nLab
hypergroupoid

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.

Remark

Properties

References

The term hypergroupoid is due to Duskin

  • John Duskin Higher-dimensional torsors and the cohomology of topoi: the abelian theory in Applications of sheaves, number 753 in Lecture Notes in Mathematics, pages 255–279. Springer-Verlag, 1979.

and his student, Paul Glenn:

  • Paul G. Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25, 1982, no. 1, 33-105, MR83j:18016

The term exact nn-type is used in

  • Tibor Beke, Higher Čech theory , K-Theory 32, 2004, 293-322.

Presentation of higher stacks (higher geometric stacks) by hypergroupoid objects is in

Revised on February 24, 2014 13:14:58 by Anonymous Coward (134.76.62.130)