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{{Redirect|Natural operation|the natural sum and natural product on ordinals|Ordinal arithmetic#Natural operations}}
{{About|natural transformations in category theory|the natural competence of bacteria to take up foreign DNA|
{{short description|Central object of study in category theory}}In [[category theory]], a branch of [[mathematics]], a '''natural transformation''' provides a way of transforming one [[functor]] into another while respecting the internal structure (i.e., the composition of [[morphism]]s) of the [[Category (mathematics)|categories]] involved. Hence, a natural transformation can be considered to be a "morphism of functors".
Indeed, this intuition can be formalized to define so-called [[functor category|functor categories]]. Natural transformations are, after categories and functors, one of the most fundamental notions of [[category theory]] and consequently appear in the majority of its applications.
==Definition==
If <math> F</math> and <math> G</math> are [[functor]]s between the categories <math> \mathcal{C}</math> and <math> \mathcal{D} </math> (both from <math> \mathcal{C}</math> to <math> \mathcal{D}</math>), then a '''natural transformation''' <math> \eta</math> from <math> F</math> to <math> G</math> is a family of morphisms that satisfies two requirements.
# The natural transformation must associate, to every object <math> X</math> in <math> \mathcal{C}</math>, a [[morphism]] <math>\eta_X : F(X) \to G(X)</math> between objects of <math> \mathcal{D} </math>. The morphism <math> \eta_X</math> is called "the '''component''' of <math> \eta</math> at <math> X </math>" or "the <math> X </math> component of <math>\eta</math>."
# Components must be such that for every morphism <math> f :X \to Y</math> in <math> \mathcal{C}</math> we have:
:::<math>\eta_Y \circ F(f) = G(f) \circ \eta_X</math>
The last equation can conveniently be expressed by the [[commutative diagram]].
[[File:Natural Transformation between two functors.svg|center|This is the commutative diagram which is part of the definition of a natural transformation between two functors.]]
If both <math> F</math> and <math> G</math> are [[contravariant functor|contravariant]], the vertical arrows in
If, for every object <math> X</math> in <math> \mathcal{C} </math>, the morphism <math> \eta_X</math> is an [[isomorphism]] in <math> \mathcal{D} </math>, then <math> \eta</math> is said to be a '''{{visible anchor|natural isomorphism}}''' (or sometimes '''natural equivalence''' or '''isomorphism of functors'''). Two functors <math> F </math> and <math> G</math> are called ''naturally isomorphic'' or simply ''isomorphic'' if there exists a natural isomorphism from <math> F</math> to <math> G</math> in their category.
An '''infranatural transformation''' <math> \eta
==Examples==
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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},
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
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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>]], where <math>[G,G]</math> denotes the commutator subgroup of <math>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
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=== Hurewicz homomorphism ===
Functors and natural transformations abound in [[algebraic topology]], with the [[Hurewicz theorem|Hurewicz homomorphisms]] serving as examples. For any [[
: <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>
===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>
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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
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===
===Finite calculus===
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{{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,
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> !)
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== Unnatural isomorphism ==
{{see also|Canonical map}}
The notion of a natural transformation is categorical, and states (informally) that a particular map between functors can be done consistently over an entire category. Informally, a particular map (esp. an isomorphism) between individual objects (not entire categories) is referred to as a "natural isomorphism", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors; formalizing this intuition was a motivating factor in the development of category theory.
Conversely, a particular map between particular objects may be called an '''unnatural isomorphism''' (or " This is similar (but more categorical) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is "not natural", or rather "not unique", as automorphisms exist that do not preserve the direct sum decomposition – see {{section link|Structure theorem for finitely generated modules over a principal ideal ___domain|Uniqueness}} for example.
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===Example: dual of a finite-dimensional vector space===
Every finite-dimensional vector space is isomorphic to its dual space, but there may be many different isomorphisms between the two spaces. There is in general no natural isomorphism between a finite-dimensional vector space and its dual space.<ref>{{harv|
The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field. However, in the absence of additional constraints (such as a requirement that maps preserve the chosen basis), the map from a space to its dual is not unique, and thus such an isomorphism requires a choice, and is "not natural". On the category of finite-dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space (say, by choosing a basis for every vector space and taking the corresponding isomorphism), but this will not define a natural transformation. Intuitively this is because it required a choice, rigorously because ''any'' such choice of isomorphisms will not commute with, say, the zero map; see {{harv|
Starting from finite-dimensional vector spaces (as objects) and the identity and dual functors, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from "all linear maps" to "linear maps that respect this structure". Explicitly, for each vector space, require that it comes with the data of an isomorphism to its dual, <math>\eta_V\colon V \to V^*</math>. In other words, take as objects vector spaces with a [[nondegenerate bilinear form]] <math>b_V\colon V \times V \to K</math>. This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps <math>T\colon V \to U</math> that commute with the isomorphisms: <math>T^*(\eta_{U}(T(v))) = \eta_{V}(v)</math> or in other words, preserve the bilinear form: <math>b_{U}(T(v),T(w))=b_V(v,w)</math>. (These maps define the ''naturalizer'' of the isomorphisms.) The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual (each space has an isomorphism to its dual, and the maps in the category are required to commute). Viewed in this light, this construction (add transforms for each object, restrict maps to commute with these) is completely general, and does not depend on any particular properties of vector spaces.
