Editing Natural transformation (section)

==Examples==

===Opposite group===
{{details|Opposite group}}
Statements such as
:"Every group is naturally isomorphic to its [[opposite group]]"
abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category 
<math>\textbf{Grp}</math> of all [[group (mathematics)|group]]s with [[group homomorphism]]s as morphisms. If <math>(G, *)</math> is a group, we define 
its opposite group <math>(G^\text{op}, {*}^\text{op})</math> as follows: <math>G^\text{op}</math> is the same set as <math>G</math>, and the operation <math>*^\text{op}</math> is defined 
by <math>a *^\text{op} b = b * a</math>. All multiplications in <math>G^{\text{op}}</math> are thus "turned around". Forming the [[Opposite category|opposite]] group becomes 
a (covariant) functor from <math>\textbf{Grp}</math> to <math>\textbf{Grp}</math> if we define <math>f^{\text{op}} = f</math> for any group homomorphism <math>f: G \to H</math>. Note that 
<math>f^\text{op}</math> is indeed a group homomorphism from <math>G^\text{op}</math> to <math>H^\text{op}</math>:
:<math>f^\text{op}(a *^\text{op} b) = f(b * a) = f(b) * f(a) = f^\text{op}(a) *^\text{op} f^\text{op}(b).</math>
The content of the above statement is:
:"The identity functor <math>\text{Id}_{\textbf{Grp}}: \textbf{Grp} \to \textbf{Grp}</math> is naturally isomorphic to the opposite functor <math>{\text{op}}: \textbf{Grp} \to \textbf{Grp}</math>"
To prove this, we need to provide isomorphisms <math>\eta_G: G \to G^{\text{op}}</math> for every group <math>G</math>, such that the above diagram commutes. 
Set <math> \eta_G(a) = a^{-1}</math>.
The formulas <math>(a * b)^{-1} = b^{-1}*a^{-1}= a^{-1}*^{\text{op}} b^{-1}</math> and <math> (a^{-1})^{-1} = a</math>
show that <math>\eta_G</math> is a group homomorphism with inverse <math> \eta_{G^\text{op}}</math>. To prove the naturality, we start with a group homomorphism 
<math>f : G \to H</math> and show <math>\eta_H \circ f = f^{\text{op}} \circ \eta_G</math>, i.e. 
<math> (f(a))^{-1} = f^\text{op}(a^{-1})</math> for all <math>a</math> in <math>G</math>. This is true since 
<math>f^{\text{op}} = f</math> and every group homomorphism has the property <math>(f(a))^{-1} = f(a^{-1})</math>.

===Modules===
Let <math> \varphi:M \longrightarrow M^{\prime} </math> be an <math>R </math>-module homomorphism of right modules. For every left module <math> N </math> there is a natural map <math> \varphi \otimes N: M \otimes_{R} N \longrightarrow M^{\prime} \otimes_{R} N</math>, form a natural transformation <math>\eta: M \otimes_{R} - \implies M' \otimes_{R} - </math>. For every right module <math> N </math> there is a natural map <math> \eta_{N}: \text{Hom}_{R}(M',N) \longrightarrow \text{Hom}_{R}(M,N) </math> defined by <math> \eta_{N}(f) = f\varphi</math>, form a natural transformation <math> \eta:\text{Hom}_{R}(M',-) \implies \text{Hom}_{R}(M,-) </math>.

===Abelianization===
Given a group <math>G</math>, we can define its [[abelianization]] <math>G^{\text{ab}} = G/</math> [[Commutator subgroup#Definition|<math>[G,G]</math>]].  Let 
<math>\pi_G: G \to G^{\text{ab}}</math> denote the projection map onto the cosets of <math>[G,G]</math>. This homomorphism is "natural in 
<math>G</math>", i.e., it defines a natural transformation,  which we now check.  Let <math>H</math> be a group.  For any homomorphism <math>f : G \to H</math>, we have that 
<math>[G,G]</math> is contained in the kernel of <math>\pi_H \circ f</math>, because any homomorphism into an abelian group kills the commutator subgroup.  Then 
<math>\pi_H \circ f</math> factors through <math>G^{\text{ab}}</math> as <math>f^{\text{ab}} \circ \pi_G = \pi_H \circ f</math> for the unique homomorphism 
<math>f^{\text{ab}} : G^{\text{ab}} \to H^{\text{ab}}</math>.  This makes <math>{\text{ab}} : \textbf{Grp} \to \textbf{Grp}</math> a functor and 
<math>\pi</math> a natural transformation, but not a natural isomorphism, from the identity functor to <math>\text{ab}</math>.

