Database Reference
In-Depth Information
Analogously, for the
α
-intersection of schema mappings, it holds that
Flux
α,
M
AC
α
M
BD
Flux
α, MakeOperads(
M
AC
)
=
Flux
α, MakeOperads(
M
BD
)
.
∩
The schema separation, connection and
α
-intersection are necessary in order to
manage complex database mappings as it will be demonstrated in what follows.
However, their introduction is a reason for the introduction of a new database cate-
gory as well. This fact will be explained in the next section.
3.1.1 Introduction to Sketch Data Models
The category-theoretic data model that we use has come to be known as the
sketch
data model
.
As the sketch data model has been applied more widely, and particularly with the
development of new approaches to problems like view update problem, theoreticians
have increasingly been seeking for a way of translating between sketch data models
and other data models. Potentially, sketch data model techniques and results might
be translated into other models more familiar to practitioners.
Much of the theory of relational databases has been based around normalization,
and it has been shown that the relational states corresponding to sketch data models
(instances of sketch data model schema) are in at least third normal form.
The detailed presentation of sketches for the database mappings and their func-
torial semantics will be given in Sect.
4.1
.
Sketches are developed by Ehresmann's school, especially by R. Guitartand and
C. Lair [
3
,
6
,
9
]. Sketch is a category together with a distinguished class of cones
and cocones. A
model
of the sketch is a set-valued functor turning all distinguished
cones into limit cones, and all distinguished cocones into colimit cocones, in the
category
Set
of sets.
There is an elementary and basic connection between sketches and logic [
15
].
Given any sketch, we can consider the underlying graph of the sketch as a (many-
sorted) language, and we can write down axioms in the
-logic (the infinitary
FOL with finite quantifiers) over this language, so that the models of the axioms
become exactly the models of the sketch.
The category of models of a given sketch has models as objects and the arrows
that represent all natural transformations between the models as functors. A category
is sketchable (esquissable) or accessible iff it is equivalent to the category of set-
valued models of a small sketch.
Recall that a graph
G
L
∞
,
∞
(V
G
,E
G
)
consists of a set of vertices denoted by
V
G
and a set of arrows (edges) denoted by
E
G
together with the operators
dom, cod
=
:
E
G
→
V
G
which assigns to each arrow its source and target. (Co)cones and dia-
grams are defined for graphs in exactly the same way as they are for categories, but
commutative co(cones) and diagrams, of course, make no sense for graphs.
