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or a system of such constructs. It has been already
shown that protoforms have a universal character, and
are useful for building intelligent systems (Kacprzyk
and Zadrozny, 2005) (Yager, 2006).
Example: One of the simplest protoforms are con-
cepts. A concept car is such an example. This proto-
form identifies components of a car, and ”relations”
between them. There are many concepts related to
car, for example racing car, sport utility car, and all
of them constitute a network of concepts.
Example: Another portoform can be a single propo-
sition V is B, where V is a variable, and B is a subset
indicating the allowable values for the variable. This
protoform can be a building block of more complex
protoforms (Yager, 2006).
Example: Protoforms can be also used to represent
database query summaries (Kacprzyk and Zadrozny,
2005). In this case, a query summary such as Most
records meeting conditions B match query S can be
represented by a profotorm: Most BRs are S, where
R means records, B is a filter, and S is a query. Ev-
idently, as protoforms may form a hierarchy, higher
level (more abstract) protoforms can be defined, for
instance replacing most by a general linguistic quan-
tifier Q: QBRs are S.
3 CATEGORY THEORY
3.1 Basics of Category Theory
Category theory (Barr and Wells, 1999) is a branch
of mathematics that deals with structures and rela-
tionships among them. The structures are called ob-
jects and a relationship between two objects is called
a morphism. The essence of category theory, as stated
in (Fiadeiro, 2005), is that category theory character-
izes objects in terms of their ”social life”. This so-
cial life represents interaction of objects among them-
selves and their universe (environment).
Due to the space limitation we do not provide def-
initions of basic concepts of category theory, such as
category, morphism, functor, and universal construc-
tions (for example, pushouts). Definitions of these
concepts can be found in any category theory book,
for example (Jacobs, 2001).
3.2 Fibrations
Special types of functors, that define relationships
that exists among objects that belong to two differ-
ent categories, are called fibrations (Barr and Wells,
1999). A fibration is designed to capture collections
ϕ
−1
(i) ϕ
−1
( j)
X
i
?
j
?
I
ϕ
?
Figure 1: Fibers over i and j (each box represents a set of
objects X that are mapped into i and j respectively.
of categories varying over a base category. An exam-
ple can be collections of sets (X
i
)
i∈I
varying over a
base, or index, set I, Fig. 1. Let’s consider a functor
ϕ : X → I . The sets in the category X appear as
fibers over elements/objects of category I
ϕ
−1
(i) = {x ∈ X|ϕ(x) = i}
for each i ∈ I . In other words, a fiber is a collec-
tion of items of one category that can be mapped (via
fibration) into a single element (object) of another cat-
egory.
A formal definition of fibration is based on the
concept of cartesian morphisms
1
.
Definition 1 Let P : E → C be a functor between
categories, let f : C → D be an arrow of C , and let
P(Y) = D. An arrow u : X → Y of E is cartesian for
f and Y if (see Fig. 2 for a graphical representation):
• P(u) = f
• for any arrow v : Z → Y of E and any arrow h :
P(Z) → C of C for which f ◦ h = P(v), there is a
unique w : Z → X in E for which u ◦ w = v and
P(w) = h.
Definition 2 A functor P : E → C is a fibration if
there is a cartesian arrow for every f : C → D in C
and every object Y in E for which P(Y) = D.
If P : E → C is a fibration, one also says that E is
fibered over C . In that case, C is the base category
and E is the total category of the fibration. Some
authors represent a fibration vertically:
E
↓
C
This way of representing a fibration is very intuitive.
1
There is also a dual concept to fibration called cofibra-
tion or opfibration. We skip the formal definition of opfi-
bration, but elements of opcartesian (dual to cartesian) mor-
phism are used in Section 7 to illustrate benefits of the ap-
proach we proposed in the paper.
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