=== Hurewicz homomorphism ===
Functors and natural transformations abound in [[algebraic topology]], with the [[Hurewicz theorem|Hurewicz homomorphisms]] serving as examples.  For any [[pointed topological space]] <math>(X,x)</math> and positive integer <math>n</math> there exists a [[group homomorphism]]

: <math>h_n \colon \pi_n(X,x) \to H_n(X)</math>

from the <math>n</math>-th [[homotopy group]] of <math>(X,x)</math> to the <math>n</math>-th [[Singular homology|homology group]] of <math>X</math>.  Both <math>\pi_n</math> and <math>H_n</math> are functors from the category '''Top<sup>*</sup>''' of pointed topological spaces to the category '''Grp''' of groups, and <math>h_n</math> is a natural transformation from  <math>\pi_n</math> to <math>H_n</math>.

===Determinant===
{{See also|Determinant#Square matrices over commutative rings}}

Given [[Commutative ring|commutative rings]] <math>R</math> and <math>S</math> with a [[ring homomorphism]] <math>f : R \to S</math>, the respective groups of [[Invertible matrix|invertible]] <math>n \times n</math> matrices <math>\text{GL}_n(R)</math> and <math>\text{GL}_n(S)</math> inherit a homomorphism which we denote by <math>\text{GL}_n(f)</math>, obtained by applying <math>f</math> 
to each matrix entry. Similarly, <math>f</math> restricts to a group homomorphism <math>f^* : R^* \to S^*</math>, where <math>R^*</math> denotes the [[group of units]] of <math>R</math>. In fact, <math>\text{GL}_n</math> and <math>*</math> are functors from the category of commutative rings <math>\textbf{CRing}</math> to <math>\textbf{Grp}</math>. 
The [[determinant]] on the group <math>\text{GL}_n(R)</math>, denoted by <math>\text{det}_R</math>, is a group homomorphism 

: <math>\mbox{det}_R \colon \mbox{GL}_n(R) \to R^*</math>
which is natural in <math>R</math>: because the determinant is defined by the same formula for every ring, <math>f^* \circ \text{det}_R = \text{det}_S\circ \text{GL}_n(f)</math> holds. This makes the determinant a natural transformation from <math>\text{GL}_n</math> to <math>*</math>.

===Double dual of a vector space===
For example, if <math>K</math> is a [[field (mathematics)|field]], then for every [[vector space]] <math>V</math> over <math>K</math> we have a "natural" [[injective]] [[linear map]] <math>V \to V^{**}</math> from the vector space into its [[double dual]]. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor.

===Finite calculus===
For every abelian group <math>G</math>, the set <math>\text{Hom}_\textbf{Set}(\mathbb{Z}, U(G))</math> of functions from the integers to the underlying set of <math>G</math> 
forms an abelian group <math>V_{\mathbb{Z}}(G)</math> under pointwise addition. (Here <math>U</math> is the standard [[forgetful functor]] <math>U:\textbf{Ab} \to \textbf{Set}</math>.) 
Given an <math>\textbf{Ab}</math> morphism <math>\varphi: G \to G' </math>, the map <math>V_\mathbb{Z}(\varphi): V_\mathbb{Z}(G) \to V_\mathbb{Z}(G')</math> given by left composing <math>\varphi</math> with the elements of the former is itself a homomorphism of abelian groups; in this way we 
obtain a functor <math>V_{\mathbb{Z}}: \textbf{Ab} \to \textbf{Ab}</math>. The finite difference operator <math>\Delta_G</math> taking each function 
<math>f: \mathbb{Z} \to U(G)</math> to <math>\Delta(f): n \mapsto f(n+ 1) - f(n)</math> is a map from <math>V_{\mathbb{Z}}(G)</math> to itself, and the collection <math>\Delta</math> of such maps gives a natural transformation <math>\Delta: V_\mathbb{Z} \to V_\mathbb{Z}</math>.

===Tensor-hom adjunction===
{{further|Tensor-hom adjunction|Adjoint functors}}
Consider the [[category of abelian groups|category <math>\textbf{Ab}</math>]] of abelian groups and group homomorphisms. For all abelian groups <math>X</math>, <math>Y</math> and <math>Z</math> we have a group isomorphism
: <math>\text{Hom}(X \otimes Y, Z) \to \text{Hom}(X, \text{Hom}(Y, Z))</math>.
These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors <math>\textbf{Ab}^{\text{op}} \times \textbf{Ab}^{\text{op}} \times \textbf{Ab} \to \textbf{Ab}</math>.
(Here "op" is the [[opposite category]] of <math>\textbf{Ab}</math>, not to be confused with the trivial [[opposite group]] functor on <math>\textbf{Ab}</math> !)

This is formally the [[tensor-hom adjunction]], and is an archetypal example of a pair of [[adjoint functors]]. Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the ''unit'' and ''counit''.