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== Operations with natural transformations ==
[[File:Natural transformation composition.svg|thumb|Horizontal and vertical composition of natural transformations]]
If <math>\eta : F \to G</math> and <math>\epsilon: G \to H</math>▼
are natural transformations between functors <math>F, G, H: C \to D</math>, then we can compose them to get a natural transformation <math>\epsilon\eta: F \to H</math>. ▼
allows one to consider the collection of all functors <math>C \to D</math> itself as a category (see below under [[#Functor categories|Functor categories]]).▼
=== Vertical composition ===
This is done component-wise:
[[Image:Vertical composition of natural transformations.svg|center|300px]]
▲This vertical composition of natural transformations is [[associative]] and has an identity, and allows one to consider the collection of all functors <math>C \to D</math> itself as a category (see below under [[#Functor categories|Functor categories]]).
The identity natural transformation <math>\mathrm{id}_F</math> on functor <math>F</math> has components <math>(\mathrm{id}_F)_X = \mathrm{id}_{F(X)}</math>.<ref>{{cite web | url=https://ncatlab.org/nlab/show/identity+natural+transformation | title=Identity natural transformation in nLab }}</ref>
:For <math>\eta : F \Rightarrow G</math>, <math>\mathrm{id}_G \circ \eta = \eta = \eta \circ \mathrm{id}_F</math>.
=== Horizontal composition ===
If <math>\eta: F \
:<math>(\epsilon * \eta)_X = \epsilon_{G(X)} \circ J(\eta_X) = K(\eta_X) \circ \epsilon_{F(X)}</math>.
By using whiskering (see below), we can write
:<math>(\epsilon * \eta)_X = (\epsilon G)_X \circ (J \eta)_X = (K \eta)_X \circ (\epsilon F)_X</math>,
hence
:<math>\epsilon * \eta = \epsilon G \circ J \eta = K \eta \circ \epsilon F</math>.
[[Image:Horizontal composition of natural transformations.svg|center|400px|alt=This is a commutative diagram generated using LaTeX. The left hand square shows the result of applying J to the commutative diagram for eta:F to G on f:X to Y. The right had side shows the commutative diagram for epsilon:J to K applied to G(f):G(X) to G(Y).]]
This horizontal composition of natural transformations is also associative with identity.
This identity is the identity natural transformation on the [[identity functor]], i.e., the natural transformation that associate to each object its [[identity morphism]]: for object <math>X</math> in category <math>C</math>, <math>(\mathrm{id}_{\mathrm{id}_C})_X = \mathrm{id}_{\mathrm{id}_C(X)} = \mathrm{id}_X</math>.
:For <math>\eta: F \Rightarrow G</math> with <math>F, G: C \to D</math>, <math>\mathrm{id}_{\mathrm{id}_D} * \eta = \eta = \eta * \mathrm{id}_{\mathrm{id}_C}</math>.
As identity functors <math>\mathrm{id}_C</math> and <math>\mathrm{id}_D</math> are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.<ref>{{cite web | url=https://bartoszmilewski.com/2015/04/07/natural-transformations/ | title=Natural Transformations | date=7 April 2015 }}</ref>
=== Whiskering ===
Whiskering is an [[external binary operation]] between a functor and a natural transformation.<ref>{{cite web | url=https://proofwiki.org/wiki/Definition:Whiskering | title=Definition:Whiskering - ProofWiki }}</ref><ref>{{cite web | url=https://ncatlab.org/nlab/show/whiskering | title=Whiskering in nLab }}</ref>
▲
If on the other hand <math>K: B \to C</math> is a functor, the natural transformation <math>\eta K:
:<math>(\eta K)_X = \eta_{K(X)}</math>.
It's also a horizontal composition where one of the natural transformations is the identity natural transformation:
▲If <math>\eta: F \to G</math> is a natural transformation between functors <math>F, G: C \to D</math>, and <math>H: D \to E</math> is another functor, then we can form the natural transformation <math>H(\eta): HF \to HG</math> by defining
▲
Note that <math>\mathrm{id}_H</math> (resp. <math>\mathrm{id}_K</math>) is generally not the left (resp. right) identity of horizontal composition <math>*</math> (<math>H \eta \neq \eta</math> and <math>\eta K \neq \eta</math> in general), except if <math>H</math> (resp. <math>K</math>) is the [[identity functor]] of the category <math>D</math> (resp. <math>C</math>).
