Database Reference
In-Depth Information
DB
OP
Proof
For any object
A
, from the int
ern
aliz
ed
Yoneda emb
edd
ing
H
:
→
DB
DB
, we obtain the covariant functor
H
A
=
H(A)
such that
H
A
=
A
⊗
_
:
DB
→
DB
. Consequently,
fo
r the covariant power-view functor
T
:
DB
→
DB
, the natural
transformation
φ
:
H
A
→
T
is well defined and monomorphic if, for
every
arrow
f
:
B
→
C
in
DB
, the following diagram commutes
Notice th
at,
from
th
e fact that for each
A
∈
Ob
DB
,
A
Υ
and
H
Υ
=
Υ
⊗
_
=
T
.
Thus,
φ
:
H
A
→
H
Υ
, that is, for any object
B
,
φ
B
=
in
A
⊗
id
B
:
A
⊗
B
→
Υ
⊗
B
and
φ
C
=
Υ
is the unique monomorphism (in this case
the inclusion) of
A
into the total object
Υ
, so that the diagram above can be reduced
to the following commutative diagram in
DB
for any
f
in
A
⊗
id
c
, where
in
A
:
A
→
∈
DB
(B,C)
),
so that
Tf
◦
Φ
B
=
(id
Υ
⊗
f)
◦
(in
A
⊗
id
B
)
=
(
from functorial property of
⊗
)
=
(id
Υ
◦
in
A
)
⊗
(f
◦
id
B
)
=
in
A
⊗
f
=
(in
A
◦
id
A
)
⊗
(id
C
◦
f)
=
(in
A
⊗
id
C
)
◦
(id
A
⊗
f)
=
φ
C
◦
H
A
(f )
. Consequently,
φ
=
id
}
⊗
in
A
.
{
_
This propositio
n p
rovides the idea in which way we are able, by using the con-
travariant functor
H
DB
DB
, to define the internalized Yoneda repre-
sentation of the arrows in
DB
OP
by the natural transformations in the category
of fu
nc
tors
DB
DB
: by assuming that in Proposition
56
, the natural transformation
φ
DB
OP
:
→
H(in
OP
A
)
is just the Yoneda representation of the epimorphism
in
O
A
:
=
Υ
A
in
DB
OP
(i.e., of the monomorphism
in
A
:
A
→
Υ
in
DB
).
Corollary 30
Th
e
internalized Yoneda representation is provided by the con-
travariant functor H
DB
OP
DB
DB
,
defined
as
foll
ow
s
:
:
→
•
For each object A
,
w
e d
efine th
e e
ndofunctor H
A
=
H
(A
)
:
DB
→
DB
such that
for
any f
:
B
→
C
,
H
A
(f )
=
H(A)(f )
=
id
A
⊗
f
:
H
A
(B)
→
H
A
(C)
,
where
A
B
.
=
⊗
H
A
(B)
B
A