Database Reference
In-Depth Information
Let us show the functorial property for the identity arrows, and consider
two objects
A
=
1
≤
j
≤
m
A
j
and
B
=
1
≤
i
≤
k
B
i
, with their identity arrows
id
A
=
1
≤
j
≤
m
id
A
j
and
id
B
=
1
≤
i
≤
k
id
B
i
such that
id
A
=
1
≤
j
≤
m
id
A
j
=
1
≤
j
≤
m
TA
j
=
TA
,
id
B
=
1
≤
i
≤
k
id
B
i
=
1
≤
i
≤
k
TB
i
=
id
A
TB
, with
={
id
A
1
,
id
B
...,id
A
m
}
={
id
B
1
,...,id
B
k
}
and
. Then,
id
A
⊗
id
B
id
A
id
B
∈
id
A
j
∩
id
B
i
}=⊥
1
=
{
q
⊥
}∪{
id
R
|
R
|
id
A
j
∈
, id
B
i
∈
=
{
k
1
q
⊥
}∪{
id
R
|
R
∈
TA
j
∩
TB
i
}=⊥
|
1
≤
j
≤
m,
1
≤
i
≤
=
id
TA
j
∩
TB
i
|
k
0
,
1
TA
j
∩
TB
i
=⊥
≤
j
≤
m,
1
≤
i
≤
=
id
A
j
⊗
B
i
|
k
0
,
1
A
j
⊗
B
i
=⊥
≤
j
≤
m,
1
≤
i
≤
id
A
⊗
B
=
.
Hence, from Definition
23
,
id
A
⊗
id
B
=
id
A
⊗
B
:
A
⊗
B
→
A
⊗
B
.
is a monoidal bifunctor with natural isomorphic trans-
formations (which generate an isomorphic arrow for each object in
DB
):
•
It is easy to verify that
⊗
α
:
(
_
⊗
_
)
⊗
_
−→
_
⊗
(
_
⊗
_
)
, (associativity)
•
:
⊗
−→
β
Υ
_
I
DB
, (left identity)
•
γ
:
⊗
Υ
−→
I
DB
, (right identity)
such that
A
⊗
B
B
⊗
A
,
A
⊗
Υ
=
Υ
⊗
A
A
and
A
⊗⊥
_
0
0
. For any
⊥
morphism
f
:
A
−→
B
, from Proposition
6
we obtain
f
A
⊗
B
.
of the monoidal category
DB
is not
unique in contrast with
the Cartesian product (we can have for simple databases,
A
A tensor product
⊗
⊗
B
=
C
⊗
B
such that
0
).
C
=
A
∪
A
1
⊃
A
with
TA
1
∩
TB
=⊥
B
is a closed object (intersection of two closed objects
TA
and
TB
), and that the information flux of any morphism from
A
to
B
is a closed
object included in this maximal information flux (i.e., overlapping) between
A
and
B
. A matching of two completely disjoint simple databases is equal to the empty
zero object
Notice that each
A
⊗
0
.
⊥
Proposition 47
Each object A with the monomorphism μ
A
:
A
⊗
A
−→
A
and the epimorphism η
A
:
Υ
A compose a monoid in the monoidal category
(
DB
,
⊗
,Υ,α,β,γ)
.
Proof
We have
A
A
is a monomorphism
(from Theorem
6
in Sect.
3.2.5
). From the fact that
TA
⊆
TΥ
=
Υ
and point 2 of
Proposition
7
, we obtain a monomorphism
in
TA
:
TA
→
Υ
and hence a monomor-
phism (composition of two monic arrows)
in
⊗
A
A
and hence
μ
A
=
in
:
A
⊗
A
−→
in
id
A
=
in
TA
◦
is
A
:
A
→
Υ
with
=
.
in
OP
Consequently, by duality, its opposite arrow
η
A
=
:
Υ
A
is an epimorphism
=
id
A
η
A
=
in
OP
=
in
in
=
id
A
=
such that
TA
. It is easy to verify that
