2000 Mathematics Subject Classification. 54J05, 54A05, 54D99, 54E05, 54E15,
54E17, 54E99
Key words and phrases. algebraic general topology, quasi-uniform spaces,
generalizations of proximity spaces, generalizations of nearness spaces,
generalizations of uniform spaces
Abstract. Partial rough draft of volume 2 of Algebraic General Topology
book. This volume is meant to contain materials which refer to more advanced
prerequisites than plain ZFC (such as category theory and classical pointfree
topology). This is a very rough draft.
Contents
Chapter 1. Introduction 6
Chapter 2. Products in dagger categories with complete ordered Hom-sets 8
1. General product in partially ordered dagger category 8
2. On duality 11
3. General coproduct in partially ordered dagger category 11
4. Applying this to the theory of funcoids and reloids 13
5. Initial and terminal objects 14
6. Canonical product and subatomic product 14
7. Further plans 15
8. Cartesian closedness 15
9. Is category Rld cartesian closed? 18
Chapter 3. Equalizers and co-Equalizers in Certain Categories 19
1. Equalizers 19
2. Co-equalizers 19
3. Rest 20
Chapter 4. Categories of filters 21
Chapter 5. Power of filters 23
1. Germs of functions 23
2. Power of filters 24
Chapter 6. Matters related to tensor product 25
Chapter 7. Mappings between endofuncoids and topological spaces 27
Chapter 8. Funcoids as closed sets 30
Chapter 9. Categories related with funcoids 31
1. Draft status 31
2. Topic of this article 31
3. Category of continuous morphisms 31
4. Definition of the categories 32
5. Isomorphisms 32
6. Direct products 33
Chapter 10. Product of funcoids over a filter 35
1. More on product of reloids 36
Chapter 11. Compact funcoids 37
1. The rest 37
Chapter 12. Pointfree funcoids as a generalization of frames 41
1. Definitions 41
2. Postface 42
3
CHAPTER 1
Introduction
I remind some definitions from volume 1 [5].
I denote a set definition like
n
x∈A
P (x)
o
instead of customary {x ∈ A | P (x)} (in
order to reduce formulas size).
I denote partial order as v. I denote lattice operations as
d
,
d
, u, t.
The following generalizes monovalued morphisms in category Rel.
Let Hom-sets be complete lattices.
Definition 1955. A morphism f of a partially ordered category is metamono-
valued when (
d
G) ◦ f =
d
g∈G
(g ◦ f) whenever G is a set of morphisms with a
suitable domain and image.
Definition 1956. A morphism f of a partially ordered category is metainjec-
tive when f ◦(
d
G) =
d
g∈G
(f ◦g) whenever G is a set of morphisms with a suitable
domain and image.
Obvious 1957. Metamonovaluedness and metainjectivity are dual to each
other.
Definition 1958. A morphism f of a partially ordered category is metacom-
plete when f ◦ (
d
G) =
d
g∈G
(f ◦ g) whenever G is a set of morphisms with a
suitable domain and image.
Definition 1959. A morphism f of a partially ordered category is co-
metacomplete when (
d
G) ◦ f =
d
g∈G
(g ◦ f) whenever G is a set of morphisms
with a suitable domain and image.
Let now Hom-sets be meet-semilattices.
Definition 1960. A morphism f of a partially ordered category is weakly
metamonovalued when (g uh)◦f = (g ◦f)u(h◦f ) whenever g and h are morphisms
with a suitable domain and image.
Definition 1961. A morphism f of a partially ordered category is weakly
metainjective when f ◦ (g u h) = (f ◦ g) u (f ◦ h) whenever g and h are morphisms
with a suitable domain and image.
Let now Hom-sets be join-semilattices.
Definition 1962. A morphism f of a partially ordered category is weakly
metacomplete when f ◦ (g t h) = (f ◦ g) t (f ◦ h) whenever g and h are morphisms
with a suitable domain and image.
Definition 1963. A morphism f of a partially ordered category is weakly co-
metacomplete when (g t h) ◦ f = (g ◦ f) t (h ◦ f ) whenever g and h are morphisms
with a suitable domain and image.
Obvious 1964.
1
◦
. Metamonovalued morphisms are weakly metamonovalued.
2
◦
. Metainjective morphisms are weakly metainjective.
6
1. INTRODUCTION 7
3
◦
. Metacomplete morphisms are weakly metacomplete.
4
◦
. Co-metacomplete morphisms are weakly co-metacomplete.
Definition 1965. For a partially ordered dagger category I will call monoval-
ued morphism such a morphism f that f ◦ f
†
v 1
Dst f
.
Definition 1966. For a partially ordered dagger category I will call entirely
defined morphism such a morphism f that f
†
◦ f w 1
Src f
.
Definition 1967. For a partially ordered dagger category I will call injective
morphism such a morphism f that f
†
◦ f v 1
Src f
.
