- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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.

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.

2-Hypergroupoids are precisely the Duskin nerves of bigroupoids.

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]

Last revised on August 20, 2022 at 17:09:04. See the history of this page for a list of all contributions to it.