globular ω-groupoid

globular

ω

-groupoid

Definition 0.1

.

An

ω

-

groupoid

has a distinct meaning from that of

ω

-

category

, although certain authors restrict its definition to the latter by adding the

restriction

of

invertible

morphisms

, and thus also assimilate the

ω

-groupoid with the

∞

-groupoid. Ronald Brown and Higgins showed in 1981 that

∞

-groupoids and crossed complexes are

equivalent

. Subsequently,in 1987, these authors introduced the

tensor products

and

homotopies

for

ω

-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

ω

-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

||

≤

1

, but with the cell

structure

for

n

≥

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

∞

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

ω

-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

http://planetphysics.org/?op=getobj&from=lec&id=61

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

∞

-groupoids and crossed complexes. Cahiers Topologie G

é

om. Diff

é

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.

Title

globular

ω

-groupoid

Canonical name

Globularomegagroupoid

Date of creation

2013-03-22 19:21:02

Last modified on

2013-03-22 19:21:02

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

42

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55Q35

Classification

msc 55Q05

Classification

msc 20L05

Classification

msc 18D05

Classification

msc 18-00

Defines

filtered space

Defines

G

n

Defines

n

-globe

Defines

fundamental globular

ω

-groupoid of a filtered topological space

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