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.

Definition

Definition

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

$$(k>n)\Rightarrow \left(\begin{array}{ccc}{\Lambda }^i[k]& \to & K\\ \downarrow & {\nearrow }_{\exists !}\\ \Delta [k]\end{array}\right).$$(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). 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.

Remark

Properties

• 1-hypergroupoids are precisely the nerves of groupoids.

• 2-hypergroupoids are precisely the Duskin nerves of bigroupoids.

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 $n$-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

• Jonathan Pridham, Presenting higher stacks as simplicial schemes, Adv. Math, (arXiv:0905.4044)