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 $k \geq 4$: a 3-truncated ∞-groupoid.
Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-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 $\geq 4$ 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 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 |
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)