Database Reference
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this means that it cannot be a CCC. In what follows, we will demonstrate this fact
by a demonstrating that it has no exponentiation.
Proposition 49
DB
is not a CCC
.
Proof
Let
C
B
be the exponent object of the hom-set
DB
(C,B)
which, as a set of
the morphisms between
C
and
D
, cannot be an object of
DB
(differently from
Set
category). Hence, we have to construct this object
C
B
∈
DB
(to “internalize” the
α
∗
(
α
∗
(
hom-set
DB
(C,B)
). For any two simple objects
B
=
B
)
and
C
=
C
)
,theset
α
∗
(
M
(i
BC
)
, can
of all arrows
{
f
1
,f
2
,...
}:
B
→
C
where each simple arrow is
f
i
=
(
f
i
∈
DB
(B,C)
f
i
)
be represented by a unique arrow
g
=
:
B
→
C
(note that
g
is a
unique ptp arrow of the complex arrow
C
representing the set
of all arrows from
B
into
C
), so we define the object
C
B
to be the information flux
(from Definition
13
)ofthisarrow
g
=
α
∗
M
(i
BC
=
f
1
,f
2
,...
:
B
→
BC
:
B
→
C
MakeOperads
M
BC
|
M
(i)
(i)
α
∗
M
(i
BC
=
=
f
i
.
(i)
BC
:
B
→
C
f
i
∈
DB
(B,C)
M
From point 2 of Definition
13
,
Δ(α,
=
M
(i
BC
)
Δ(α,
M
(i
BC
)
.
(i)
BC
:
B
→
C
(i)
BC
:
B
→
C
M
M
Consequently,
Flux
α
M
T
Δ
α,
M
M
(i)
BC
M
(i)
BC
g
=
=
(i)
BC
:
B
→
C
(i)
BC
:
B
→
C
=
T
Δ
α,
M
(i
BC
=
T
T
Δ
α,
M
(i
BC
(i)
BC
:
B
→
C
(i)
BC
:
B
→
C
M
M
T
Flux
α,
M
(i
BC
T
DB
(B,C)
f
i
=
=
f
i
∈
(i)
BC
:
B
→
C
M
f
i
|
DB
(B,C)
.
=
f
i
∈
{
f
i
|
Thus, we define the hom-object
C
B
which “internalize”
the hom-sets (i.e., the
merging
of all closed objects obtained from a hom-set of
arrows from
B
to
C
), i.e., the merging of all compact elements
A
such that
A
f
i
∈
DB
(B,C)
}
B
⊗
C
.
Hence, we obtain that
T
f
i
|
DB
(B,C)
C
B
=
f
i
∈
(from Proposition
7
)