Definition 1968. For a partially ordered dagger category I will call surjective
morphism such a morphism f that f ◦ f
†
w 1
Dst f
.
Remark 1969. It is easy to show that this is a generalization of monovalued,
entirely defined, injective, and surjective functions as morphisms of the category
Rel.
Obvious 1970. “Injective morphism” is a dual of “monovalued morphism” and
“surjective morphism” is a dual of “entirely defined morphism”.
CHAPTER 2
Products in dagger categories with complete
ordered Hom-sets
FiXme: This is a rough draft. It is not yet checked for errors.
Note 1971. What I previously denoted
Q
F is now denoted
Q
(L)
F (and like-
wise for
`
). The other draft chapters referring to this chapter may be not yet
updated.
Proposition 1972. FiXme: Should we move this to volume 1?
1
◦
. Every entirely defined monovalued morphism is metamonovalued and
metacomplete.
2
◦
. Every surjective injective morphism is metainjective and co-
metacomplete.
Proof. Let’s prove the first (the second follows from duality):
Let f be an entirely defined monovalued morphism.
(
d
G) ◦ f v
d
g∈G
(g ◦ f) by monotonicity of composition.
Using the fact that f is monovalued and entirely defined:
d
g∈G
(g ◦ f)
◦ f
†
v
d
g∈G
(g ◦ f ◦ f
†
) v
d
G;
d
g∈G
(g ◦ f) v
d
g∈G
(g ◦ f)
◦ f
†
◦ f v (
d
G) ◦ f.
So (
d
G) ◦ f =
d
g∈G
(g ◦ f).
Let f be a entirely defined monovalued morphism.
f ◦ (
d
G) w
d
g∈G
(f ◦ g) by monotonicity of composition.
Using the fact that f is entirely defined and monovalued:
f
†
◦
d
g∈G
(f ◦ g)
w
d
g∈G
(f
†
◦ f ◦ g) w
d
G;
d
g∈G
(f ◦ g) w f ◦ f
†
◦
d
g∈G
(f ◦ g) w f ◦ (
d
G).
So f ◦ (
d
G) =
d
g∈G
(f ◦ g).
1. 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.
1.1. Infimum product. Let C be a dagger category, each Hom-set of which
is a complete lattice (having order agreed with the dagger).
We will designate some morphisms as principal and require that principal mor-
phisms 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 ∈ dom π (which consists of
small indexed families of objects of C) to indexed families
Q
(Q)
X → X
i
of principal
morphisms (called projections) for every i ∈ dom X.
We will denote particular morphisms as π
X
i
.
8
1. GENERAL PRODUCT IN PARTIALLY ORDERED DAGGER CATEGORY 9
Remark 1973. 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 1974. Infimum product
Q
F (such that π is defined at λj ∈ n :
Src F
j
and λj ∈ n : Dst F
j
) is defined by the formula
(L)
Y
F =
l
i∈dom F
((π
λj∈n:Dst F
j
i
)
†
◦ F
i
◦ π
λj∈n:Src F
j
i
).
This formula can be (over)simplified to:
(L)
Y
F =
l
i∈dom F
((π
Dst ◦F
i
)
†
◦ F
i
◦ π
Src ◦F
i
).
Remark 1975. (π
λj∈n:Dst F
j
i
)
†
◦ F
i
◦ π
λj∈n:Src F
j
i
∈
Hom
Q
(Q)
j∈n
Src F
j
,
Q
(Q)
j∈n
Dst F
j
are properly defined and have the same sources
and destination (whenever i ∈ dom F is), thus the meet in the formulas is properly
defined.
Remark 1976. Thus
F
0
×
(L)
F
1
= ((π
(Dst F
0
,Dst F
1
)
0
)
†
◦F
0
◦π
(Src F
0
,Src F
1
)
0
)u((π
(Dst F
0
,Dst F
1
)
1
)
†
◦F
1
◦π
(Src F
0
,Src F
1
)
1
)
that is product is defined by a pure algebraic formula.
Proposition 1977.
Q
(L)
F = max
(
Φ∈Hom
Q
(Q)
j∈n
Src F
j
,
Q
(Q)
j∈n
Dst F
j
∀i∈n:Φv(π
λj∈n:Dst F
j
i
)
†
◦F
i
◦π
λj∈n:Src F
j
i
)
.
Proof. By definition of meet on a complete lattice.
Corollary 1978.
Q
(L)
F =
d
(
Φ∈Hom
Q
(Q)
j∈n
Src F
j
,
Q
(Q)
j∈n
Dst F
j
∀i∈n:Φv(π
λj∈n:Dst F
j
i
)
†
◦F
i
◦π
λj∈n:Src F
j
i
)
.
Theorem 1979. Let π
X
i
be metamonovalued morphisms. If S ∈
P(Hom(A
0
, B
0
) × Hom(A
1
, B
1
)) for some sets A
0
, B
0
, A
1
, B
1
then
l
a ×
(L)
b
(a, b) ∈ S
=
l
dom S ×
(L)
l
im S.
