nLab (infinity,0)-category

Idea

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

So in an $(\infty,0)$-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 “$(\infty,0)$-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.

References

On model categories presenting $(\infty,0)$-categories, namely models for $\infty$-groupoids (such as the model structure on simplicial groupoids) akin to corresponding models for $(\infty,1)$-categories (such as the model structure on simplicial categories):

• Julia E. Bergner, Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy Appl. 10 2 (2008) 175-193 [doi:10.4310/HHA.2008.v10.n2.a9, doi:0710.2254, erratum:doi:10.4310/HHA.2012.v14.n1.a15]