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.
|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||(-1)-groupoid/truth value||mere proposition, h-proposition|
|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|
Formulations in homotopy type theory include
See also at category object in an (infinity,1)-category for more along these lines.