nLab
natural transformation

Idea

Just as functors is a morphism between categories, a natural transformations is a morphism between two functors.

Definition

Given categories $C$ and $D$ and functors $F,G:C\to D,$ a natural transformation $\alpha :F\Rightarrow G$, denoted

$$\begin{array}{c}\\ & \nearrow {\searrow }^F\\ C& {\Downarrow }^{\alpha }& D\\ & \searrow {\nearrow }_G\end{array},$$\array{ \\ & \nearrow \searrow^F \\ C &\Downarrow^{\alpha}& D \\ & \searrow \nearrow_G } \,,

assigns to every object $x$ in $C$ a morphism ${\alpha }_x:F(x)\to G(x)$ in $D$ (called the component of $\alpha$ at $x$) such that for any morphism $f:x\to y$ in $C$, the following diagram commutes in $D$:

(1)$$\begin{array}{ccc}F(x)& \stackrel{F(f)}{\to }& F(y)\\ {\alpha }_x\downarrow & & \downarrow {\alpha }_y\\ G(x)& \stackrel{G(f)}{\to }& G(y)\end{array}.$$\array{ F(x) & \stackrel{F(f)}{\to} & F(y) \\ \alpha_x\downarrow && \downarrow \alpha_y \\ G(x) & \stackrel{G(f)}{\to} & G(y) } \,.

Natural transformations between functors $C\to D$ compose in the obvious way and functors $F:C\to D$ with natural transformations between them form the functor category

$$[C,D]\in \mathrm{Cat}$$[C,D] \in Cat

The notation alludes to the fact that this makes Cat a closed monoidal category. Since $\mathrm{Cat}$ is in fact a cartesian closed category, another common notation is $D^C$. In fact, if we want $\mathrm{Cat}$ to be cartesian closed, the definition of natural transformation is forced (since an adjoint functor is unique).

There is also a horizontal composition of natural transformations, which makes Cat a 2-category. In fact, Cat is a 2-category (a $\mathrm{Cat}$-enriched category) because it is (cartesian) closed: closed monoidal categories are automatically enriched over themselves, via their internal hom.

An alternative but ultimately equivalent way to define a natural transformation $\alpha :F\to G$ is as an assignment to every morphism $m:x\to y$ in $C$ of a morphism $\alpha (m):F(x)\to G(y)$, in such a way as that $\alpha (m_0{m}_1{m}_2)=G(m_0)\alpha (m_1)F(m_2)$ for every ternary composition $m_0{m}_1{m}_2$ in $C$. The relation of this to the previous definition is that the commutative squares in the previous definition for any morphism $f$ give the value $\alpha (f)$, and each $\alpha ({\mathrm{id}}_x)$ gives the component ${\alpha }_x$. Composition of natural transformations can be specified directly in terms of this account as well: specifically, an $n$-ary composition ${\alpha }_1...{\alpha }_n$ of natural transformations is uniquely determined by the property that $({\alpha }_1...{\alpha }_n)(m_1...m_n)={\alpha }_1(m_1)...{\alpha }_n(m_n)$, for every $n$-ary composition $m_1...m_n$ in $C$.

in terms of the cartesian closed monoidal structure on $\mathrm{Cat}$

The definition of the functor category $[C,D]$ with morphisms being natural transformations is precisely the one that makes $\mathrm{Cat}$ a cartesian closed monoidal category.

The category Cat of all categories (regarded for the moment just as an ordinary 1-category) is a cartesian monoidal category: for every two categories $C$ and $D$ there is the cartesian product category $C\times D$, whose objects and morphisms are simply pairs of objects and morphisms in $C$ and $D$: $\mathrm{Mor}(C\times D)=\mathrm{Mor}(C)\times \mathrm{Mor}(D)$.

It therefore makes sense to ask if there is for each category $C\in \mathrm{Cat}$ an internal hom functor $[C,-]:\mathrm{Cat}\to \mathrm{Cat}$ that would make Cat into a closed monoidal category in that for $A,B,C\in \mathrm{Cat}$ we have natural isomorphisms of sets of functors

$$\mathrm{Funct}(A\times C,B)\simeq \mathrm{Funct}(A,[C,B]).$$Funct(A \times C , B) \simeq Funct(A, [C,B]) \,.

This is precisely the case for $[C,B]$ being the functor category with functors $C\to B$ as objects and natural transformations, as defined above, as morphisms.

Since $\mathrm{Cat}$ here is cartesian closed, one often uses the exponential notation $B^C:=[B,C]$ for the functor category.

