Higher-dimensional algebra V: 2-Groups

John C. Baez ; Aaron D. Lauda

Theory and applications of categories, Tome 12 (2004), pp. 423-491

Résumé

A 2-group is a `categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m ; G x G -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 G has been replaced by a functor.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 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 \tensor y \iso 1 \iso y \tensor x. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x' and isomorphisms i_x : 1 -> x \tensor x', e_x : x' \tensor x -> 1 forming an adjunction. We describe 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-equivalence of 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 this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G_h (h in 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_h (h in R) described in a companion paper.

Détail

Classification : 18D05, 18D10, 20J06, 20L05, 22A22, 22E70
Keywords: 2-group, categorical group, Chern-Simons theory, group cohomology