Proof.
l
a × b
(a, b) ∈ S
=
l
(
((π
(Dst a,Dst b)
0
)
†
◦ a ◦ π
(Src a,Src b)
0
) u ((π
(Dst a,Dst b)
1
)
†
◦ b ◦ π
(Src a,Src b)
1
)
(a, b) ∈ S
)
=
l
(
(π
(Dst a,Dst b)
0
)
†
◦ a ◦ π
(Src a,Src b)
0
a ∈ dom S
)
u
l
(
(π
(Dst a,Dst b)
1
)
†
◦ b ◦ π
(Src a,Src b)
1
b ∈ im S
)
=
(π
(Dst a,Dst b)
0
)
†
◦
l
n
a
a ∈ dom S
o
◦ π
(Src a,Src b)
0
u
(π
(Dst a,Dst b)
1
)
†
◦
l
b
b ∈ im S
◦ π
(Src a,Src b)
1
=
(π
(Dst a,Dst b)
0
)
†
◦
l
dom S
◦ π
(Src a,Src b)
0
u
(π
(Dst a,Dst b)
1
)
†
◦
l
im S
◦ π
(Src a,Src b)
1
=
l
dom S ×
l
im S.
Corollary 1980. (a
0
×
(L)
b
0
) u (a
1
×
(L)
b
1
) = (a
0
u a
1
) ×
(L)
(b
0
u b
1
).
Corollary 1981. a
0
×
(L)
b
0
6 a
1
×
(L)
b
1
⇔ a
0
6 a
1
∧ b
0
6 b
1
.
1. GENERAL PRODUCT IN PARTIALLY ORDERED DAGGER CATEGORY 10
1.2. 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
X
i
be a monovalued entirely defined morphism (for each i ∈ dom F).
Then
Q
(L)
F =
d
i∈dom F
((π
λj∈n:Ob F
j
i
)
†
◦ F
i
◦ π
λj∈n:Ob F
j
i
) (if π is defined at
λj ∈ n : Ob F
j
).
Abbreviate π
i
= π
λj∈n:Ob F
j
i
.
So
Q
(L)
F =
d
i∈dom F
((π
i
)
†
◦ F
i
◦ π
i
).
Q
(L)
F = max
(
Φ∈End
Q
(Q)
j∈n
Ob F
j
∀i∈n:Φv(π
i
)
†
◦F
i
◦π
i
)
.
Taking into account that π
i
is a monovalued entirely defined morphism, we get:
Obvious 1982.
Q
(L)
F = max
(
Φ∈End
Q
(Q)
j∈n
Ob F
j
∀i∈n:π
i
∈C(Φ,F
i
)
)
.
Remark 1983. 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 1984. π
i
∈ C
Q
(L)
F, F
i
for every i ∈ dom F .
1.3. Category of continuous morphisms. Let π
i
= π
X
i
(for i ∈ 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
i∈n
(π
i
◦ f ) = f .
Definition 1985. 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 ∈ 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, 1
C
a
).
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 1986. 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 1987. π
cont(C)
i
=
Q
(L)
F, F
i
, π
i
.
Proposition 1988. π
i
are continuous, that is π
cont
(C)
i
are morphisms.
3. GENERAL COPRODUCT IN PARTIALLY ORDERED DAGGER CATEGORY 11
Proof. We need to prove π
i
∈ C
Q
(L)
F, F
i
but that was proved above.
Lemma 1989. f ∈ Hom
cont(C)
Y,
Q
(L)
F
is continuous iff all π
i
◦ f are con-
tinuous.
Proof.
⇒. Let f ∈ Hom
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
cont(C)
i
◦ f ∈ Hom
cont(C)
(Y, F
i
); π
cont(C)
i
◦
f ◦ Y v F
i
◦ π
cont(C)
i
◦ f. We need to prove Y v f
†
◦
Q
(L)
F
◦ f that is
Y v f
†
◦
l
i∈n
((π
i
)
†
◦ F
i
◦ π
i
) ◦ f
for what is enough (because f is metamonovalued)
Y v
l
i∈n
(f
†
◦ (π
i
)
†
◦ F
i
◦ π
i
◦ f )
what follows from Y v
d
i∈n
(f
†
◦ (π
i
)
†
◦ π
i
◦ f ◦ Y ) what is obvious.
Theorem 1990.
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
2. 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
dual
←−→ B when two formulas A and B are
dual with this duality.
Proposition 1991. f ∈ C(µ, ν)
dual
←−→ f
†
∈ C(ν
†
, µ
†
).
Proof. f ∈ C(µ, ν) ⇔ f ◦ µ v ν ◦ f
dual
←−→ µ
†
◦ f
†
w f
†
◦
†
ν
−1
⇔ f
†
∈
C(ν
†
, µ
†
).
f is entirely defined ⇔ f
†
◦ f w 1
Src f
dual
←−→ f
†
◦ f v 1
Src f
⇔ f is injective ⇔
f
†
is monovalued.
f is monovalued ⇔ f ◦ f
†
v 1
Dst f
dual
←−→ f ◦ f
†
w 1
Dst f
⇔ f is surjective ⇔
f
†
is entirely defined.
