Database Reference
In-Depth Information
Corollary 31
The following strong relationship between the relational-algebra sig-
nature endofunctor Σ
D
,
translated into the database category
DB
,
and the closure
endofunctor T holds
:
T
=
Lan
J
◦
K
(Σ
D
)
:
DB
→
DB
.
As we can see the translation of the relational-algebra signature
Σ
R
is provided
by the power-view endofunctor
Lan
J
◦
K
(Σ
D
)
=
:
→
T
DB
DB
, as informally pre-
sented in the introduction.
Consequently, the endofunctor
(A
Lan
J
◦
K
(Σ
D
))
=
(A
T)
:
DB
→
DB
, from
Proposition
59
,isthe
ω
-cocontinuous endofunctor
(A
T)
:
DB
→
DB
, with a
chain
0
(A
0
1
(A
0
2
···
0
T)
2
T)
ω
,
⊥
T)
⊥
⊥
(A
where
0
0
⊥
=⊥
(A
T)
A,
=
(A
T)
⊥
A
=⊥
(A
T)
2
0
0
0
⊥
A
TA,
T)
⊥
TA
T)
3
0
0
(A
⊥
=
(A
A
0
T
2
A
0
=⊥
A
TA
=⊥
A
TA
TA,
ω
TA
.
T)
ω
0
=⊥
and we obtain that the colimit of this diagram in
DB
is
(A
A
0
),
0
From the fact that for the coproduct (and initial object
⊥
⊥
B
B
for any
B
,
ω
TA
. This colimit is a least
T)
ω
then we can take as the colimit
(A
=
A
fixpoint of the monotone operator
(A
T)
in the complete lattice of databases in
DB
(by Knaster-Tarski theorem for least fixpoints,
(A
T)
ω
).
Notice that the coproduct of two databases
A
and
B
in the
DB
category [
16
,
19
]
corresponds to completely disjoint databases, in the way that it is not possible to
use the relations from both these databases in the
same
query: hence
T(A
B)
=
TA
T)
ω
)
T )((A
(A
TB
, that is, the set of all views of a coproduct
A
B
is a disjoint union of
views of
A
and views of
B
.
In fact,
T
A
TA
T
ω
T)
(A
T)
ω
=
(A
A
=
A
TA
TA
ω
=
A
TA
TTA
=
A
TA
TA
ω
ω
T)
ω
.
=
A
TA
=
(A
ω
We can denote this isomorphic arrow in the
DB
category, which is the initial
(A
ω
TA)
ω
TA)
.
T)
-algebra, by
[
inl
A
, inr
A
]:
(A
T )(A
→
(A
Consequently, the variable injection
inl
A
:
A
→
T
P
A
in
Set
is translated into
:
A
→
(A
ω
TA)
in the
1
the corresponding monomorphism
inl
A
=
id
A
,
⊥
DB
category, with the information flux
inl
A
=
0
TA
⊥
TA
. The right inclusion
