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T f i | f i
C
=
B
T f i | f i
C
=
B
(from algebraic lattice property, Sect. 8.1.5 )
=
T(B
C)
=
B
C.
To have a Cartesian closed DB category (from Theorem 5 in Sect. 3.2.4 ,
×≡
), for
any complex arrow f
=[
f 1 ,f 2 ]:
A
B
−→
C in DB , we must have the exponential
C B
is a simple derived morphism by “abstraction” Λ and the complex arrow eval B,C =
[
diagram commutativity, i.e., f
=
eval B,C
(Λ(f )
id B ) , where Λ(f )
:
A
−→
C B
e 1 ,e 2 ]:
×
−→
C is a universal “application” morphism. Thus, from the com-
mutativity in DB of such an exponential diagram, we obtain f 2 = e 2 : B C and
f 1 =
B
C , thus f 1 =
e 1 Λ(f ) . However, f 1
TC while
e 1 Λ(f ) (T A TC) TB , so that there are the morphisms f 1 for which
f 1
e 1
Λ(f )
:
A
TA
e 1 Λ(f ) and hence the exponential diagram is not commutative. That is,
DB has no exponentiation and hence is not a CCC.
Remark From point 1 of Proposition 46 and the proof of this proposition, the “ex-
ponential” object C B which “internalizes” in DB the hom-set of its morphisms
DB (B,C) is equal to the information flux of the principal morphism from B into C .
Thus, from the fact that DB is not a CCC, we deduce that it is not a standard
topos. But it still can be a kind of a weak topos , if we substitute the exponentiation
with a kind of weak exponentiation (by replacing a (strong) product with a kind of
(weak) tensorial product) and define a proper subobject classifier in DB (which has
to be a kind of special pullback diagram, specified in dedicated Sect. 9.1.2 ).
8.1.4 Universal Algebra Considerations
In what follows, we will consider the signature Σ R of the SPRJU Codd's relational
algebra (in Definition 31 , Sect. 5.1 ) which is used to define the views in the SOtgd
implications for the schema database mappings. In order to explore the universal
algebra properties of the category DB [ 18 ], where a morphism is not a function but
a nonempty set of functions (this fact significatively complicates a definition of the
mappings from DB -morphisms into the homomorphisms of the category of Σ R -
algebras), we will use an, equivalent to DB , a “sets-like” category DB sk such that
its arrows can be seen as total functions.
Proposition 50 Let us denote the full skeletal subcategory of DB composed of
closed objects only by DB sk .
Such a category is equivalent to the category DB , i . e ., there exist an adjunction
of a surjective functor T sk :
DB
−→
DB sk and an inclusion functor In sk :
DB sk −→
DB such that T sk In sk =
Id DB sk and In sk T sk
Id DB .
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