3. 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
FiXme: What is ι? are entirely defined monovalued morphisms to an
object Z.
Let ι
i
dual
←−→ π
i
that is ι
i
= (π
i
)
†
. We have the above equivalent to π
i
being
monovalued and entirely defined.
3. GENERAL COPRODUCT IN PARTIALLY ORDERED DAGGER CATEGORY 12
3.1. Supremum coproduct. Let C be a dagger category, each Hom-set of
which is a complete lattice (having order agreed with the dagger).
We will designate some morphisms as principal and require that principal mor-
phisms 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 ∈ dom ι (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 ∈ dom X.
Definition 1992. Supremum coproduct
`
(L)
F (such that ι is defined at λj ∈
n : Dst F
j
and λj ∈ n : Src F
j
) is defined by the formula
(L)
a
F =
l
i∈dom F
(ι
λj∈n:Src F
j
i
◦ F
†
i
◦ (ι
λj∈n:Dst F
j
i
)
†
).
This formula can be (over)simplified to:
(L)
a
F =
l
i∈dom F
(ι
Src ◦F
i
◦ F
†
i
◦ (ι
Dst ◦F
i
)
†
).
Remark 1993. ι
λj∈n:Src F
j
i
◦ F
i
◦ (ι
λj∈n:Dst F
j
i
)
†
∈
Hom
`
(Q)
j∈n
Src F
j
,
`
(Q)
j∈n
Dst F
j
are properly defined and have the same sources
and destination (whenever i ∈ dom F is), thus the meet in the formulas is properly
defined.
Remark 1994. Thus
F
0
q
(L)
F
1
= (ι
(Src F
0
,Src F
1
)
0
◦F
†
0
◦(ι
(Dst F
0
,Dst F
1
)
0
)
†
)t(ι
(Src F
0
,Src F
1
)
1
◦F
†
1
◦(ι
(Dst F
0
,Dst F
1
)
1
)
†
)
that is coproduct is defined by a pure algebraic formula.
Proposition 1995.
`
(L)
F = min
(
Φ∈End
`
(Q)
j∈n
Ob F
j
∀i∈n:Φwι
λj∈n:Src F
j
i
◦F
†
i
◦(ι
λj∈n:Dst F
j
i
)
†
)
.
Proof. By definition of meet on a complete lattice.
Corollary 1996.
`
(L)
F =
d
(
Φ∈End
`
(Q)
j∈n
Ob F
j
∀i∈n:Φwι
λj∈n:Src F
j
i
◦F
†
i
◦(ι
λj∈n:Dst F
j
i
)
†
)
.
Theorem 1997. Let π
X
i
be metainjective morphisms. If S ∈
P(Hom(A
0
, B
0
) × Hom(A
1
, B
1
)) for some sets A
0
, B
0
, A
1
, B
1
then
l
a ×
(L)
b
(a, b) ∈ S
=
l
dom S ×
(L)
l
im S.
Corollary 1998. (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 1999. a
0
q
(L)
b
0
≡ a
1
q
(L)
b
1
⇔ a
0
≡ a
1
∧ b
0
≡ b
1
.
3.2. 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 ∈ dom F).
Definition 2000.
`
(L)
F =
d
i∈dom F
(ι
λj∈n:Ob F
j
i
◦ F
†
i
◦ (ι
λj∈n:Ob F
j
i
)
†
) (if ι is
defined at λj ∈ n : Ob F
j
). (I call it supremum coproduct).
4. APPLYING THIS TO THE THEORY OF FUNCOIDS AND RELOIDS 13
Abbreviate ι
i
= ι
λj∈n:Ob F
j
i
.
So
`
F =
d
i∈dom F
(ι
i
◦ F
†
i
◦ (ι
i
)
†
).
`
F = min
(
Φ∈End
`
(Q)
j∈n
Ob F
j
∀i∈n:Φwι
i
◦F
†
i
◦(ι
i
)
†
)
.
Taking into account that ι
i
is a monovalued entirely defined morphism, we get:
Obvious 2001.
`
(L)
= min
(
Φ∈End
`
(Q)
j∈n
Ob F
j
∀i∈n:ι
i
∈C(F
†
i
,Φ)
)
.
Corollary 2002. ι
i
∈ C
F
i
,
`
(L)
F
for every i ∈ dom F .
3.3. Category of continuous morphisms. Let ι
i
(for i ∈ dom F ) be en-
tirely defined monovalued and metacomplete morphisms.
Let
L
of an indexed family of morphisms is a morphism; (
L
f) ◦ ι
i
= f
i
;
L
i∈n
(f ◦ ι
i
) = f (a dual of the above).
