Contents

category theory

# Contents

## Idea

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

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

## Definition

### Explicit definition

###### Definition

(natural transformations)
Given categories $C$ and $D$ and functors $F,G \colon C \to D,$ a natural transformation $\alpha \colon F \Rightarrow G$ between them, denoted

is an assignment to every object $x$ in $C$ of 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 $D$:

###### Remark

(composition of natural transformations) 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

(1)$[C,D] \;\in\; Cat$

The notation alludes to the fact that this makes Cat a closed monoidal category. (Since $Cat$ is in fact a cartesian closed category, another common notation is $D^C$, cf. exponential objects.)

In fact, if we want $Cat$ 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 $Cat$-enriched category) because it is (cartesian) closed: closed monoidal categories are automatically enriched over themselves, via their internal hom.

###### Remark

(categories of presheaves)
The functor categories (1) are also called categories of presheaves, in particular if they are of the form

$PSh(C) \,\coloneqq\, [C^{op}, Set] \,,$

hence if they are categories whose

Similarly, functor categories of the form

$CPSh(C) \,\coloneqq\, [C, Set]$

are also called categories of copresheaves.

### In terms of morphismwise components

An alternative but ultimately equivalent way to define a natural transformation $\alpha : F \rightarrow G$ is as an assignment to every morphism $m : x \rightarrow y$ in $C$ of a morphism $\alpha(m) : F(x) \rightarrow G(y)$, in such a way as that $G(m_1)\alpha(m_0) = \alpha(m_1)F(m_0)$ for every binary composition $m_1 m_0$ in $C$ (or equivalently $\alpha(m_2 m_1 m_0) = G(m_2) \alpha(m_1) F(m_0)$ for every ternary composition $m_2m_1m_0$ 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 $\alpha(f) = G(f) \circ \alpha_x = \alpha_y \circ F(f)$, and the identity morphisms for any object $x$ give the component $\alpha_x = \alpha(id_x)$.

Vertical 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 $C$.

Horizontal composition is even easier, as the horizontal composite of $\alpha_1, ..., \alpha_n$ is just $\alpha_1 ... \alpha_n$.

### In terms of the cartesian closed monoidal structure on $Cat$

The definition of the functor category $[C,D]$ with morphisms being natural transformations is precisely the one that makes $Cat$ 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 $D$: $Mor(C \times D) = Mor(C) \times Mor(D)$.

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

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

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 $Cat$ here is cartesian closed, one often uses the exponential notation $C^B := [B,C]$ 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 $I \to [C,B]$ are. But by the defining property of $[C,B]$ as an internal hom, such functors are in natural bijection to functors $I \times C \to B$.

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

But, as mentioned above, we know 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{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

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

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:

$\array{ F(c_1) & \stackrel{F(f)}{\to} & F(c_2) \\ \alpha_x\downarrow && \downarrow \alpha_y \\ G(c_1) & \stackrel{G(f)}{\to} & G(c_2) } \,.$

### 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 $Cat(C,(\square D,\circ_1))$. But then $\circ_2$ induces a category structure on this and so we get $CAT(C,D)$.

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.

## Examples

###### Example

(unit of double-dualization)
For any ground field $\mathbb{K}$, the canonical linear map from a vector space $V$ to its double dual $\big(V^{\ast}\big)^\ast$

(2)

is a natural transformation

$Id \longrightarrow \big((-)^{\ast}\big)^\ast$

from the identity functor $Id$ on $\mathbb{K} Vect$ to the composition $\big((-)^{\ast}\big)^\ast$ of the linear dual-endofunctor with itself.

This example was the motivating example of Eilenberg & MacLane 1945 (right on the first pages) for introducing the notion of natural transformation (and with it category theory) in the first place.

The conceptual subtlety that these authors sought to resolve here is that for any finite-dimensional vector space there exists also an isomorphism from $V$ to its single-dual vector space:

$V \in \mathbb{K} Vect^{fdim} \;\;\;\;\;\; \vdash \;\;\;\;\;\; \exists \; V \xrightarrow{\sim} V^\ast \,.$

But these linear maps are conceptually different from (2) in that they are not natural in the technical sense that they do not form the components of a natural transformation between the evident functors. Instead they involve an arbitrary choice equivalent to that of an (possibly indefinite) inner product on $V$, which is not preserved by general linear isomorphisms (but just by the corresponding isometries).

###### Example

The determinant is a natural transformation $det\colon GL_n\rightarrow (-)^\times$ from the general linear group to the group of units of a ring, which are both functors from Ring to Grp.

###### Example

The Frobenius homomorphism is a natural transformation $Frob_p\colon Id_{Ring_p}\Rightarrow Id_{Ring_p}$ from the identity functor on the full subcategory of Ring containing all rings with characteristic $p$ to itself.

###### Example

The Hurewicz homomorphism is a natural transformation $h_n\colon\pi_n\Rightarrow H_n(-;\mathbb{Z})$ from the homotopy group to singular homology, which are both functors from Top to Grp.

###### Example

The inversion $G \rightarrow G^{op},g\mapsto g^{-1}$ for every group $G$ yields a natural transformation $\Id_{Grp}\Rightarrow (-)^{op}$ from the identity functor on Grp to the opposite group-assigning functor.

###### Example

The coprojection $G\rightarrow G^{ab},g\mapsto[g]$ for every group $G$ yields a natural transformation $\Id_{Grp}\Rightarrow -^{ab}$ from the identity functor on $Grp$ to the abelianization functor.

###### Example

(homomorphisms of diagrams)
By Remark , every category identified with a category of presheaves or copresheaves has its morphisms identified with natural transformations.

For instance, the category of directed graphs (digraphs) may be identified with the category of copresheaves on the diagram shape $\big\{ V \rightrightarrows E\big\}$, and under this identification the natural transformations between functors $\big\{ V \rightrightarrows E \big\} \longrightarrow Set$ are identified with digraph homomorphisms.

## 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.

## References

The notion of natural transformations between functors is due to

where it served as the motivation for the definition of categories and functors in the first place.

Textbook accounts: