Database Reference
In-Depth Information
Consequently, the sketch
Sch
(G)
will have an
atomic
arrow (in Definition
17
),
M
AB
=
, which for an R-algebra
α
will have the in-
formation flux
Flux(α,
M
AB
)
, and hence
α
∗
(
M
AB
)
MakeOperads(
{
Φ
}
)
:
A
→
B
α
∗
(
α
∗
(
:
A
)
→
B
)
is (from The-
orem
1
) an atomic morphism in
DB
.
It is easy to verify that for each of these cases
α
∗
(
M
AB
)
Eval(o
i
)
. The iden-
tity arrow
o
i
is mapped into the identity morphism in
DB
and, from these sketch's
interpretations, the functorial composition of the arrows is valid: For any composed
arrow
o
k
◦
=
o
i
, where
M
BC
is the sketch's arrow for
o
k
and
M
AB
is the sketch's arrow
for
o
i
, for the atomic arrow composition
M
BC
◦
M
AB
in a sketch category,
Eval(o
k
◦
o
i
)
=
α
∗
(
M
BC
◦
M
AB
)
from the functorial property of
α
∗
α
∗
(
M
BC
)
α
∗
(
M
AB
)
=
◦
=
Eval(o
k
)
◦
Eval(o
i
).
Thus, the functor
Eval
is valid for composed (non atomic) arrows as well.
Example 34
In what follows, we will present the three basic atomic update op-
erations, updating (i), inserting (ii) and deleting (iii) of tuples in a given instance
database
α(
, and we will show that each of them is representable
in
DB
by a morphism
f
:
α(
A
)
→
α
1
(
A
)
where
α
1
(
A
)
is the resulting database
instance obtained after such an update. This result is important because it demon-
strates not only the denotational property of
DB
, but also its operational capability
to represent the atomic database-updating transactions by the atomic morphisms of
DB
category:
(i) Let us consider the following representation in
DB
category of an update of
a given relation
r
A
)
of a schema
A
α
∗
(
∈
S
A
of a database instance
A
=
A
)
of a schema
A
=
(S
A
,
∅
)
, with
α(r)
=
r
#
the extension of this relation before this update such
that
r
WHERE
C
#
is a nonempty
set of tuples:
