A $2$-groupoid is
a 2-category in which all morphisms are equivalences
an n-groupoid for $n = 2$
a $2$-truncated ∞-groupoid.
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.
There are various objects that model the abstract notion of $2$-groupoid.
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.
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.
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.
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.
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 2-groupoid of a topological space.
The fundamental $\infty$-groupoid of a topological space that is itself a homotopy 2-type.
The path 2-groupoid of a smooth manifold.
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 |
Last revised on January 19, 2023 at 11:43:55. See the history of this page for a list of all contributions to it.