=== Interchange law ===
▲:<math> (H \eta)_X = H \eta_X. </math>
{{Main|Interchange law}}
▲If on the other hand <math>K: B \to C</math> is a functor, the natural transformation <math>\eta K: FK \to GK</math> is defined by
The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations <math>\alpha, \alpha', \beta, \beta'</math> as shown on the image to the right, then the following identity holds:
:<math> (\beta' \circ \alpha') * (\beta \circ \alpha) = (\beta' * \beta) \circ (\alpha' * \alpha)</math>.
Vertical and horizontal compositions are also linked through identity natural transformations:
:for <math>F: C \to D</math> and <math>G: D \to E</math>, <math>\mathrm{id}_G * \mathrm{id}_F = \mathrm{id}_{G \circ F}</math>.<ref>https://arxiv.org/pdf/1612.09375v1.pdf, p. 38</ref>
As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of <math>\eta: F \Rightarrow G</math> and <math>\epsilon: J \Rightarrow K</math> without having to analyze components and the commutative diagram:
▲:<math> (\eta K)_X = \eta_{K(X)}. </math>
:<math>\begin{align}
\epsilon * \eta & = (\epsilon \circ \mathrm{id}_J) * (\mathrm{id}_G \circ \eta) = (\epsilon * \mathrm{id}_G) \circ (\mathrm{id}_J * \eta) = \epsilon G \circ J \eta \\
& = (\mathrm{id}_K \circ \epsilon) * (\eta \circ \mathrm{id}_F) = (\mathrm{id}_K * \eta) \circ (\epsilon * \mathrm{id}_F) = K \eta \circ \epsilon F
\end{align}</math>.
==Functor categories==
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If <math>C</math> is any category and <math>I</math> is a [[small category]], we can form the [[functor category]] <math>C^I</math> having as objects all functors from <math>I</math> to <math>C</math> and as morphisms the natural transformations between those functors. This forms a category since for any functor <math>F</math> there is an identity natural transformation <math>1_F: F \to F</math> (which assigns to every object <math>X</math> the identity morphism on <math>F(X)</math>) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.
The [[isomorphism]]s in <math>C^I</math> are precisely the natural isomorphisms. That is, a natural transformation <math>\eta: F \to G</math> is a natural isomorphism [[if and only if]] there exists a natural transformation <math>\epsilon: G \to F</math> such that <math>\eta\epsilon = 1_G</math> and <math>\epsilon\eta = 1_F</math>.
The functor category <math>C^I</math> is especially useful if <math>I</math> arises from a [[directed graph]]. For instance, if <math>I</math> is the category of the directed graph {{nobreak|1=• → •}}, then <math>C^I</math> has as objects the morphisms of <math>C</math>, and a morphism between <math>\phi: U \to V</math> and <math>\psi: X \to Y</math> in <math>C^I</math> is a pair of morphisms <math>f: U \to X</math> and <math>g: V \to Y</math> in <math>C</math> such that the "square commutes", i.e. <math>\psi \circ f = g \circ \phi</math>.
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=== More examples ===
Every [[Limit (category theory)|limit]] and colimit provides an example for a simple natural transformation, as a [[cone (category theory)|cone]] amounts to a natural transformation with the [[diagonal functor]] as ___domain.
==Yoneda lemma==
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*[[Universal property]]
*[[Higher category theory]]
*[[Modification (mathematics)]]
==Notes==
{{Reflist|group=lower-alpha}}
== References ==
{{reflist}}
{{refbegin}}
*{{citation| first = Saunders | last = Mac Lane |
* {{citation|first1=Saunders|last1=
* {{cite book | last = Awodey | first = Steve | title = Category theory | url = https://archive.org/details/categorytheoryse00awod | url-access = limited | publisher = Oxford University Press | ___location = Oxford New York | year = 2010 | isbn =
* {{cite book | last = Lane | first = Saunders | title = Sheaves in geometry and logic : a first introduction to topos theory | url = https://archive.org/details/sheavesgeometryl00macl_937 | url-access = limited | publisher = Springer-Verlag | ___location = New York | year = 1992 | isbn = 0387977104 | page = [https://archive.org/details/sheavesgeometryl00macl_937/page/n12 13] }}
{{refend}}
==External links==
* [http://ncatlab.org/nlab nLab], a wiki project on mathematics, physics and philosophy with emphasis on the ''n''-categorical point of view
* J. Adamek, H. Herrlich, G.
▲* J. Adamek, H. Herrlich, G. Stecker, [http://katmat.math.uni-bremen.de/acc/acc.pdf Abstract and Concrete Categories-The Joy of Cats]
* [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/category-theory/ Category Theory]"—by Jean-Pierre Marquis. Extensive bibliography.
* Baez, John, 1996,"[http://math.ucr.edu/home/baez/week73.html The Tale of ''n''-categories.]" An informal introduction to higher categories.
{{Category theory}}
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