# nLab n-groupoid

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

An $n$-groupoid is an

## Definitions

In terms of a known notion of (n,r)-category, we can define an $n$-groupoid explicitly as an $\infty$-category such that:

• every $j$-morphism (at any level) is an equivalence;
• every parallel pair of $j$-morphisms is equivalent, for $j\gt n$.

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.

## Models

### As Kan complexes

A general model for ∞-groupoids is provided by Kan complexes (the fibrant objects in the classical model structure on simplicial sets which presents ∞Grpd).

In this context an $n$-groupoid in the general sense is modeled by a Kan complex all of whose homotopy groups vanish in degree $k \gt n$. In this generality one also speaks of a homotopy $n$-type or an $n$-truncated object of ∞Grpd.

Every such $n$-type is equivalent to a “small” model, an $(n+1)$-coskeletal Kan complex (see there): one in which every $k$-sphere $\partial \Delta^{k+1}$ for $k \geq n+1$ has a unique filler.

Yet a bit smaller than this are Kan complexes that is an $n$-hypergroupoid, where in addition to these sphere-fillers also the horn fillers in degree $n+1$ are unique.

(For 1-groupoids/1-hypergroupoids this situation is further spelled out here).

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-$\infty$-groupoid

## References

Last revised on February 14, 2023 at 14:03:55. See the history of this page for a list of all contributions to it.