Database Reference
In-Depth Information
5
6
We have
f,g
◦[
k,l
]=
f
◦[
k,l
]
,g
◦[
k,l
]=
[
f
◦
k,f
◦
l
]
,
[
g
◦
k,g
◦
l
]
and
6
l
=
5
.
f,g
◦[
k,l
]=
f,g
◦
k,
f,g
◦
f
◦
k,g
◦
k
,
f
◦
l,g
◦
l
We can show that it is true by considering the ptp arrows. In fact,
h
=
[
f
◦
k,f
◦
l
]
,
[
g
◦
k,g
◦
l
]
=
[
f
◦
k,g
◦
k
,
f
◦
l,g
◦
l
]
={
f
◦
k
:
A
1
→
C
1
,g
◦
k
:
A
1
→
C
2
,f
◦
l
:
A
2
→
C
1
,g
◦
l
:
A
2
→
C
2
}
.
2. Let us show that
f
=
id
C
, id
C
◦[
id
C
, id
C
]=[
id
C
, id
C
]
,
[
id
C
, id
C
]:
C
C
→
C
C
. In fact, in order
to define the ptp arrows of these two morphisms, we have to use the 'indexing
by position' for the source database
A
=
A
1
A
2
and the target database
B
=
B
1
→
C
C
is different from
g
=
id
C
id
C
:
C
C
→
C
B
1
where
A
1
=
A
2
=
B
1
=
B
2
=
C
. Thus,
id
C
, id
C
◦[
id
C
, id
C
]
={
f
11
=
id
C
:
A
1
→
B
1
,f
12
=
id
C
:
A
1
→
B
2
,
f
21
=
id
C
:
A
2
→
B
1
,f
22
=
id
C
:
A
2
→
B
2
}
id
C
id
C
={
g
11
=
id
C
:
A
1
→
B
1
,g
22
=
id
C
:
A
2
→
B
2
}=
.
Consequently, by point 3 of Definition
23
, the identity arrow
id
C
id
C
=
id
C
, id
C
◦[
id
C
, id
C
]
.
In fact,
id
A
=
id
A
1
id
A
2
because for any
f
:
A
→
X
we have from Definition
22
that
f
i
2
◦
f
◦
(id
A
1
id
A
2
)
f
=
id
|
f
i
2
∈
, id
∈{
id
A
1
, id
A
2
}
cod(id)
id
A
1
id
A
2
=
, dom(f
i
2
)
=
f
i
2
|
f
f
=
f
i
2
∈
=
so that, by point 3 of Definition
23
,
f
=
f
◦
(id
A
1
id
A
2
)
and analogously
f
=
(id
A
1
id
A
2
)
◦
f
, so that
(id
A
1
id
A
2
)
is the identity arrow of
A
=
A
1
A
2
. That
id
A
2
.
Let us consider the associativity of composition of complex morphisms (that
DB
is well defined for complex arrows as well):
3. For
k
is,
id
A
=
id
A
1
A
2
=
id
A
1
:
A
1
→
B,l
:
A
2
→
B,f
:
B
→
C
1
,g
:
B
→
C
2
,j
:
C
1
→
D
and
m
:
C
2
→
D
, we consider the composition
[
j,m
]◦
f,g
◦[
k,l
]:
A
1
A
2
→
B
→
C
1
C
2
→
D
. Let us consider the two different orders of composition:
