Algebraic General Topology. Vol 1: Paperback / E-book || Axiomatic Theory of Formulas: Paperback / E-book

Products in dagger categories with complete ordered
Mor-sets
by Victor Porton
Email: porton@narod.ru
Web: http://www.mathematics21.org
June 1, 2015
[TODO: This is a rough draft. It is not yet checked for errors.]
1 Prerequisites
To understand the section on funcoids and reloids you need ﬁrst read my book [1].
Note 1. What I previosly denoted
Q
F is now denoted
Q
(L)
F (and likewise for
`
). The other
2 Some terminology
Some terminology from my book [1]:
Let f be a morphism f in a dagger category whose Mor-sets are complete lattices (with an
order v and lattice operations
F
and
d
), having the order agreed with the dagger.
Deﬁnition 2. The morphism f is:
monovalued iﬀ f f
y
v 1
Dst f
;
injective iﬀ f
y
f v 1
Src f
;
entirely deﬁned iﬀ f
y
f w 1
Src f
;
surjective iﬀ f f
y
w 1
Dst f
.
Deﬁnition 3. A morphism f of a partially ordered category is metamonovalued when (
d
G) f =
d
g2G
(g f) whenever G is a set of funcoids with a suitable domain and image.
Deﬁnition 4. A morphism f of a partially ordered category is metainjective when f (
d
G) =
d
g2G
(f g) whenever G is a set of funcoids with a suitable domain and image.
Deﬁnition 5. A morphism f of a partially ordered category is metacomplete when f (
F
G) =
F
g2G
(f g) whenever G is a set of funcoids with a suitable domain and image.
Deﬁnition 6. A morphism f of a partially ordered category is co-metacomplete when (
F
G) f =
F
g2G
(g f) whenever G is a set of funcoids with a suitable domain and image.
Obvious 7. Every function f 2 Mor
Set
(A; B) considered as a morphism f 2 Mor
Rel
(A; B) is
monovalued, entirely deﬁned, metamonovalued, metacomplete, and co-metacomplete.
Obvious 8. Every morphism f 2 Mor
Rel
(A; B) is metacomplete and co-metacomplete.
Deﬁnition 9. Let µ and ν are endomorphisms. A monovalued entirely deﬁned morphism f 2
Mor(Ob µ; Ob ν) is continuous (denoted f 2 C(µ; ν)) iﬀ f µ v ν f.
In the book [1] is proved:
1
Proposition 10. f 2 C(µ; ν) , µ v f
y
ν f , f µ f
y
v ν for every monovalued entirely deﬁned
morphism f.
Proposition 11. [TODO: Check correct usage of these implications below!]
1. Every entirely deﬁned monovalued morphism is metamonovalued and metacomplete.
2. Every surjective injective morphism is metainjective and co-metacomplete.
Proof. 1. Let f be an entirely deﬁned monovalued morphism.
(
d
G) f v
d
g2G
(g f) by monotonicity of composition.
Using the fact that f is monovalued and entirely deﬁned:
¡
d
g2G
(g f )
f
y
v
d
g2G
(g f f
y
) v
d
G;
d
g2G
(g f) v
¡
d
g2G
(g f)
f
y
f v (
d
G) f .
So (
d
G) f =
d
g2G
(g f).
Let f be a entirely deﬁned monovalued morphism.
f (
F
G) w
F
g2G
(f g) by monotonicity of composition.
Using the fact that f is entirely deﬁned and monovalued:
f
y
¡
F
g2G
(f g)
w
F
g2G
(f
y
f g) w
d
G;
F
g2G
(f g) w f f
y
F
g2G
(f g) w f (
F
G).
So f (
F
G) =
F
g2G
(f g).
2. By duality.
3 General product in partially ordered dagger category
To understand the below better, you can restrict your imagination to the case when C is the
category Rel.
3.1 Inﬁmum product
Let C be a dagger category, each Mor-set of which is a complete lattice (having order agreed with
the dagger).
We will designate some morphisms as principal and require that principal morphisms are both
metacomplete and co-metacomplete. (For a particular example of the category Rel, all morphisms
are considered principal.)
Let
Q
(Q)
X be an object for each indexed family X of objects.
