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.
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