nLab 3-group



Group Theory

(,1)(\infty,1)-Category theory



A 3-group is equivalently

  1. a 2-truncated ∞-group;

  2. a 2-groupoid GG equipped with the structure of a loop space object of a connected 3-groupoid BG\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.


Presentation by crossed complexes

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

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

Last revised on August 9, 2019 at 08:23:30. See the history of this page for a list of all contributions to it.