Database Reference
In-Depth Information
It is easy to verify that for each pair of objects (i.e., databases)
A
and
B
in
DB
there exists
f
AB
∈
DB
(B
A
,(TB)
TA
)
, called an
action
of
T
on
B
A
, such that for all
g
∈
DB
(A,B)
the following is valid:
Λ
g
β(A)
=
Λ
T(g)
β(TA)
:
(T B)
TA
,
f
AB
◦
◦
◦
Υ
−→
where
β
I
DB
is a left identity natural transformation of a monoid
(
DB
,
⊗
,Υ,α,β,γ)
, thus
β(A)
=
:
Υ
⊗
_
−→
is
−
A
:
Υ
⊗
A
→
A
(where
Υ
⊗
A
=
TA
from
Theorem
12
) and
β(TA)
=
id
TA
. In fact, we take
f
AB
=
id
B
A
(the action of
T
TB
with
Tf
=
f
), which is an iden-
on
f
:
A
→
B
is equal to
Tf
:
TA
→
tity action; moreover,
B
A
TB
TA
. Hence, by consider-
A
⊗
B
=
TA
⊗
TB
=
1
≤
j
≤
m
,m
=
1
≤
i
≤
k
,k
1 and
is
−
A
:
ing that for
A
≥
1,
B
≥
TA
→
A
with
is
−
1
A
is
−
1
={
A
j
:
TA
j
→
A
j
|
1
≤
j
≤
m
}
, we obtain (see the proof of Theorem
15
):
T(g
ji
)
Λ(T (g))
g
=
|
(g
ji
:
A
j
→
B
i
)
∈
and
g
ji
|
g
ji
◦
is
−
A
)
is
−
1
A
A
j
:
TA
j
→
B
i
∈
Λ(g
◦
g
◦
, g
ji
=
is
−
1
is
−
1
A
j
=
g
ji
◦
g
ji
|
g
ji
∈
g
, g
ji
=
g
ji
∩
is
−
1
=
A
j
g
ji
|
TA
j
g
, g
ji
=
=
g
ji
∈
g
ji
∩
(from
g
ji
⊆
A
j
⊗
B
i
=
TA
j
∩
TB
i
⊆
TA
j
)
g
ji
|
g
ji
.
g
, g
ji
=
=
g
ji
∈
g
,
g
ji
=
T(g
ji
)
, i.e.,
g
ji
=
Thus, for each
g
ji
∈
g
ji
=
T(g
ji
)
, and consequently,
Λ
g
is
−
A
=
Λ
T(g)
.
◦
(8.3)
is
−
A
)
is
−
A
)
(
1
)
Λ(T (g))
Hence,
f
AB
◦
Λ(g
◦
β(A))
=
id
B
A
◦
Λ(g
◦
=
Λ(g
◦
=
=
Λ(T (g)
β(TA))
.
Consequently,
T
is a closed endofunctor.
The composition law
m
A,B,C
may be equivalently represented by a natural trans-
formation
m
◦
id
TA
)
=
Λ(T (g)
◦
:
(B
⊗
_
)
⊗
(
_
⊗
B)
−→ ⊗
and an identity element
j
A
by natural trans-
:
−→ ⊗◦
:
−→
×
formation
j
Y
, where
DB
DB
DB
is a diagonal functor, while
Y
:
id
Υ
for
any arrow
f
in
DB
. It is easy to verify that two coherent diagrams (associativity
and unit axioms) commute, and hence
DB
is enriched over itself V-category (as, for
example,
Set
category).
DB
−→
DB
is a constant endofunctor,
Y(A)
Υ
for any
A
and
Y(f)
