Database Reference
In-Depth Information
and
id
A
=
α
∗
(
=
1
≤
i
≤
k
A
i
, where
TA
. For a complex object
A
=
A
1
†
···
†
A
k
)
A
i
=
α
∗
(
A
i
)
, its identity arrow is
id
A
=
1
≤
i
≤
k
id
A
i
with simple atomic identity
=
1
≤
i
≤
k
id
A
i
=
1
≤
i
≤
k
TA
i
which
A
i
. Moreover,
id
A
=
arrows
id
A
i
:
A
i
→
TA
is a closed object as well.
Let
M
AB
=
MakeOperads(
M
AB
)
={
q
1
,...,q
m
,
1
r
∅
}
and
=
q
1
,...,q
n
,
1
r
∅
.
M
BC
=
MakeOperads(
M
BC
)
Then
α
∗
(
M
BC
◦
M
AB
)
=
α
∗
q
i
·
q
i
1
,...,q
i
k
i
|
≤
j
≤
k
i
1
≤
i
≤
n
and 1
≤
i
j
≤
m
for 1
1
r
∅
}
(
from definition of R-algebras in Definition
10
)
=
α
q
i
α
q
i
1
×···×
∪{
1
r
∅
·
α
q
i
k
i
|
k
i
≤
≤
≤
i
j
≤
≤
≤
1
i
n
and 1
m
for 1
j
q
⊥
)
=
α
q
1
,...,α
q
n
,q
⊥
◦
α
q
1
,...,α
q
M
,q
⊥
=
α
∗
q
1
,...,q
n
,
1
r
∅
◦
α
∗
q
1
,...,q
m
,
1
r
∅
=
∪{
q
⊥
}
(
from
α(
1
r
∅
)
=
α
∗
(
M
BC
)
α
∗
(
M
AB
).
◦
α
∗
(
M
AB
)
α
∗
(
M
BC
)
Thus, for given two morphisms
f
=
:
A
→
B
and
g
=
:
B
→
C
their composition is equal to:
g
◦
f
=
α
∗
(
M
BC
)
◦
α
∗
(
M
AB
)
=
α
∗
(
M
BC
◦
M
AB
)
:
A
→
C.
Let as show the associativity of composition of morphisms in
DB
, with
h
=
α
∗
(
M
CD
)
:
h
◦
(g
◦
f)
M
AB
)
(
from the associativity of the operad-mapping composition in Corollary
3
)
=
α
∗
(
M
CD
◦
◦
α
∗
(
M
BC
)
α
∗
(
M
AB
)
=
α
∗
M
CD
◦
α
∗
(
M
CD
)
=
◦
(
M
BC
◦
M
BC
)
◦
α
∗
(
M
AB
)
=
α
∗
(
M
CD
)
◦
α
∗
(
M
BC
)
◦
α
∗
(
M
AB
)
=
(h
◦
g)
◦
f.
-atomic arrows in Definition
15
are based on
component-to-component composition of atomic arrows and are specified in detail
in the rest of this section. Thus, the properties above hold for the
The left and right compositions of
-atomic arrows
as well. Consequently,
DB
is a well-defined category.
0
0
The object composed of the empty relation only is denoted by
⊥
and
T
⊥
=
0
⊥
. We will show in Lemma
10
that the empty database (a database with only
empty relations) is isomorphic to this bottom object
={⊥}
0
. Notice that this statement
⊥
