Remarks on 2-Groups

A 2-group is a categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G x G -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 G has been replaced by a functor. A number of precise definitions of this notion have already been explored, but a full treatment of their relationships is difficult to extract from the literature. Here we describe the relation between two of the most important versions of this notion, 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 and construct an `improvement’ 2-functor which turns weak 2-groups into coherent ones; using this one can show that these 2-categories are biequivalent. We also internalize the concept of a coherent 2-group. This gives a way of defining topological 2-groups, Lie 2-groups, and the like.