Database Reference
In-Depth Information
In fact, each monoidal closed category is itself a V-category: hom-sets from
A
to
B
are defined as “internalized” hom-objects (cotensors)
B
A
. The composition
is given by the image of the bijection
Λ
:
DB
(D,C
A
)
, where
DB
(D
⊗
A,C)
C
B
B
A
, of the arrow
ε
B
◦
D
=
⊗
(id
C
B
⊗
ε
A
)
◦
α
C
B
,B
A
,A
and hence
m
A,B,C
=
Λ(ε
B
◦
α
C
B
,B
A
,A
)
, that
is, a monomorphism. The identities are given by the image of the isomorphism
β
A
:
(id
C
B
⊗
ε
A
)
◦
α
C
B
,B
A
,A
)
=
Λ(eval
B,C
◦
(id
C
B
⊗
eval
A,B
)
◦
DB
(Υ,A
A
)
and hence
Υ
⊗
A
−→
A
, under the bijection
Λ
:
DB
(Υ
⊗
A,A)
j
A
=
Λ(β
A
)
:
Υ
A
A
(
j
A
that is an epimorphism).
Moreover, for a V-category
DB
, the following isomorphism (which extends
the tensor-cotensor isomorphism
Λ
of exponential diagram in Theorem
15
)is
valid in all enriched Lawvere theories [
23
],
DB
(D
⊗
A,C)
DB
(D,C
A
)
DB
(A,
DB
(D,C))
.
Finally, from the fact that
DB
is an lfp category enriched over the lfp symmetric
monoidal closed category with a tensor product
(i.e., the matching operator for
databases), and the fact that
T
is a finitary enriched monad on
DB
, from Kelly-
Power theorem we have that
DB
admits a presentation by operations and equations,
what was implicitly assumed in the definition of this power-view operator (see also
[
16
,
19
]).
⊗
8.2.2 Internalized Yoneda Embedding
As we have seen in Sect.
8.2.1
, the fact that a monoidal structure is closed means that
we have an internal hom-functor
(
_
)
(
_
)
DB
OP
:
×
DB
→
DB
which “internalizes”
DB
OP
the external hom-functor,
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)
=
DB
(A,B)
(the set of all morphisms from
A
to
B
). We may apply the Yoneda embed-
ding to the
DB
category by the contravariant functor
H
=
DB
OP
Set
DB
, such that
:
→
for any object
A
we obtain a functor
H
A
=
DB
(A,
_
)
:
DB
→
Set
and hence for any
object
B
in
DB
,
H
A
(B)
=
DB
(A,B)
is a hom-set of all morphisms
in
DB
between
A
and
B
, and for any morphism
f
hom(A,B)
=
:
→
B
C
in
DB
the function
H
A
(f )
_.
The internalized Yoneda embedding can be obtained by replacing the base
Set
category with the base databa
se
category
DB
and hence this embedding is provided
by the contravariant functor
H
=
DB
(A,f )
:
DB
(A,B)
→
DB
(A,C)
is a composition
H
A
(f )
=
f
◦
DB
OP
DB
DB
. So, w
e
internalize a hom-set
:
→
H
A
(B)
=
DB
(A,B)
in
Set
by the (co)tensor
H
A
(B)
=
A
⊗
B
in
DB
,
and we internalize a composition f
un
ction
H
A
(f )
hom(A,B)
=
=
f
◦
_in
Set
by the (co)tensor
composition of morphisms in
DB
,
H
A
(f )
C
.
This internalized Yoneda embedding for
DB
category satisfies the following
property:
=
id
A
⊗
f
:
A
⊗
B
→
A
⊗
Proposition 56
For any o
bje
ct A in
DB
with a monomorphisms in
A
:
A
→
Υ
,
the
natural transformation φ
:
H
A
→
T is monomorphic and such that φ
=
in
A
⊗
id
.
{
_
}
