Database Reference
In-Depth Information
each arrow
f
in
DB
I
represents the PO relation
dom(f )
cod(f )
. Let us show
that each PO relation
A
B
is represented by a monomorphism in
DB
I
as well: if
0
⊥
A
then it is represented by a composition of the isomorphism (monic as well)
0
and initial arrow from
0
into
B
which is monic as well. Otherwise
A
is
B
is generally representable as
A
A
S
B
S
B
where
A
S
is a strictly-complex
object obtained by eliminating all objects
:
A
→⊥
⊥
0
from
A
(analogously for
B
S
). Thus
A
B
is represented in
DB
I
as a composition of these two isomorphisms (that are
also monic) and a monomorphism in
S
I
between two strictly-complex objects (the
composition of monic arrows is monic as well). Consequently, between any two
given objects in
DB
I
there can exist at most one arrow, so this is a PO category. The
power-view operator
T
satisfies the following:
(i)
TA
⊥
=
T(TA)
, as explained in the introduction, Sect.
1.4.1
.
(ii)
A
⊆
B
implies
TA
⊆
TB
:
T
is a monotonic operator w.r.t.
⊆
(the free
SPRJU algebra of terms
L
A
is monotonic with respect to its carrier set
A
;
see Sect.
1.4.1
).
(iii)
A
⊆
TA
, each element of
A
is also a view of
A
.
Thus,
T
is a closure operator and an object
A
such that
A
=
TA
is a closed
object.
For any two simple morphisms
f,g
B
such that
f
:
A
→
⊆
g
, it is easy to show
:
f
T
e
(f
OP
f
OP
in
=
◦
f
ep
;
f
in
◦
→
that there are monomorphisms
β
)
g
. In fact, for
ep
ep
◦
f
ep
:
A
→
A
with
h
1
=
f
OP
ep
∩
f
ep
=
f
∩
f
=
f
and
h
2
=
f
in
◦
f
OP
h
1
=
f
OP
in
:
B
with
h
2
=
f
in
∩
f
OP
f
∩
f
=
f
, we obtain that
h
2
◦
=
f
∩
f
=
f
and
B
→
f
in
g
∩
f
=
f
(for
f
◦
h
1
=
g
⊆
g
), so that
h
2
◦
f
=
g
◦
h
1
, and, consequently, the
:
f
morphism
T
e
(h
1
;
T
e
(f
OP
f
OP
in
h
2
)
=
ep
◦
f
ep
;
f
in
◦
)
→
g
is well defined. Thus,
β
=
T
e
(f
OP
ep
◦
f
ep
;
f
in
◦
f
OP
in
OP
g
◦
T(f
in
◦
f
OP
in
)
=
in
◦
f)
◦
in
f
, so that (from
T f
OP
in
=
T f
in
=
T f
=
f
),
β
=
in
OP
g
∩
T f
in
∩
T f
OP
in
∩
Tf
∩
in
f
=
g
∩
f
=
f
,so
T
e
(f
OP
f
OP
that
β
in
)
is a monomorphism (from Corollary
9
).
It is easy to verify that
DB
is a 2-category with 0-cells (its objects), 1-cells
(its ordinary morphisms) and 2-cells (“inclusions” arrows between mappings). The
horizontal and vertical composition of 2-cells is just the comp
os
ition of PO re-
lations. Given
f,g,h
:
A
−→
B
w
ith 2-cells
√
β
:
f
−→
g
,
√
δ
:
g
−→
h
, their
vertical composition is
√
γ
=
ep
◦
f
ep
;
f
in
◦
=
√
δ
◦
√
β
f
−→
:
h
.Given
f,g
:
A
−→
B
and
g
,
√
δ
C
with 2-cells
√
β
f
−→
h
−→
h,l
:
B
−→
:
:
l
, for a given composition
functor
•:
DB
(A,B)
×
DB
(B,C)
−→
DB
(A,C)
, their horizontal composition is
√
δ
•
√
β
√
γ
f
=
:
◦
−→
◦
h
l
g
.
For example, the equivalence
q
A
i
≈
q
B
j
(i.e.,
q
A
i
q
B
j
and
q
B
j
q
A
i
)oftwo
view mappings
q
A
i
:
A
−→
TA
and
q
B
j
:
B
−→
TB
, for the simple databases
A
and
B
, is obtained when they produce the same view.
The categorial symmetry operator
T
e
J
:
Ob
DB
for any morphism
f
in
DB
produces its information flux
f
(i.e., the “conceptualized” database of this
mapping). Consequently, we can define a “mapping between mappings” (which are
Mor
DB
−→