Let π be a partial function mapping elements X 2 dom π (which consists of small indexed fam-
ilies of objects of C) to indexed families
Q
(Q)
X ! X
i
of principal morphisms (called projections)
for every i 2 dom X.
We will denote particular morphisms as π
i
X
.
Remark 12. In some important examples the function π is entire, that is dom π is the set of all
small indexed families of objects of C. However there are also some important examples where it
is partial.
Deﬁnition 13. Inﬁmum product
Q
F (such that π is deﬁned at λj 2 n: Src F
j
and λj 2 n: Dst F
j
)
is deﬁned by the formula
Y
(L)
F =
l
i2dom F
¡
π
i
λj 2n:Dst F
j
y
F
i
π
i
λj 2n:Src F
j
:
This formula can be (over)simpliﬁed to:
Y
(L)
F =
l
i2dom F
¡¡
π
i
DstF
y
F
i
π
i
SrcF
:
2 Section 3
Remark 14.
¡
π
i
λj 2n:Dst F
j
y
F
i
π
i
λj 2n:Src F
j
2 Mor
Q
j2n
(Q)
Src F
j
;
Q
j 2n
(Q)
Dst F
j
are properly
deﬁned and have the same sources and destination (whenever i 2 dom F is), thus the meet in the
formulas is properly deﬁned.
Remark 15. Thus
F
0
×
(L)
F
1
=

π
0
(Dst F
0
;Dst F
1
)
y
F
0
π
0
(Src F
0
;Src F
1
)
u

π
1
(Dst F
0
;Dst F
1
)
y
F
1
π
1
(Src F
0
;Src F
1
)
that is product is deﬁned by a pure algebraic formula.
Proposition 16.
Q
(L)
F = max
n
Φ 2 Mor
Q
j 2n
(Q)
Src F
j
;
Q
j 2n
(Q)
Dst F
j
j 8i 2 n: Φ v
¡
π
i
λj 2n:Dst F
j
y
F
i
π
i
λj 2n:Src F
j
o
.
Proof. By deﬁnition of meet on a complete lattice.
Corollary 17.
Q
(L)
F =
F
n
Φ 2 Mor
Q
j 2n
(Q)
Src F
j
;
Q
j2n
(Q)
Dst F
j
j 8i 2 n: Φ v
¡
π
i
λj 2n:Dst F
j
y
F
i
π
i
λj 2n:Src F
j
o
.
Theorem 18. Let π
i
X
be metamonovalued morphisms. If S 2 P (Mor(A
0
; B
0
) × Mor(A
1
; B
1
)) for
some sets A
0
, B
0
, A
1
, B
1
then
l
a ×
(L)
b j (a; b) 2 S
=
l
dom S ×
(L)
l
im S:
Proof.
d
fa × b j (a; b) 2 S g =
d
n
π
0
(Dst a;Dst b)
y
a π
0
(Src a;Src b)
u

π
1
(Dst a;Dst b)
y
b
π
1
(Src a;Src b)
j (a; b) 2S
o
=
d
n
π
0
(Dst a;Dst b)
y
a π
0
(Src a;Src b)
j a 2dom S
o
u
d
n
π
1
(Dst a;Dst b)
y
b π
1
(Src a;Src b)
j b 2im S
o
=

π
0
(Dst a;Dst b)
y
d
fa j a 2 domS g π
0
(Src a;Src b)
u

π
1
(Dst a;Dst b)
y
d
fb j b 2 im S g π
1
(Src a;Src b)
=

π
0
(Dst a;Dst b)
y
(
d
dom S) π
0
(Src a;Src b)
u

π
1
(Dst a;Dst b)
y
(
d
im S) π
1
(Src a;Src b)
=
d
dom S ×
d
im S.
Corollary 19.
¡
a
0
×
(L)
b
0
u
¡
a
1
×
(L)
b
1
= (a
0
u a
1
) ×
(L)
(b
0
u b
1
).
Corollary 20. a
0
×
(L)
b
0
/ a
1
×
(L)
b
1
, a
0
/ a
1
^ b
0
/ b
1
.
3.2 Inﬁmum product for endomorphisms
Let F is an indexed family of endomorphisms of C.
