Database Reference
In-Depth Information
:
A
→
A
where
M
A
A
={
, which
represents such an operation from a database
A
into the closed database
TA
will
have a finite cardinality of
∂
0
(f
o
i
)
q
i
,
1
∅
}=
MakeOperads(
{∀
x
i
(q
i
(
x
i
)
⇒
r
q
(
x
i
))
}
)
ar(o
i
)
, so that
we can restrict
Σ
D
to finite databases only. The extension of
Σ
D
to
all
databases,
as infinite databases which are not closed objects (i.e., compact objects in
DB
), can
be successively obtained by left Kan extension of this finite restriction as will be
demonstrated in what follows.
First of all, we have to demonstrate the existence of the
ω
-cocontinuous end-
ofunctors for the
DB
category which can be used for a construction of the initial
algebra based on morphisms of the
DB
category. We have demonstrated in Propo-
⊆
ω
A
, with cardinality
|
∂
0
(f
o
i
)
|=
sition
53
that the “merging with
A
” endofunctor
A
=
DB
is
ω
-cocontinuous, i.e., a monotone function which preserves l.u.bs of
ω
-chains.
Let us define another
ω
-cocontinuous endofunctor:
A
⊕
_
:
DB
−→
Proposition 59
For each object A in the category
DB
with coproduct
+
,
equal
from Theorem
5
to the disjoint union
,
the endofunctor A
+
T _
=
A
T _
:
DB
−→
DB
is ω-cocontinuous
.
Proof
Constant endofunctor
A
DB
is an
ω
-cocontinuous endofunctor, iden-
tity endofunctors are
ω
-cocontinuous, colimit functors (thus coproduct
:
DB
→
+
, which is,
by Theorem
5
in Sect.
3.3.2
, equal to disjoint union
)are
ω
-cocontinuous (be-
cause of the standard “interchange of colimits”). Since
ω
-cocontinuity is preserved
by functor composition
T
it is
enough to show that
T
is an
ω
-cocontinuous endofunctor. In fact, let us consider the
following diagram obtained by an iterative application of the endofunctor
T
◦
, for the second endofunctor
A
+
T
_
=
(A
+
Id
_
)
◦
0
T
⊥
0
1
T
2
0
2
···
T
ω
0
0
,
⊥
⊥
⊥
0
is the initial object in
DB
and all objects
T
n
0
0
and hence all arrows
where
⊥
⊥
=⊥
T
ω
0
0
, and
TColimJ
in this chain are identities. Thus,
ColimJ
=
⊥
=⊥
=
ColimTJ
=
0
, so that
T
is an
ω
-cocontinuous endofunctor.
⊥
In what follows, we will make the translation of the inductive principle from the
Set
into
DB
category, for any given database schema
A
and R-algebra
α
, based on
the following considerations:
•
The object (i.e., a set of relational symbols of a database schema)
A
in
Set
is
considered as set of variables
X
=
A
, while an instance of this schema
A
=
α
∗
(
A
)
, obtained by R-algebra
α
:
X
→
TA
, is an object in
DB
. Analogously, the
set of terms with variables
T
P
X
used in the
Set
category is translated (by the
surjective function
α
#
:
T
P
X
TA
, obtained as a unique homomorphism from
the initial
Σ
R
-algebra of terms
T
P
X
in
Set
) into the set
TA
of all views (which
are relations obtained by computation of these terms with variables in
X
=
A
).
, while in
Set
it is considered as
aset
of relations) is not a carrier set for the initial
(A
However,
TA
(in
DB
it is an
instance database
of a schema
A
Σ
D
)
-
algebra in
DB
for the endofunctor
Σ
D
=
T
:
DB
→
DB
(see bellow) because,
