Database Reference
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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 D )
:
Σ D
formation η
Lan K 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 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 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 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 D )(A)
Σ D B
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