Following the terminology of (n,r)-categories, an (,0)(\infty,0)-category is an ∞-category in which every jj-morphism (for j>0j \gt 0) is an equivalence.

So in an (,0)(\infty,0)-category every morphism is an equivalence. 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 “(,0)(\infty,0)-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.

Last revised on June 8, 2019 at 04:22:43. See the history of this page for a list of all contributions to it.