Database Reference
In-Depth Information
'in
DB
are translated into in-
jective functions (i.e., the monomorphisms in
Set
) and hence the directed-colimit
diagram in
DB
with
ColimF
equal to the l.u.b. of all objects in its cocone-base
F
is translated by the functor
H
Thus, the PO arrows (the monomorphisms) '
=
DB
(Υ,
_
)
into the direct-colimit diagram (with re-
⊆
spect to inclusion-PO '
') in
Set
where the
H(ColimF)
is the l.u.b. with respect
to partial ordering
. Consequently, any other
E
that makes this diagram commu-
tative with the same cocone base
HF
(composed of all injective functions) from
H(B
i
)
into
E
for all 1
⊆
≤
i
≤
n
must be a bigger set then l.u.b.
H(ColimF)
and
with unique injective function
k
:
H(ColimF)
→
E
. All arrows of the cocone
E
,
l
i
:
(
{
f
|
f
TB
i
}
,
)
−→
E
, must be injective functions (however, with different
domains) in order to preserve the commutativity of this colimiting cocone
E
. Thus,
the function
k
:
(S,
)
−→
E
is a unique function such that, for any
v
∈
S,
,
{
f
|
f
{
f
|
f
k(v)
=
l
i
(v)
for some
l
i
:
(
TB
i
}
,
)
−→
E
and
v
∈
(
TB
i
}
,
)
.
From
HColim
S
we can conclude that there is a unique ar-
row in
Set
from
H ColimF
into
E
. Consequently,
HColim
is a colimit in
Set
, i.e.,
H
=
DB
(Υ, ColimF)
=
DB
(Υ,
_
)
preserves directed colimits, and hence
Υ
is finitely presentable.
Remark
We emphasize the fact that
Υ
is an fp object for a more general consid-
erations of the theory of enriched categories, which will be elaborated in Sect.
8.2
as, for example, the demonstration that the monad based on the power-view endo-
functor
T
DB
is an enriched monad. The Kelly-Power theory applies in the
case of a symmetric monoidal closed category, which is an lfp and a closed category:
it is equivalent to demanding that the underlying ordinary category is lfp and that
the monoidal structure on this ordinary category restricts to one on its fp objects.
For details see [
10
,
11
], and consider particularly that the unit
Υ
must be finitely
presentable.
:
DB
→
A locally finitely presentable category [
7
] is the category of models for an essen-
tially
algebraic
theory, which allows operations whose domain is an equationally
defined subset of some product of the previously defined domains (the canonical
example is a composition in a category, which is defined only on composable, not
arbitrary pairs of arrows).
In fact, we deduce from the algebraic (complete and compact) lattice
(Ob
DB
,
)
=
{
}=
S
(we recall that
|
⊆
ω
A
⊕
that for any simple object
A
,
A
TA
TB
B
is a generalization in
DB
of the union operation
∪
for sets and
X
⊕
Y
=
TX
⊕
TY
),
where the set
S
is upward directed, i.e., for any two finite
B
1
,B
2
⊆
ω
A
there is
C
=
B
1
∪
B
2
∈
S
such that
B
1
C
and
B
2
C
, with
TC
=
B
1
⊕
B
2
.
That is, any object in
DB
is generated from finite objects and this generated object
is just a directed colimit of these fp objects.
An important consequence of this freedom is that we can express
conditional
equations in the logic for databases.
Another important result, from the fact that
DB
is a complete and cocomplete lfp
category, is that it can be used as the category of models for essentially algebraic
theory [
15
,
23
] as a relational database theory. Thus, it is a category of models for
a finite limit sketch, where sketches are called graph-based logic [
3
,
12
], and it is
={
B
|
B
⊆
ω
A
}
