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Certain ca tegories are cartesian closed

by Victor Porton

Email: porton@narod.ru

Web: http://www.mathematics21.org

November 25, 2013

Abstract

I prove that the category of continuous maps between endofuncoids is cartesian closed.

Whether the category of continuous maps between endoreloids is cartesian closed is yet an

open problem.

This is a rough draft. There are errors!

Cartesian closed categories

Deﬁnition 1. A category is cartesian closed iﬀ :

• It has ﬁnite products.

• For each objects A, B is given an object MOR(A;B) (exponentiation) and a morphism ε

A,B

Dig

:

MOR(A; B) × A → B.

• For e ach morphism f: Z × A → B there is given a morphism (expon ential transpose) ∼f:

Z → MOR(A; B).

• ε ◦ (∼f × 1

A

) = f .

• ∼(ε ◦ (g × 1

A

)) = g.

Our puspose is to prove ( or disprove) that categories Dig, Fcd, and Rld are cartes ian closed.

Note that they have ﬁnite (and even inﬁnite) pro ducts is already proved in http://www.mathe-

matics21.org/binaries/product.pdf

Deﬁnitions of our categories

Categories Dig, Fcd, and Rld are respectively categories of:

1. discretely continuous maps between digraphs;

2. (proximally) continuous map s between endofuncoids;

3. (uniformly) continuous maps between endoreloids.

Deﬁnition 2. Digraph is an endomorphism of the category Rel.

Deﬁnition 3. Category Dig of digraphs is the category whose objects are digraphs and morphisms

are disc retely continuous maps between digraphs. That is morphisms from a digraph µ to a digraph

ν are functions (or more precisely morphisms of Set) f such that f ◦ µ ⊑ ν ◦ f (or eq uivalently

µ ⊑ f

−1

◦ ν ◦ f or equivalently f ◦ µ ◦ f

−1

⊑ ν).

1

Remark 4. Category of digraphs is sometimes deﬁned in an other (non equivalent) way, allowing

multiple edges between two given vertices.

Deﬁnition 5. Category Fcd of continuous maps between endofuncoids is the category who se

objects are endofunco ids and morphisms are proximally continuous maps between endofuncoids.

That is morphism s f rom an endofuncoid µ to an endofuncoid ν are functions (or more precisely

morphisms of Set) f such that ↑

FCD

f ◦ µ ⊑ ν ◦ ↑

FCD

f (or equivalently µ ⊑ ↑

FCD

f

−1

◦ ν ◦ ↑

FCD

f or

equivalently ↑

FCD

f ◦ µ ◦ ↑

FCD

f

−1

⊑ ν).

Deﬁnition 6. Category Rld of continuous maps between endoreloids is the category whose objects

are endoreloids and morphisms are uniformly continuous maps between endoreloids. That is mor-

phisms from an endoreloid µ to an endoreloid ν are functions (or more precisely morphisms of

Set) f such that ↑

RLD

f ◦ µ ⊑ ν ◦ ↑

RLD

f (or equivalently µ ⊑ ↑

RLD

f

−1

◦ ν ◦ ↑

RLD

f or equivalently

↑

RLD

f ◦ µ ◦ ↑

RLD

f

−1

⊑ ν).

Category of digraphs is cartesian closed

Category of digraphs is the simplest of our three categories and it is easy to demonstrate that it

is cartes ian close d. I demonstrate cartes ian close dness of Dig mainly with the purpose to show a

pattern similarly to which we may probably demonstrate our two other categories are cartesian

closed.

Let G and H be g raphs :

• Ob MO R(G; H) = (Ob H)

Ob G

;

• (f; g) ∈ GR MOR(G; H) ⇔ ∀(v; w)∈ GR G: (f(v); g(w)) ∈ GR H for every f , g ∈ ObMOR(G;

H) = (Ob H)

Ob G

;

GR 1

MOR(B;C)

= id

Ob MOR(B;C)

= id

(Ob H)

Ob G

Equivalently

(f; g) ∈ GR MOR(G; H) ⇔ ∀(v; w) ∈ GR G: g ◦ {(v; w)} ◦ f

−1

⊆ GR H

(f; g) ∈ GR MOR(G; H) ⇔ g ◦ (GR G) ◦ f

−1

⊆ GR H

(f; g) ∈ GR MOR(G; H) ⇔

f ×

(C)

g

GR G ⊆ GR H

The transposition (the isomorphism) is uncurrying.

∼f = λa ∈ Zλy ∈ A: f (a; y) that is (∼f )(a)(y) = f (a; y).

(−f)(a; y) = f(a)(y)

If f : A × B → C then ∼f : A → MOR(B; C)

Proposition 7. Transposition and its inverse are morphisms of Dig.

