Database Reference
In-Depth Information
T
TC
l
∩
k
jil
|
k
in
=
1
≤
l
≤
n,k
jil
∈
and
in
li
∈
T
k
in
k
jil
|
=
1
≤
l
≤
n,k
jil
∈
and
in
li
∈
T
(from (iii))
=
{
g
jl
|
l
∈
L
ji
}
=
f
ji
(from (i))
=
τ
−
1
J(f
ji
)
.
There is another important property of this weak monoidal topos
DB
which is
valid in any standard topos:
Proposition 66
The following topos properties are valid in
DB
:
•
Coproduct preserve pullbacks
.
That is
,
if
are pullbacks in the
DB
category
,
then so is the following diagram
•
Pullbacks preserve epics
.
Proof
Claim 1. We recall that by duality
A
+
B
is equal to
A
×
B
(equal to disjoint
B
and
l
2
:
union
A
B
). Let us show that for any two arrows
l
1
:
E
→
B
+
E
→
C
[
h,h
]◦
l
1
=
k
◦
l
2
, then there is a unique arrow
e
1
:
E
→
A
+
A
such
that the following diagram commutes:
it holds that
