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n-groupoid (Definition)
Definition 0.1   An $n$- groupoid is an $n$-category such that, for all

$\displaystyle 0 < m \leq n,$
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).
Remark 0.1  

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 NAQAT (or NA-QAT).

In particular, a 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 multiple groupoid is not at all an obvious generalization or extension of the concept of double groupoid.



"n-groupoid" is owned by bci1.

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See Also: 2-category, groupoid, 2-groupoid, n-groupoid, R-algebroid, superalgebroids in higher dimensions, homotopy double groupoid of a Hausdorff space, n-groupoids, category theory

Other names:  n-category
Also defines:  2-groupoid, HDA, higher dimensional algebra, (m-1) arrows, $(m-1)$- morphism, $infinity$-groupoid
Keywords:  n-category with invertible morphisms, HDA, NAAT, NA-QAT, n-groupoids, 2-category, double groupoid, multiple groupoid

Cross-references: generalized Van Kampen theorem, theorems, representations, two-dimensional, concepts, morphisms, 2-category, NAAT, non-Abelian algebraic topology, non-commutative, double groupoids, cohomology theories, non-Abelian, objects, groupoid
There are 24 references to this object.

This is version 13 of n-groupoid, born on 2009-01-29, modified 2009-01-31.
Object id is 448, canonical name is NGroupoid.
Accessed 2236 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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