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 $\mathbf{B}G$ (its delooping);

3. a monoidal 2-category in which every object has an weak inverse under the tensor product, every 1-morphism has a weak inverse, and every 2-morphism has an inverse.

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

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/​unit type/​contractible type
h-level 1	(-1)-truncated	contractible-if-inhabited	(-1)-groupoid/​truth value	(0,1)-sheaf/​ideal	mere proposition/​h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/​set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/​groupoid	(2,1)-sheaf/​stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf/​2-stack	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf/​3-stack	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf/​n-stack	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/​∞-stack	h-$\infty$-groupoid