Let F
i
∈ End
`
(Q)
j∈n
Ob F
j
for all i ∈ n (where n is some index set) (a self-dual
of the above).
Definition 2003. ι
cont(C)
i
=
`
(L)
F, F
†
i
, ι
i
.
Proposition 2004. ι
i
are continuous, that is ι
cont(C)
i
are morphisms.
Lemma 2005. f ∈ Hom
cont(C)
`
(L)
F, Y
FiXme: What is Y ? is continuous
iff all f ◦ ι
cont(C)
are continuous.
Theorem 2006.
`
(L)
together with
L
is a (partial) coproduct in the category
cont(C).
4. Applying this to the theory of funcoids and reloids
4.1. Funcoids.
Definition 2007. 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
X
i
∈ 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
X
i
∈ 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 2008. π
i
◦
N
f = f
i
;
N
i∈n
(π
i
◦ f ) = f .
Obvious 2009. (
L
f) ◦ ι
i
= f
i
;
L
i∈n
(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 co-
products and thus both
Q
(L)
F and
`
(L)
F are defined.
6. CANONICAL PRODUCT AND SUBATOMIC PRODUCT 14
4.2. Reloids.
Definition 2010. 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
X
i
∈ 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
X
∈ 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 2011. π
i
◦
N
f = f
i
;
N
i∈n
(π
i
◦ f ) = f .
Obvious 2012. (
L
f) ◦ ι
i
= f
i
;
L
i∈n
(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 co-
products 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 uniformilty.
5. 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 2013. Terminal objects of Fcd are exactly ↑
F
{∗}×
FCD
↑
F
{∗} =↑
FCD
{(∗, ∗)} where ∗ is an arbitrary point.
Proof. In order for a function f : X →↑
FCD
{(∗, ∗)} be a morphism, it is
required exactly f ◦ X v↑
FCD
{(∗, ∗)} ◦ f
f ◦ X v (f
−1
◦ ↑
FCD
{(∗, ∗)})
−1
; f ◦ X v ({∗} ×
FCD
hf
−1
i{∗})
−1
; f ◦ X v
hf
−1
i{∗} ×
FCD
{∗} what true exactly when f is a constant function with the value
∗.
If n = ∅ then Z = {∅};
Q
(L)
∅ = max FCD(Z, Z) =↑
F
{∅}×
FCD
↑
F
{∅} =↑
FCD
{(∅, ∅)}.
FiXme: Initial and terminal objects of Rld.
6. Canonical product and subatomic product
FiXme: Confusion between filters on products and multireloids.
Proposition 2014. Pr
RLD
i
|
F(Z)
= hπ
i
i for every index i of a cartesian product
Z.
Proof. If X ∈ F(Z) then (Pr
RLD
i
|
F(Z)
)X = Pr
RLD
i
X =
d
F
hPr
i
i
∗
X =
d
hπ
i
i up X = hπ
i
iX .
Proposition 2015.
Q
(A)
F =
d
i∈n
π
FCD
Q
i∈n
Dst F
i
−1
◦ F
i
◦ π
FCD
Q
i∈n
Src F
i
!
.
8. CARTESIAN CLOSEDNESS 15
Proof. a
h
Q
(A)
F
i
b ⇔ ∀i ∈ dom F : Pr
RLD
i
a [F
i
] Pr
RLD
i
b ⇔
∀i ∈ dom F :
*
π
FCD
Q
i∈n
Dst F
i
−1
+
[F
i
]
π
FCD
Q
i∈n
Src F
i
⇔
∀i ∈ dom F : a
"
π
FCD
Q
i∈n
Dst F
i
−1
◦ F
i
◦ π
FCD
Q
i∈n
Src F
i
#
b ⇔
a
"
d
i∈n
π
FCD
Q
i∈n
Dst F
i
−1
◦ F
i
◦ π
FCD
Q
i∈n
Src F
i
!#
b for ultrafilters a and
b.
Corollary 2016.
Q
(L)
F =
Q
(A)
F is F is a small indexed family of funcoids.
7. Further plans
Does the formula
Q
(L)
i∈n
(g
i
◦ f
i
) =
Q
(L)
g ◦
Q
(L)
f hold?
Coordinate-wise continuity.
8. Cartesian closedness
We are not only to prove (or maybe disprove) that our categories are cartesian
closed, but also to find (if any) explicit formulas for exponential transpose and
evaluation.
”Definition” A category is //cartesian closed// iff:
1
◦
. It has finite products.
2
◦
. For each objects A, B is given an object MOR(A, B) (//exponentiation//)
and a morphism ε
Dig
A,B
: MOR(A, B) × A → B.
3
◦
. For each morphism f : Z × A → B there is given a morphism (//expo-
nential transpose//) ∼ f : Z → MOR(A, B).
4
◦
. ε
B,C
◦ (∼ f × 1
A
) = f for f : A → B × C.
