Database Reference
In-Depth Information
d(B,A)
and hence, from Corollary
14
in Sect.
3.4.1
, we obtain the metric symmetry
property
d(A,B)
≈
d(B,A)
.
0
(which is the terminal and initial object in the
DB
category; also called an infi-
nite object) we obtain:
d(
From the definition of the distance we have that for the infinite distance
⊥
0
,
0
)
Υ
(zero distance is the total object in the
DB
category), and for any other database
A
⊥
⊥
=
0
, the distance from
A
to the infinite object (database) is
infinite
. Thus, the bottom element
0
,
d(A,
0
)
⊥
⊥
=⊥
0
and the top
element
Υ
in the database lattice are, for this database metric system, the infinite
and zero distances (closed objects), respectively. Let us make a comparison between
this database metric space in Proposition
63
and the general metric space (Frechet
axioms):
⊥
Frechet axioms
DB
metric space
d(A,B)
+
d(B,C)
≥
d(A,C)
d(A,B)
⊗
d(B,C)
d(A,C)
0
≤
d(A,A)
Υ
d(A,A)
if
d(A,B)
=
0 then
A
=
B
if
d(A,B)
≈
Υ
then
A
≈
B
0
d(A,B)<
∞
d(A,B)
⊥
d(A,B)
=
d(B,A)
d(A,B)
≈
d(B,A)
Note that by Definition
19
in Sect.
3.2
of the PO relation '
',
A
≈
Υ
only when
A
Υ
and, from Proposition
63
above,
A
≈
B
. Thus, a database metric space
DB
met
, where the distances are the
closed
objects (databases), is a subcategory of
DB
OP
I
=
Υ
. Thus,
d(A,B)
≈
Υ
implies
d(A,B)
=
composed of only closed objects of
DB
: each arrow is an epimorphism
in
OP
:
:
→
A
B
(for a monomorphism
in
B
A
in
DB
I
defined by Theorem
6
in Sect.
3.2.5
from
B
A
) which corresponds to
the distance relation
A
B
. Hence, we can say that a database metric space is
embedded in the
DB
category, where the distances are the closed databases and the
distance relations are the epimorphisms between closed databases.
9.1.2 Subobject Classifier
The concept of a
subobject
is the categorial version of the set-theoretical
subset
.The
main idea is to regard a subset
B
of a given object
A
as a monomorphism from
B
into
A
. Every subset
B
⊆
A
in the category
Set
can be described by its characteristic
function
C
f
:
−→
=
∈
A
Ω
, such that
C
f
(x)
1 (True) if
x
B
, 0 (False) otherwise,
where
Ω
is the set of truth values (see Definition
1
in Sect.
1.3
). The
subobject classifier for
Set
is the object
Ω
=
2
={
0
,
1
}
=
2
with the arrow
true
:{∗}−→
Ω
(with
true(
∗
)
=
1), and a constant function
t
B
:
B
→{∗}
from
B
into a terminal object
{∗}
(a singleton) that satisfies the
Ω
-axiom:
For each monomorphism
in
B
:
B
→
A
there is one and only one characteristic
arrow
C
in
B
:
−→
A
Ω
such that the diagram on the left
