Database Reference
In-Depth Information
where ψ is a statement
is the number of at-
tributes of the relation
(
view
)
R
,
and π
k
is a kth projection
.
We define T
w
(f ) for
a complex arrow f by the set of ptp arrows
T
w
(f )
∀
1
≤
k
≤|
R
|
∀
(d
i
∈
π
k
(R))Va l(d
i
),
|
R
|
f
={
|
f
ji
∈
}
T
w
(f
ji
)
.
−→
2.
There exist the natural transformations ξ
:
T
w
T
(
natural monomorphism
)
−→
and ξ
−
1
:
T
T
w
(
natural epimorphism
)
such that for any simple object A
,
inc
A
is a monomorphism and ξ
−
1
(A)
inc
OP
A
ξ(A)
=
=
is an epimorphism
,
which
ξ
−
1
(A)
.
are equivalent
,
i
.
e
.,
ξ(A)
≈
Proof
Let us show that the morphism
T
w
(f )
:
T
w
(A)
→
T
w
(B)
, for a given mor-
α
∗
(
M
AB
)
phism
f
B
, corresponds to the definition of arrows in
DB
cat-
egory, specified by Theorem
1
, that is, there is an SOtgd
Φ
of a schema mapping
such that
T
w
(f )
=
:
A
→
α
∗
(MakeOperads(
))
. Let us define the set of relational sym-
bols in the instance-database
T
w
(A)
by
S
T
w
A
={
=
{
Φ
}
r
i
|
r
i
∈R
and
α(r
i
)
∈
T
w
(A)
}
, and
α
∗
(
C
=
∅
=
C
its schema
(S
T
w
A
,
)
, so that
T
w
(A)
)
(for the instance-database
T
w
(B)
,
D
=
∅
we define analogously the schema
(S
T
w
B
,
)
). Let us define the set of relational
∈
f
and
symbols
S
T
w
f
={
r
i
|
r
i
∈R
and
R
=
α(r
i
)
∀
1
≤
k
≤|
R
|
∀
(d
i
∈
π
k
(R))Va l (d
i
)
}
.
Then we define the SOtgd
Φ
byaformula
{∀
x
i
(r
i
(
x
i
)
⇒
r
i
(
x
i
))
|
r
i
∈
S
T
w
f
}
.
In fact,
MakeOperads(
{
Φ
}
)
={
1
r
i
∈
O(r
i
,r
i
)
|
r
i
∈
S
T
w
f
}∪{
1
r
∅
}
is a sketch's
mapping from the schema
C
into
D
and
α
∗
MakeOperads
{
}
T
w
(f )
=
Φ
α
∗
1
r
i
∈
S
T
w
f
∪{
1
r
∅
}
=
O(r
i
,r
i
)
|
r
i
∈
=
id
r
i
:
S
T
w
f
∪{
α(r
i
)
→
α(r
i
)
∈
O(r
i
,r
i
)
|
r
i
∈
q
⊥
}
.
For any identity arrow
id
A
:
A
→
A
,
=
id
R
:
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
∈
id
A
and
→
|
T
w
(id
A
)
R
R
R
=
id
R
:
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
,
R
→
R
|
R
∈
TA
and
T
w
(id
A
)
with
={
R
|
R
∈
TA
and
∀
1
≤
k
≤|
R
|
∀
(d
i
∈
π
k
(R))Va l(d
i
)
}=
T
w
(A)
, and
hence we obtain the identity arrow
T
w
(id
A
)
T
w
(A)
.Itisex-
tended to identity arrows of complex objects analogously (the disjoint union of sim-
ple identity arrows).
It is easy to verify that for any two simple arrows
f
=
id
T
w
A
:
T
w
(A)
→
:
A
−→
B
and
g
:
B
−→
C
,
T
w
(g
◦
f)
=
id
R
:
R
→
R
|
R
∈
g
◦
f
and
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
=
id
R
:
R
→
R
|
R
∈
g
∩
f
and
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
.
Thus,
=
R
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
∩
f
and
T
w
(g
◦
f)
|
R
∈
g
=
R
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
|
R
∈
g
and
