Theorem 22.

Φ

f

=

g

.

Proof.

Because

Φ

f

|

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}

=

g

|

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}

and the lemma.

Theorem 23.

1.

ϕ

A

= Φ

|

h

f

i{

x

X

A

dom

A

|

ρ

A

x

=

A

}

for every small indexed family

A

of forms.

2.

Φ

is the union of all functions

ϕ

A

where

A

is a small indexed family of forms.

Proof.

Both are trivial from the above.

Definition 24.

A product-projection system is a quasi-cartesian function together with a function

Pr

whose values are indexed families, such that for every

x

dom

f

:

x

ZC

0

Pr

fx

=

x.

[TODO: Also:

f

(

Pr

y

) =

y

if

y

im

f

.]

[TODO: Particular product-projection systems.]

Some examples of quasi-cartesian situations and functions

Definition 25.

Let

C

is a category with zero morphisms. The corresponding quasi-cartesian sit-

uation is:

Forms are pairs of objects.

Arguments are morphisms.

Form of an argument

x

is

(

Src

x

;

Dst

x

)

.

Zero for form

(

A

;

B

)

is the zero morphism

0

A B

.

Let us prove it is really a quasi-cartesian situation.

Proof.

We need to prove

ρ

Z

ρ

=

ρ

. Really, let

f

is an argument. Then

ρ Z ρf

=

ρ Z

(

Src

f

;

Dst

f

) =

ρ

0

Src

f ,

Dst

f

= (

Src

f

;

Dst

f

) =

ρf

.

The above definition immediately gives rise of quasi-cartesian situations for binary relations (the
category

Rel

), pointfree funcoids (the category of small pointfree funcoids), reloids (the category

of small reloids).

Definition 26.

The quasi-cartesian situation of anchored relations:

Forms

F

are small indexed families of sets.

Arguments are small anchored relations.

Form of an argument is the arity of anchored relation.

5
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