n-category = (n,n)-category
n-groupoid = (n,0)-category
A simple and very useful incarnation of -groupoids is available using a geometric definition of higher categories in the form of simplicial sets that are Kan complexes: the -cells of the underlying simplicial set are the k-morphisms of the -groupoid, and the Kan horn-filler conditions encode the fact that adjacent -morphisms have a (non-unique) composite -morphism and that every -morphism is invertible with respect to this composition. See Kan complex for a detailed discussion of how these incarnate -groupoids.
One may turn this geometric definition into an algebraic definition of ∞-groupoids by choosing horn-fillers . The resulting notion is that of an algebraic Kan complex that has been shown by Thomas Nikolaus to yield an equivalent (∞,1)-category of -groupoids.
There are various model categories which are Quillen equivalent to . For instance the standard model structure on topological spaces, a model structure on marked simplicial sets and many more. All these therefore present ∞Grpd.
Moreover, the corresponding homotopy category of an (∞,1)-category , hence a category whose objects are homotopy types of -groupoids, is given by the homotopy category of the category of presheaves over any test category. See there for more details.
Every other algebraic definition of omega-categories is supposed to yield an equivalent notion of -groupoid when restricted to -categories all whose k-morphisms are invertible. This is the statement of the homotopy hypothesis, which is for Kan complexes and algebraic Kan complexes a theorem and for other definitions regarded as a consistency condition.
Notably in Pursuing Stacks and the earlier letter to Larry Breen, Alexander Grothendieck initiated the study of -groupoids and the homotopy hypothesis with his original definition of Grothendieck weak infinity-groupoids, that has recently attracted renewed attention.
These are presented by simplicial groups. Notably abelian simplicial groups are therefore a model for abelian -groupoids. Under the Dold-Kan correspondence these are equivalent to non-negatively graded chain complexes, which therefore also are a model for abelian -groupoids. This way much of homological algebra is secretly the study of special -groupoids.
h-level 2 | 0-truncated | discrete space | 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 | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level | -truncated | homotopy n-type | n-groupoid | | h--groupoid | h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid
Formulations in homotopy type theory include
See also at category object in an (infinity,1)-category for more along these lines.