nLab
3-group

Definition

A 3-group is equivalently

1. a 2-truncated ∞-group;

2. a 2-groupoid $G$ equipped with the structure of a loop space object of a connected 3-groupoid $BG$ (its delooping);

3. a monoidal 2-category in which every object has an weak inverse under the tensor product.

Properties

Presentation by crossed complexes

Some classes of 3-groups are modeled by 2-crossed modules or crossed squares.

Related concepts

• group

• 2-group, crossed module, differential crossed module

• 3-group, 2-crossed module / crossed square, differential 2-crossed module

  • braided 3-group, sylleptic 3-group, symmetric 3-group

• n-group

• ∞-group, simplicial group, crossed complex, hypercrossed complex