double groupoid with connection

double groupoid with connection

1

Double Groupoid with Connection

1.1

Introduction: Geometrically defined

double groupoid with connection

In the setting of a

geometrically defined double groupoid with connection

, as in

[

2

]

, (resp.

[

3

]

), there is an appropriate notion of

geometrically thin

square. It was proven in

[

2

]

,
(

Theorem

5.2 (resp.

[

3

]

, Proposition 4)), that in the cases there specified

geometrically and algebraically

thin squares

coincide

.

1.2

Basic definitions

1.2.1

Double Groupoids

Definition 1.1

.

Generally, the

geometry

of squares and their compositions lead to a common

representation

, or

definition

of a

double groupoid

in the following form:

𝒟

=

\xymatrix

@=3pc S

\ar

@¡1ex¿ [r] ^s^1

\ar

@¡-1ex¿ [r]
_t^1

\ar

@¡1ex¿ [d]^  t_2

\ar

@¡-1ex¿ [d]_s_2 & H

\ar

[l]

\ar

@¡1ex¿ [d]^ t

\ar

@¡-1ex¿ [d]_s

V

\ar

[u]

\ar

@¡1ex¿ [r] ^s

\ar

@¡-1ex¿ [r] _t & M

\ar

[l]

\ar

[u]

,

(1.1)

where

M

is a set of ‘points’,

H

,

V

are ‘horizontal’ and ‘vertical’

groupoids

, and

S

is a set of
‘squares’ with two compositions.

The laws for a double groupoid are also defined, more generally, for any

topological space

𝕋

, and make it also describable as a groupoid internal to the

category of groupoids

.

Definition 1.2

.

A map

Φ

:

|

K

|

⟶

|

L

|

where

K

and

L

are
(finite)

simplicial complexes

is

PWL

(

piecewise linear

) if
there exist subdivisions of

K

and

L

relative to which

Φ

is simplicial.

1.3

Remarks

We briefly recall here the related concepts involved:

Definition 1.3

.

A

square

u

:

I

2

⟶

X

in a topological space

X

is

thin

if there
is a factorisation of

u

,

u

:

I

2

⟶

Φ

u

J

u

⟶

p

u

X

,

where

J

u

is a

tree

and

Φ

u

is piecewise linear (PWL, as defined next) on the

boundary

∂

⁡

I

2

of

I

2

.

Definition 1.4

.

A

tree

, is defined here as the

underlying space

|

K

|

of a
finite

1

-

connected

1

-dimensional simplicial complex

K

boundary

∂

⁡

I

2

of

I

2

.

References

1

Ronald Brown:

Topology and Groupoids

, BookSurge LLC (2006).

2

Brown, R., and Hardy, J.P.L.:1976,

Topological groupoids

I:
universal constructions,

Math. Nachr.

, 71: 273–286.

3

Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The

homotopy
double groupoid

of a

Hausdorff space

.,

Theory and pplications of

Categories

10

, 71–93.

4

Ronald Brown R, P.J. Higgins, and R. Sivera.:

Non-Abelian

algebraic topology

,(

in preparation

),(2008).

http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf

(available here as PDF)
,

http://www.bangor.ac.uk/ mas010/publicfull.htm

see also other available, relevant papers at this website.

5

R. Brown and J.–L. Loday: Homotopical excision, and Hurewicz theorems, for

n

–cubes of spaces,

Proc. London Math. Soc.

, 54:(3), 176–192,(1987).

6

R. Brown and J.–L. Loday:

Van Kampen Theorems

for

diagrams

of spaces,

Topology

, 26: 311–337 (1987).

7

R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths
(

Preprint

), 1986.

8

R. Brown and C.B. Spencer: Double groupoids and crossed modules,

Cahiers Top. Géom. Diff.

, 17 (1976), 343–362.

Title

double groupoid with connection

Canonical

name

DoubleGroupoidWithConnection

Date of creation

2013-03-22 19:19:40

Last modified on

2013-03-22 19:19:40

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

11

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U40

Classification

msc 18E05

Classification

msc 18D05

Defines

connection

Defines

double groupoid

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