Database Reference
In-Depth Information
l.u.b. of the first component, i.e., of the hom-object
DB
I
(B,A)
=
DB
I
(B,T A)
=
in
B
, for an inclusion arrow
in
B
:
B
→
TA
.
Hence, the component
Σ
D
B
in
DB
I
(B,A)
⊗
Σ
D
B
means that the significant
computation of the object
Lan
K
(Σ
D
)(A)
DB
is done only for the
finite
objects
B
in
DB
, while the component
DB
I
(B,A)
means that such objects (i.e., the instance
databases) have to be the subobjects of the (also infinite) database
A
. That is,
B
A
,
and hence
B
has to satisfy the condition
B
∈
ω
A
.
Thus, the meaning of the natural transformation
β
is to represent the monomor-
phism
β
B
:
ω
A
. That is,
β
represents the
set of all monomorphisms from
finite
objects into a given (also infinite) object
A
in
DB
I
category: it is
a cocone
of the object
A
in
DB
I
(with a monomorphism
β
B
=
B
→
A
in
DB
for each finite object
B
0
in
∅
:⊥
→
A
, when
DB
I
(B,A)
is the empty hom-set).
Notice that the same result can be obtained by considered the colimits in
DB
and hence by considering the object
Lan
K
(Σ
D
)(A)
∈
DB
as a colimit (point-to-
P
Σ
D
DB
f
DB
(where
P
is a
point) such that the composition
(K
↓
A)
functor-projection, mapping
B,K(B)
→
A
→
B
), for each (also infinite) object
=
−−−→
A
∈
DB
I
, is a colimit in
DB
, with cocone
β
, denoted by
Lan
K
(A)
Colim((K
↓
P
Σ
D
DB
)
, which is the dinatural transformation
β
used for a com-
putation of the tensor functor product. Here, instead, this cocone
β
(a dinatural
transformation) is represented by the comma category
(K
DB
f
A)
↓
A)
(where the objects
are the monomorphisms from the finite objects
B
i
∈
=
B
i
∈
DB
f
with
K(B
i
)
DB
I
into a fixed object
A
∈
DB
I
), that is, to the set
β
of monomorphisms
{
in
i
:
K(B
i
)
. The functor
Lan
K
(Σ
D
)
maps this cocone in
DB
I
into the colimit cocone represented by the corresponding set of monomorphisms
{
i
:
→
A
∈
DB
I
|
B
i
∈
DB
f
}
→
∈
|
B
i
∈
=
}
. Thus,
the colimit object
Lan
K
(Σ
D
)(A)
is just the union of all fluxes of the monomor-
phisms from the finite objects
B
i
into it, that is,
Σ
D
(B
i
)
Lan
K
(Σ
D
)(A)
DB
DB
f
so that
Σ
D
(B
i
)
T(B
i
)
in
i
|
in
i
:
K(B
i
)
→
A
∈
β
Lan
K
(Σ
D
)(A)
=
in
i
|
β
=
(in
i
:
B
i
)
→
A)
∈
TB
i
|
β
=
(in
i
:
B
i
)
→
A)
∈
=
{
TB
i
|
B
i
ω
A
}
=
{
B
i
|
B
i
ω
A
}
=
TA
in the complete
algebraic
lattice
of closed objects in
DB
, by Corollary
28
.
Consequently, we obtain that
Lan
K
(Σ
D
)
, the conservative extension of
Σ
D
to
all (also infinite non-closed) objects in
DB
, is equal (up to isomorphism) to the
endofunctor
T
.
That is, formally we obtain:
C
