On the importance of natural transformations

In p. 18 of Categories for the working mathematician (2d ed.), Mac Lane remarks that

..."category" has been defined in order to define "functor" and "functor" has been defined in order to define "natural transformation".

For a concept to warrant so much preliminary machinery it must yield huge benefits, but somehow in the case of natural transformations I still don't see them.

For example, one of the few examples of natural transformations that I can fully follow among those that Mac Lane offers is the determinant natural transformation, $\det$ (on p.16). His description goes something like this.

Let $\mathbf{CRng}$ and $\mathbf{Grp}$ be, respectively, the categories of commutative rings and of groups; fix $n$ to be an arbitrary natural number; for any commutative ring $K$, let $\mathrm{GL}_nK$ be the group of all non-singular $n \times n$ matrices with entries in $K$, and let $K\text{*}$ be the group of invertible elements (aka units) of $K$. Then, $\mathrm{GL}_n$ and $\text{*}$ are functors $\mathbf{CRng}\to\mathbf{Grp}$, and, moreover, the following diagram commutes:

$$ \require{AMScd} \begin{CD} \mathrm{GL}_nK @>\det_K>> K\text{*}\\ @V\mathrm{GL}_nfVV @VVf\text{*}V\\ \mathrm{GL}_nK^\prime @>\det_{K^\prime}>> K^\prime\text{*} \end{CD} $$

IOW, $\det$ is a natural transformation $\mathrm{GL}_n \dot{\to} \text{*}$.

It's a very nice, very clear example, but after I follow all the definitions and chase all the arrows, I'm left asking myself "so what"?

More specifically, before working out the diagram, I knew pretty much all the information expressed by its arrows, and compositions thereof. Even the commutativity between "taking the determinant" and "applying $f$" is not hard to see. So, at least in this case, the concept of natural transformation does not seem to be contributing much to what I already knew, and it's thus hard for me to justify all the apparatus needed to define it.

What am I missing?

Solution 1:

The naturality squares of a natural transformation are: $$\require{AMScd} \begin{CD} FA @>{\eta_A}>> GA\\ @V{Ff}VV @VV{Gf}V\\ FB @>>{\eta_B}> GB \end{CD}$$ where we ask the square to commute for any choice of $f\in \mathsf{Hom}(A,B)$.

To me, the significance of this definition is somewhat philosophical (and has hitherto given me good intuition for naturality):

By the Yoneda lemma, $A\cong B\iff H_A\cong H_B$ where $H_A=\mathsf{Hom}(-,A)$. This means that objects are determined by the arrows into them. Asking the naturality squares to commute for every appropriate $f$ is basically saying "this square commutes regardless of what $\mathsf{Hom}(A,B)$ looks like", which by Yoneda is the same as saying "this square commutes regardless of the properties of $A,B$".

Thus, to say some arrow $\eta _A$ is natural in $A$ is to say that the specific object $A$ plays no role in defining it, hence it can be extended to a global construction between functors.

The most elegant example I can think of where naturality really simplifies a proof is the homotopy invariance of singular homology. In this instance, naturality literally enables you to solve a global problem (for all topological spaces) by solving it only for very simple spaces (the standard simplices). This same principle is the heart of the acyclic models theorem, which itself can be used to prove many other wonderful theorems.

Solution 2:

Say you have a real matrix and want to consider its eigenvalues. The real ones won't suffice for your purpose, so you also consider the complex eigenvalues. These are the eigenvalues of the correspondig complex matrix. Why? Because $\det$ is natural with respect to $\mathbb{R}[t] \hookrightarrow \mathbb{C}[t]$. Naturality is used all the time, starting from the first day learning higher mathematics. (Of course, nobody tells you.)

Solution 3:

First of all, I don't think categories and functors count as "much preliminary machinery", they are just two (most basic) definitions of one mathematical theory. So natural transformation are still a very elementary notion, and simple examples of them tend to be a bit obtuse. That $\det : \mathrm{GL}_n → (-)^*$ is natural means that it's a nice way to transform a general linear group into a group of units. That's a simple idea, and of course you already knew that, so there shouldn't be anything flashy going on. Compare it if you will to proving that squaring is a continuous real function.

The importance is not in ability to give a formal label to things you already intuitively knew are "transformations of constructions" (although it's certainly a good thing!), it's that now given two functorial constructions, you know what every transformation should satisfy, and that natural transformations between them are exactly the thing you want to focus on. Considering the ubiquity of functorial constructions, this is an important realization.

In fact, many mathematical objects can themselves be described as functors. Representations of a group $G$ are functors from the single object category $G$: to $\mathrm{Set}$ for group actions, and to $\mathrm{Vect}$ for (linear) representations, and natural transformations between these functors are exactly the morphisms of $G$-sets and intertwinings respectively. Similarly $R$-modules correspond to additive functors from the single-object additive category $R$ to $\mathrm{Ab}$, and module morphisms are again natural transformations. In an other direction, presheaves on a topological space $(X, \mathcal T)$ are functors $(\mathcal T, ⊇) → \mathrm{Set}$.

Again, each of these claims is obvious, and every mathematician would know what a morphism of modules ought to be, with or without category theory. The unifying paradigm however could certainly prove useful if you try to generalize or study a new kind of objects that are just functors in disguise.

Finally, since categories, functors and natural transformations are (among) the most basic definitions of category theory, almost everything else uses (abstract) natural transformations either implicitly or explicitly, so they are an integral part of why the entire category theory is important (not that this isn't something that's often considered doubtful too).