5
◦
. ∼ (ε
B,C
◦ (g × 1
A
)) = g for g : A → MOR(B, C).
We will also denote f 7→ (−f) the reverse of the bijection f 7→ (∼ f).
Our purpose is to prove (or disprove) that categories Dig, Fcd, and Rld are
cartesian closed. Note that they have finite (and even infinite) products is already
proved.
Alternative way to prove: you can prove that the functor − × B is left adjoint
to the exponentiation −
B
where the counit is given by the evaluation map.
8.1. Definitions. Categories Dig, Fcd, and Rld are respectively categories
of:
1
◦
. discretely continuous maps between digraphs;
2
◦
. (proximally) continuous maps between endofuncoids;
3
◦
. (uniformly) continuous maps between endoreloids.
”Definition” //Digraph// is an endomorphism of the category Rel.
For a digraph A we denote Ob A the set of vertexes or A and GR A the set of
edges or A.
”Definition” Category Dig of digraphs is the category whose objects are di-
graphs and morphisms are discretely continuous maps between digraphs. That is
morphisms from a digraph µ to a digraph ν are functions (or more precisely mor-
phisms of Set) f such that f ◦ µ v ν ◦ f (or equivalently µ v f
−1
◦ ν ◦ f or
equivalently f ◦ µ ◦ f
−1
v ν).
”Remark” Category of digraphs is sometimes defined in an other (non equiva-
lent) way, allowing multiple edges between two given vertices.
8. CARTESIAN CLOSEDNESS 16
8.2. Conjectures.
Conjecture 2017. 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.
Our purpose is to prove (or disprove) that categories Dig, Fcd, and Rld are
cartesian closed. Note that they have finite (and even infinite) products is already
proved.
Alternative way to prove: you can prove that the functor − × B is left adjoint
to the exponentiation −
B
where the counit is given by the evaluation map.
See http://www.springer.com/us/book/9780387977102 for another way to
prove Cartesian closedness.
8.3. Category Dig is cartesian closed. Category of digraphs is the sim-
plest of our three categories and it is easy to demonstrate that it is cartesian closed.
I demonstrate cartesian closedness of Dig mainly with the purpose to show a pat-
tern similarly to which we may probably demonstrate our two other categories are
cartesian closed.
Let G and H be graphs:
• Ob MOR(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 ∈ Ob MOR(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) ⇔ hf ×
(C)
gi 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” Transposition and its inverse are morphisms of Dig.
”Proof” It follows from 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 ∧ vBw ⇒ ((∼ f)xv, (∼ f )yw) ∈ 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 defined by the formula:
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” Evaluation is a morphism of Dig.
”Proof” Because ε
B,C
(p, q) = −(1
MOR(B,C)
).
It remains to prove: * ε
B,C
◦ (∼ f × 1
A
) = f for f : A → B × C; * ∼
(ε
B,C
◦ (g × 1
A
)) = g for g : A → MOR(B, C).
”Proof” ε
B,C
(∼ f × 1
A
)(a, p) = ε
B,C
((∼ f )a, p) = (∼ f )ap = f(a, p). So
ε
B,C
◦ (∼ f × 1
A
) = f.
∼ (ε
B,C
◦ (g × 1
A
))(p)(q) = (ε
B,C
◦ (g × 1
A
))(p, q) = ε
B,C
(g × 1
A
)(p, q) =
ε
B,C
(gp, q) = g(p)(q). So ∼ (ε
B,C
◦ (g × 1
A
)) = g.
8.4. By analogy with the proof that Dig is cartesian closed. The most
obvious way for proof attempt that Fcd is cartesian closed is an analogy with the
proof that Dig is cartesian closed.
8. CARTESIAN CLOSEDNESS 17
Consider the long formula above. The proof would arise if we replace x and y in
this formula with filters and operations and relations on set element with operations
and relations on filters.
This proof could be simplified in either of two ways:
• replace x and y with ultrafilters, see [[Proof for Fcd using ultrafilters]];
• replace x and y with sets (principal filter), see [[Proof for Fcd using sets]].
This is not quite easy however, because we need to calculate uncurrying for
a entirely defined monovalued principal funcoid (what is essentially the same as a
function of a Set-morphisms) taking either ultrafilters or principal filters as argu-
ments. Such (generalized) uncurrying is not quite easy.
To sum what we need to prove:
• Transposition is a morphism.
• Evaluation is a morphism.
• ε
B,C
◦ (∼ f × 1
A
) = f for f : A → B × C.
• ∼ (ε
B,C
◦ (g × 1
A
)) = g for g : A → MOR(B, C).
8.5. Attempt to describe exponentials in Fcd.
• Exponential object HOM(A, B) is the following endofuncoid:
• – Object Ob HOM(A, B) = (Ob B)
Ob A
;
– Graph is GR HOM(A, B) =↑
FCD
n
(f,g)
f,g∈Hom
Set
(Ob A,Ob B)∧↑
FCD
g◦A◦↑
FCD
f
−1
vB
o
.
