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:
Search WWH ::




Custom Search