Natural transformations

When I first read the above quote, I couldn't quite wrap my head around how this could be case. Out of all of the foundations of basic category theory, natural transformations initially seemed to me the least intuitive concept. I've since come to the understanding, however, that this is precisely why we need category theory.

After all, once you've spent times studying vector spaces (with the linear transformations between them), groups (with the group morphisms between them), rings (with the ring morphisms between them) and topological spaces (with the continuous maps between them), you can't help but notice a general pattern: there always seems to be some type of algebraic object (i.e., a set with additional structure) and structure-preserving maps between those structures. So the basic idea of a category (objects and arrows) seems inevitable.

And it doesn't take long to make the jump to connections between categories, i.e., functors. If this were where the story ended, you'd have a strong argument to suggest that functors should have been simply called morphisms between categories (and you still could). Category theory would also have been a lot less powerful.

But there's more to this story. Anyone who has spent enough time studying some type of algebraic structure eventually encounters constructions that seem "natural," in the sense that they're "the same for every object." A popular example of this is the determinant, the function that assigns to each square matrix a number with some very nice properties. We've all memorized the formula for computing the determinant, and that formula doesn't depend on the type of entries in the matrix. In other words, regardless of whether the entries of your square matrix are integers, real numbers, or real-valued functions, the formula for the determinant of that matrix is always "the same," namely a signed-sum over all possible products of entries (with exactly one entry selected from each row and column). No one ever said "This is how you compute the determinant if your entries are real numbers, but this is how you compute the determinant if your entries are real-valued functions."

There are many (many!) more examples, and we'll see a bunch of them soon, but historically it took time to pin down exactly what was meant mathematically by "natural" or "the same for all objects." It was this issue that likely delayed the development of category theory, at least a bit. And it's the fact that category theory finally enables us to be precise about "naturality" that gives the theory it's first real chance to shine. So let's get to it!

Mathematical naturality

At long last, here is the definition:

A natural transformation $\tau :F\Rightarrow G$ can be viewed as a way to compare the images of two functors, since it directly connects the image of each object under $F$ with the image (of the same object) under $G$; the naturality condition guarantees these comparisons are compatible with arrow composition. In other words, the natural transformation connects the image of the functor $F$ to the image of the functor $G$ in a "natural way," i.e., consistent with the full categorical structures of $C$ and $D$.

Examples

Moving forward from this point, we will see many examples. What follows is just an initial sampling.

The identity natural transformation

For any functor $F:C\to D$, we always have the identity natural transformation $1_F:F\Rightarrow F$, with component arrows given by the identity arrows, i.e., for every $c\in C$ we have $(1_F{)}_c=1_{F(c)}:F(c)\to F(c)$.

The determinant

Even though it's actually far from the simplest and most intuitive example, our first nontrivial example should probably be the determinant, since it was an early motivator for this new idea. For simplicity, we will mainly restrict to the situation of the determinant of an invertible square matrix.

To that end, let $R$ be a commutative ring. The determinant is a function that assigns to each $n\times n$ matrix $M$ with entries in $R$ a very specific element in $R$. This function has the following properties:

1. $M$ is invertible exactly when its determinant is a unit; and
2. The determinant function is multiplicative
   It follows that the determinant defines a group morphism ${\mathrm{GL}}_n(R)\to R^{\times }$. Since this function ostensibly depends on the ring $R$, we should denote it $\underset{R}{det}:{\mathrm{GL}}_n(R)\to R^{\times }$. However, it is common to simply denote the determinant function by $det$, usually with the acknowledgement that the formula for the determinant is "the same over every ring." The precise meaning of this is that for every ring morphism $f:R\to S$ we have the following commutative diagram in $\mathbf{G}\mathbf{r}\mathbf{p}$:

The vertical arrow on the left is the group morphism that takes each $n\times n$ matrix with entries in $R$ and applies $f:R\to S$ to each entry to produce an $n\times n$ matrix with entries in $S$. The right vertical arrow is simply the ring morphism $f:R\to S$ restricted to the multiplicative units of $R$. (It is an easy result in ring theory that if $r\in R$ is a unit, then $f(r)\in S$ is also a unit; i.e., $f(r)\in S^{\times }$.)

The above diagram is a fancy way of expressing the fact that, given an invertible $n\times n$ matrix $M$ with entries in $R$, we can either compute its determinant and then apply $f$ to the answer, or first apply $f$ to move that matrix to one over $S$ and then compute the determinant. In other words, the computation of the determinant is "the same for all rings" in the only sense in which such a statement can be made precise: if we were to compare the computations in two different rings (using a ring morphism $f:R\to S$ to enable the comparison), we get "the same thing either way."

In any case, the above diagram exactly says that $det:{\mathrm{GL}}_n\Rightarrow {\bullet }^{\times }$ is a natural transformation between the two functors ${\mathrm{GL}}_n,{\bullet }^{\times }:\mathbf{\text{CRing}}\to \mathbf{\text{Grp}}$.

Maps between diagrams

We can use functors to formalize the notion of "diagrams" in a given category, and then use natural transformations to formalize the notion of "maps between diagrams" of "the same shape".

Suppose $J$ and $C$ are two categories. A diagram of shape $J$ in $C$ is simply a functor $F:J\to C$. A map between diagrams is then simply a map between such functors, i.e., a natural transformation between such functors.

We can visualize these natural transformations as cylinders with cross-sections given by the shape category. For example, suppose the category $J$ looks like

Suppose $F,G:J\to C$ are two such diagrams in $C$. Then a natural transformation $\tau :F\Rightarrow G$ can be visualized as

The vertical green arrows are the components of the natural transformation. The naturality condition here is the requirement that each of the three "vertical" rectangles commutes.

