Database Reference
In-Depth Information
For a given complex arrow
f
:
A
⊗
B
→
C
, we define the mapping
Λ
such that
f
→{
f
and
f
jli
=
f
jil
}
f
jli
:
A
j
→
C
l
⊗
B
i
|
(f
jil
:
A
j
⊗
B
i
→
C
l
)
∈
, and its
inverse
Λ
−
1
such that
f
jil
:
,
C
l
|
f
jli
:
B
i
∈
Λ(f )
Λ(f )
and
f
jil
=
f
jli
→
A
j
⊗
B
i
→
A
j
→
C
l
⊗
and hence
Λ
is a bijection.
Let us show the commutativity
f
=
in
◦
(Λ(f )
⊗
id
B
)
of the diagram, i.e.,
f
in
◦
(Λ(f )
⊗
id
B
)
(f
jli
⊗
that
=
. In fact, for each
(in
li
◦
id
B
i
)
:
A
j
⊗
B
i
→
C
l
)
∈
id
B
i
)
=
(T C
l
∩
TB
i
)
∩
(f
jli
∩
TB
i
)
=
(
from
f
jli
⊆
in
◦
(Λ(f )
⊗
id
B
)
in
li
◦
(f
jli
⊗
,
=
f
jli
=
f
jil
, for the corresponding
(f
jil
:
f
TA
j
∩
TB
i
∩
TC
l
)
A
j
⊗
B
i
→
C
l
)
∈
.
Consequently,
in
li
◦
(f
jli
⊗
id
B
i
)
=
f
jil
for each ptp arrow in
in
◦
(Λ(f )
⊗
id
B
)
and
=
◦
⊗
=
eval
B,C
◦
⊗
vice versa. Hence,
f
id
B
)
.
Consequently,
DB
is closed and symmetric, that is, a biclosed category.
in
(Λ(f )
id
B
)
(Λ(f )
We have seen that all objects in
DB
are finitely representable (from Corol-
lary
29
and Proposition
54
). Let us denote the representable functor
DB
(Υ,
_
)
by
V
C
B
=
DB
(Υ,
_
)
:
DB
−→
Set
. By putting
A
=
Υ
in
Λ(f )
:
A
→
of the 'expo-
nent' diagram in Theorem
15
, and by using the isomorphism
β
:
Υ
⊗
B
B
, we get
DB
(Υ,C
B
)
.
Then
C
B
is exhibited as a lifting through
V
of the hom-set
DB
(B,C)
(i.e., in the
case when
B
and
C
are the simple objects, the hom-object
C
B
is a set of all views
which gives a possibility to pass from a “state”
B
to a “state”
C
). It is called the
internal hom
of
B
and
C
.
By putting
B
=
Υ
in
Λ(f )
:
A
→
C
B
of the 'exponent' diagram in Theorem
15
,
and by using the isomorphism
γ
V(C
B
)
a natural isomorphism in
Set
(a bijection
Λ
),
DB
(B,C)
=
:
⊗
A
Υ
A
, we deduce a natural isomorphism
C
Υ
(it is obvious, from the fact that
C
Υ
i
C
).
The fact that a monoidal structure is closed means that we have an internal Hom
functor,
(
_
)
(
_
)
:
C
C
⊗
Υ
DB
OP
:
×
DB
→
DB
, which 'internalizes' the external Hom functor,
DB
OP
:
×
→
Hom
DB
Set
, such that for any two objects
A
and
B
, the hom-object
B
A
(
_
)
(
_
)
(A,B)
represents the hom-set
Hom(A,B)
(i.e., the set of all morphisms
from
A
to
B
).
Monoidal closed categories generalize the Cartesian closed ones in that they also
possess exponent objects
B
A
which “internalize” the hom-sets. One may then ask
if there is a way to “internally” describe the behavior of functors on morphisms.
That is, given a monoidal closed category
C
and a functor
F
=
C
, consider,
say,
f
∈
C(A,B)
, then
F(f)
∈
C(F(A),F(B))
. Since the hom-object
B
A
and
F(B)
F(A)
represent hom-sets
C(A,B)
and
C(F(A),F(B))
in
C
, one may study
the conditions under which
F
is “represented” by a morphism in
C(B
A
,F(B)
F(A)
)
,
for each
A
and
B
.
:
C
−→
Theorem 16
The endofunctor T
:
DB
−→
DB
is closed
.
