Some Research On2-Groups

A2-group can be viewed as a“categorical”version of a group. Weuse a category instead of the underlying set G, and a functor instead of themultiplication map m: G×G�'G in it. In this paper, two versions ofthe concept of2-groups which we call“weak”and“coherent”2-groupswill be introduced in detail. A weak2-group is a weak monoidal category inwhich every morphism is invertible and for every object x there is a“weakinverse”, which is an object y such that x y1y x. A“coherent”2-group is a weak2-group, with every object x equiped with a specified weakinverse xˉand isomorphisms ix: together formingan adjunction. We describe the2-categories of weak2-groups and coherent2-groups. Then an“improvement”2-functor which turns weak2-groups intocoherent2-groups will be given and we will prove that it is a2-equivalencebetween these2-categories. We internalize the concept of coherent2-groupand thus get a quick way to define Lie2-groups. In the end we will give someexamples of Lie2-groups.