Natural transformation explained

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 morphisms) of the categories 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 and consequently appear in the majority of its applications.

Definition

If

F

and

G

are functors between the categories

C

and

D

(both from

C

to

D

), then a natural transformation

η

from

F

to

G

is a family of morphisms that satisfies two requirements.
  1. The natural transformation must associate, to every object

X

in

C

, a morphism

ηX:F(X)\toG(X)

between objects of

D

. The morphism

ηX

is called the component of

η

at

X

.
  1. Components must be such that for every morphism

f:X\toY

in

C

we have:

ηY\circF(f)=G(f)\circηX

The last equation can conveniently be expressed by the commutative diagram

If both

F

and

G

are contravariant, the vertical arrows in the right diagram are reversed. If

η

is a natural transformation from

F

to

G

, we also write

η:F\toG

or

η:FG

. This is also expressed by saying the family of morphisms

ηX:F(X)\toG(X)

is natural in

X

.

If, for every object

X

in

C

, the morphism

ηX

is an isomorphism in

D

, then

η

is said to be a (or sometimes natural equivalence or isomorphism of functors). Two functors

F

and

G

are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from

F

to

G

.

An infranatural transformation

η

from

F

to

G

is simply a family of morphisms

ηX:F(X)\toG(X)

, for all

X

in

C

. Thus a natural transformation is an infranatural transformation for which

ηY\circF(f)=G(f)\circηX

for every morphism

f:X\toY

. The naturalizer of

η

, nat

(η)

, is the largest subcategory of

C

containing all the objects of

C

on which

η

restricts to a natural transformation.

Examples

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

bf{Grp}

of all groups with group homomorphisms as morphisms. If

(G,*)

is a group, we define its opposite group

(Gop,{*}op)

as follows:

Gop

is the same set as

G

, and the operation

*op

is defined by

a*opb=b*a

. All multiplications in

Gop

are thus "turned around". Forming the opposite group becomes a (covariant) functor from

bf{Grp}

to

bf{Grp}

if we define

fop=f

for any group homomorphism

f:G\toH

. Note that

fop

is indeed a group homomorphism from

Gop

to

Hop

:

fop(a*opb)=f(b*a)=f(b)*f(a)=fop(a)*opfop(b).

The content of the above statement is:

"The identity functor

Idbf{Grp

}: \textbf \to \textbf is naturally isomorphic to the opposite functor

{op

}: \textbf \to \textbf"To prove this, we need to provide isomorphisms

ηG:G\toGop

for every group

G

, such that the above diagram commutes. Set

ηG(a)=a-1

.The formulas

(a*b)-1=b-1*a-1=a-1*opb-1

and

(a-1)-1=a

show that

ηG

is a group homomorphism with inverse
η
Gop
. To prove the naturality, we start with a group homomorphism

f:G\toH

and show

ηH\circf=fop\circηG

, i.e.

(f(a))-1=fop(a-1)

for all

a

in

G

. This is true since

fop=f

and every group homomorphism has the property

(f(a))-1=f(a-1)

.

Modules

Let

\varphi:M\longrightarrowM\prime

be an

R

-module homomorphism of right modules. For every left module

N

there is a natural map

\varphiN:MRN\longrightarrowM\primeRN

, form a natural transformation

η:MR-\impliesM'R-

. For every right module

N

there is a natural map

ηN:HomR(M',N)\longrightarrowHomR(M,N)

defined by

ηN(f)=f\varphi

, form a natural transformation

η:HomR(M',-)\impliesHomR(M,-)

.

Abelianization

Given a group

G

, we can define its abelianization

Gab=G/

[G,G]

. Let

\piG:G\toGab

denote the projection map onto the cosets of

[G,G]

. This homomorphism is "natural in

G

", i.e., it defines a natural transformation, which we now check. Let

H

be a group. For any homomorphism

f:G\toH

, we have that

[G,G]

is contained in the kernel of

\piH\circf

, because any homomorphism into an abelian group kills the commutator subgroup. Then

\piH\circf

factors through

Gab

as

fab\circ\piG=\piH\circf

for the unique homomorphism

fab:Gab\toHab

. This makes

{ab

} : \textbf \to \textbf a functor and

\pi

a natural transformation, but not a natural isomorphism, from the identity functor to

ab

.

Hurewicz homomorphism

(X,x)

and positive integer

n

there exists a group homomorphism

hn\colon\pin(X,x)\toHn(X)

from the

n

-th homotopy group of

(X,x)

to the

n

-th homology group of

X

. Both

\pin

and

Hn

are functors from the category Top* of pointed topological spaces to the category Grp of groups, and

hn

is a natural transformation from

\pin

to

Hn

.

Determinant

R

and

S

with a ring homomorphism

f:R\toS

, the respective groups of invertible

n x n

matrices

GLn(R)

and

GLn(S)

inherit a homomorphism which we denote by

GLn(f)

, obtained by applying

f

to each matrix entry. Similarly,

f

restricts to a group homomorphism

f*:R*\toS*

, where

R*

denotes the group of units of

R

. In fact,

GLn

and

*

are functors from the category of commutative rings

bf{CRing}

to

bf{Grp}

. The determinant on the group

GLn(R)

, denoted by

detR

, is a group homomorphism

detR\colonGLn(R)\toR*

which is natural in

R

: because the determinant is defined by the same formula for every ring,

f*\circdetR=detS\circGLn(f)

holds. This makes the determinant a natural transformation from

GLn

to

*

.

