nLab (infinity,0)-category

Redirected from "(∞,0)-categories".
Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

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.

References

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

Last revised on April 25, 2023 at 08:18:49. See the history of this page for a list of all contributions to it.