n-groupoid in nLab
source link: https://ncatlab.org/nlab/show/n-groupoid
Go to the source link to view the article. You can view the picture content, updated content and better typesetting reading experience. If the link is broken, please click the button below to view the snapshot at that time.
Contents
1. Idea
An n-groupoid is an
2. Definitions
In terms of a known notion of (n,r)-category, we can define an n-groupoid explicitly as an ∞-category such that:
- every j-morphism (at any level) is an equivalence;
- every parallel pair of j-morphisms is equivalent, for j>n.
Or we define an n-groupoid abstractly as an n-truncated object in the (∞,1)-category ∞Grpd.
The n-groupoids form an (n+1,1)-category, nGrpd.
3. Models
As Kan complexes
A general model for ∞-groupoids is that of Kan complexes. In this context an n-groupoid in the general sense is modeled by a Kan complex all of whose homotopy groups vanish in degree k>n. In this generality one also speaks of a homotopy n-type.
Every such n-type is equivalent to a “small” model, an (n+1)-coskeletal Kan complex: one in which every k-sphere ∂Δk+1 for k≥n+1 has a unique filler.
Even a bit smaller than this is a Kan complex that is an n-hypergroupoid, where in addition to these spheres also the horn fillers in degree n+1 are unique.
4. Related entries
Last revised on June 4, 2020 at 07:16:25. See the history of this page for a list of all contributions to it.
Recommend
About Joyk
Aggregate valuable and interesting links.
Joyk means Joy of geeK