Database Reference
In-Depth Information
The most accessible example of a category is the category
Set
of sets, where the
objects are sets and the arrows are functions from one set to another. However, it
is important to note that the objects of a category need not be sets or the arrows
functions; any way of formalizing a mathematical concept such that it meets the
basic conditions on the behavior of objects and arrows is a valid category, and all
the results of category theory will apply to it.
Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane
in 1942-45 [
36
], in connection with algebraic topology. The study of categories is
an attempt to axiomatically capture what is commonly found in various classes of
related mathematical structures by relating them to the structure-preserving func-
tions between them. A systematic study of category theory then allows us to prove
general results about any of these types of mathematical structures from the axioms
of a category.
A category
C
consists of the following three mathematical entities:
•
Aclass
Ob
C
whose elements are called objects;
•
Aclass
Mor
C
whose elements are called morphisms or maps, or arrows. Each
morphism
f
has a unique source object
A
and target object
B
. The expres-
sion (arrow)
f
B
would be verbally stated as “
f
is a morphism from
A
to
B
”. The expression
hom(A,B)
—alternatively expressed as
hom
C
(A,B)
,
or
mor(A,B)
,or
C
(A,B)
—denotes the hom-class of all morphisms from
A
to
B
. If for all
A,B
this hom-class is a set, this category
C
is called
locally small
.
:
A
→
•
A binary operation
, called composition of morphisms, such that for any
three objects
A,B
, and
C
, we have a function
◦
◦:
hom(A,B)
×
hom(B,C)
→
hom(A,C)
. Each morphism
f
∈
hom(A,B)
has a domain
A
=
dom(f )
and
codomain
B
cod(f )
which are objects.
The composition of
f
=
:
A
→
B
and
g
:
B
→
C
is written as
g
◦
f
and is
governed by two axioms:
-
Associativity
.If
f
:
A
→
B,g
:
B
→
C
and
h
:
C
→
D
then
h
◦
(g
◦
f)
=
(h
◦
g)
◦
f
, and
-
Identity
. For every object
A
, there exists a morphism
id
A
:
A
→
A
called the
identity morphism for
A
, such that for every morphism
f
:
A
→
B
,wehave
f
.
From these axioms, it can be proved that there is exactly one identity morphism for
every object. Because of that we can identify each object with its identity morphism.
Relations among morphisms are often depicted using commutative diagrams, with
nodes representing objects and arrows representing morphisms, for example,
g
id
B
◦
f
=
f
◦
id
A
=
◦
f
=
h
is graphically represented by the following commutative diagram:
Examples:
