Database Reference
In-Depth Information
8
The Properties of DB Category
8.1
Expressive Power of the DB Category
This chapter is a direct continuation of Chap. 3 , dedicated to the definition of the DB
category and its basic properties. In fact, in Chap. 3 , we demonstrated the duality
and categorial symmetry properties of the DB category, the semantics for different
types of (iso, mono)morphisms between its objects (databases), and basic set of op-
erators: the power-view operator, the data separation (isomorphic to the (co)product
and the data federation (connection). In this chapter, we will define the matching
and, dual to it, the merging operator for databases. Then we will investigate another
fundamental properties of the DB category and, especially, its (co)completeness, the
lattice property for its objects, the enrichments, and also its topological properties.
First of all, we recall that DB is not a topos (from Proposition 5 ) because, gener-
ally, an arrow which is monic and epic is not an isomorphism in DB (the converse is
valid in all categories, i.e., an isomorphism is always monic and epic). Let us explain
the reason for that. In fact, for any simple morphism f
B , so that both A and
B are non-separation-composed objects, the following is true (from Corollary 9 ):
1. f is monic iff its information flux f
:
A
=
TA ;
2. f is epic iff its information flux f
=
TB ;
3. f is an isomorphism iff f
TB .
Consequently, each simple arrow, which is both monic and epic, is an isomorphism.
But it does not hold for complex arrows, as used in Proposition 5 , k
=
TA
=
=[
id C , id C ]:
k
A 1
A 2
B 1 , with A 1 =
A 2 =
B 1 =
C and
={
id C :
A 1
B 1 , id C :
A 2
B 1 }
and hence h
= id C id C =
C) . Thus, from Proposition 8 ,itis
monic and, from Proposition 5 , it is also epic, however, it is not an isomorphism .
Consequently, only for the complex arrows we have the situation that a given
morphism which is both monic and epic is not also an isomorphism. Thus, DB is
not a topos.
Now, in what follows, we will explain what exactly the particular condition is,
when a complex arrow which is monic and epic is also an isomorphism in the DB
category.
TC
TC
=
T(C
 
 
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