Database Reference
In-Depth Information
DB
(B,C)
,
f
j
⊆
for each simple arrow
f
j
∈
B
⊗
C
. Thus, based on the complete
{
f
j
|
,
⊕
lattice
L
DB
=
(Ob
DB
,
,
⊗
)
in Proposition
51
,theset
f
j
∈
DB
(B,C)
}
is a sublattice with the l.u.b. equal to
B
C
, so that the merging (join operator
of this lattice) of all elements in this set is equal to its l.u.b. That is, we obtain
that
C
B
⊗
=
{
f
j
|
T(
{
f
j
|
T(
{
f
j
|
f
j
⊆
f
j
∈
}=
f
j
∈
}
=
DB
(B,C)
DB
(B,C)
)
B
⊗
C
=
TB
∩
TC
}
)
=
T(TB
∩
TC)
=
T(TB)
∩
T(TC)
=
TB
∩
TC
=
B
⊗
C
.
Consequently, the hom-set
DB
(B,C)
in
Set
of all morphisms
B
→
C
can be
'internalized' into the hom-object
C
B
in
DB
category by merging of compact el-
ements
A
C
is the “distance” between
B
and
C
, following
Lawvere's idea, as follows from the definition of metric space for the
DB
category
in Sect.
9.1.1
).
The aim to 'internalize' the hom-sets is justified by the necessity to substitute
the base category
Set
by the base database category
DB
in the cases as Yoneda
embeddings for categories by the contravariant functor
H
:
B
⊗
C
(where
B
⊗
Set
C
introduced
C
OP
→
in technical preliminaries (Sect.
1.5
).
For any two objects B and C
,
C
B
B
C
Lemma 18
B
⊗
C
.
Proof
From Definition
58
,
h
(see the remark above for simple objects)
C
B
=
f
il
|
DB
(B
i
,C
l
)
f
il
∈
1
≤
i
≤
k,
1
≤
l
≤
n
=
B
i
⊗
C
l
1
≤
i
≤
k,
1
≤
l
≤
n
(from Definition
19
)
=
B
⊗
C.
=
1
≤
j
≤
m
A
j
,m
Remark
From duality, for any two objects
A
≥
1 and
B
=
1
≤
i
≤
k
B
i
,k
≥
1 there is the bijection _
OP
:
DB
(A,B)
→
DB
(B,A)
such that for
B
,
f
OP
_
OP
(f )
any morphism
f
:
A
→
=
:
B
→
A
, where
f
OP
,
f
OP
=
f
ji
|
f
ji
:
A
j
→
B
i
∈
with equal information fluxes
f
OP
ji
=
f
ji
(the flux transmitted from simple objects
A
j
into
B
i
can be reflexed from
B
i
to
A
j
as well). Consequently, by Lemma
18
,
