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 ${\displaystyle {\textbf {Grp}}}$ of all groups with group homomorphisms as morphisms. If ${\displaystyle (G,*)}$ is a group, we define its opposite group ${\displaystyle (G^{\text{op}},{*}^{\text{op}})}$ as follows: ${\displaystyle G^{\text{op}}}$ is the same set as ${\displaystyle G}$, and the operation ${\displaystyle *^{\text{op}}}$ is defined by ${\displaystyle a*^{\text{op}}b=b*a}$. All multiplications in ${\displaystyle G^{\text{op}}}$ are thus "turned around". Forming the opposite group becomes a (covariant) functor from ${\displaystyle {\textbf {Grp}}}$ to ${\displaystyle {\textbf {Grp}}}$ if we define ${\displaystyle f^{\text{op}}=f}$ for any group homomorphism ${\displaystyle f:G\to H}$. Note that ${\displaystyle f^{\text{op}}}$ is indeed a group homomorphism from ${\displaystyle G^{\text{op}}}$ to ${\displaystyle H^{\text{op}}}$:

${\displaystyle f^{\text{op}}(a*^{\text{op}}b)=f(b*a)=f(b)*f(a)=f^{\text{op}}(a)*^{\text{op}}f^{\text{op}}(b).}$

The content of the above statement is:

"The identity functor ${\displaystyle {\text{Id}}_{\textbf {Grp}}:{\textbf {Grp}}\to {\textbf {Grp}}}$ is naturally isomorphic to the opposite functor ${\displaystyle {\text{op}}:{\textbf {Grp}}\to {\textbf {Grp}}}$"

To prove this, we need to provide isomorphisms ${\displaystyle \eta _{G}:G\to G^{\text{op}}}$ for every group ${\displaystyle G}$, such that the above diagram commutes. Set ${\displaystyle \eta _{G}(a)=a^{-1}}$. The formulas ${\displaystyle (a*b)^{-1}=b^{-1}*a^{-1}=a^{-1}*^{\text{op}}b^{-1}}$ and ${\displaystyle (a^{-1})^{-1}=a}$ show that ${\displaystyle \eta _{G}}$ is a group homomorphism with inverse ${\displaystyle \eta _{G^{\text{op}}}}$. To prove the naturality, we start with a group homomorphism ${\displaystyle f:G\to H}$ and show ${\displaystyle \eta _{H}\circ f=f^{\text{op}}\circ \eta _{G}}$, i.e. ${\displaystyle (f(a))^{-1}=f^{\text{op}}(a^{-1})}$ for all ${\displaystyle a}$ in ${\displaystyle G}$. This is true since ${\displaystyle f^{\text{op}}=f}$ and every group homomorphism has the property ${\displaystyle (f(a))^{-1}=f(a^{-1})}$.

Modules

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

Abelianization

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

Hurewicz homomorphism

Functors and natural transformations abound in algebraic topology, with the Hurewicz homomorphisms serving as examples. For any pointed topological space ${\displaystyle (X,x)}$ and positive integer ${\displaystyle n}$ there exists a group homomorphism

${\displaystyle h_{n}\colon \pi _{n}(X,x)\to H_{n}(X)}$

from the ${\displaystyle n}$-th homotopy group of ${\displaystyle (X,x)}$ to the ${\displaystyle n}$-th homology group of ${\displaystyle X}$. Both ${\displaystyle \pi _{n}}$ and ${\displaystyle H_{n}}$ are functors from the category Top^* of pointed topological spaces to the category Grp of groups, and ${\displaystyle h_{n}}$ is a natural transformation from ${\displaystyle \pi _{n}}$ to ${\displaystyle H_{n}}$.

Determinant

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

${\displaystyle {\mbox{det}}_{R}\colon {\mbox{GL}}_{n}(R)\to R^{*}}$

which is natural in ${\displaystyle R}$: because the determinant is defined by the same formula for every ring, ${\displaystyle f^{*}\circ {\text{det}}_{R}={\text{det}}_{S}\circ {\text{GL}}_{n}(f)}$ holds. This makes the determinant a natural transformation from ${\displaystyle {\text{GL}}_{n}}$ to ${\displaystyle R^{*}}$.

Double dual of a vector space

Given an endofunctor, that is, a functor ${\displaystyle F}$ that maps a category to that same category, then there is a natural transformation ${\displaystyle \alpha }$ from the identity functor to that functor, with components ${\displaystyle \alpha _{C}=F(C)}$.