I will denote Ob f the object (source and destination) of an endomorphism f .
Let also π
i
X
be a monovalued entirely deﬁned morphism (for each i 2 dom F ).
Then
Q
(L)
F =
d
i2dom F
¡
π
i
λj 2n:Ob F
j
y
F
i
π
i
λj 2n:Ob F
j
(if π is deﬁned at λj 2 n: Ob F
j
).
Abbreviate π
i
= π
i
λj 2n:Ob F
j
.
So
Q
(L)
F =
d
i2dom F
((π
i
)
y
F
i
π
i
).
Q
(L)
F = max
n
Φ 2 End
Q
j2n
(Q)
Ob F
j
j 8i 2 n: Φ v (π
i
)
y
F
i
π
i
o
.
Taking into account that π
i
is a monovalued entirely deﬁned morphism, we get:
Obvious 21.
Q
(L)
F = max
n
Φ 2 End
Q
j 2n
(Q)
Ob F
j
j 8i 2 n: π
i
2 C(Φ; F
i
)
o
.
Remark 22. The above formula may allow to deﬁne the product for non-dagger categories (but
only for endomorphisms). In this writing I don't introduce a notation for this, however.
Corollary 23. π
i
2 C
Q
(L)
F ; F
i
for every i 2 dom F .
General product in partially ordered dagger category 3
3.3 Category of continuous morphisms
Let π
i
= π
i
X
(for i 2 dom F ) be entirely deﬁned monovalued morphisms (we suppose it is deﬁned
at X).
Let
N
of an indexed family of morphisms is a morphism; π
i
N
f = f
i
;
N
i2n
(π
i
f ) = f.
Deﬁnition 24. The category cont(C) is deﬁned as follows:
Objects are endomorphisms of the category C.
Morphisms are triples (f ; a; b) where a and b are objects and f: Ob a ! Ob b is an entirely
deﬁned monovalue principal morphism of the category C such that f 2 C(a; b) (in other
words, f a v b f ).
Composition of morphisms is deﬁned by the formula (g; b; c) (f; a; b) = (g f; a; c) .
Identity morphisms are (a; a; id
a
C
).
It is really a category:
Proof. We need to prove that: composition of morphisms is a morphism, composition is associative,
and identity morphisms can be canceled on the left and on the right.
That composition of morphisms is a morphism by properties of generalized continuity.
That composition is associative is obvious.
That identity morphisms can be canceled on the left and on the right is obvious.
Remark 25. The “physical” meaning of this category is:
Objects (endomorphisms of C) are spaces.
Morphisms are continuous functions between spaces.
f a v b f intuitively means that f combined with an inﬁnitely small is less than inﬁnitely
small combined with f (that is f is continuous).
Deﬁnition 26. π
i
cont(C)
=
Q
(L)
F ; F
i
; π
i
.
Proposition 27. π
i
are continuous, that is π
i
cont(C)
are morphisms.
Proof. We need to prove π
i
2 C
Q
(L)
F ; F
i
but that was proved above.
Lemma 28. f 2 Mor
cont(C)
Y ;
Q
(L)
F
is continuous iﬀ all π
i
f are continuous.
Proof.
). Let f 2 Mor
cont(C)
Y ;
Q
(L)
F
. Then f Y v
Q
(L)
F
f ; π
i
f Y v π
i
Q
(L)
F
f ;
π
i
f Y v
Q
(L)
F
π
i
f . Thus π
i
f is continuous.
(. Let all π
i
f be continuous. Then π
i
cont(C)
f 2 Mor
cont(C)
(Y ; F
i
); π
i
cont(C)
f Y v
F
i
π
i
cont(C)
f. We need to prove Y v f
y
Q
(L)
F
f that is
Y v f
y
l
i2n
((π
i
)
y
F
i
π
i
) f
for what is enough (because f is metamonovalued)
Y v
l
i2n
(f
y
(π
i
)
y
F
i
π
i
f )
what follows from Y v
d
i2n
(f
y
(π
i
)
y
π
i
f Y ) what is obvious.
4 Section 3
Theorem 29.
Q
(L)
together with
N
is a (partial) product in the category cont(C).
Proof. Obvious.
