category of groupoids

category of groupoids

1

Category of Groupoids

1.1

Properties

The

category of groupoids

,

G

p

⁢

d

, has several important properties distinct from those of the category of groups,

G

p

, although it does contain the category of groups as a

full subcategory

. One such important property is that

G

p

⁢

d

is

cartesian closed

. Thus, if

J

and

K

are two

groupoids

, one can form a groupoid

G

⁢

P

⁢

D

⁢

(

J

,

K

)

such that if

G

also is a groupoid then there exists a

natural equivalence

G

p

⁢

d

⁢

(

G

×

J

,

K

)

→

G

p

⁢

d

⁢

(

G

,

G

⁢

P

⁢

D

⁢

(

J

,

K

)

)

.

Other important properties of

G

p

⁢

d

are:

1.

The

category

G

p

⁢

d

also has a unit interval object

I

, which is the groupoid with two objects

0

,

1

and exactly one arrow

0

→

1

;

2.

The groupoid

I

has allowed the

development

of a useful

Homotopy Theory

(

http://planetmath.org/http://planetmath.org/encyclopedia/HomotopyCategory2.html

) for groupoids that leads to

analogies

between groupoids and spaces or manifolds; effectively, groupoids may be viewed as “adding the spatial notion of a ‘place’ or location” to that of a group;

3.

Groupoids extend the notion of

invertible

operation

by comparison with that available for groups; such invertible operations also

occur in

the theory of

inverse semigroups

. Moreover, there are interesting

relations

beteen inverse semigroups and ordered groupoids. Such

concepts

are thus applicable to sequential machines and

automata

whose

state spaces

are

semigroups

. Interestingly, the category of finite automata, just like

G

p

⁢

d

is also

cartesian closed

;

4.

The category

G

p

⁢

d

has a

variety

of

types of morphisms

, such as:

quotient

morphisms

,

retractions

,

covering

morphisms,

fibrations

,

universal morphisms

, (in contrast to only the

epimorphisms

and monomorphisms of group theory);

5.

A monoid object,

E

⁢

N

⁢

D

⁢

(

J

)

=

G

⁢

P

⁢

D

⁢

(

J

,

J

)

, also exists in the category of groupoids, that contains a

maximal subgroup

object denoted here as

A

⁢

U

⁢

T

⁢

(

J

)

. Regarded as a

group object

in the category groupoids,

A

⁢

U

⁢

T

⁢

(

J

)

is

equivalent

to a crossed module

C

M

, which in the case when

J

is a group is the traditional crossed module

J

→

A

⁢

u

⁢

t

⁢

(

J

)

, defined by the

inner automorphisms

.

References

1

May, J.P. 1999,

A Concise Course in Algebraic Topology.

, The University of Chicago Press: Chicago

2

R. Brown and G. Janelidze.(2004). Galois theory and a new

homotopy double groupoid

of a map of spaces.(2004).

Applied

Categorical

Structures

,

12

: 63-80. Pdf file in arxiv: math.AT/0208211

3

P. J. Higgins. 1971.

Categories and Groupoids.

, Originally published by: Van Nostrand Reinhold, 1971
Republished in:

Reprints in Theory and Applications of Categories

, No. 7 (2005) pp 1-195:
http://www.tac.mta.ca/tac/reprints/articles/7/tr7.pdf

Title

category of groupoids

Canonical name

CategoryOfGroupoids

Date of creation

2013-03-22 19:15:54

Last modified on

2013-03-22 19:15:54

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U05

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Related topic

GroupoidCategory

Related topic

HomotopyCategory

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