In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a natural transformation provides a way of transforming one
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
into another while respecting the internal structure (i.e., the composition of
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s) of the
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
*Categories (Aristotle), ''Categories'' (Aristotle)
*Category (Kant)
...
involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.
Indeed, this intuition can be formalized to define so-called
functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
and consequently appear in the majority of its applications.
Definition
If
and
are
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s between the categories
and
, then a natural transformation
from
to
is a family of morphisms that satisfies two requirements.
# The natural transformation must associate, to every object
in
, a
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
between objects of
. The morphism
is called the component of
at
.
# Components must be such that for every morphism
in
we have:
:::
The last equation can conveniently be expressed by the
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
If both
and
are
contravariant, the vertical arrows in the right diagram are reversed. If
is a natural transformation from
to
, we also write
or
. This is also expressed by saying the family of morphisms
is natural in
.
If, for every object
in
, the morphism
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
in
, then
is said to be a (or sometimes natural equivalence or isomorphism of functors). Two functors
and
are called ''naturally isomorphic'' or simply ''isomorphic'' if there exists a natural isomorphism from
to
.
An infranatural transformation
from
to
is simply a family of morphisms
, for all
in
. Thus a natural transformation is an infranatural transformation for which
for every morphism
. The naturalizer of
, nat
, is the largest
subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...
of
containing all the objects of
on which
restricts to a natural transformation.
Examples
Opposite group
Statements such as
:"Every group is naturally isomorphic to its
opposite group
In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action.
Monoids, groups, rings, and algebras can be viewed as categor ...
"
abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category
of all
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
s with
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
s as morphisms. If
is a group, we define
its opposite group
as follows:
is the same set as
, and the operation
is defined
by
. All multiplications in
are thus "turned around". Forming the
opposite group becomes
a (covariant) functor from
to
if we define
for any group homomorphism
. Note that
is indeed a group homomorphism from
to
:
:
The content of the above statement is:
:"The identity functor
is naturally isomorphic to the opposite functor
"
To prove this, we need to provide isomorphisms
for every group
, such that the above diagram commutes.
Set
.
The formulas
and
show that
is a group homomorphism with inverse
. To prove the naturality, we start with a group homomorphism
and show
, i.e.
for all
in
. This is true since
and every group homomorphism has the property
.
Abelianization
Given a group
, we can define its
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
. Let
denote the projection map onto the cosets of