Check http://math.stackexchange.com/questions/102632/how-to-check-whether-it-is-a-direct-
product/102677#102677
4 On duality
We will consider duality where both the category C and orders on Mor-sets are replaced with their
dual. I will denote A
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dual
B when two formulas A and B are dual with this duality.
Proposition 30. f 2 C(µ; ν)
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dual
f
y
2 C(ν
y
; µ
y
).
Proof. f 2 C(µ; ν) , f µ v ν f
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dual
µ
y
f
y
w f
y
y
ν
¡1
, f
y
2 C(ν
y
; µ
y
).
f is entirely deﬁned,f
y
f w 1
Src f
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dual
f
y
f v 1
Src f
, f is injective,f
y
is monovalued.
f is monovalued,f f
y
v 1
Dst f
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dual
f f
y
w 1
Dst f
, f is surjective,f
y
is entirely deﬁned.
5 General coproduct in partially ordered dagger category
The below is the dual of the above, proofs are omitted as they are dual.
Let ι
i
[TODO: What is i?] are entirely deﬁned monovalued morphisms to an object Z.
Let ι
i
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dual
π
i
that is ι
i
= (π
i
)
y
. We have the above equivalent to π
i
being monovalued and
entirely deﬁned.
5.1 Supremum coproduct
Let C be a dagger category, each Mor-set of which is a complete lattice (having order agreed with
the dagger).
We will designate some morphisms as principal and require that principal morphisms are both
metacomplete and co-metacomplete. (For a particular example of the category Rel, all morphisms
are considered principal.)
Let
`
(Q)
X be an object for each indexed family X of objects.
Let ι be a partial function mapping elements X 2 domι (which consists of small indexed families
of objects of C) to indexed families X
i
!
`
(Q)
X of principal morphisms (called injections ) for
every i 2 dom X.
Deﬁnition 31. Supremum coproduct
`
(L)
F (such that ι is deﬁned at λj 2 n: Dst F
j
and λj 2 n:
Src F
j
) is deﬁned by the formula
a
(L)
F =
G
i2dom F
ι
i
λj 2n:Src F
j
F
i
y
¡
ι
i
λj 2n:Dst F
j
y
:
This formula can be (over)simpliﬁed to:
a
(L)
F =
G
i2dom F
¡
ι
i
SrcF
F
i
y
(ι
i
DstF
)
y
:
Remark 32. ι
i
λj 2n:Src F
j
F
i
¡
ι
i
λj 2n:Dst F
j
y
2 Mor
`
j 2n
(Q)
Src F
j
;
`
j 2n
(Q)
Dst F
j
are properly
deﬁned and have the same sources and destination (whenever i 2 dom F is), thus the meet in the
formulas is properly deﬁned.
General coproduct in partially ordered dagger category 5
Remark 33. Thus
F
0
q
(L)
F
1
=
ι
0
(Src F
0
;Src F
1
)
F
0
y
ι
0
(Dst F
0
;Dst F
1
)
y
t
ι
1
(Src F
0
;Src F
1
)
F
1
y
ι
1
(Dst F
0
;Dst F
1
)
y
that is coproduct is deﬁned by a pure algebraic formula.
Proposition 34.
`
(L)
F = min
n
Φ 2 End
`
j 2n
(Q)
Ob F
j
j 8i 2 n: Φ w ι
i
λj 2n:Src F
j
F
i
y
¡
ι
i
λj 2n:Dst F
j
y
o
.
Proof. By deﬁnition of meet on a complete lattice.
Corollary 35.
`
(L)
F =
d
n
Φ 2 End
`
j 2n
(Q)
Ob F
j
j 8i 2 n: Φ w ι
i
λj 2n:Src F
j
F
i
y
¡
ι
i
λj 2n:Dst F
j
y
o
.
Theorem 36. Let π
i
X
be metainjective morphisms. If S 2 P (Mor(A
0
; B
0
) × Mor(A
1
; B
1
)) for some
sets A
0
, B
0
, A
1
, B
1
then
G
a ×
(L)
b j (a; b) 2 S
=
G
dom S ×
(L)
G
im S:
Corollary 37.
¡
a
0
q
(L)
b
0
t
¡
a
1
q
(L)
b
1
= (a
0
u a
1
) q
(L)
(b
0
u b
1
).
