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# globular $\omega$-groupoid

###### Definition 0.1.

An $\omega$-groupoid has a distinct meaning from that of $\omega$-category, although certain authors restrict its definition to the latter by adding the restriction of invertible morphisms, and thus also assimilate the $\omega$-groupoid with the $\infty$-groupoid. Ronald Brown and Higgins showed in 1981 that $\infty$-groupoids and crossed complexes are equivalent. Subsequently,in 1987, these authors introduced the tensor products and homotopies for $\omega$-groupoids and crossed complexes. “It is because the geometry of convex sets is so much more complicated in dimensions $>1$ than in dimension $1$ that new complications emerge for the theories of higher order group theory and of higher homotopy groupoids.”

However, in order to introduce a precise and useful definition of globular $\omega$-groupoids one needs to define first the $n$-globe $G^{n}$ which is the subspace of an Euclidean $n$-space $R^{n}$ of points $x$ such that that their norm $||x||\leq 1$, but with the cell structure for $n\geq 1$ specified in Section 1 of R. Brown (2007). Also, one needs to consider a filtered space that is defined as a compactly generated space $X_{{\infty}}$ and a sequence of subspaces $X_{*}$. Then, the $n$-globe $G^{n}$ has a skeletal filtration giving a filtered space ${G^{n}}_{*}$.

Thus, a fundamental globular $\omega$-groupoid of a filtered (topological) space is defined by using an $n$-globe with its skeletal filtration (R. Brown, 2007 available from: arXiv:math/0702677v1 [math.AT]). This is analogous to the fundamental cubical omega–groupoid of Ronald Brown and Philip Higgins (1981a-c) that relates the construction to the fundamental crossed complex of a filtered space. Thus, as shown in R. Brown (2007: http://arxiv.org/abs/math/0702677), the crossed complex associated to the free globular omega-groupoid on one element of dimension $n$ is the fundamental crossed complex of the $n$-globe.

more to come… entry in progress

###### 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).

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.

# References

- 1 Brown, R. and Higgins, P.J. (1981). The algebra of cubes. J. Pure Appl. Alg. 21 : 233–260.
- 2 Brown, R. and Higgins, P. J. Colimit theorems for relative homotopy groups. J.Pure Appl. Algebra 22 (1) (1981) 11–41.
- 3 Brown, R. and Higgins, P. J. The equivalence of $\infty$-groupoids and crossed complexes. Cahiers Topologie G$\'{e}$om. Diff$\'{e}$rentielle 22 (4) (1981) 371–386.
- 4
Brown, R., Higgins, P. J. and R. Sivera,: 2011. “Non-Abelian Algebraic Topology”, EMS Publication.

http://www.bangor.ac.uk/ mas010/nonab-a-t.html ;

http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf - 5
Brown, R. and G. Janelidze: 2004. Galois theory and a new homotopy double groupoid of a map of spaces,
*Applied Categorical Structures*12: 63-80.

## Mathematics Subject Classification

55Q35*no label found*55Q05

*no label found*20L05

*no label found*18D05

*no label found*18-00

*no label found*

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