Database Reference
In-Depth Information
DB
is a V-category enriched over itself
,
with the composition law monomorphism
m
A,B,C
:
C
B
B
A
C
A
and identity element
(
epimorphism
)
j
A
:
A
A
which
⊗
→
Υ
“picks up” the identity in A
A
.
The monad (T,η,μ) is an enriched monad, and hence
DB
is an object of V-cat
and endofunctor T
:
DB
→
DB
is an arrow of V-cat
.
Proof
Let us show that the arrows introduced by this proposition satisfy the mor-
phisms in the definition of the
DB
category; see Theorem
1
in Sect.
3.2
.Fora
given database schemas
and an R-algebra
α
such that
A
=
α
∗
(
A
,
B
and
C
A
),B
=
α
∗
(
α
∗
(
A
α
C
B
=
C
),C
)
, we have that there exist the
α
-intersection schemas
and
A
α
C
)
α
B
so that
α
∗
(
A
α
C
C
A
and
(
)
=
A
⊗
C
=
α
∗
α
B
A
α
C
C
B
B
A
.
=
TA
∩
TC
∩
TB
=
A
⊗
C
⊗
B
=
⊗
A
α
C
)
α
B
→
A
α
C
Thus, we may define a schema mapping
M
={
Φ
}:
(
, where
Ψ
is a SOtgd
{∀
x
(r(
x
)
⇒
r(
x
))
|
r
∈
(
A
α
C
)
α
B
}
, and
1
r
|
r
∈
α
B
A
α
C
M
=
MakeOperads(
M
)
=
∪{
1
r
∅
}
and hence we obtain a monomorphism
α
∗
(
M
)
m
A,B,C
=
1
R
:
R
→
R
|
R
∈
TA
∩
TC
∩
TB
}∪{
q
⊥
}:
C
B
⊗
B
A
→
C
A
.
={
In the case of complex objects
A,B
and
C
, this monomorphism
m
A,B,C
is uniquely
determined by the PO relation
C
B
B
A
C
A
⊗
as specified by Theorem
6
in
Sect.
3.2.5
. In fact, from
C
B
B
A
B)
,
C
B
B
A
⊗
(B
⊗
C)
⊗
(A
⊗
⊗
≈
(B
⊗
C)
⊗
(A
⊗
B)
≈
(C
⊗
B)
⊗
(B
⊗
A)
≈
(
from
B
⊗
B
≈
B)
≈
C
⊗
B
⊗
A
≈
B
⊗
(A
⊗
,
⊕
C)
(
from the complete lattice
L
DB
=
(Ob
DB
,
,
⊗
)
in Proposition
51
)
C
A
.
The identity element
j
A
is defined by a schema mapping
C
A
)
A
⊗
C(
from
A
⊗
C
≈
}:
Υ
M
1
={
Ψ
→
, where
Ψ
is a SOtgd
{∀
A
α
A
x
(r(
x
)
⇒
r(
x
))
|
r
∈
A
α
A
}
and hence
α
∗
MakeOperads(
M
1
)
={
A
A
.
j
A
=
1
R
:
R
→
R
|
R
∈
TA
}∪{
q
⊥
}:
Υ
In the case when
A
=
1
≤
j
≤
m
A
j
,m
≥
2 is a complex object, with
A
A
=
A
⊗
A
with
j
A
j
A
j
A
id
A
⊗
id
A
so that
j
A
is defined by the set of ptp arrows,
=
=
id
A
⊗
id
A
=
A
⊗
A
, and consequently, by Proposition
8
in Sect.
3.2
,
j
A
is an epi-
morphism.
Analogously, for all other morphisms used in this proof a schema mapping can
be provided from which they are derived by an R-algebra
α
.