Corollary 38. a
0
q
(L)
b
0
a
1
q
(L)
b
1
, a
0
a
1
^ b
0
b
1
.
5.2 Supremum coproduct for endomorphisms
Let F be an indexed family of endomorphisms of C.
I will denote Ob f the object (source and destination) of an endomorphism f .
Let also ι
i
be a monovalued entirely deﬁned morphism (for each i 2 dom F ).
Deﬁnition 39.
`
(L)
F =
F
i2dom F
ι
i
λj 2n:Ob F
j
F
i
y
¡
ι
i
λj 2n:Ob F
j
y
(if ι is deﬁned at λj 2 n:
Ob F
j
). (I call it supremum coproduct ).
Abbreviate ι
i
= ι
i
λj 2n:Ob F
j
.
So
`
F =
F
i2dom F
¡
ι
i
F
i
y
(ι
i
)
y
.
`
F = min
n
Φ 2 End
`
j 2n
(Q)
Ob F
j
j 8i 2 n: Φ w ι
i
F
i
y
(ι
i
)
y
o
.
Taking into account that ι
i
is a monovalued entirely deﬁned morphism, we get:
Obvious 40.
`
(L)
=min
n
Φ 2 End
`
j 2n
(Q)
Ob F
j
j 8i 2 n: ι
i
2 C
¡
F
i
y
; Φ
o
.
Corollary 41. ι
i
2 C
F
i
;
`
(L)
F
for every i 2 dom F .
5.3 Category of continuous morphisms
[TODO: What is X?]
Let ι
i
(for i 2 dom F ) be entirely deﬁned monovalued and metacomplete morphisms.
Let
L
of an indexed family of morphisms is a morphism; (
L
f) ι
i
= f
i
;
L
i2n
(f ι
i
) = f (a
dual of the above).
Let F
i
2 End
`
j 2n
(Q)
Ob F
j
for all i 2 n (where n is some index set) (a self-dual of the above).
Deﬁnition 42. ι
i
cont(C)
=
`
(L)
F ; F
i
y
; ι
i
.
Proposition 43. ι
i
are continuous, that is ι
i
cont(C)
are morphisms.
6 Section 5
Lemma 44. f 2 Mor
cont(C)
`
(L)
F ; Y
is continuous iﬀ all f ι
cont(C)
are continuous.
Theorem 45.
`
(L)
together with
L
is a (partial) coproduct in the category cont(C).
6 Applying this to the theory of funcoids and reloids
6.1 Funcoids
Deﬁnition 46. Fcd =
def
cont(FCD).
Let F be a family of endofuncoids.
The cartesian product
Q
(Q)
X =
def
Q
X.
I deﬁne π
i
= π
i
X
2 FCD(
Q
X; X
i
) as the principal funcoid corresponding to the i-th projection.
(Here π is entirely deﬁned.)
The disjoint union
`
(Q)
X =
def
`
X.
I deﬁne ι
i
= ι
i
X
2 FCD(X
i
;
`
X) as the principal funcoid corresponding to the i-th canonical
injection. (Here ι is entirely deﬁned.)
Let
N
and
L
be deﬁned in the same way as in category Set.
Obvious 47. π
i
N
f = f
i
;
N
i2n
(π
i
f) = f.
Obvious 48. (
L
f) ι
i
= f
i
;
L
i2n
(f ι
i
) = f.
It is easy to show that π
i
is entirely deﬁned monovalued, and ι
i
is metacomplete and co-
metacomplete.
Thus we are under conditions for both canonical products and canonical coproducts and thus
both
Q
(L)
F and
`
(L)
F are deﬁned.
6.2 Reloids
Deﬁnition 49. Rld =
def
cont(RLD).
Let F be a family of endoreloids.
The cartesian product
Q
(Q)
X =
def
Q
X.
I deﬁne π
i
= π
i
X
2 RLD(
Q
X; X
i
) as the principal reloid corresponding to the i-th projection.
(Here π is entirely deﬁned.)
The disjoint union
`
(Q)
X =
def
`
X.
I deﬁne ι
i
= ι
i
FCD(Z
1
)
2 RLD(X
i
;
`
X) as the principal reloid corresponding to the i-th canonical
injection. (Here ι is entirely deﬁned.)
