Database Reference
In-Depth Information
in
g
2
◦
in
f
2
,
T(k
1
◦
f
1
)
◦
in
f
1
, in
g
2
◦
T(k
1
◦
f
1
)
◦
in
g
1
◦
in
f
2
,
T(k
2
◦
f
2
)
◦
in
f
1
, in
g
1
◦
T(k
2
◦
f
2
)
◦
in
g
2
◦
in
f
1
,T
e
(h
2
,k
2
)
:
f
1
+
f
2
→
T(k
2
◦
f
2
)
◦
g
1
+
g
2
.
3.2.5 Partial Ordering for Databases: Top and Bottom Objects
Let us consider the “observational” Partial Order (PO) relation introduced in Defi-
nition
19
, its extension to morphisms and a 2-category property of
DB
:
Theorem 6
PO subcategory
DB
I
⊆
DB
is defined by Ob
DB
I
=
Ob
DB
and Mor
DB
I
m
⊥
0
→
1
≤
j
≤
m
A
j
1
,...,
⊥
1
is a union of all isomorphisms and initial arrows
:⊥
for two strictly-complex objects A
=
1
≤
j
≤
m
A
j
B
=
and S
I
={
in
:
A
→
B
|
1
≤
i
≤
k
B
i
with a mapping (from Definition
19
)σ
:{
1
,...,m
}→{
1
,...,k
}
and
in
.
This ordering can be extended by categorial symmetry property to morphisms as
well
:
={
in
jσ(j)
:
A
j
→
B
σ(j)
|
j
=
1
,...,m
}}
f
f
g
iff
g.
The power-view endofunctor T
:
DB
−→
DB
is a
2
-endofunctor and a closure op-
erator
.
DB
is a
2
-category
,
and
1
-cells are its ordinary morphisms
,
while
2
-cells
(
de-
noted by
√
_
)
are the arrows between ordinary morphisms
:
for any two morphisms
f,g
:
A
−→
B such that f
g
,
a
2
-cell arrow is the “inclusion”
√
β
:
f
−→
g
.
Such a
2
-cell arrow is represented by an ordinary monomorphism in
DB
,
β
:
f
→
g
,
with β
=
T
e
(f
OP
in
)
,
where f
ep
=
τ(J(f))
:
A
f is an epimor-
phism
,
and the arrow f
in
=
τ
−
1
(J(f ))
:
f
→
B is a monomorphism
.
ep
◦
f
ep
;
f
in
◦
f
OP
Proof
DB
I
is well defined: for any object
A
we have that its identity arrow
id
A
:
A
id
A
),
thus an arrow in
DB
I
. Each isomorphism is a monomorphism as well, and any initial
arrow is monic as well, so that all arrows and their compositions in
DB
I
are monic
arrows. For each isomorphism
A
→
A
is a monomorphism (we have
A
A
with
σ
identity function, and
in
=
B
represented by arrow
is
:
A
→
B
in
DB
I
,we
is
0
;
define
σ
:{
1
,...,m
}→{
1
,...,k
}
such that
σ(j)
=
i
if
(is
ji
:
A
j
→
B
i
)
∈
=⊥
0
). Thus for each
j
1 otherwise (when
A
j
=⊥
=
1
,...,m
,
TA
j
⊆
TB
σ(j)
, hence
m
⊥
→
1
≤
j
≤
m
A
j
and any
σ
1
,...,
1
0
A
B
. For each initial arrow
⊥
:⊥
:{
1
}→
0
0
0
{
1
,...,m
}
,T
⊥
=⊥
⊆
TA
σ(
1
)
, thus it represents the PO relation
⊥
A
.Any
monomorphism in
S
I
represents a PO relation
A
B
between two strictly-complex
objects. Each composed arrow by the monic arrows above is a composition of PO
relations of each of its components and hence a PO relation as well. Consequently,
