category theory

# Contents

## Idea

Just as a functor is a morphism between categories, a natural transformations is a 2-morphism between two functors.

Natural transformations are the 2-morphisms in the 2-category Cat.

## Definition

### Explicit definition

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

$\begin{array}{c}\\ & ↗{↘}^{F}\\ C& {⇓}^{\alpha }& D\\ & ↘{↗}_{G}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \\ & \nearrow \searrow^F \\ C &\Downarrow^{\alpha}& D \\ & \searrow \nearrow_G } \,,

is an assignment to every object $x$ in $C$ of a morphism ${\alpha }_{x}:F\left(x\right)\to G\left(x\right)$ 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)$\begin{array}{ccc}F\left(x\right)& \stackrel{F\left(f\right)}{\to }& F\left(y\right)\\ {\alpha }_{x}↓& & ↓{\alpha }_{y}\\ G\left(x\right)& \stackrel{G\left(f\right)}{\to }& G\left(y\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ F(x) & \stackrel{F(f)}{\to} & F(y) \\ \alpha_x\downarrow && \downarrow \alpha_y \\ G(x) & \stackrel{G(f)}{\to} & G(y) } \,.

### Composition

Natural transformations between functors $C\to D$ and $D\to E$ compose in the obvious way to natural transformations $C\to E$ (this is their vertical composition in the 2-category Cat) and functors $F:C\to D$ with natural transformations between them form the functor category

$\left[C,D\right]\in \mathrm{Cat}$[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). This is discussed in a section below.

There is also a horizontal composition of natural transformations, which makes Cat a 2-category: the Godement product. See there for details.

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 $\alpha \left(m\right):F\left(x\right)\to G\left(y\right)$, in such a way as that $\alpha \left({m}_{0}{m}_{1}{m}_{2}\right)=G\left({m}_{0}\right)\alpha \left({m}_{1}\right)F\left({m}_{2}\right)$ 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 $\alpha \left(f\right)$, and each $\alpha \left({\mathrm{id}}_{x}\right)$ 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 $\left({\alpha }_{1}...{\alpha }_{n}\right)\left({m}_{1}...{m}_{n}\right)={\alpha }_{1}\left({m}_{1}\right)...{\alpha }_{n}\left({m}_{n}\right)$, 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 $\left[C,D\right]$ 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 $C×D$, whose objects and morphisms are simply pairs of objects and morphisms in $C$ and $D$: $\mathrm{Mor}\left(C×D\right)=\mathrm{Mor}\left(C\right)×\mathrm{Mor}\left(D\right)$.

It therefore makes sense to ask if there is for each category $C\in \mathrm{Cat}$ an internal hom functor $\left[C,-\right]:\mathrm{Cat}\to \mathrm{Cat}$ 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

$\mathrm{Funct}\left(A×C,B\right)\simeq \mathrm{Funct}\left(A,\left[C,B\right]\right)\phantom{\rule{thinmathspace}{0ex}}.$Funct(A \times C , B) \simeq Funct(A, [C,B]) \,.

This is precisely the case for $\left[C,B\right]$ 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 ${B}^{C}:=\left[B,C\right]$ for the functor category.

To derive from this the definition of natural transformations above, it is sufficient to consider the interval category $A:=I:=\left\{a\to b\right\}$. 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 $\left[C,B\right]$ is by checking what functors $I\to \left[C,D\right]$ are. But by the defining property of $\left[C,D\right]$ as an internal hom, such functors are in natural bijection to functors $I×C\to B$.

$\mathrm{Funct}\left(I,\left[C,B\right]\right)\simeq \mathrm{Funct}\left(I×C,B\right)\phantom{\rule{thinmathspace}{0ex}}.$Funct(I, [C,B]) \simeq Funct(I \times C, B) \,.

But, as mentioned above, we know what the category $I×C$ 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

$\begin{array}{rl}\left({c}_{1},a\right)\stackrel{\left(f,\left(a\to b\right)\right)}{\to }\left({c}_{2},b\right)& =\left({c}_{1},a\right)\stackrel{\left(f,\mathrm{Id}\right)}{\to }\left({c}_{2},a\right)\stackrel{\left(\mathrm{Id},\left(a\to b\right)}{\to }\left({c}_{2},b\right)\\ & =\left({c}_{1},a\right)\stackrel{\left(\mathrm{Id},\left(a\to b\right)\right)}{\to }\left({c}_{1},b\right)\stackrel{\left(f,\mathrm{Id}}{\to }\left({c}_{2},b\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}\left({c}_{1},a\right)& \stackrel{\left(f,\mathrm{Id}\right)}{\to }& \left({c}_{2},a\right)\\ {↓}^{\left(\mathrm{Id},\left(a\to b\right)\right)}& & {↓}^{\left(\mathrm{Id},\left(a\to b\right)\right)}\\ \left({c}_{1},b\right)& \stackrel{\left(f,\mathrm{Id}\right)}{\to }& \left({c}_{2},b\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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:

$\begin{array}{ccc}F\left({c}_{1}\right)& \stackrel{F\left(f\right)}{\to }& F\left({c}_{2}\right)\\ {\alpha }_{x}↓& & ↓{\alpha }_{y}\\ G\left({c}_{1}\right)& \stackrel{G\left(f\right)}{\to }& G\left({c}_{2}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\square D,{\circ }_{1},{\circ }_{2}\right)$ be the double category of commutative squares in $D$. Then the class of natural transformations of functors $C\to D$ can be described as $\mathrm{Cat}\left(C,\left(\square D,{\circ }_{1}\right)\right)$. But then ${\circ }_{2}$ induces a category structure on this and so we get $\mathrm{CAT}\left(C,D\right)$.

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 $C⇉D$, see extranatural transformation and dinatural transformation.

Revised on April 4, 2013 14:24:17 by Urs Schreiber (82.169.65.155)