• Transposition is uncurrying.
• Evaluation is ε
A,B
x = hPr
(A)
0
xi Pr
(A)
1
x.
We need to prove that the above defined are really an exponential and an
evaluation.
Possible ways to prove that Fcd is cartesian closed follow:
NEW IDEA: Prove GR HOM(A, B) =↑
FCD
(Pr
(A)
0
p,Pr
(A)
1
p)
p∈??∧hpiAvB
(what’s about
other kinds of projections?)
8.6. Proof for Fcd using sets. Currying for sets is hfi(X × Y ) =
S
hh∼
fiXiY (as it’s easy to prove). This simple formula gives hope, but...
It does not work with sets because an analogy for sets of the last equality of
the above mentioned long formula would be:
∀X, Y, V, W ∈ P Ob A : (X × V [A × B]
∗
Y × W ⇒ hf i(X × V ) [C]
∗
hfi(Y × W )) ⇒
f : A × B → C
but this implication seems false.
The most obvious way for proof attempt that Fcd is cartesian closed is an
analogy with the proof that Dig is cartesian closed.
Use the exponential object, transposition, and evaluation as defined in [[this
page|Is category Fcd cartesian closed?]]
8.7. Reducing to the fact that Dig is cartesian closed. It is probably a
simpler way to prove that Fcd is cartesian closed by embedding it into Dig (which
is [[already known to be cartesian closed|Category Dig is cartesian closed]]).
Fcd can be embedded into Dig by the formulas:
• A 7→ hAi;
• f 7→ hfi.
That this really maps a morphism of Fcd into a morphism of Dig follows from
the fact that hg ◦ fi = hgi ◦ hfi.
Obviously this embedding (denote it T ) is an injective (both on objects and
morphisms) functor.
9. IS CATEGORY RLD CARTESIAN CLOSED? 18
We will define:
• ε
Fcd
A,B
= T
−1
ε
Dig
T A,T B
;
• ∼
Fcd
f = T
−1
∼
Dig
T f.
Due to functoriality and injectivity of T it is enough to prove that above defined
ε
Fcd
A,B
and ∼
Fcd
f exist and are morphisms of Fcd.
ε
Dig
T A,T B
6= T ε
Fcd
A,B
because ε
Dig
T A,T B
accepts ordered pairs as the argument and
T ε
Fcd
A,B
accepts sets as the argument. So this is a dead end. Can the proof idea be
salvaged?
9. Is category Rld cartesian closed?
We may attempt to prove that Rld is cartesian closed by embedding it into
supposedly cartesian closed category Fcd by the function ρ:
hρfix = f ◦ x and hρf
−1
iy = f
−1
◦ y.
TODO: More to write on this topic.
CHAPTER 3
Equalizers and co-Equalizers in Certain Categories
It is a rough draft. Errors are possible.
FiXme: Change notation
Q
→
Q
(L)
.
1. Equalizers
Categories cont(C) are defined above.
I will denote W the forgetful functor from cont(C) to C.
In the definition of the category cont(C) take values of ↑ as principal morphisms.
FiXme: Wording.
Lemma 2018. Let f : X → Y be a morphism of the category cont(C) where C is
a concrete category (so W f =↑ ϕ for a Rel-morphism ϕ because f is principal) and
im ϕ = A ⊆ Ob Y . Factor it ϕ = E
Ob Y
◦ u where u : Ob X → A using properties of
Set. Then u is a morphism of cont(C) (that is a continuous function X → ι
A
Y ).
Proof. (E
Ob Y
)
−1
◦ ϕ = (E
Ob Y
)
−1
◦ E
Ob Y
◦ u;
(E
Ob Y
C
)
−1
◦ ↑ ϕ = (E
Ob Y
C
)
−1
◦ E
Ob Y
C
◦ ↑ u;
(E
Ob Y
C
)
−1
◦ ↑ ϕ =↑ u;
X v (↑ u)
−1
◦ π
A
Y ◦ ↑ u ⇔ X v (↑ ϕ)
−1
◦ E
Ob Y
C
◦ π
A
Y ◦ (E
Ob Y
C
)
−1
◦ ↑ ϕ ⇔
X v (↑ ϕ)
−1
◦ E
Ob Y
C
◦ (E
Ob Y
C
)
−1
◦ Y ◦ E
Ob Y
C
◦ (E
Ob Y
C
)
−1
◦ ↑ ϕ ⇔ X v (↑ ϕ)
−1
◦ Y ◦ ↑
ϕ ⇔ X v (W f)
−1
◦ Y ◦ W f what is true by definition of continuity.
Equational definition of equalizers:
http://nforum.mathforge.org/comments.php?DiscussionID=5328/
Theorem 2019. The following is an equalizer of parallel morphisms f, g : A →
B of category cont(C):
• the object X = ι
{
x∈Ob A
f x=gx
}
A;
• the morphism E
Ob X,Ob A
considered as a morphism X → A.
