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Idea

A $2$-groupoid is

• a (2,0)-category

• a 2-category in which all morphisms are equivalences

• an n-groupoid for $n = 2$

• a $2$-truncated ∞-groupoid.

Definition

Fix a meaning/model of ∞-groupoid, however weak or strict you wish. Then a $2$-groupoid is an $\infty$-groupoid such that all parallel pairs of $j$-morphisms are equivalent for $j \geq 3$. Thus, up to equivalence, there is no point in mentioning anything beyond $2$-morphisms, except whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $2$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $2$-morphisms as equality.

Specific models

There are various objects that model the abstract notion of $2$-groupoid.

Bigroupoids

A bigroupoid is a bicategory in which all morphisms are equivalences.

Bigroupoids may equivalently be thought of in terms of their Duskin nerves. These are precisely the Kan complexes that are 2-hypergroupoids.

2-Hypergroupoids

A $2$-hypergroupoid is a model for a 2-groupoid. This is a simplicial set, whose vertices, edges, and 2-simplices we identify with the objects, morphisms and 2-morphisms of the form

in the 2-groupoid, respectively.

Moreover, the 3-simplices in the simplicial set encode the composition operation: given three composable 2-simplex faces of a tetrahedron (a 3-horn)

the unique composite of them is is a fourth face $\kappa$ and a 3-cell $comp$ filling the resulting hollow tetrahedron:

The 3-coskeletal-condition says that every boundary of a 4-simplex made up of five such tetrahedra has a unqiue filler. This is the associativity coherence law on the comoposition operation:

This says that any of the possible ways to use several of the 3-simpleces to compose a bunch of compsable 2-morphisms are actually equal.

Homotopy 2-types

More generally one may consider a Kan complex that are just homotopy equivalent to a $3$-coskeletal one as a $2$-groupoid – precisely: as representing the same homotopy type, namely a homotopy 2-type.

Strict $2$-groupoids

The general notion of $2$-groupoid above is also called weak $2$-groupoid to distinguish from the special case of strict 2-groupoids. A strict $2$-groupoid is a strict 2-category in which all morphisms are strictly invertible. This is equivalently a certain type of Grpd-enriched category.

Examples

• For $A$ an abelian group, there is its double delooping 2-groupoid $\mathbf{B}^2 A$. This is given by the strict? 2-groupoid that comes from the crossed complex $A \to 0 \stackrel{\to}{\to} 0$. As a Kan complex this is the image under the Dold-Kan correspondence of the chain complex $[A \to 0 \to 0]$.

• The fundamental infinity-groupoid 2-groupoid of a topological space . that is itself a homotopy 2-type.

• The fundamental $\infty$-groupoid of a topological space that is itself a homotopy 2-type.

• The path 2-groupoid of a smooth manifold.

Related concepts