n-category = (n,n)-category
n-poset = (n−1,n)-category
n-groupoid = (n,0)-category
algebraic definition of higher category
Grothendieck weak ∞-groupoid?
The notion of -groupoid is the generalization of that of group and groupoids to higher category theory:
an -groupoid – equivalently an (∞,0)-category – is an ∞-category in which all k-morphisms for all are equivalences.
The collection of all -groupoids forms the (∞,1)-category ∞Grpd.
Special cases of -groupoids include
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.
The (∞,1)-category of all -groupoids is presented along these lines by the Quillen model structure on simplicial sets, whose fibrant-cofibrant objects are precisely the Kan complexes:
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.
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-groupoid?s, that has recently attracted renewed attention.
One may also consider entirely strict -groupoids, usually called -groupoids or strict ω-groupoids. These are equivalent to crossed complexes of groups and groupoids.
-groupoids with a single object are the delooping of ∞-groups. These are equivalently modeled 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.