Double dual of a vector space

For example, if

K

is a field, then for every vector space

V

over

K

we have a "natural" injective linear map

V\toV**

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

G

, the set

Hombf{Set}(Z,U(G))

of functions from the integers to the underlying set of

G

forms an abelian group

VZ(G)

under pointwise addition. (Here

U

is the standard forgetful functor

U:bf{Ab}\tobf{Set}

.) Given an

bf{Ab}

morphism

\varphi:G\toG'

, the map

VZ(\varphi):VZ(G)\toVZ(G')

given by left composing

\varphi

with the elements of the former is itself a homomorphism of abelian groups; in this way we obtain a functor

VZ:bf{Ab}\tobf{Ab}

. The finite difference operator

\DeltaG

taking each function

f:Z\toU(G)

to

\Delta(f):n\mapstof(n+1)-f(n)

is a map from

VZ(G)

to itself, and the collection

\Delta

of such maps gives a natural transformation

\Delta:VZ\toVZ

.

Tensor-hom adjunction

Consider the category

bf{Ab}

of abelian groups and group homomorphisms. For all abelian groups

X

,

Y

and

Z

we have a group isomorphism

Hom(XY,Z)\toHom(X,Hom(Y,Z))

.These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors

bf{Ab}op x bf{Ab}op x bf{Ab}\tobf{Ab}

.(Here "op" is the opposite category of

bf{Ab}

, not to be confused with the trivial opposite group functor on

bf{Ab}

!)

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.

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 "an isomorphism that is not natural") if the map cannot be extended to a natural transformation on the entire category. Given an object

X,

a functor

G

(taking for simplicity the first functor to be the identity) and an isomorphism

η\colonX\toG(X),

proof of unnaturality is most easily shown by giving an automorphism

A\colonX\toX

that does not commute with this isomorphism (so

η\circAG(A)\circη

). More strongly, if one wishes to prove that

X

and

G(X)

are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism

η

, there is some

A

with which it does not commute; in some cases a single automorphism

A

works for all candidate isomorphisms

η

while in other cases one must show how to construct a different

Aη

for each isomorphism. The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance.

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 for example.

Some authors distinguish notationally, using

\cong

for a natural isomorphism and

for an unnatural isomorphism, reserving

=

for equality (usually equality of maps).

Example: fundamental group of torus

As an example of the distinction between the functorial statement and individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus.

The homotopy groups of a product space are naturally the product of the homotopy groups of the components,

\pin((X,x0) x (Y,y0))\cong\pin((X,x0)) x \pin((Y,y0)),

with the isomorphism given by projection onto the two factors, fundamentally because maps into a product space are exactly products of maps into the components – this is a functorial statement.

However, the torus (which is abstractly a product of two circles) has fundamental group isomorphic to

Z2

, but the splitting

\pi1(T,t0)Z x Z

is not natural. Note the use of

,

\cong

, and

=

:

\pi1(T,t0)

1,x
\pi
0)

x

1,y
\pi
0)

\congZ x Z=Z2.

This abstract isomorphism with a product is not natural, as some isomorphisms of

T

do not preserve the product: the self-homeomorphism of

T

(thought of as the quotient space

R2/Z2

) given by

\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)

(geometrically a Dehn twist about one of the generating curves) acts as this matrix on

Z2

(it's in the general linear group

GL(Z,2)

of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product

(T,t0)=

1,x
(S
0)

x

1,y
(S
0)
– equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components".

Naturality is a categorical notion, and requires being very precise about exactly what data is given – the torus as a space that happens to be a product (in the category of spaces and continuous maps) is different from the torus presented as a product (in the category of products of two spaces and continuous maps between the respective components).

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. However, related categories (with additional structure and restrictions on the maps) do have a natural isomorphism, as described below.

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 for detailed discussion.

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,

ηV\colonV\toV*

. In other words, take as objects vector spaces with a nondegenerate bilinear form

bV\colonV x V\toK

. This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps

T\colonV\toU

that commute with the isomorphisms:
*(η
T
U

(T(v)))=ηV(v)

or in other words, preserve the bilinear form:

bU(T(v),T(w))=bV(v,w)

. (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.

In this category (finite-dimensional vector spaces with a nondegenerate bilinear form, maps linear transforms that respect the bilinear form), the dual of a map between vector spaces can be identified as a transpose. Often for reasons of geometric interest this is specialized to a subcategory, by requiring that the nondegenerate bilinear forms have additional properties, such as being symmetric (orthogonal matrices), symmetric and positive definite (inner product space), symmetric sesquilinear (Hermitian spaces), skew-symmetric and totally isotropic (symplectic vector space), etc. – in all these categories a vector space is naturally identified with its dual, by the nondegenerate bilinear form.

