nLab n-groupoid

Idea

An $n$-groupoid is an

• n-category in which all k-morphisms are equivalences;

• (n,0)-category;

• $n$-truncated ∞-groupoid (a homotopy n-type).

Definitions

In terms of a known notion of (n,r)-category, we can define an $n$-groupoid explicitly as an $\infty$-category such that:

• every $j$-morphism (at any level) is an equivalence;
• every parallel pair of $j$-morphisms is equivalent, for $j\gt n$.

Or we define an $n$-groupoid abstractly as an n-truncated object in the (∞,1)-category ∞Grpd.

The $n$-groupoids form an (n+1,1)-category, nGrpd.

Models

As Kan complexes

A general model for ∞-groupoids is provided by Kan complexes (the fibrant objects in the classical model structure on simplicial sets which presents ∞Grpd).

In this context an $n$-groupoid in the general sense is modeled by a Kan complex all of whose homotopy groups vanish in degree $k \gt n$. In this generality one also speaks of a homotopy $n$-type or an $n$-truncated object of ∞Grpd.

Every such $n$-type is equivalent to a “small” model, an $(n+1)$-coskeletal Kan complex (see there): one in which every $k$-sphere $\partial \Delta^{k+1}$ for $k \geq n+1$ has a unique filler.

Yet a bit smaller than this are Kan complexes that is an $n$-hypergroupoid, where in addition to these sphere-fillers also the horn fillers in degree $n+1$ are unique.

(For 1-groupoids/1-hypergroupoids this situation is further spelled out here).

Related entries

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

• homotopy n-type / n-groupoid / n-truncated object in an (∞,1)-category

• model structure for homotopy n-types

• n-truncation modality

References

Discussion of homotopy $n$-types/$n$-truncated objects in homotopy type theory:

• Univalent Foundations Project, §7 of: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]

• Nicolai Kraus, Truncation levels in Homotopy Type Theory, Nottingham (2015) [pdf, eprints:28986]

• Felix Cherubini, Egbert Rijke, Thm. 3.10 of: Modal Descent, Mathematical Structures in Computer Science , 31 4 (2021) 363-391 [doi:10.1017/S0960129520000201, arXiv:2003.09713]