Higher category theory
higher category theory
Extra properties and structure
An -hypergroupoid is a model for an n-groupoid: it is an Kan complex that is like the nerve of a groupoid (), bigroupoid () etc.
An -hypergroupoid is a Kan complex in which the horn-fillers are unique in dimension greater than :
(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 has lower dimensional horn fillers. In Beke 04 these are called exact -types instead. For a review on the definition see (Pridham 09, section 2).
Equivalently, this are those Kan complexes which are -coskeletal and such that the -horns and -horns have unique fillers.
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 -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