 CHAPTER 2

Unfixed categories

FiXme

: This is a draft not thoroughly checked for errors.

1. Axiomatics for unfixed morphisms

Definition

2056

.

Category with restricted identities

is defined axiomatically:

Restricted identity

id

C

(

A,B

)

X

and

projection

A

7→

[

A

] are described by the axioms:

1

.

C

is a category with the set of objects

Z

;

2

. every Hom-set

C

(

A, B

) is a lattice;

3

.

Z

and

A

are lattices;

4

.

A

[

A

] is a lattice embedding from

C

(

A, B

) to

A

whenever

A

ranges a

Hom-set

C

(

A, B

);

5

. id

C

(

A,B

)

X

Hom

C

(

A, B

) whenever

A

3

X

v

[

A

]

u

[

B

];

6

. id

C

(

A,A

)

[

A

]

= 1

C

A

;

7

. id

C

(

B,C

)

Y

id

C

(

A,B

)

X

= id

C

(

A,C

)

X

u

Y

whenever

A

3

X

v

[

A

]

u

[

B

] and

A

3

Y

v

[

B

]

u

[

C

];

8

.

A

A

B

Z

:

A

v

[

B

].

For a

partially ordered category with restricted identities

axiom

X

v

Y

id

C

(

A,B

)

X

v

id

C

(

A,B

)

Y

.

For

dagger categories with restricted identities

id

C

(

A,B

)

X

= id

C

(

B,A

)

X

.

Definition

2057

.

I call a category with restricted identities

injective

when the

axiom

X

6

=

Y

id

C

(

A,B

)

X

6

= id

C

(

A,B

)

Y

whenever

X, Y

v

[

A

]

u

[

B

] holds.

Definition

2058

.

Define

E

A,B

C

= id

C

(

A,B

)

[

A

]

u

[

B

]

.

Proposition

2059

.

1

. If [

A

]

v

[

B

] then

E

A,B

C

is a monomorphism.

2

. If [

A

]

w

[

B

] then

E

A,B

C

is an epimorphism.

Proof.

We’ll prove only the first as the second is dual.

Let

E

A,B

C

f

=

E

A,B

C

g

. Then

E

B,A

C

◦ E

A,B

C

f

=

E

B,A

C

◦ E

A,B

C

g

; 1

A

f

= 1

A

g

;

f

=

g

.

Proposition

2060

.

E

B,C

C

◦ E

A,B

C

=

E

A,C

C

if

B

w

A

u

C

(for every sets

A

,

B

,

C

).

Proof.

E

B,C

C

◦ E

A,B

C

=

E

A,C

C

is equivalent to:

id

C

(

B,C

)

B

u

C

id

C

(

A,B

)

A

u

B

= id

C

(

A,C

)

A

u

C

what is obviously true.

2. Rectangular embedding-restriction

Definition

2061

.

ι

B

0

,B

1

f

=

E

Dst

f,B

1

C

f

◦ E

B

0

,

Src

f

C

for

f

Hom

C

(

A

0

, A

1

).

For brevity

ι

B

f

=

ι

B,B

f

.

6