Database Reference
In-Depth Information
=
1
≤
j
≤
m
A
j
,m
=
1
≤
i
≤
k
B
i
,k
For any two given objects
A
≥
1 and
B
≥
1 and
a complex arrow
f
between them,
f
OP
f
f
OP
={
ji
|
f
ji
∈
}
, so that
f
OP
.
f
OP
◦
f
f
=
ji
◦
f
ji
:
A
j
→
A
j
|
f
ji
:
A
j
→
B
i
∈
Thus,
f
ji
◦
f
OP
f
◦
◦
f
f
=
h
:
A
j
→
B
i
|
f
ji
:
A
j
→
B
i
∈
,
f
OP
◦
f
dom(f
ji
)
=
A
j
=
cod(h),h
∈
f
ji
◦
f
OP
f
ji
:
f
=
ji
◦
A
j
→
B
i
|
f
ji
:
A
j
→
B
i
∈
f
ji
:
f
=
A
j
→
B
i
|
f
ji
∈
from
(
i
)
and fact that
f
ji
are simple arrows
f
=
.
f
OP
=
◦
◦
Consequently, from Definition
23
,
f
f
f
is valid for complex morphisms
as well.
From point 1 of Proposition
7
, we have for any morphism
f
:
A
→
B
the
in
B
:
f
→
TB
and
in
in
A
:
f
→
TA
with
monomorphisms
in
f
=
f
OP
=
in
ji
={
,
in
A
=
f
∈
f
ji
}:
f
ji
id
R
|
→
TA
j
|
(f
ji
:
A
j
→
∈
R
B
i
)
in
ji
={
.
in
B
=
f
id
R
|
R
∈
f
ji
}:
f
ji
→
TB
i
|
(f
ji
:
A
j
→
B
i
)
∈
Thus,
in
OP
in
OP
f
=
f
∈
f
ji
}:
TB
i
f
ji
|
ji
={
id
R
|
R
(f
ji
:
A
j
→
B
i
)
∈
and
in
OP
in
OP
f
in
f
OP
◦
∈
f
ji
}:
=
ji
◦
in
ji
={
id
R
◦
id
R
|
R
TB
i
f
ji
→
TA
j
|
f
(f
ji
:
A
j
→
B
i
)
∈
