background image

15.4 Other

Conjecture 181.

Values

x

i

(for every

i

dom

x

) can be restored from the value of

Q

(

C

)

x

provided

that

x

is an indexed family of non-zero reloids.

Conjecture 182.

Values

x

i

(for every

i

dom

x

) can be restored from the value of

Q

(

DP

)

x

provided that

x

is an indexed family of non-zero funcoids.

Definition 183.

Let

f

P

Z

`

Y

where

Z

is a set and

Y

is a function.

Pr

k

(

D

)

f

=

Pr

k

{

curry

z

|

z

f

}

.

Proposition 184.

Pr

k

(

D

)

Q

(

D

)

F

=

F

k

for every indexed family

F

of non-empty relations.

Proof.

Obvious.

Corollary 185.

GR Pr

k

(

D

)

Q

(

D

)

F

=

GR

F

k

and form Pr

k

(

D

)

Q

(

D

)

F

=

form

F

k

for every indexed

family

F

of non-empty anchored relations.

16 Coordinate-wise continuity

Theorem 186.

Let

µ

and

ν

are indexed (by some index set

n

) families of endo-morphisms for a

partially ordered dagger category with star-morphisms, and

f

i

Hom

(

Ob

µ

i

;

Ob

ν

i

)

for every

i

n

.

Then:

1.

i

n

:

f

i

C

(

µ

i

;

ν

i

)

Q

(

C

)

f

C

Q

(

C

)

µ

;

Q

(

C

)

ν

;

2.

i

n

:

f

i

C

(

µ

i

;

ν

i

)

Q

(

C

)

f

C

Q

(

C

)

µ

;

Q

(

C

)

ν

;

3.

i

n

:

f

i

C

′′

(

µ

i

;

ν

i

)

Q

(

C

)

f

C

′′

Q

(

C

)

µ

;

Q

(

C

)

ν

.

Proof.

Using the corollary 129:

1.

i

n

:

f

i

C

(

µ

i

;

ν

i

)

⇔ ∀

i

n

:

f

i

µ

i

ν

i

f

i

Q

i

n

(

C

)

(

f

i

µ

i

)

Q

i

n

(

C

)

(

ν

i

f

i

)

Q

(

C

)

f

Q

(

C

)

µ

Q

(

C

)

ν

Q

(

C

)

f

Q

(

C

)

f

C

Q

(

C

)

µ

;

Q

(

C

)

ν

.

2.

i

n

:

f

i

C

(

µ

i

;

ν

i

)

⇔ ∀

i

n

:

µ

i

f

i

ν

i

f

i

Q

(

C

)

µ

Q

i

n

(

C

)

f

i

ν

i

f

i

Q

(

C

)

µ

Q

i

n

(

C

)

f

i

Q

i

n

(

C

)

ν

i

Q

i

n

(

C

)

f

i

Q

(

C

)

µ

Q

i

n

(

C

)

f

i

Q

i

n

(

C

)

ν

i

Q

i

n

(

C

)

f

i

Q

(

C

)

f

C

Q

(

C

)

µ

;

Q

(

C

)

ν

.

3.

i

n

:

f

i

C

′′

(

µ

i

;

ν

i

)

⇔ ∀

i

n

:

f

i

µ

i

f

i

ν

i

Q

i

n

(

C

)

f

i

µ

i

f

i

Q

i

n

(

C

)

ν

i

Q

i

n

(

C

)

f

i

Q

i

n

(

C

)

µ

i

Q

i

n

(

C

)

f

i

Q

i

n

(

C

)

ν

i

Q

i

n

(

C

)

f

i

Q

i

n

(

C

)

µ

i

Q

i

n

(

C

)

f

i

Q

i

n

(

C

)

ν

i

Q

i

n

(

C

)

f

i

C

′′

Q

(

C

)

µ

;

Q

(

C

)

ν

.

Theorem 187.

Let

µ

and

ν

are indexed (by some index set

n

) families of endo-funcoids, and

f

i

FCD

(

Ob

µ

i

;

Ob

ν

i

)

for every

i

n

. Then:

1.

i

n

:

f

i

C

(

µ

i

;

ν

i

)

Q

(

A

)

f

C

Q

(

A

)

µ

;

Q

(

A

)

ν

;

2.

i

n

:

f

i

C

(

µ

i

;

ν

i

)

Q

(

A

)

f

C

Q

(

A

)

µ

;

Q

(

A

)

ν

;

3.

i

n

:

f

i

C

′′

(

µ

i

;

ν

i

)

Q

(

A

)

f

C

′′

Q

(

A

)

µ

;

Q

(

A

)

ν

.

Proof.

Similar to the previous theorem.

30

Section 16