral version of quantum sl(2). Here we construct for each positive integer N a representation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N + 1)-dimensional representation of quantum sl(2).
This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topo-logical quantum field theory. Our chronology ends around 2000, with just a… (More)
In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum… (More)
We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a… (More)
We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetriz-able Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded… (More)