For example, if ${\displaystyle K}$ is a field, then for every vector space ${\displaystyle V}$ over ${\displaystyle K}$ we have a "natural" injective linear map ${\displaystyle V\to V^{**}}$ 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 ${\displaystyle G}$, the set ${\displaystyle {\text{Hom}}_{\textbf {Set}}(\mathbb {Z} ,U(G))}$ of functions from the integers to the underlying set of ${\displaystyle G}$ forms an abelian group ${\displaystyle V_{\mathbb {Z} }(G)}$ under pointwise addition. (Here ${\displaystyle U}$ is the standard forgetful functor ${\displaystyle U:{\textbf {Ab}}\to {\textbf {Set}}}$.) Given an ${\displaystyle {\textbf {Ab}}}$ morphism ${\displaystyle \varphi :G\to G'}$, the map ${\displaystyle V_{\mathbb {Z} }(\varphi ):V_{\mathbb {Z} }(G)\to V_{\mathbb {Z} }(G')}$ given by left composing ${\displaystyle \varphi }$ with the elements of the former is itself a homomorphism of abelian groups; in this way we obtain a functor ${\displaystyle V_{\mathbb {Z} }:{\textbf {Ab}}\to {\textbf {Ab}}}$. The finite difference operator ${\displaystyle \Delta _{G}}$ taking each function ${\displaystyle f:\mathbb {Z} \to U(G)}$ to ${\displaystyle \Delta (f):n\mapsto f(n+1)-f(n)}$ is a map from ${\displaystyle V_{\mathbb {Z} }(G)}$ to itself, and the collection ${\displaystyle \Delta }$ of such maps gives a natural transformation ${\displaystyle \Delta :V_{\mathbb {Z} }\to V_{\mathbb {Z} }}$.

Tensor-hom adjunction

Consider the category ${\displaystyle {\textbf {Ab}}}$ of abelian groups and group homomorphisms. For all abelian groups ${\displaystyle X}$, ${\displaystyle Y}$ and ${\displaystyle Z}$ we have a group isomorphism

${\displaystyle {\text{Hom}}(X\otimes Y,Z)\to {\text{Hom}}(X,{\text{Hom}}(Y,Z))}$.

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors ${\displaystyle {\textbf {Ab}}^{\text{op}}\times {\textbf {Ab}}^{\text{op}}\times {\textbf {Ab}}\to {\textbf {Ab}}}$. (Here "op" is the opposite category of ${\displaystyle {\textbf {Ab}}}$, not to be confused with the trivial opposite group functor on ${\displaystyle {\textbf {Ab}}}$ !)

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.

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, ${\displaystyle \pi _{n}((X,x_{0})\times (Y,y_{0}))\cong \pi _{n}((X,x_{0}))\times \pi _{n}((Y,y_{0})),}$ 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 ${\displaystyle Z^{2}}$, but the splitting ${\displaystyle \pi _{1}(T,t_{0})\approx \mathbf {Z} \times \mathbf {Z} }$ is not natural. Note the use of ${\displaystyle \approx }$, ${\displaystyle \cong }$, and ${\displaystyle =}$:^{[lower-alpha 1]}

${\displaystyle \pi _{1}(T,t_{0})\approx \pi _{1}(S^{1},x_{0})\times \pi _{1}(S^{1},y_{0})\cong \mathbf {Z} \times \mathbf {Z} =\mathbf {Z} ^{2}.}$

This abstract isomorphism with a product is not natural, as some isomorphisms of ${\displaystyle T}$ do not preserve the product: the self-homeomorphism of ${\displaystyle T}$ (thought of as the quotient space ${\displaystyle R^{2}/\mathbb {Z} ^{2}}$) given by ${\displaystyle \left({\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}\right)}$ (geometrically a Dehn twist about one of the generating curves) acts as this matrix on ${\displaystyle \mathbb {Z} ^{2}}$ (it's in the general linear group ${\displaystyle {\text{GL}}(\mathbb {Z} ,2)}$ 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 ${\displaystyle (T,t_{0})=(S^{1},x_{0})\times (S^{1},y_{0})}$ – 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.^[1] 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 (Mac Lane & Birkhoff 1999, §VI.4) 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, ${\displaystyle \eta _{V}\colon V\to V^{*}}$. In other words, take as objects vector spaces with a nondegenerate bilinear form ${\displaystyle b_{V}\colon V\times V\to K}$. This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps ${\displaystyle T\colon V\to U}$ that commute with the isomorphisms: ${\displaystyle T^{*}(\eta _{U}(T(v)))=\eta _{V}(v)}$ or in other words, preserve the bilinear form: ${\displaystyle b_{U}(T(v),T(w))=b_{V}(v,w)}$. (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.

Vertical composition

If ${\displaystyle \eta :F\Rightarrow G}$ and ${\displaystyle \epsilon :G\Rightarrow H}$ are natural transformations between functors ${\displaystyle F,G,H:C\to D}$, then we can compose them to get a natural transformation ${\displaystyle \epsilon \circ \eta :F\Rightarrow H}$. This is done componentwise:

${\displaystyle (\epsilon \circ \eta )_{X}=\epsilon _{X}\circ \eta _{X}}$.

