nLab 3-groupoid



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



The notion of 3-groupoid is the next higher generalization in higher category theory of groupoid and 2-groupoid.


A 3-groupoid is an ∞-groupoid such that all parallel pairs of k-morphism are equivalent for k4k \geq 4: a 3-truncated ∞-groupoid.

Thus, up to equivalence, there is no point in mentioning anything beyond 33-morphisms, except whether two given parallel 33-morphisms are equivalent. This definition may give a concept more general than your preferred definition of 33-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of 33-morphisms as equality.

See also n-groupoid.


A general 3-groupoid is geometrically modeled by a 4-coskeletal Kan complex. Equivalently – via the homotopy hypothesis-theorem – by a homotopy 3-type.

A small model of this is a 3-hypergroupoid, where all horn-filelrs in dimension 4\geq 4 are unique .

A 3-groupoid is algebraically modeled by a tricategory in which all morphisms are invertible, and by a 3-truncated algebraic Kan complex.

A semistrict algebraic model for 3-groupoids is provided by the notion of Gray-groupoid. These in turn are encoded by 2-crossed modules.

An entirely strict algebraic model for 3-groupoids (which no longer models all homotopy 3-types) is a 3-truncated strict omega-groupoid.

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


  • Simona Paoli, Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids, Journal of Pure and Applied Algebra 211 (2007), 801-820. (arXiv)

  • Carlos Simpson, Homotopy types of strict 3-groupoids (arXiv)

On the homotopy hypothesis for Grothendieck 3-groupoids:

Last revised on August 25, 2023 at 17:00:05. See the history of this page for a list of all contributions to it.