Higher-Dimensional Algebra V: 2-Groups

Higher-Dimensional Algebra V: 2-Groups

[John Baez]

Here's a new paper that studies categorified groups and
Lie groups:

Higher-Dimensional Algebra V: 2-Groups
John C. Baez and Aaron D. Lauda

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underlying set G has been replaced by a category and the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 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 G has been replaced by a functor.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 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*, e_x: x* tensor x -> 1 forming an
adjunction.  We define 2-categories of weak and coherent
2-groups, construct an "improvement" 2-functor which turns
weak 2-groups into coherent ones, and prove this 2-functor
is a 2-equivalence of 2-categories.  We also internalize
the concept of coherent 2-group, which gives a way to define
topological 2-groups, Lie 2-groups, affine 2-group schemes,
and the like.  We conclude with a tour of examples.
Diagrammatic methods are emphasized throughout - especially
string diagrams.

This paper will soon appear on the mathematics arXiv, but
their computer seems unable to draw some of the pictures
correctly, so I urge you to try this PDF version instead:

http://math.ucr.edu/home/baez/hda5.pdf

The next paper in this series will study categorified Lie algebras.