| Higher-dimensional algebra V: 2-groups, available as math-QA/0307200 (2008) | |||||||||||||||
Abstract | |||||||||||||||
| Abstract. A 2-group is a `categorified ' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G * G! G has beenreplaced by a functor.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 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a `weak inverse': an object y such that x \Omega y, = 1, = y \Omega x.A coherent 2-group is a weak 2-group in which every object x is equipped with a specifiedweak inverse _ x and isomorphisms ix: 1! x\Omega _x, ex: _x\Omega x! 1 forming an adjunction. Wedescribe 2-categories of weak and coherent 2-groups and an `improvement ' 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalenceof 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the `fundamental 2-group'of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using thisclassification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G ~ ( ~ 2 Z) having G as its group of objects and U(1)as the group of automorphisms of its identity object. These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g ~ ( ~ 2 R) describedin a companion paper. | |||||||||||||||
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