Database Reference
In-Depth Information
We have shown in Corollary
2
that the schema-level mappings have the catego-
rial morphism's properties (an associative composition, with an identity mapping
for each schema) and now we will show that the information fluxes of the schema
mappings have the categorial morphism's properties at the instance-mapping level
as well:
Corollary 7
The information fluxes of the schema mappings and the composition
(
i
.
e
.,
the set intersection
)
of information fluxes satisfy categorial properties of the
morphisms between instance-databases as objects of such a category
.
Proof
For a given atomic schema mapping
M
AB
:
A
→
B
and an instance
A
=
)
, the information flux
f
α
∗
(
A
=
Flux(α, MakeOperads(
M
AB
))
⊆
TA
may repre-
α
∗
(
sent the instance-mapping from the instance
A
into an instance
B
=
B
)
.
Note that if
(A,B)
∈
Inst(
M
AB
)
(that is, when the instances
A
and
B
satisfy the
M
AB
) then
all
information contained in this information flux
f
is
transferred from
A
into
B
.
It is easy to verify that for an identity schema mapping (which is always satisfied)
Id
A
:
A
→
A
schema mapping
, defined in Lemma
5
,
Flux(α, MakeOperads(Id
A
))
=
TA
, so that
Flux
α, MakeOperads(Id
A
)
M
AB
)
◦
MakeOperads(
Flux
α, MakeOperads(Id
A
)
∩
Flux
α, MakeOperads(
M
AB
)
=
Flux
α, MakeOperads(
M
AB
)
=
∩
TA
Flux
α, MakeOperads(
M
AB
)
,
=
and hence the property of the categorial composition with identity morphisms (rep-
resented here by the information flux of an identity schema mapping) is satisfied.
The set intersection
is an associative operation so that, together with the iden-
tity property above, it satisfies the categorial properties for composition of mor-
phisms.
∩
BasedonCorollary
2
and Corollary
7
, it is possible to define the categorial se-
mantics for the schema mappings, by defining a functor from the sketch category
of a given schema-mapping graph into an instance-level category where the objects
are the database-instances and the morphisms are characterized by the information
fluxes of the schema-mappings.
The formalization of the schema mappings by means of operads, as it will be
demonstrated in next two chapters, is useful in order to be able to extend each
R-algebra
α
to a functor from the category sketch (obtained from a graph of (in-
ter)schema mappings) into the base
DB
category, which represents the denotational
semantics of this schema database mapping graph.
That is, each schema mapping
M
AB
transformed into its mapping-operad
M
AB
=
may be seen as a mor-
phism of a sketch category, and hence it will be mapped by a functor (R-algebra)
α
,
which satisfies Definition
11
(i.e., such that it is a
mapping-interpretation
), into the
MakeOperads(
M
AB
)
={
q
1
,...,q
n
,
1
r
∅
}:
A
→
B