The center of a group

Recall that to each group $G$ we can associate the category $\mathbf{B}G$, the so-called delooping groupoid. One functor that certainly exists is the identity functor on this category, which we'll denote $I:\mathbf{B}G\to \mathbf{B}G$. We claim that the set of natural transformations from this identity functor to itself is in bijection with the elements of the center of $G$.

Indeed, first suppose $\tau :I\Rightarrow I$ is a natural transformation from $I$ to itself. Since the category $\mathbf{B}G$ has only a single object $\star$, this natural transformation is determined entirely by the single component arrow ${\tau }_{\star }:I(\star )\to I(\star )$. But since $I(\star )=\star$, this is nothing more than an arrow in $\mathbf{B}G$, i.e., an element $z\in G$.

Now let's investigate the naturality condition. Take any arrow in $\mathbf{B}G$, say $g:\star \to \star$ where $g\in G$. Then the diagram below must commute:

The vertical arrows correspond to the element $z\in G$, so the composition $g\circ {\tau }_{\star }$ corresponds to the element $gz$. Similarly, the composition ${\tau }_{\star }\circ g$ corresponds to the element $zg$. The commutativity of the above diagram is therefore the requirement $gz=zg$, for all $g\in G$. This is exactly the condition for $z$ to be in the center of $G$.

It's not hard to show the converse, namely that every element of the center induces a corresponding natural transformation.

Even better, this bijection can be upgraded to an isomorphism of groups! There is a composition operation on natural transformations (see below), and this can be used to put a group structure on the set of these natural transformations $I\Rightarrow I$. One can prove this group is indeed isomorphic to the center of $G$.

Free vector spaces and "insertion of generators"

Let $k$ be a field.^[1] For each set $X$, the free $k$-vector space on $X$ is the vector space $F(X)$ "generated" by the elements of $X$. It is usually described as consisting of all formal finite $k$-linear combinations of elements in $X$, i.e., it consists of formal sums of the form ${\displaystyle \sum _{x\in X}{c}_{x}x}$ where $c_x\in k$, all but finitely many zero. (A more formal definition is that $F(X)$ consists of all set maps $f$ from the set $X$ to the (set of elements of the) field $k$, such that $f(x)=0$ for all but finitely many $x\in X$.) One can show that this defines the object function of a functor $F:\mathbf{\text{Set}}\to {\mathbf{\text{Vec}}}_k$.

Let $U:{\mathbf{\text{Vec}}}_k\to \mathbf{\text{Set}}$ be the usual forgetful functor and consider the composition $UF:\mathbf{\text{Set}}\to \mathbf{\text{Set}}$. This composition sends each set $X$ to the set of elements of the vector space $F(X)$. There is evidently a set map ${\eta }_X:X\to UF(X)$ that sends each element $x\in X$ to the same element in $F(X)$, only now considered as a formal linear combination of the elements of $X$ (that just happens not to involve any of the other elements). (In the more precise formal definition of $F(X)$ mentioned parenthetically above, the map ${\eta }_X$ sends each element $x\in X$ to the characteristic function of $x$; i.e., $f(x)=1$ and $f(x^{\prime })=0$ for all $x^{\prime }\ne x$.) This map is sometimes called the insertion of generators. It is immediate to verify this defines a natural transformation from the identity functor on $\mathbf{\text{Set}}$ to the composition $UF$.

Soon we will see that the functors $U$ and $F$ are adjoints. This natural transformation will be an important part of that relationship.

Abelianization and forgetting

For each group $G$ the projection ${\pi }_G:G\to G/[G,G]$ defines a natural transformation from the identity functor on $\mathbf{\text{Grp}}$ to the abelianization-forgetful composition functor $\mathbf{\text{Grp}}\to \mathbf{\text{Ab}}\to \mathbf{\text{Grp}}$. This transformation is natural, since for every group morphism $f:G\to H$ you can check that the diagram below commutes:

Similar to the previous example, we will soon see that the abelianization and forgetful functors are an adjoint pair.

Composition of natural transformations

There are two ways one can compose natural transformations, which visually can be thought of as
"vertical composition" and "horizontal composition."

Vertical composition of natural transformations

Suppose we have three functors $F,G,H:C\to D$ and a pair of natural transformations $\tau :F\Rightarrow G$ and $\eta :G\Rightarrow H$. Visually we might represent this information as follows:

We can define a natural transformation $\eta \tau :F\Rightarrow H$ by composing the component arrows of $\tau$ with the component arrows of $\eta$:

The naturality condition for $\eta \tau$ follows immediately from the "stacked" naturality conditions of $\tau$ and $\eta$:

This vertical composition will allow us to define an entire category of functors between two fixed categories, i.e., so-called functor categories. In fact, you should check that for each functor $F:C\to D$ the identity natural transformation $1_F:F\Rightarrow F$ is the identity for vertical composition of natural transformations from $F$ to itself.

Horizontal composition of natural transformations

You can probably see this one coming, but now suppose we had successive natural transformations between pairs of functors, as illustrated below:

We can combine the components of the natural transformations $\tau$ and $\eta$ to product a natural transformation $F_2{F}_1\Rightarrow G_2{G}_1$. Can you see how?

I'm purposely not filling in the details for this composition for two reasons:

1. We will not need this composition anytime soon.
2. If we proceed any further, we'll need to decide how to distinguish the two composition operations. Should we use different symbols for each, like $\tau \circ \eta$ and $\tau ◻\eta$? Should we leave it as $\tau \eta$ and make it context-specific? Rather than decide, let's avoid the issue for as long as we can. Maybe forever!

Natural isomorphisms

Now that we have a way to compose natural transformations we can consider the idea of "invertible" natural transformations and natural "isomorphisms."

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