variable groupoid

variable groupoid

\xyoption

curve

Definition 0.1

.

A

variable groupoid

is defined as a family of groupoids

{

𝖦

λ

}

indexed by

a parameter

λ

∈

T

, with

T

being either an

index set

or a class (which may be a time parameter, for

time-dependent or dynamic

groupoids

). If

λ

belongs to a set

M

, then we may consider simply a

projection

𝖦

×

M

⟶

M

, which is an
example of a

trivial

fibration

. More generally, one can consider a

fibration of groupoids

𝖦

↪

Z

⟶

M

(Higgins and Mackenzie, 1990) as defining a non-trivial

variable groupoid

.

Remarks

An indexed family or class of

topological groupoids

[

𝖦

i

]

with

i

∈

I

in the

category

Grpd

of groupoids
with additional axioms, rules, or properties of the underlying topological groupoids,
that specify an indexed family of topological groupoid

homomorphisms

for each variable groupoid

structure

.

Besides systems modelled in terms of a

fibration of groupoids

,
one may consider a

multiple

groupoid

defined as a set of

N

groupoid structures, any distinct pair of which

satisfy

an

interchange law

which can be formulated as follows.
There exists a unique expression with the following

content

:

[

x

y

z

w

]

\objectmargin

=

0

p

t

\xy

(

0

,

4

)

*

+

=

"

a

"

,

(

0

,

-

2

)

*

+

i

=

"

b

"

,

(

7

,

4

)

*

+

j

=

"

c

"

\ar

@

-

>

"

a

"

;

"

b

"

\ar

@

-

>

"

a

"

;

"

c

"

\endxy

,

(0.1)

where

i

and

j

must be distinct for this

concept

to be well defined.
This uniqueness can also be represented by the

equation

(

x

∘

j

y

)

∘

i

(

z

∘

j

w

)

=

(

x

∘

i

z

)

∘

j

(

y

∘

i

w

)

.

(0.2)

Remarks

This illustrates the principle that a 2-dimensional

formula

may be
more comprehensible than a linear one.

Brown and Higgins, 1981a, showed that certain multiple groupoids
equipped with an extra structure called

connections

were

equivalent

to another structure called a

crossed complex

which had already occurred in

homotopy theory

. such as

double, or multiple

groupoids (Brown, 2004; 2005).
For example, the notion of an

atlas

of structures should,
in principle, apply to a lot of interesting, topological and/or
algebraic, structures: groupoids, multiple groupoids,

Heyting
algebras

,

n

-valued

logic algebras

and

C

*

-convolution
-

algebras

. Such examples occur

frequently in

Higher Dimensional Algebra

(HDA).

Title

variable groupoid

Canonical name

VariableGroupoid

Date of creation

2013-03-22 18:15:45

Last modified on

2013-03-22 18:15:45

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

17

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55U05

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Synonym

variable topology

Related topic

VariableCategory

Related topic

HigherDimensionalAlgebra

Related topic

GroupoidCDynamicalSystem

Related topic

HDA

Related topic

VariableTopology

Related topic

HigherDimensionalAlgebraHDA

Related topic

Supercategories3

Related topic

2Category2

Defines

family of groupoids

Defines

GroupoidCDynamicalSystem

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