Database Reference
In-Depth Information
function
inr
A
:
→
T
P
A
in
Set
is translated into an isomorphism (which is
also a monomorphism)
inr
A
:
Σ
R
T
P
A
ω
TA)
ω
TA)
in the
DB
category,
Σ
D
(A
(A
based on the fact that
T
A
TA
T
ω
=
TA
TA
=
TA
TA
A
TA.
ω
ω
ω
ω
TA))
.
Moreover, by this translation, any
Σ
R
algebra
h
:
Σ
R
B
→
B
in
Set
is translated
into an isomorphism in
DB
,
h
D
=
ω
TA
ω
TA)
So that
inr
A
=
TA
=
T(A
∩
T(Σ
D
(A
is
−
B
:
TB
→
B
with
h
D
=
TB
.
Consequently, the initial algebra for a given simple database
A
with a set of
views in
TA
,ofthe
ω
-cocontinuous endofunctor
(A
DB
comes with
an induction principle, which we can rephrase follows: For every finite object
B
(for
which
Σ
D
=
T)
:
DB
−→
T
) and
Σ
D
-algebra structure
h
D
:
Σ
D
B
−→
B
(which must be an iso-
morphism) and every simple mapping
f
:
A
−→
B
(between the simple objects
A
ω
TA
and
B
) there exists a unique arrow
f
#
:
A
−→
B
such that the following
diagram in
DB
inr
O
A
]
commutes, where
Σ
D
=
T
and
f
#
=[
f,h
D
◦
Σ
D
(f
#
)
◦
is the unique
induc-
tive extension of h
D
along the mapping f
.
It is easy to verify that it is valid. From the fact that
inl
A
=
f,h
D
◦
inr
O
A
◦
id
A
,
1
f
#
◦
Σ
D
(f
#
)
◦
⊥
inr
O
A
◦⊥
1
1
=
f
◦
id
A
,h
D
◦
Σ
D
(f
#
)
◦
=
f,
⊥
,
inl
A
=
=
f
f.
0
f
#
◦
⊥
1
⊥
f,
(by Lemma
9
)
Thus, by Definition
23
, the left triangle commutes, i.e.,
f
inl
A
.
From Proposition
7
,
f
#
⊆
T(A
ω
TA)
∩
TB
=
inr
A
∩
TB
, and
Σ
D
f
#
=
T f
#
=
f
#
, and
h
D
=
is
−
B
=
=
f
#
◦
TB
. Consequently,
=
inr
A
=
f
#
∩
inr
A
=
f
#
=
f
#
∩
f
#
◦
TB
h
D
◦
Σ
D
(f
#
)
and hence, form Definition
23
, the commutativity of the right square holds, i.e.,
h
D
◦
Σ
D
(f
#
)
=
f
#
◦
inr
A
.
The diagram above can be equivalently represented by the following unique mor-
phism between initial
(A
Σ
D
)
-algebra and any other
(A
Σ
D
)
-algebra: