A mapping from one
category
into another that is compatible with the category structure. More precisely, a
covariant functor
from a category
into a category
or, simply, a functor from
into
,
is a pair of mappings
,
usually denoted by the same letter, for example
(
),
subject to the conditions:
1)
for every
;
2)
for all morphisms
,
.
A functor from the category
dual to
into the category
is called a
contravariant functor
from
into
.
Thus, for a contravariant functor
,
condition 1) must be satisfied as before, and condition 2) is replaced by: 2*)
for all morphisms
,
.
An
-place functor
from categories
into
that is covariant in the arguments
and contravariant in the remaining arguments is a functor from the Cartesian product
into
,
where
for
and
for the remaining
.
Two-place functors that are covariant in both arguments are called
bifunctors.
Examples of functors.
1)
The identity mapping of a category
onto itself is a covariant functor, called the
identity functor
of the category and denoted by
or
.
2)
Let
be an arbitrary locally small category, let
be the category of sets, and let
be a fixed object of
.
If one associates to each
the set
and to each morphism
the mapping
,
where
for each
,
one obtains a functor from
into
.
This functor is called the
covariant representable functor
from
into
with
representing object
.
Similarly, if one associates to an object
the set
and to a morphism
the mapping
,
where
,
one obtains the
contravariant representable functor
from
into
with
representing object
.
These functors are denoted by
and
,
respectively. If
is the category of vector spaces over a field
,
then
takes a space
to its dual space of linear functionals
.
In the category of topological Abelian groups, the functor
,
where
is the quotient group of the real numbers by the
integers, associates to each group its group of characters.
3)
If one associates to each pair of objects
and
of an arbitrary category the set
,
and to each pair of morphisms
and
the mapping
defined by the equation
for any
,
one obtains a two-place functor into the category
that is contravariant in the first argument and covariant in the second.
In any category with finite products, the product can be regarded as an
-place
functor that is covariant in all arguments, for any natural number
.
As a rule, a construction that may be defined for any object of
a category or for any sequence of objects of a fixed
length, independently of the individual properties of the objects, is likely to be
functorial. Examples of this are the construction of free algebras in some
variety of universal algebras, which can be uniquely associated to each
object of the category of sets; the construction of the fundamental
group of a topological space, the construction of
homology and cohomology groups of various dimensions; etc.
Any functor
defines a mapping of each set
into
which associates to a morphism
the morphism
.
The functor
is called
faithful
if these mappings are all injective, and
full
if they are all surjective. For every
small category
,
the assignment
can be extended to a full faithful functor
from
into the category
of diagrams (cf.
Diagram)
with scheme
over the category of sets
.