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
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:
Search WWH ::




Custom Search