Higher-Dimensional Algebra V: 2-Groups

By John C. Baez, Aaron D. Lauda. Theory and Applications of Categories, 2003

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A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor.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 and y tensor x are isomorphic to 1. 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* and 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_hbar (for integral values of hbar) 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_hbar (for real hbar) described in a companion paper.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ace550ca3b2fe26034e7d5ee23e57b23b3bd8673

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Citations in the corpus

Twisted Differential String and Fivebrane Structures (2009, background): “It is well known from the theory of higher groups that such cocycles can be regarded as higher representations on shifted vector spaces (see [11] and references within).”
Integrating $L_\infty $-algebras (2006, methodology): “So the notion of weak Lie 2-group presented in [3] is maybe not weak enough.”
Exploring 2-group global symmetries (2018, background): “15) to symmetries was pointed out in [11], where the underlying mathematical structure was identified as a 2-group [12] (see sections 1.”
Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces (2004, methodology): “It seems this also was first proved in Sinh’s thesis [28] (a proof can also be found in [2], Sec.”
Membrane sigma-models and quantization of non-geometric flux backgrounds (2012, methodology): “Integration/differentiation between strict Lie 2-algebras and strict Lie 2-groups is described in [5, 6]; a general procedure for integrating L∞-algebras is described in [30, 36].”
n-Groupoids and Stacky Groupoids (2008, background): “…the various kinds of 2-group(oid)s or double groupoids in [5, 8] (see [20, Appendix] for an explanation of the relation between our 2-group and the one in [5]), and are not exactly the same as in [30], as he requires a choice of composition and strict units; however, they are the same as in [14].”
Higher symmetry and gapped phases of gauge theories (2013, background): “Gapped phases of matter are described at long distances by unitary Topological Quantum Field Theories.”
Higher gauge theory (2005, background): “Just as ordinary electromagnetism is described by a connection on a U(1) bundle, usually called the ‘vector potential’ and denoted A, the stringy analogue of electromagnetism is described by a connection on a U(1) gerbe, called the B field.”