Database Reference
In-Depth Information
, we obtain
p
A
p
B
In fact, when
B
=
A
and
S
A
=
S
B
=∅
=
with limit object
=
S
E
corresponding to the equalizer.
From the fact that
DB
is an
extended symmetric
category, for any morphism
LimP
f
:
A
→
B
(i.e., each object
J(f)
=
A,B,f
in
DB
↓
DB
) the following diagram
commutes (see the proof of Theorem
4
in Sect.
3.2.3
):
such that
f
=
T
e
J(f)
=
f
f
ji
,
f
f
ji
∈
τ
J(f
ji
)
:
,
τ
J(f)
=
f
A
j
f
ji
|
(f
ji
:
A
j
→
B
i
)
∈
and
τ
−
1
.
τ
−
1
J(f)
=
f
J(f
ji
:
f
ji
→
B
i
|
(f
ji
:
A
j
→
B
i
)
∈
f
and one monic
This decomposition of each morphism
f
into one epic
τ
J(f)
:
A
τ
−
1
J(f)
:
f
→
B
arrows will be used to demonstrate the existence of a limit in
DB
for
each diagram
D
composed of a single morphism
f
:
A
→
B
in
DB
, by the pullback
id
B
f
B
of the diagram
B
A
(when
C
=
B
and
g
=
id
B
). Here we will
consider the following particular cases for
f
:
Corollary 27
For any given morphism f
:
A
→
B such that for each its ptp
=
1
≤
j
≤
M
A
j
,M
arrow f
ji
:
A
j
→
B
i
,
TA
j
⊆
TB
i
,
with A
≥
1
and B
=
1
≤
j
≤
K
B
i
,K
id
B
f
B
≥
1,
and a diagram B
A
,
the cone with the ver-
=
(f
ji
:
A
j
→
B
i
)
∈
(A
j
×
f
ji
)
,
and with outgoing arrows
p
B
τ
−
1
tex Lim(f )
={
J(f
ji
)
:
f
τ
−
1
J(f)
and
p
A
f
}=
id
A
f
ji
→
B
i
|
f
ji
∈
=
,
defines a pullback diagram
,
This diagram can be simplified by substituting the limit object Lim(f ) with the ob-
ject A
×
f isomorphic to it
.
