 6. OPERATIONS ON THE SET OF UNFIXED MORPHISMS

15

The above proposition allows to define:

Definition

2112

.

DOM

f

= DOM

F

and IM

f

= IM

F

for

F

f

.

6.5. Rectangular restriction.

Proposition

2113

.

ι

A,B

f

=

ι

A,B

g

if

f

g

.

Proof.

Let

f

g

. Then

g

=

ι

Src

g,

Dst

g

f

. So

ι

A,B

g

=

ι

A,B

ι

Src

g,

Dst

g

f

v

(proposition

2090

)

v

ι

A,B

f

. Similarly,

ι

A,B

f

v

ι

A,B

g

. So

ι

A,B

f

=

ι

A,B

g

.

The above proposition allows to define:

Definition

2114

.

ι

A,B

F

=

ι

A,B

f

for an unfixed morphism

F

and arbitrary

f

F

.

Definition

2115

.

F

A,B

= [

ι

A,B

F

] for every unfixed morphism

F

.

Proposition

2116

.

F

A,B

= id

B

F

id

A

for every unfixed morphism

F

and

objects

A

and

B

.

Proof.

Take

f

F

.

F

A,B

=

[

ι

A,B

F

]

=

[

ι

A,B

f

]

=

[

E

Dst

f,B

f

◦ E

A,

Src

f

] = [id

C

(Dst

f,B

)

B

u

Dst

f

f

id

C

(

A,

Src

f

)

A

u

Src

f

] = [id

C

(Dst

f,B

)

B

id

C

(Dst

f,

Dst

f

)

Dst

f

f

id

C

(Src

f,

Src

f

)

Src

f

id

C

(

A,

Src

f

)

A

] = [id

C

(Dst

f,B

)

B

f

id

C

(

A,

Src

f

)

A

] = [id

C

(Dst

f,B

)

B

]

[

f

]

[id

C

(

A,

Src

f

)

A

] = id

B

F

id

A

.

Proposition

2117

.

f

A

0

,B

0

A

1

,B

1

=

f

A

0

u

A

1

,A

1

u

B

1

.

Proof.

From the previous

f

A

0

,B

0

A

1

,B

1

= id

B

1

id

B

0

f

id

A

0

id

A

1

=

id

B

0

u

B

1

f

id

A

0

u

A

1

=

f

A

0

u

A

1

,A

1

u

B

1

.

Definition

2118

.

f

|

X

=

f

id

X

for every unfixed morphism

f

and

X

A

.

Obvious

2119

.

(

f

|

X

)

|

Y

=

f

X

u

Y

.

6.6. Algebraic properties of the lattice of unfixed morphisms.

The

following proposition allows to easily prove algebraic properties (cf. distributivity)
of the poset of unfixed morphisms:

Theorem

2120

.

The following are mutually inverse bijections:

1

. Let

A

and

B

be objects.

f

7→

[

f

] and

F

7→

ι

A,B

F

are mutually inverse

order isomorphisms between

n

f

unfixed morphisms

A

DOM

f,B

IM

f

o

and

C

(

A, B

). If

A

=

B

they are also semigroup isomorphisms.

2

. Let

T

be an unfixed morphism.

f

7→

[

f

] and

F

7→

ι

Src

t,

Dst

t

F

are mutually

inverse order isomorphisms between the lattice

DT

and

Dt

whenever

t

T

.

Proof.

We will prove that these functions are mutually inverse bijections.

That they are order-preserving is obvious.

1

.

ι

A,B

F

∈ C

(

A, B

) is obvious.

We need to prove that [

f

]

n

f

unfixed morphisms

A

DOM

f,B

IM

f

o

. For this it’s enough to prove

A

DOM[

f

]

B

IM[

f

] what is the same as

A

DOM

f

B

IM

f

what follows

from proposition

2071

.

Because

f

7→

[

f

] is an injection, it is enough

1

to prove that

ι

A,B

[

f

] =

f

. Really,

ι

A,B

[

f

] =

ι

A,B

f

=

f

.

That they are semigroup isomorphisms follows from the already proved formula

[

g

f

] = [

g

]

[

f

].

1

https://math.stackexchange.com/a/3007051/4876