Database Reference
In-Depth Information
Byasketchmemeana4-tuple
(G,u,D,C)
where
G
is a graph,
u
:
V
G
→
E
G
is
a function which takes each vertex (node)
,
D
is a
class of diagrams in
G
, and
C
is a class of (co)cones in
G
. Each (c)cone in
G
goes
(to)from some vertex (from)to some diagram; that diagram need not be in
D
and, in
fact, it is necessary to allow diagrams which are not in
D
as bases of (co)cones.
Notice that, differently from the work dedicated to categorical semantics of
Entity-Relationship internal relational database models where nodes of sketches are
single relations, here, at a higher level of abstraction, the nodes are whole databases.
Consequently, in such a framework, we do not use commutative database mapping
systems and hence
D
is an empty set. In fact, in a database mapping system, the
(co)cone diagrams above will never be used in practical representations of database
mapping systems. Instead of that, they will be alternatively used only for their
self-consistent parts, as a first diagram above, or, equivalently, as a single arrow
M
:
A
A
in
V
G
to an arrow from
A
to
A
B
→
C
.
However, for the introduced schema composition operator †, the above cone and
cocone diagrams have to be presented in
C
for our sketches.
Consequently, we obtain the following fundamental lemma for the categorial
modeling of database mappings:
†
Lemma 6
The
Set
cannot be used as the base category for the models of database-
mapping sketches
.
Proof
Let
E
(G,u,D,C)
be a database sketch where
C
is a set of (co)cones of
the two diagrams introduced for the database schema composition operator †, and
a model of this sketch be a functor
F
:
=
is a base category. Then all
cones in
C
have to be functorially translated into limit commutative diagrams in
E
→B
where
B
B
and all cocones in
C
have to be functorially translated into colimit commutative
diagrams in
, i.e., the cocone in the figure in Sect.
3.1
has to be translated into a
coproduct diagram and the cone has to be translated into a product diagram in
B
B
.
Consequently, the object
F(
A
†
B
)
has to be both the product
A
×
B
and coproduct
A
+
B
, where
A
=
F(
A
)
and
B
=
F(
B
)
are two objects in
B
;
×
and
+
are the
product and coproduct operators in
. However, this cannot be done in
Set
. In fact,
the product
A
×
B
in
Set
is the Cartesian product of these two sets
A
and
B
, while
the coproduct
A
B
+
×
+
B
is not an isomor-
phism in
Set
. Another reason is that any morphism in
Set
is a single function, while
in our case it is a set of functions (or a
binary relation of tuples
obtained by union
of graphs of this set of functions).
B
is the disjoint union. Hence,
A
B
A
Remark
The fundamental consequence of this lemma is that we need to define a
new base category for the categorial semantics of database mappings.
In fact, this is the first task that has to be done and we will define this new base
category
, denoted by
DB
(DataBase) category, and will show that it satisfies the
duality property where the product and coproduct diagrams are dual diagrams and
B
