Database Reference
In-Depth Information
3.2.4 (Co)products
We recall that for a tuple of objects (i.e., instance-databases) in
DB
category
(S
1
,...,S
n
)
,
n
≥
2, we define its disjoint union (see Definition
19
)as
(S
1
,...,S
n
)
(i,a)
S
i
,
S
i
=
=
|
a
∈
1
≤
i
≤
n
1
≤
i
≤
n
the union
indexed by positions
of the sets in a given tuple, denoted also as
S
1
S
2
···
S
n
. It will be demonstrated that the disjoint union of two objects is isomorphic
to separation-composition of these objects.
Theorem 5
There exists an idempotent coproduct bifunctor
+:
DB
×
DB
−→
DB
which is a disjoint union
for objects and arrows in
DB
.
The category
DB
is cocartesian with initial
(
and terminal
)
object
0
⊥
and for
+
every pair of objects A and B the object A
B is a coproduct with monomorphisms
1
, id
B
:
B
→
A
+
B
.
Based on duality property
,
DB
is also a Cartesian category with a terminal object
1
(
injections
)
in
A
=
id
A
,
⊥
:
A
→
A
+
B and in
B
=⊥
0
.
For each pair of objects A and B there exists a categorial product A
⊥
×
B with
in
O
A
:
in
O
B
:
epimorphisms
(
projections
)
p
A
=
A
×
B
A and p
B
=
A
×
B
B
.
Thus
,
the product bifunctor is equal to the coproduct bifunctor
,
i
.
e
.,
×≡+
.
is just the disjoint union
, so that:
1. For any identity arrow
(id
A
, id
B
)
in
DB
Proof
Note that the coproduct
+
×
DB
where
id
A
and
id
B
are the
id
A
+
B
id
A
+
id
B
identity arrows of
A
and
B
, respectively,
=
={
id
A
, id
B
}
. Thus,
1
(id
A
, id
B
)
B
.
2. For any
k
:
A
−→
A
1
,
k
1
:
A
1
−→
A
2
,
l
:
B
−→
B
1
,
l
1
:
B
1
−→
B
2
, it holds
+
=
id
A
+
id
B
=
id
A
+
B
is an identity arrow of the object
A
+
1
(k
1
,l
1
)
1
(k,l)
1
(k
1
◦
+
◦+
B
2
}=
+
k,l
1
◦
l)
={
k
1
◦
k
:
A
→
A
2
,l
1
◦
l
:
B
→
.
Thus, from point 3 of Definition
23
, the functorial property for composition of
morphisms
1
(k,l)
is valid.
3. Let us demonstrate the coproduct property of this bifunctor. For any two simple
arrows
f
1
(k
1
◦
1
(k
1
,l
1
)
+
k,l
1
◦
l)
=+
◦+
:
A
−→
C
and
g
:
B
−→
C
, there exists a unique arrow
k
=[
f,g
]:
1
A
+
B
−→
C
such that
f
=
k
◦
in
A
and
g
=
k
◦
in
B
where
in
A
=
id
A
,
⊥
:
1
, id
B
:
A
→
A
+
B
and
in
B
=⊥
B
→
A
+
B
are the monomorphisms (injec-
tions). In fact, for any two arrows
h,k
:
D
→
A
,if
in
A
◦
h
=
in
A
◦
k
then
h
=
k
1
1
1
1
(indeed
in
A
◦
h
=
id
A
,
⊥
◦
h
=
id
A
◦
h,
⊥
◦
h
=
h,
⊥
=
in
A
◦
k
=
h,
⊥
,
so that
h
k
). Analogously, it holds also for
in
B
. That is, the following coprod-
uct diagram in
DB
commutes
Assume that, for given schema
=
A
,
B
and
C
, we have the sketch's mappings
M
AC
:
α
∗
(
A
→
C
and
M
BC
:
B
→
C
and an interpretation
α
such that
A
=
A
),B
=
α
∗
(
α
∗
(
α
∗
(
M
AC
)
and
g
α
∗
(
M
BC
)
. Then we define the mor-
B
=
C
=
=
),C
),f
phisms
in
A
=
α
∗
(
1
and
in
B
=
α
∗
(
{
M
AA
,
{
1
r
∅
}
)
=
id
A
,
⊥
1
r
∅
}
,
M
BB
)
=
1
, id
B
⊥
where
M
AA
={
1
r
|
r
∈
A
}∪{
1
r
∅
}
and
M
BB
={
1
r
|
r
∈
B
}∪{
1
r
∅
}
.
