Database Reference
In-Depth Information
the fact that
DB
I
and
DB
have the same objects). Thus it is enough to analyze only
the first left Kan extension (left one) given by the following commutative diagrams:
DB
DB
f
DB
DB
I
That is, we have the functor
Lan
K
:
→
which is a left adjoint to
DB
DB
I
DB
DB
f
, and hence the left Kan extension of
the functor
G
=
_
◦
K
:
→
DB
DB
f
DB
DB
I
, and natural trans-
Σ
D
∈
∈
along
K
is given by functor
Lan
K
(Σ
D
)
:
Σ
D
→
◦
formation
η
Lan
K
(Σ
D
)
K
is an universal arrow. That is, for any other
:
DB
I
→
:
Σ
D
→
◦
functor
S
DB
and a natural transformation
α
S
K
, there is a
unique natural transformation
σ
:
Lan
K
(Σ
D
)
→
S
such that
α
=
σK
•
η
(where
•
is a vertical composition for natural transformations).
This adjunction is denoted by a tuple
(Lan
K
,G,ε,η)
, where
W
Lan
K
(C)
η
C
:
C
→
=
is a universal arrow for each object
C
(let us consider the case when
C
Σ
D
). In
fact, for any object (i.e., a functor)
S
in
DB
DB
I
and a morphism (i.e., a natural trans-
formation)
α
:
Σ
D
→
G(S)
, there exists a unique morphism
σ
:
Lan
K
(Σ
D
)
→
S
such that the following two adjoint diagrams commute:
so that
η
Σ
D
is a universal arrow from
Σ
D
into
G
and, dually,
ε
S
is a couniversal
arrow from
Lan
K
into
S
. Consequently, in this adjunction, the unit
η
generates a
universal arrow for each object in
DB
DB
f
and the counit
ε
generates a couniversal
arrow for each object in
DB
DB
I
.
From the well know theorem for left Kan extensions, when we have a tenso-
rial product
B
∈
DB
f
DB
I
(K(B),A)
⊗
Σ
D
B
for every
A
∈
DB
I
, i.e.,
A
∈
DB
,
the function for the objects of the functor
Lan
K
(Σ
D
)
(here
B
ω
A
means that
B
∈
DB
f
&
B
A
, and we take the case when
A/
∈
DB
f
is an
infinite
database
which is not a closed object in
DB
) is defined by:
B
∈
DB
f
DB
I
K(B),A
⊗
Lan
K
(Σ
D
)(A)
Σ
D
B