PlanetPhysics/N Groupoids

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An $n$- [[../QuantumOperatorAlgebra5/|groupoid]] is an $n$-category such that, for all

$0<m\le n,$ each $m$-arrow is invertible with respect to the $(m-1)$--composition; in the case of an infinite groupoid, the notation $\mathrm{\infty }$-groupoid is used in the literature (rather than $\omega$-groupoid that has a distinct meaning from that of $\omega$-category).

An important reason for studying $n$--categories, and especially $n$-groupoids, is to use them as coefficient [[../TrivialGroupoid/|objects]] for [[../AbelianCategory3/|non-Abelian]] [[../NoncommutativeGeometry4/|cohomology theories]]. Thus, some [[../WeakHomotopy/|double groupoids]] defined over Hausdorff spaces that are non-Abelian (or [[../AbelianCategory3/|non-commutative]]) are relevant to [[../ModuleAlgebraic/|non-Abelian algebraic topology]] ([[../NAQAT2/|NAAT]]) and NAQAT (or NA-QAT).

In particular, a 2-groupoid is a [[../2Category/|2-category]] whose [[../TrivialGroupoid/|morphisms]] are all invertible ones.

One needs to distinguish between a 2-groupoid and a double-groupoid as the two [[../PreciseIdea/|concepts]] are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true [[../CoriolisEffect/|two-dimensional]] geometric [[../CategoricalGroupRepresentation/|representations]] with special properties that allow generalizations of important [[../Formula/|theorems]] in algebraic topology and higher dimensional algebra, such as the [[../SingularComplexOfASpace/|generalized Van Kampen theorem]] with significant consequences that cannot be obtained through Abelian means.

Furthermore, whereas the definition of an $n$-groupoid is a straightforward generalization of a 2-groupoid, the notion of a multiple groupoid is not at all an obvious generalization or extension of the concept of double groupoid.