# nLab hypergroupoid

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

An $n$-hypergroupoid is a model for an n-groupoid: it is an Kan complex that is like the nerve of a groupoid ($n= 1$), bigroupoid ($n = 2$) etc.

## Definition

###### Definition

An $n$-hypergroupoid is a Kan complex $K$ in which the horn-fillers are unique in dimension greater than $n$:

$(k \gt n) \Rightarrow \left( \array{ \Lambda^i[k] &\to& K \\ \downarrow & \nearrow_{\exists !} \\ \Delta[k] } \right) \,.$

(The lower dimensional horn fillers of course also exist, but are not in general unique.)

This is due to (Duskin 79, Glenn 82), however their definition does not ask $K$ has lower dimensional horn fillers. In Beke 04 these are called exact $n$-types instead. For a review on the definition see (Pridham 09, section 2).

Equivalently, this are those Kan complexes which are $(n+1)$-coskeletal and such that the $(n+1)$-horns and $(n+2)$-horns have unique fillers.

## Properties

###### Example

1-Hypergroupoids are precisely the nerves of groupoids, see also the example here.

###### Example

2-Hypergroupoids are precisely the Duskin nerves of bigroupoids.

## References

The term hypergroupoid is due to

• John Duskin, Higher-dimensional torsors and the cohomology of topoi: the abelian theory, p. 255-279 in: Applications of sheaves, Lecture Notes in Mathematics 753, Springer (1979) [doi:10.1007/BFb0061822]

and

The term exact $n$-type is used in

On presentation of higher stacks (higher geometric stacks) by hypergroupoid objects:

Last revised on August 20, 2022 at 17:09:04. See the history of this page for a list of all contributions to it.