Database Reference
In-Depth Information
3.1.
h
1
=[
j,m
]◦
f,g
◦[
k,l
]
=
[
j,m
]◦
[
f
◦
k,f
◦
l
]
,
[
g
◦
k,g
◦
l
]
5
4
[
]
:
=
j
◦
f
◦
k,j
◦
f
◦
l
]
,
[
m
◦
g
◦
k,m
◦
g
◦
l
A
1
A
2
→
D
;
3.2.
h
2
=
[
j,m
]◦
f,g
◦[
k,l
]=
4
j
◦
f,m
◦
g
◦[
k,l
]
7
[
]
:
=
j
◦
f
◦
k,j
◦
f
◦
l
]
,
[
m
◦
g
◦
k,m
◦
g
◦
l
A
1
A
2
→
D,
both of them with the
same set
of two fused point-to-point arrows
=
(j
h
1
=
h
2
◦
f
◦
k)
∪
(m
◦
g
◦
k)
:
A
1
→
D,
D
,
(j
◦
f
◦
l)
∪
(m
◦
g
◦
l)
:
A
2
→
so that
h
1
=
h
2
, that is, this composition of complex arrows is associative as
well.
4. For
k
:
A
→
B
1
,l
:
A
→
B
2
,f
:
B
1
→
C,g
:
B
2
→
C,j
:
C
→
D
1
and
m
:
C
→
D
2
, we consider the compositions
j,m
◦[
f,g
]◦
k,l
:
A
→
B
1
B
2
→
C
→
D
1
D
2
. Let us consider the two different orders of composition:
4.1.
◦
[
=
4
h
1
=
]◦
◦
◦
◦
j,m
f,g
k,l
j,m
f
k,g
l
7
:
=
j
◦
f
◦
k,j
◦
g
◦
l
,
m
◦
f
◦
k,m
◦
g
◦
l
A
→
D
1
D
2
,
4.2.
h
2
=
j,m
◦[
f,g
]
◦
k,l
=
6
j
◦
f,m
◦
f
,
j
◦
g,m
◦
g
◦
k,l
4
:
=
j
◦
f
◦
k,m
◦
f
◦
k
,
j
◦
g
◦
l,m
◦
g
◦
l
A
→
D
1
D
2
.
They are equal,
h
1
=
h
2
, because
=
(j
◦
f
◦
k)
∪
(j
◦
g
◦
l)
:
A
→
D
1
,
(m
h
1
=
h
2
D
2
.
◦
f
◦
k)
∪
(m
◦
g
◦
l)
:
A
→
From the examples above, we have seen that there are different structural rep-
resentations for the same arrow between two complex objects. That is, for a given
set of ptp arrows between two complex objects, we can have a number of different
structural representations that are mutually equal. Thus, it is natural to fix a canon-
ical representation of such equivalent classes of arrows and to use them whenever
possible. The fact is that each arrow in
DB
can be substituted by such canonical ar-
row, in an analogous way as any object can be substituted by an isomorphic object.