Let
N
and
L
are deﬁned in the same way as in category Set.
Obvious 50. π
i
N
f = f
i
;
N
i2n
(π
i
f) = f.
Obvious 51. (
L
f) ι
i
= f
i
;
L
i2n
(f ι
i
) = f.
It is easy to show that π
i
is entirely deﬁned monovalued, and ι
i
is metacomplete and co-
metacomplete.
Thus we are under conditions for both canonical products and canonical coproducts and thus
both
Q
(L)
F and
`
(L)
F are deﬁned.
It is trivial that for uniform spaces inﬁmum product of reloids coincides with product uni-
formilty.
Applying this to the theory of funcoids and reloids 7
6.3 Cross-composition of pointfree funcoids
[TODO: This section is partially written.]
Let now C be the category of all small pointfree funcoids. Let principal morphisms be principal
funcoids.
Deﬁne π
i
X
= Pr
i
(RLD)
in
X whenever X is a funcoid. [TODO: We should generalize it for
multifuncoids or staroids.]
7 Initial and terminal objects
Initial object of Fcd is the endofuncoid "
FCD(;;;)
;. It is initial because it has precisely one morphism
o (the empty set considered as a function) to any object Y . o is a morphism because o "
FCD(;;;)
;v
Y o.
Proposition 52. Terminal objects of Fcd are exactly "
F
f∗g ×
FCD
"
F
f∗g = "
FCD
f(; )g where is
an arbitrary point.
Proof. In order for a function f: X ! "
FCD
f(; )g be a morphism, it is required exactly f X v
"
FCD
f(; )g f
f X v (f
¡1
"
FCD
f(; )g)
¡1
; f X v (f∗g ×
FCD
hf
¡1
if∗g)
¡1
; f X v hf
¡1
if∗g ×
FCD
f∗g what
true exactly when f is a constant function with the value .
If n = ; then Z = f;g;
Q
(L)
; = max FCD(Z; Z) = "
F
f;g ×
FCD
"
F
f;g = "
FCD
f(;; ;)g.
[TODO: Initial and terminal objects of Rld.]
8 Canonical product and subatomic product
[TODO: Confusion between ﬁlters on products and multireloids.]
Proposition 53. Pr
i
RLD
j
F(Z)
=hπ
i
i for every index i of a cartesian product Z.
Proof. If X 2 F(Z) then (Pr
i
RLD
j
F(Z)
)X = Pr
i
RLD
X =
d
h"ihPr
i
iX =
d
hπ
i
iup X = hπ
i
iX .
Proposition 54.
Q
(A)
F =
d
i2n
π
i
FCD
¡
Q
i2n
Dst F
¡1
F
i
π
i
FCD
¡
Q
i2n
Src F
.
Proof. a
h
Q
(A)
F
i
b , 8i 2 dom F : Pr
i
RLD
a [F
i
] Pr
i
RLD
b , 8i 2 dom F :
D
π
i
FCD
¡
Q
i2n
Dst F
¡1
E
[F
i
]
D
π
i
FCD
¡
Q
i2n
Src F
E
, 8i 2 dom F : a
h
π
i
FCD
¡
Q
i2n
Dst F
¡1
F
i
π
i
FCD
¡
Q
i2n
Src F
i
b ,
a
h
d
i2n
π
i
FCD
¡
Q
i2n
Dst F
¡1
F
i
π
i
FCD
¡
Q
i2n
Src F
i
b for ultraﬁlters a and b.
Corollary 55.
Q
(L)
F =
Q
(A)
F is F is a small indexed family of funcoids.
9 Further plans
Does the formula
Q
i2n
(L)
(g
i
f
i
) =
Q
(L)
g
Q
(L)
f hold?
Conjecture 56. The categories Fcd and Rld are cartesian closed (actually two conjectures).
http://mathoverﬂow.net/questions/141615/how-to-prove-that-there-are-no-exponential-
object-in-a-category suggests to investigate colimits to prove that there are no exponential object.
Coordinate-wise continuity.
Bibliography
[1] Victor Porton. Algebraic General Topology. Volume 1. 2013.
8 Section