Proof. It follows fr om the equivalence ∼f: A → MOR(B; C) ⇔ ∀x, y: (xAy ⇒ (∼f) x (MOR(B;

C)) (∼f ) y) ⇔ ∀x, y: (xAy ⇒ ∀(v; w) ∈ B: ((∼f) xv; (∼f ) yw) ∈ C) ⇔ ∀x, y, v, w: (xAy ∧ v Bw ⇒

((∼f ) x v; (∼f) y w) ∈ C) ⇔ ∀x, y, v, w: ((x; v) (A × B) (y; w) ⇒ (f (x; v); f(y; w)) ∈ C) ⇔ f:

A × B → C.

Evaluation ε: MOR(G; H) × G → H is deﬁ ne d by the formula:

2

Then evaluation is ε

B,C

= −(1

MOR(B;C)

).

So ε

B,C

(p; q) = (−(1

MOR(B;C)

))(p; q) = (1

MOR(B;C)

)(p)(q) = p(q).

Proposition 8. E va luation is a morphism of Dig.

Proof. Because ε

B,C

(p; q) = −(1

MOR(B;C)

).

It remains to prove:

[FIXME: ε

X ,Y

. What are X and Y ?]

• ε ◦ (∼f × 1

A

) = f ;

• ∼(ε ◦ (g × 1

A

)) = g.

Proof. ε(∼f × 1

A

)(a; p) = ε((∼f)a; p) = (∼f )ap = f(a; p). So ε ◦ (∼f × 1

A

) = f .

∼(ε ◦ (g × 1

A

))(p)(q) = (ε ◦ (g × 1

A

))(p; q) = ε(g × 1

A

)(p; q) = ε(gp; q) = g(p)(q). So

∼(ε ◦ (g × 1

A

)) = g.

Exponentials in category Fcd

Deﬁne ∼

Fcd

f = ↑

FCD

∼

Dig

f

Deﬁnition 9. A category is cartesian closed iﬀ:

• ε ◦ (∼f × 1

A

) = f .

• ∼(ε ◦ (g × 1

A

)) = g.

But this follows from functoriality of ↑

FCD

.

??

Embed Fcd into Dig by the formulas:

A

λX ∈ POb A: hAiX

f

hf i

Obviously this embedding (denote it T) is an injective (both on objects and morphisms) functor.

ε

A,B

Fcd

(p × q) = hpiq

[TODO: Shoul d p and q be atomic?]

∼

Rld

is induced by ∼

Dig

.

Due its injectivity and functoriality, it i s e nough to prove:

1. binary products are preserved

2. ε

TA,TB

Dig

= Tε

A,B

Fcd

3. that ∼

Dig

T f = T∼

Fcd

f for every f : A → B

(Tε

A,B

Fcd

)(p × q) = hε

A,B

Fcd

i(p × q) = hpiq

ε

TA,TB

Dig

X = (TB)

TA

X = (λY ∈ POb B: hB iY )

λX ∈POb A :hAiX

X

??

3

Due its injectivity and functoriality, it i s e nough to prove:

1. binary products are preserved

2. for every ε

TA,TB

Dig

there exist ε

A,B

Fcd

such that ε

TA,TB

Dig

= Tε

A,B

Fcd

3. for every f: TA → TB there exists g: A → B that ∼

Dig

f = T∼

Fcd

g

Consider ε

TA,TB

Dig

. Then ε

TA,TB

Dig

X = (TB)

TA

X = (λX ∈ P Ob B: hBiX)

λX ∈POb A :hAiX

X ∈ (λX ∈

POb B: hBiX) for as suitab le X. Thus ?? ε

TA,TB

Dig

0 = 0 and ε

TA,TB

Dig

(I ∪ J) = ε

TA,TB

Dig

I ∪ ε

TA,TB

Dig

J.

Consequently ε

A,B

Fcd

exists.

Consider f : TA → TB.

??

Then f ∈ (TB)

TA

and f ∈ C(TA; TB).

fX = ??

(∼

Dig

f)(p; q) = f(p)(q) =

Thus ??

??

Binary products are subatomic products and so are compatible with products of graphs.

A try t o prove this directly:

Proposition 10. Transposition and its inverse are morphisms of Fcd.

Proof. ??

[TODO: Use below sets instead of ultraﬁlters.]

It follows from the e quivalence (??is it an equivale nce? the last step seems just an implication)

∼f : A → MOR(B; C) ⇔ ∀x, y ∈ atoms

F

: (x [A] y ⇒ h∼f i x [MOR(B; C)] h∼f iy ) ⇔ ∀x,

y ∈ atoms

F

: (x [A] y ⇒ ∀(v; w) ∈ atoms B: (h∼f ix v ×

FCD

h∼f iyw) ∈ atoms C) ⇔ ∀x, y,

v, w: (x [A] y ∧ v [B] w ⇒ (h∼f ix v ×

FCD

h∼f iy w) ∈ atoms C) ⇔ ∀x, y, v, w ∈ atoms

F

:

(x ×

RLD

v [A × B] y ×

RLD

w ⇒ (f(x; v); f (y; w)) ∈ C) ⇔ f: A × B → C.

Expo nentials in category Rld

TODO

4