Aaron D. Lauda

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A 2-group is a ‘categorified’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor.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)
We categorify Lusztig’s U̇ – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2-category U̇ is constructed whose split Grothendieck ring is isomorphic to the algebra U̇. The indecomposable morphisms of this 2-category lift Lusztig’s canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form(More)
We categorify one-half of the quantum group associated to an arbitrary Cartan datum. Cartan data and algebras Af . A Cartan datum (I, ·) consists of a finite set I and a symmetric bilinear form on Z[I] taking values in Z, subject to conditions • i · i ∈ {2, 4, 6, . . .} for any i ∈ I, • 2 i·j i·i ∈ {0,−1,−2, . . .} for any i 6= j in I. We set dij = −2 i·j(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)
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 topological quantum field theory. Our chronology ends around 2000, with just 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 symmetrizable 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 modules(More)