To derive from this the definition of natural transformations above, it is sufficient to consider the interval category $A:=I:=\{a\to b\}$. For any category $E$, a functor $I\to E$ is precisely a choice of morphism in $E$. This means that we can check what a morphism in the internal hom category $[C,B]$ is by checking what functors $I\to [C,D]$ are. But by the defining property of $[C,D]$ as an internal hom, such functors are in natural bijection to functors $I\times C\to B$.

$$\mathrm{Funct}(I,[C,B])\simeq \mathrm{Funct}(I\times C,B).$$Funct(I, [C,B]) \simeq Funct(I \times C, B) \,.

But, as mentioned above, we kown what the category $I\times C$ is like: its morphisms are pairs of morphisms in $I$ and $C$, subject to the obvious composition law, which says in particular that for $f:c_1\to c_2$ any morphism in $C$ we have

$$\begin{array}{rl}(c_1,a)\stackrel{(f,(a\to b))}{\to }(c_2,b)& =(c_1,a)\stackrel{(f,\mathrm{Id})}{\to }(c_2,a)\stackrel{(\mathrm{Id},(a\to b)}{\to }(c_2,b)\\ & =(c_1,a)\stackrel{(\mathrm{Id},(a\to b))}{\to }(c_1,b)\stackrel{(f,\mathrm{Id}}{\to }(c_2,b)\end{array}.$$\begin{aligned} (c_1,a) \stackrel{(f,(a \to b))}{\to} (c_2,b) & = (c_1,a) \stackrel{(f, Id)}{\to} (c_2,a) \stackrel{(Id, (a \to b)}{\to} (c_2, b) \\ &= (c_1,a) \stackrel{(Id, (a\to b))}{\to} (c_1,b) \stackrel{(f,Id}{\to} (c_2, b) \end{aligned} \,.

Here the right side is more conveniently depicted as a commuting square

$$\begin{array}{ccc}(c_1,a)& \stackrel{(f,\mathrm{Id})}{\to }& (c_2,a)\\ {\downarrow }^{(\mathrm{Id},(a\to b))}& & {\downarrow }^{(\mathrm{Id},(a\to b))}\\ (c_1,b)& \stackrel{(f,\mathrm{Id})}{\to }& (c_2,b)\end{array}.$$\array{ (c_1,a) &\stackrel{(f,Id)}{\to}& (c_2,a) \\ \downarrow^{\mathrlap{(Id,(a \to b))}} && \downarrow^{\mathrlap{(Id, (a \to b))}} \\ (c_1,b) &\stackrel{(f,Id)}{\to}& (c_2,b) } \,.

So a natural transformation between functors $C\to D$ is given by the images of such squares in $D$. By tracing back the way the hom-isomorphism works, one finds that the image of such a square in $D$ for a natural transformation $\alpha :F\to G$ is the naturality square from above:

$$\begin{array}{ccc}F(c_1)& \stackrel{F(f)}{\to }& F(c_2)\\ {\alpha }_x\downarrow & & \downarrow {\alpha }_y\\ G(c_1)& \stackrel{G(f)}{\to }& G(c_2)\end{array}.$$\array{ F(c_1) & \stackrel{F(f)}{\to} & F(c_2) \\ \alpha_x\downarrow && \downarrow \alpha_y \\ G(c_1) & \stackrel{G(f)}{\to} & G(c_2) } \,.

In terms of double categories

There is a nice way of describing these structures due to Charles Ehresmann. For a category $D$ let $(\square D,{\circ }_1,{\circ }_2)$ be the double category of commutative squares in $D$. Then the class of natural transformations of functors $C\to D$ can be described as $\mathrm{Cat}(C,(\square D,{\circ }_1))$. But then ${\circ }_2$ induces a category structure on this and so we get $\mathrm{CAT}(C,D)$.

An advantage of this approach is that it applies to the case of topological categories and groupoids (working in a convenient category of spaces).

An analogous approach works for strict cubical $\omega$-categories with connections, using the good properties of cubes, so leading to a monoidal closed structure for these objects. This yields by an equivalence of categories a monoidal closed structure on strict globular omega-categories, where the tensor product is the Crans-Gray tensor product.

Variations

For functors between higher categories, see lax natural transformation etc.

A transformation which is natural only relative to isomorphisms may be called a canonical transformation.

For functors with more complicated shapes than $C\rightrightarrows D$, see extranatural transformation and dinatural transformation.

Discussion

• See natural transformation (discussion) for an informal discussion about natural transformations.