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given two arrows
f
:
B
→
C
and
g
:
C
→
D
, we have the compositional endofunc-
tors property:
id
A
rn
g
jl
id
A
rn
f
ij
A
⊕
(g)
◦
A
⊕
(f )
g
◦
f
=
◦
|
g
jl
◦
f
ij
∈
id
A
rn
(g
jl
◦
g
◦
f
=
f
ij
)
|
g
jl
◦
f
ij
∈
A
⊕
(g
◦
f)
=
.
Consequently,
A
⊕
(g)
◦
A
⊕
(f )
=
A
⊕
(g
◦
f)
.
Moreover,
A
⊕
B
=
B
⊕
A
(if
B
is simple database as well),
A
⊕
Υ
=
Υ
,
0
⊕⊥
⊕
=
⊕
A
A
,
A
TA
TA
and
A
A
A
.
=
1
≤
j
≤
m
A
j
,
m
2, can be
obtained from the object-component of the functor defined in the theorem above,
and it will be considered in more details in Sect.
8.1.5
, by obtaining the symmetric
operator '
⊕
'The merging with a complex-object
A
', when
A
≥
', such that for any two complex objects
A
and
B
the isomorphism
A
⊕
B
B
⊕
A
is valid. Matching '
' and (generalized) merging '
⊕
' operators are
dual operators in the category
DB
: in fact, they are also two dual lattice operators
(meet and join, respectively) w.r.t. the database ordering
⊗
, as we will show in
Sect.
8.1.5
.
Remark
Notice that
A
rn
B
B
, that is, from the behavioral point of view, the
data federation is equivalent (isomorphic) to the data merging. That is, for any query
over the data federation
A
rn
B
, which returns a view
R
, there exists a query over
the data merging
A
A
⊕
⊕
B
which returns the same view
R
; and vice versa.
We generalize the merging operator for objects for the set
S
={
A
1
,...,A
n
}
of
more than two simple databases by
⊕
S
=
T(
∪
S)
=
T(A
1
∪···∪
A
n
)
.
8.1.3 (Co)Limits and Exponentiation
Previously, in Sect.
3.3
, we introduced the (co)products in the
DB
category that are
operators used in order to make the separation-composition of the databases. Here,
instead, we will investigate the rest of the limits and colimits in the
DB
category in
order to establish the general properties of this category, used as a base category for
the functorial semantics of schema database mapping systems.
In order to explain these concepts in another way, we can see the limits and
colimits as a left and a right adjunction for the diagonal functor
DB
J
:
DB
−→
DB
J
for a small index category (i.e., a diagram)
J
. For any colimit functor
F
:
−→
DB
J
DB
we have a left adjunction to the diagonal functor,
(F,
,η
C
,ε
C
)
:
−→
DB
J
DB
, with the colimit object
F(D)
for any object (i.e., a diagram)
D
∈
and
