nLab hypergroupoid

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.

Remark

Beware that, while groups are equivalently pointed groupoids with a single object, hypergroups are not hypergroupoids with a single object. For this reason some sources refer to the objects discussed here as Duskin-Glenn hypergroupoids.

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

• Paul G. Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25 1 (1982) 33-105 [doi:10.1016/0022-4049(82)90094-9]

The term exact $n$-type is used in

• Tibor Beke, Higher Čech theory, K-Theory 32 4 (2004) 293-322 [K:0646]

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

• Jonathan Pridham, Presenting higher stacks as simplicial schemes, Adv. Math. 238 (2013) 184-245 [arXiv:0905.4044, doi:10.1016/j.aim.2013.01.009]