nLab hypergroupoid

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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

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

Definition

Definition

An nn-hypergroupoid is a Kan complex KK in which the horn-fillers are unique in dimension greater than nn:

(k>n)(Λ i[k] K ! Δ[k]). (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 KK has lower dimensional horn fillers. In Beke 04 these are called exact nn-types instead. For a review on the definition see (Pridham 09, section 2).

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