n-groupoidNGroupoid2009-01-29 23:55:522009-01-30 00:26:35Definition2-groupoid(m-1) arrows$(m-1)$- morphism$infinity$-groupoidn-category with invertible morphismsHDANAATNA-QATn-groupoids2-categorydouble groupoidmultiple groupoid% this is the default PlanetPhysics preamble. as your knowledge
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\def\D{\mathsf{D}}\begin{definition}
An \emph{$n$- groupoid} 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 objects for non-Abelian Cohomology theories. Thus, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) are relevant to non-Abelian Algebraic Topology (NAAT) and NA-QAT (or NAQAT).
In particular, a {\em 2-groupoid} is a 2-category whose morphisms are all invertible
ones.
One needs to distinguish between a 2-groupoid and a double-groupoid as the two concepts are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true two-dimensional geometric representations with special properties that allow generalizations of important theorems in algebraic topology and higher dimensional algebra, such as the 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 \emph{multiple groupoid} is not at all an obvious generalization or extension of the concept of double groupoid.
\end{remark}