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%%% Primary Title: n-groupoids
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%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
An \emph{$n$- \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}} is an $n$-category such that, for all
$$0 < m \leq n,$$ each $m$-arrow is invertible with respect to the $(m-1)$--composition; in the case of an infinite groupoid, the notation $\infty$-groupoid is used in the literature
(rather than $\omega$-groupoid that has a distinct meaning from that of $\omega$-category).
\end{definition}
\begin{remark}
An important reason for studying $n$--categories, and especially
$n$-groupoids, is to use them as coefficient \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} for \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} \htmladdnormallink{cohomology theories}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}. Thus, some \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/WeakHomotopy.html} defined over Hausdorff spaces that are non-Abelian (or \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}) are relevant to \htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} (\htmladdnormallink{NAAT}{http://planetphysics.us/encyclopedia/NAQAT2.html}) and \htmladdnormallink{NAQAT (or NA-QAT)}{http://planetphysics.org/?op=getobj&from=lec&id=61}.
In particular, a {\em 2-groupoid} is a \htmladdnormallink{2-category}{http://planetphysics.us/encyclopedia/2Category.html} whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are all invertible
ones.
One needs to distinguish between a 2-groupoid and a double-groupoid as the two \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} geometric \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} with special properties that allow generalizations of important \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} in algebraic topology and higher dimensional algebra, such as the \htmladdnormallink{generalized Van Kampen theorem}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} 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 \emph{multiple groupoid} is not at all an obvious generalization or extension of the concept of double groupoid.
\end{remark}
\end{document}