This vertical composition of natural transformations is associative and has an identity, and allows one to consider the collection of all functors ${\displaystyle C\to D}$ itself as a category (see below under Functor categories). The identity natural transformation ${\displaystyle \mathrm {id} _{F}}$ on functor ${\displaystyle F}$ has components ${\displaystyle (\mathrm {id} _{F})_{X}=\mathrm {id} _{F(X)}}$.^[2]

For ${\displaystyle \eta :F\Rightarrow G}$, ${\displaystyle \mathrm {id} _{G}\circ \eta =\eta =\eta \circ \mathrm {id} _{F}}$.

Horizontal composition

If ${\displaystyle \eta :F\Rightarrow G}$ is a natural transformation between functors ${\displaystyle F,G:C\to D}$ and ${\displaystyle \epsilon :J\Rightarrow K}$ is a natural transformation between functors ${\displaystyle J,K:D\to E}$, then the composition of functors allows a composition of natural transformations ${\displaystyle \epsilon *\eta :J\circ F\Rightarrow K\circ G}$ with components

${\displaystyle (\epsilon *\eta )_{X}=\epsilon _{G(X)}\circ J(\eta _{X})=K(\eta _{X})\circ \epsilon _{F(X)}}$.

By using whiskering (see below), we can write

${\displaystyle (\epsilon *\eta )_{X}=(\epsilon G)_{X}\circ (J\eta )_{X}=(K\eta )_{X}\circ (\epsilon F)_{X}}$,

hence

${\displaystyle \epsilon *\eta =\epsilon G\circ J\eta =K\eta \circ \epsilon F}$.

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 ${\displaystyle X}$ in category ${\displaystyle C}$, ${\displaystyle (\mathrm {id} _{\mathrm {id} _{C}})_{X}=\mathrm {id} _{\mathrm {id} _{C}(X)}=\mathrm {id} _{X}}$.

For ${\displaystyle \eta :F\Rightarrow G}$ with ${\displaystyle F,G:C\to D}$, ${\displaystyle \mathrm {id} _{\mathrm {id} _{D}}*\eta =\eta =\eta *\mathrm {id} _{\mathrm {id} _{C}}}$.

As identity functors ${\displaystyle \mathrm {id} _{C}}$ and ${\displaystyle \mathrm {id} _{D}}$ are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.^[3]

Whiskering

Whiskering is an external binary operation between a functor and a natural transformation.^[4][5]

If ${\displaystyle \eta :F\Rightarrow G}$ is a natural transformation between functors ${\displaystyle F,G:C\to D}$, and ${\displaystyle H:D\to E}$ is another functor, then we can form the natural transformation ${\displaystyle H\eta :H\circ F\Rightarrow H\circ G}$ by defining

${\displaystyle (H\eta )_{X}=H(\eta _{X})}$.

If on the other hand ${\displaystyle K:B\to C}$ is a functor, the natural transformation ${\displaystyle \eta K:F\circ K\Rightarrow G\circ K}$ is defined by

${\displaystyle (\eta K)_{X}=\eta _{K(X)}}$.

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

${\displaystyle H\eta =\mathrm {id} _{H}*\eta }$ and ${\displaystyle \eta K=\eta *\mathrm {id} _{K}}$.

Note that ${\displaystyle \mathrm {id} _{H}}$ (resp. ${\displaystyle \mathrm {id} _{K}}$) is generally not the left (resp. right) identity of horizontal composition ${\displaystyle *}$ (${\displaystyle H\eta \neq \eta }$ and ${\displaystyle \eta K\neq \eta }$ in general), except if ${\displaystyle H}$ (resp. ${\displaystyle K}$) is the identity functor of the category ${\displaystyle D}$ (resp. ${\displaystyle C}$).

Interchange law

The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations ${\displaystyle \alpha ,\alpha ',\beta ,\beta '}$ as shown on the image to the right, then the following identity holds:

${\displaystyle (\beta '\circ \alpha ')*(\beta \circ \alpha )=(\beta '*\beta )\circ (\alpha '*\alpha )}$.

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

for ${\displaystyle F:C\to D}$ and ${\displaystyle G:D\to E}$, ${\displaystyle \mathrm {id} _{G}*\mathrm {id} _{F}=\mathrm {id} _{G\circ F}}$.^[6]

As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of ${\displaystyle \eta :F\Rightarrow G}$ and ${\displaystyle \epsilon :J\Rightarrow K}$ without having to analyze components and the commutative diagram:

${\displaystyle {\begin{aligned}\epsilon *\eta &=(\epsilon \circ \mathrm {id} _{J})*(\mathrm {id} _{G}\circ \eta )=(\epsilon *\mathrm {id} _{G})\circ (\mathrm {id} _{J}*\eta )=\epsilon G\circ J\eta \\&=(\mathrm {id} _{K}\circ \epsilon )*(\eta \circ \mathrm {id} _{F})=(\mathrm {id} _{K}*\eta )\circ (\epsilon *\mathrm {id} _{F})=K\eta \circ \epsilon F\end{aligned}}}$.

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.