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
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