Proof. Denote e = E
Ob X,Ob A
.
Let f ◦ z = g ◦ z for some morphism z.
Let’s prove e ◦ u = z for some u : Src z → X. Really, as a morphism of Set it
exists and is unique.
Consider z as as a generalized element.
f(z) = g(z). So z ∈ X (that is Dst z ∈ X). Thus z = e ◦ u for some u (by
properties of Set). The generalized element u is a cont(C)-morphism because of
the lemma above. It is unique by properties of Set.
We can (over)simplify the above theorem by the obvious below:
Obvious 2020.
n
x∈Ob A
fx=gx
o
= dom(f ∩ g).
2. Co-equalizers
http://math.stackexchange.com/questions/539717/
how-to-construct-co-equalizers-in-mathbftop
Let ∼ be an equivalence relation. Let’s denote π its canonical projection.
19
3. REST 20
Definition 2021. f/ ∼=↑ π ◦ f◦ ↑ π
−1
for every morphism f.
Obvious 2022. Ob(f/ ∼) = (Ob f)/r.
Obvious 2023. f/ ∼= h↑
FCD
π×
(C)
↑
FCD
πif for every morphism f.
To define co-equalizers of morphisms f and g let ∼ be is the smallest equivalence
relation such that fx = gx.
Lemma 2024. Let f : X → Y be a morphism of the category cont(C) where C
is a concrete category (so W f =↑ ϕ for a Rel-morphism ϕ because f is principal)
such that ϕ respects ∼. Factor it ϕ = u ◦ π where u : Ob(X/ ∼) → Ob Y using
properties of Set. Then u is a morphism of cont(C) (that is a continuous function
X/ ∼→ Y ).
Proof. f ◦ X ◦ f
−1
v Y ; ↑ u◦ ↑ π ◦ X◦ ↑ π
−1
◦ ↑ u
−1
v Y ; ↑ u ∈ C(↑ π ◦ X◦ ↑
π
−1
, Y ) = C(X/ ∼, Y ).
Theorem 2025. The following is a co-equalizer of parallel morphisms f, g :
A → B of category cont(C):
• the object Y = f/ ∼;
• the morphism π considered as a morphism B → Y .
Proof. Let z ◦ f = z ◦ g for some morphism z.
Let’s prove u ◦ π = z for some u : Y → Dst z. Really, as a morphism of Set it
exists and is unique.
Src z ∈ Y . Thus z = u ◦ π for some u (by properties of Set). The function u is
a cont(C)-morphism because of the lemma above. It is unique by properties of Set
(π obviously respects equivalence classes).
3. Rest
Theorem 2026. The categories cont(C) (for example in Fcd and Rld) are
complete. FiXme: Note that small complete category is a preorder!
Proof. They have products and equalizers.
Theorem 2027. The categories cont(C) (for example in Fcd and Rld) are
co-complete.
Proof. They have co-products and co-equalizers.
Definition 2028. I call morphisms f and g of a category with embeddings
equivalent (f ∼ g) when there exist a morphism p such that Src p v Src f, Src p v
Src g, Dst p v Dst f, Dst p v Dst g and ι
Src f,Dst f
p = f and ι
Src g,Dst g
p = g.
Problem 2029. Find under which conditions:
1
◦
. Equivalence of morphisms is an equivalence relation.
2
◦
. Equivalence of morphisms is a congruence for our category.
CHAPTER 4
Categories of filters
In [1] two categories, whose objects are related with filters on sets, are defined
and researched.
Accordingly [1] infinite product is defined just in the first (denoted F there)
of these two categories. So we will for now consider the first category. (Usefulness
of the second category for our research is questionable.)
Let f : A → B be a function, A be a filter on A.
Proposition 2030.
n
Y ∈PB
hf
−1
i
∗
Y ∈A
o
is a filter.
Proof. That it is an upper set is obvious.
Let Y
0
, Y
1
∈
n
Y ∈PB
hf
−1
i
∗
Y ∈A
o
. Then
f
−1
∗
Y
0
∈ A and
f
−1
∗
Y
1
∈ A. We have
f
−1
∗
(Y
0
∩ Y
1
) =
f
−1
∗
Y
0
∩
f
−1
∗
Y
1
∈ A
since f is monovalued. Thus Y
0
∩ Y
1
∈
n
Y ∈PB
hf
−1
i
∗
Y ∈A
o
.
Theorem 2031. FiXme: Should be moved above in the book.
n
Y ∈PB
hf
−1
i
∗
Y ∈A
o
is
equal to the filter generated by the filter base
hfi
∗
∗
A, for every filter A.
Proof. Denote B =
n
Y ∈PB
hf
−1
i
∗
Y ∈A
o
, C =
hfi
∗
∗
A.
Let Y ∈ C. Then Y = hfi
∗
A where A ∈ A. Then
f
−1
∗
hfi
∗
A ⊇ A and so