category of paths on a graph

category of paths on a graph

A nice class of illustrative examples of some notions of

category theory

is provided by

categories

of paths on a graph.

Let

G

be an

undirected graph

. Denote the set of vertices of

G

by “

V

”
and denote the set of edges of

G

by “

E

”.

A path of the graph

G

is an

ordered tuplet

of vertices

(

x

1

,

x

2

,

…

⁢

x

n

)

such that, for all

i

between

1

and

n

-

1

, there exists an edge connecting

x

i

and

x

i

+

i

. As a special case, we allow

trivial

paths which consist
of a single vertex — soon we will see that these in fact play an important
role as

identity elements

in our category.

In our category, the vertices of the graph will be the objects and the

morphisms

will be paths; given two of these objects

a

and

b

, we set

Hom

⁡

(

a

,

b

)

to be the set of all paths

(

x

1

,

x

2

,

…

⁢

x

n

)

such that

x

1

=

a

and

x

n

=

b

. Given an object

a

, we set

1

a

=

(

a

)

,
the trivial path mentioned above.

To finish specifying our category, we need to specify the composition operation.
This

operation

will be the

concatenation

of paths, which is defined as follows:
Given a path

(

x

1

,

x

2

,

…

,

x

n

)

∈

Hom

⁡

(

a

,

b

)

and a path

(

y

1

,

y

2

,

…

,

y

m

)

∈

Hom

⁡

(

a

,

b

)

, we set

a

∘

b

=

(

x

1

,

x

2

,

…

⁢

x

n

,

y

2

,

…

,

y

m

)

.

(Remember that

x

n

=

y

1

=

b

.) To have a bona fide category, we need to
check that this choice

satisfies

the

defining properties

(A1 - A3 in the
entry

http://planetmath.org/node/965

category). This is rather easily verified.

A1:

Given a morphism

(

x

1

,

x

2

,

…

⁢

x

n

)

, it can only
belong to

Hom

⁡

(

a

,

b

)

if

x

1

=

a

and

x

n

=

b

, hence

Hom

⁡

(

a

,

b

)

∪

Hom

⁡

(

c

,

d

)

=

∅

unless

a

=

c

and

b

=

d

.

A2:

Suppose that we have four objects

a

,

b

,

c

,

d

and three
morphisms,

(

x

1

,

x

2

,

…

⁢

x

n

)

∈

Hom

⁡

(

a

,

b

)

,

(

y

1

,

y

2

,

…

⁢

y

m

)

∈

Hom

⁡

(

b

,

c

)

, and

(

z

1

,

z

2

,

…

⁢

z

k

)

∈

Hom

⁡

(

c

,

d

)

. Then,
by the

definition

of the operation

∘

given above,

(

(

x

1

,

x

2

,

…

,

x

n

)

∘

(

y

1

,

y

2

,

…

,

y

m

)

)

∘

(

z

1

,

z

2

,

…

,

z

k

)

=

(

x

1

,

x

2

,

…

,

x

n

,

y

2

,

…

,

y

m

)

∘

(

z

1

,

z

2

,

…

,

z

k

)

=

(

x

1

,

x

2

,

…

,

x

n

,

y

2

,

…

,

y

m

,

z

2

,

…

,

z

k

)

(

x

1

,

x

2

,

…

,

x

n

)

∘

(

(

y

1

,

y

2

,

…

,

y

m

)

∘

(

z

1

,

z

2

,

…

,

z

k

)

)

=

(

x

1

,

x

2

,

…

,

x

n

)

∘

(

y

1

,

y

2

,

…

,

y

m

,

z

2

,

…

,

z

k

)

=

(

x

1

,

x

2

,

…

,

x

n

,

y

2

,

…

,

y

m

,

z

2

,

…

,

z

k

)

.

Since these two quantities are equal, the operation is

associative

.

A3:

It is easy enough to check that paths with a single
vertex act as identity elements:

(

x

1

)

∘

(

x

1

,

x

2

,

…

,

x

n

)

=

(

x

1

,

x

2

,

…

,

x

n

)

(

x

1

,

x

2

,

…

,

x

n

)

∘

(

x

n

)

=

(

x

1

,

x

2

,

…

,

x

n

)

It is also possible to consider the

equivalence class

of paths
modulo retracing. To introduce this category, we first define
a

binary relation

≈

on the class of paths as follows:
Let

A

and

B

be any two paths such that the right

endpoint

of

A

is the same as the left endpoint of

B

, i.e.

A

∈

Hom

⁡

(

a

,

b

)

and

B

∈

Hom

⁡

(

b

,

c

)

for some vertices

a

,

b

,

c

of our graph. Let

d

be any vertex
which shares an edge with

d

. Then we set

A

∘

B

≈

A

∘

(

c

,

d

,

c

)

∘

B

.

Let

∼

be the smallest

equivalence relations

which contains

≈

. We will call this equivalence relation

retracing

.

As defined above, it may not intuitively

obvious

what this

equivalence

amounts to. To this end, we may consider a different
description. Define the

reversal

of a path to be the
path obtained by reversing the order of the vertices traversed:

(

x

1

,

x

2

,

…

,

x

n

-

1

,

x

n

)

-

1

=

(

x

n

,

x

n

-

1

,

…

,

x

2

,

x

1

)

Then we may show that two paths are

equivalent

under retracing
if they may both be obtained from a third path by inserting
terms of the form

X

⁢

X

-

1

. In symbols, we claim that

A

∼

B

if there exists an

integer

n

⁢

Ã

⁢

¯

⁢

Â

⁢

¼

⁢

Å

⁢

¾

⁢

0

and paths

X

1

,

…

⁢

X

n

+

1

,

Y

1

,

…

⁢

Y

n

-

1

,

Z

1

,

…

⁢

Z

n

such that

A

=

X

1

∘

X

1

-

1

∘

Z

1

∘

X

2

∘

X

2

-

1

∘

⋯

∘

X

n

-

1

∘

X

n

-

1

-

1

∘

Z

n

∘

X

n

∘

X

n

-

1

∘

Z

n

∘

X

n

+

1

∘

X

n

+

1

-

1

and

B

=

Y

1

∘

Y

1

-

1

∘

Z

1

∘

Y

2

∘

Y

2

-

1

∘

⋯

∘

Y

n

-

1

∘

Y

n

-

1

-

1

∘

Z

n

∘

Y

n

∘

Y

n

-

1

∘

Z

n

∘

Y

n

+

1

∘

Y

n

+

1

-

1

This

characterization

explains the choice of the term “retracing” —
we do not change the equivalence class of the path if we happen to
wander off somewhere in the course of following the path but then
backtrack and pick the path up again where we left off on our
digression.

Rather than presenting a detailed formal proof, we will sketch how
the two definitions may be shown to be equivalent.

Title

category of paths on a graph

Canonical

name

CategoryOfPathsOnAGraph

Date of creation

2013-03-22 16:45:54

Last modified on

2013-03-22 16:45:54

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

18

Author

rspuzio (6075)

Entry type

Example

Classification

msc 20L05

Classification

msc 18B40

Related topic

IndexOfCategories

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