Operations with natural transformations

Vertical composition

If

η:FG

and

\epsilon:GH

are natural transformations between functors

F,G,H:C\toD

, then we can compose them to get a natural transformation

\epsilon\circη:FH

. This is done componentwise:

(\epsilon\circη)X=\epsilonX\circηX

.

This vertical composition of natural transformations is associative and has an identity, and allows one to consider the collection of all functors

C\toD

itself as a category (see below under Functor categories).The identity natural transformation

idF

on functor

F

has components

(idF)X=idF(X)

.[1]

For

η:FG

,

idG\circη=η=η\circidF

.

Horizontal composition

If

η:FG

is a natural transformation between functors

F,G:C\toD

and

\epsilon:JK

is a natural transformation between functors

J,K:D\toE

, then the composition of functors allows a composition of natural transformations

\epsilon*η:J\circFK\circG

with components

(\epsilon*η)X=\epsilonG(X)\circJ(ηX)=K(ηX)\circ\epsilonF(X)

.By using whiskering (see below), we can write

(\epsilon*η)X=(\epsilonG)X\circ(Jη)X=(Kη)X\circ(\epsilonF)X

,hence

\epsilon*η=\epsilonG\circJη=Kη\circ\epsilonF

.

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

X

in category

C

,
(id
idC

)X=

id
idC(X)

=idX

.

For

η:FG

with

F,G:C\toD

,
id
idD

*η=η=η*

id
idC
.As identity functors

idC

and

idD

are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.[2]

Whiskering

Whiskering is an external binary operation between a functor and a natural transformation.[3] [4]

If

η:FG

is a natural transformation between functors

F,G:C\toD

, and

H:D\toE

is another functor, then we can form the natural transformation

Hη:H\circFH\circG

by defining

(Hη)X=H(ηX)

.If on the other hand

K:B\toC

is a functor, the natural transformation

ηK:F\circKG\circK

is defined by

(ηK)X=ηK(X)

.

It's also an horizontal composition where one of the natural transformations is the identity natural transformation:

Hη=idH*η

and

ηK=η*idK

.Note that

idH

(resp.

idK

) is generally not the left (resp. right) identity of horizontal composition

*

(

Hηη

and

ηKη

in general), except if

H

(resp.

K

) is the identity functor of the category

D

(resp.

C

).

Interchange law

The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations

\alpha,\alpha',\beta,\beta'

as shown on the image to the right, then the following identity holds:

(\beta'\circ\alpha')*(\beta\circ\alpha)=(\beta'*\beta)\circ(\alpha'*\alpha)

.Vertical and horizontal compositions are also linked through identity natural transformations:

for

F:C\toD

and

G:D\toE

,

idG*idF=idG

.[5]

As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of

η:FG

and

\epsilon:JK

without having to analyze components and the commutative diagram:

\begin{align} \epsilon*η&=(\epsilon\circidJ)*(idG\circη)=(\epsilon*idG)\circ(idJ*η)=\epsilonG\circJη\\ &=(idK\circ\epsilon)*(η\circidF)=(idK*η)\circ(\epsilon*idF)=Kη\circ\epsilonF \end{align}

.

Functor categories

See main article: Functor category. If

C

is any category and

I

is a small category, we can form the functor category

CI

having as objects all functors from

I

to

C

and as morphisms the natural transformations between those functors. This forms a category since for any functor

F

there is an identity natural transformation

1F:F\toF

(which assigns to every object

X

the identity morphism on

F(X)

) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.

The isomorphisms in

CI

are precisely the natural isomorphisms. That is, a natural transformation

η:F\toG

is a natural isomorphism if and only if there exists a natural transformation

\epsilon:G\toF

such that

η\epsilon=1G

and

\epsilonη=1F

.

The functor category

CI

is especially useful if

I

arises from a directed graph. For instance, if

I

is the category of the directed graph, then

CI

has as objects the morphisms of

C

, and a morphism between

\phi:U\toV

and

\psi:X\toY

in

CI

is a pair of morphisms

f:U\toX

and

g:V\toY

in

C

such that the "square commutes", i.e.

\psi\circf=g\circ\phi

.

bf{Cat}

whose

C

and

D

are the functors from

C

to

D

,

F:C\toD

and

G:C\toD

are the natural transformations from

F

to

G

.The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category

CI

is then simply a hom-category in this category (smallness issues aside).

More examples

Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category.

Yoneda lemma

See main article: Yoneda lemma. If

X

is an object of a locally small category

C

, then the assignment

Y\mapstoHomC(X,Y)

defines a covariant functor

FX:C\tobf{Set}

. This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of

X

). The natural transformations from a representable functor to an arbitrary functor

F:C\tobf{Set}

are completely known and easy to describe; this is the content of the Yoneda lemma.

Historical notes

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly would be isomorphic to those of the singular theory. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.

See also

References

External links

Notes and References

  1. Web site: Identity natural transformation in nLab .
  2. Web site: Natural Transformations . 7 April 2015 .
  3. Web site: Definition:Whiskering - ProofWiki .
  4. Web site: Whiskering in nLab .
  5. https://arxiv.org/pdf/1612.09375v1.pdf, p. 38