nLab 2-groupoid

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

$$\array{ && y \\ & \nearrow &\Downarrow& \searrow \\ x &&\stackrel{}{\to}&& z }$$

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)

$$\array{ y &\to& &\to& z \\ \uparrow &\seArrow& &\nearrow& \downarrow \\ \uparrow &\nearrow& &\Downarrow& \downarrow \\ x &\to&&\to& w } \;\;\; \;\;\; \array{ y &\to& &\to& z \\ &\searrow& &\swArrow& \downarrow \\ && &\searrow& \downarrow \\ &&&& w }$$

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

$$\array{ y &\to& &\to& z \\ \uparrow &\seArrow& &\nearrow& \downarrow \\ \uparrow &\nearrow& &\Downarrow& \downarrow \\ x &\to&&\to& w } \;\;\; \stackrel{comp}{\to} \;\;\; \array{ y &\to& &\to& z \\ \uparrow &\searrow& &\swArrow& \downarrow \\ \uparrow &{}_\kappa\Downarrow& &\searrow& \downarrow \\ x &\to&&\to& w } \,.$$

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:

$$\array{\arrayopts{\rowalign{center}} \array{\begin{svg} [[!include monoidal category > pentagonator]] \end{svg}} }$$

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 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.

Related concepts

• Grpd-enriched category

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