Following the terminology of (n,r)-categories, an -category is an ∞-category in which every -morphism (for ) is invertible.
So in an -category every morphism is invertible. Such ∞-categories are usually called ∞-groupoids.
This is directly analogous to how a 0-category is equivalent to a set, a (1,0)-category is equivalent to a groupoid, and so on. (In general, an (n,0)-category is equivalent to an n-groupoid.)
The term “-category” is rarely used, but does for instance serve the purpose of amplifying the generalization from Kan complexes, which are one model for ∞-groupoids, to quasi-categories, which are a model for (∞,1)-categories.
Revised on July 3, 2009 19:02:29
by Toby Bartels