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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 )
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