Database Reference
In-Depth Information
1
,p
B
hence from Proposition
8
the arrow
[⊥
]:
LimP
→
B
is epic as well, so that
pullbacks preserve epics.
Since a standard topos is a Cartesian closed category (CCC) with subobject clas-
sifier (and with terminal object
1
), it is supposed to be '
Set
-like', its initial object
0
ought to behave like the empty set
, and have no elements. Otherwise, if
0
has
elements, then there is a unique arrow from the terminal object
1
into initial ob-
ject
0
(and its unique opposite arrow from the initial object into
1
) which has to be
an isomorphism, so that CCC degenerates (all objects become isomorphic, that is,
'equal'). Thus, in a non-degenerate standard topos,
0
has no elements.
In
DB
, terminal and initial objects are equal to the zero object
∅
0
, and
from the fact that it is a non-degenerate category we can see that it is not a standard
topos.
The question of the existence of
elements of objects
relates to the notion of
ex-
tensionality principle
, the principle that sets with the same elements are identical.
For the arrows in
Set
(the functions), this principle takes the following form: if two
parallel arrows
f,g
⊥
={⊥}
:
A
→
B
are
distinct
arrows, then there is an
element x
:
1
→
A
of
A
such that
f
x
.
A non-degenerate standard topos that satisfies this extensionality principle for
arrows is called
well-pointed
.
Let us now consider the topos properties which are not satisfied in the
DB
cate-
gory.
◦
x
=
g
◦
Proposition 67
The following properties distinguish the
DB
category from a
topos
:
1.
DB
category has no power objects
.
2.
DB
category is not well-pointed
.
P
Proof
Claim 1. By definition, the power object of
A
(if it exists) is an object
(A)
×
:
→
which represents the contravariant functor
Sub(
_
A)
DB
Set
, where for any
={
f
}={
f
|
f
object
B
,
Sub(B)
is the set of all sub-
objects (monomorphic arrows with the target object
B
)of
B
. Let us show that for
any simple object
A
|
f
is a subobject of
A
⊆
TA
}
0
⊥
there is no the power object
P
(A)
such that the bijection
DB
(
_
,
P
(A))
Sub(
_
×
A)
holds in
Set
. In fact,
Sub(A
×
A)
=
Sub(A
+
A)
=
{
f
|
f
}={
f
1
,f
2
|
f
1
f
2
+
=
+
}
T(A
A)
TA
TA
TA
TA
. Hence,
|=|{
f
|
f
}|⊆|{
f
|
f
|
P
⊆
∩
P
⊆
}|⊆|
×
|
DB
(A,
(A))
TA
T(
(A))
TA
Sub(A
A)
.
:
→
Claim 2. The extensionality principle for arrows “if
f,g
A
B
is a pair of
0
distinct parallel arrows, then there is an element
x
:⊥
→
A
of
A
such that
f
◦
x
=
x
” does not hold because
f
0
g
◦
◦
x
=
g
◦
x
=⊥
for the (unique) element (arrow)
0
0
.
x
:⊥
→
A
such that
x
=⊥
