Database Reference
In-Depth Information
Inverted arrow of any identity arrow is equal to the identity arrow. For the com-
position of
f
with
g
:
B
→
C
we have
is
−
A
◦
T(g
f)
OP
S
1
(g
◦
f)
=
◦
◦
is
C
is
−
A
◦
Tf)
OP
=
(T g
◦
◦
is
C
is
−
A
◦
(Tf )
OP
(T g)
OP
=
◦
◦
is
C
is
−
A
◦
(Tf )
OP
is
−
B
◦
(T g)
OP
=
◦
is
B
◦
◦
is
C
S
1
(f )
S
1
(g).
=
◦
Thus, this contravariant functor is well defined.
It is convenient to represent this contravariant functor as a covariant functor
S
:
DB
OP
DB
or a covariant functor
S
OP
DB
OP
. It is easy to verify that
−→
:
−→
DB
for compositions of these covariant functors
SS
OP
=
I
DB
and
S
OP
S
=
I
DB
OP
w.r.t.
DB
OP
S,S
OP
,φ
the adjunction
DB
, where
φ
is a bijection: for each pair
of objects
A,B
in
DB
we have the bijection of hom-sets,
φ
A,B
:
:
−→
DB
(A,S(B))
DB
OP
(S
OP
(A),B)
, i.e.,
φ
A,B
:
DB
(A,B)
DB
(B,A)
, such that for any arrow
DB
(A,B)
,
φ
A,B
(f )
=
S
1
(f )
=
f
OP
. The unit and counit of this adjunction
are the identity natural transformations,
η
OP
:
f
∈
SS
OP
and
OP
:
S
OP
S
−→
I
DB
OP
, respectively, such that for any object
A
they return by its identity arrow.
Thus, from this adjunction, we obtain that
DB
is isomorphic to
DB
OP
.
I
DB
−→
Let us show that the morphism
(Tf )
OP
:
TB
→
TA
, for a given morphism
f
=
α
∗
(
M
AB
)
:
A
→
B
with
M
AB
=
, is well defined
due to Theorem
1
, that is, there exists an SOtgd
Φ
of a schema mapping such that
(Tf )
OP
MakeOperads(
M
AB
)
:
A
→
B
α
∗
(MakeOperads(
))
.
Let us define the set of relational symbols
S
TB
={
=
{
Φ
}
r
i
|
r
i
∈ R
and
α(r
i
)
∈
⊆
f
A
α
A
B
α
B
Δ(α,
M
AB
)
}
and the
α
-intersection schemas
and
with
TA
=
α
∗
(
A
α
A
α
∗
(
B
α
B
)
and
TB
=
)
(in Definition
16
). Then we can define the SOtgd
Φ
by a conjunctive formula
{∀
x
i
(r
i
(
x
i
)
⇒
r
i
(
x
i
))
|
r
i
∈
S
TB
}
.
In fact,
MakeOperads(
{
Φ
}
)
={
1
r
i
∈
O(r
i
,r
i
)
|
r
i
∈
S
TB
}∪{
1
r
∅
}
is a sketch's
mappings between schemas
B
and
A
⊥∈
f
. Conse-
. Note that
r
∅
∈
S
TB
because
quently,
α
∗
MakeOperads
{
}
=
α
∗
1
r
i
∈
S
TB
∪{
1
r
∅
}
(Tf )
OP
=
Φ
O(r
i
,r
i
)
|
r
i
∈
=
id
R
|
R
∈
Δ(α,
M
AB
)
⊆
f
∪{
q
⊥
}
:
TB
→
TA.
is
−
A
◦
Thus, both inverted arrows
(Tf )
OP
and
f
OP
(Tf )
OP
is
B
(the demon-
stration is equal to the demonstration for arrow
Tf
after the proof of Theorem
2
)
are morphisms in the
DB
category as specified in Theorem
1
.
=
◦
Example 22
Let us consider the inverted arrows of the isomorphic arrows
is
A
:
A
→
TA
and
is
−
A
:
TA
→
A
.
For
is
A
(here
B
=
TA
) we have from Theorem
3
the following: