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Edit: As an attempt to make the question somewhat more precise, I'll describe my intuitions in more detail here. Yes, I'm changing the details of question slightly, but the spirit of it remains the same.

First, there's the basic idea of viewing real numbers simultaneously as "points", "adders", and "multipliers", to use the terminology in the linked video.

If we consider "adders" (is there a better term?) as morphisms in some category, then the operation $+$ becomes simply composition in that category. Likewise for multipliers and $\times$. But we want more than this: $\mathbb{R}$ is actually a field, although I'm not exactly sure how much of the field structure is necessary to define $\exp$.

Now the second idea is the "naturality" of $\exp$. Provided we can interpret $+$ and $\times$ as (bi-)functors in a manner consistent with the necessary structure, we find that $\exp$ does indeed obey a kind of naturality: we have $\times \circ (\exp, \exp) = \exp \circ +$.

Finally, there is a possible application of the Yoneda lemma. The Yoneda lemma states that a natural transformation $\eta:\hom(A,-)\to F$ from a representable functor is completely determined by where it sends the identity map $id_A$; the element $\eta(id_A)$ is called a "universal element" for $\eta$. Each element of $F(A)$ determines a unique such $\eta$.

Now consider $\square+a$,$\square\times a$, and $a^\square$ as families of functions parameterized by $a$: notice how a function in each family is completely, uniquely determined by its value at the respective identity elements, $0+a=1\times a = a^1 = a$; and every real $a$ determines a unique function in each family. This directly parallels the Yoneda lemma as stated above, and I wonder if there is any connection.

Furthermore, where there is nontrivial representability, we typically have interesting universal properties. Indeed, the idea that $e$ is the "most natural" base suggests a universal property of some sort.

What I am asking is whether there is any kind of precise category-theoretic backing for any of these four informal observations.

End edit. The original question follows.

Yesterday, I came across [this video] (https://www.youtube.com/watch?v=F_0yfvm0UoU) by "3Blue1Brown" about Euler's identity $e^{i \pi} = -1$.

By itself, the video is not MathOverflow material - its subject is far from "research-level". But it did strike a chord with me that I believe is worth following up on here. For many months now I've been trying to teach myself category theory; right now I'm trying to get a handle on universal properties, (co)limits, the Yoneda lemma, etc.

So the first 'aha' moment for me came at just under 2 minutes into the video. Here 3Blue talks about addition of reals this way:

Think of "adders" [real functions $f_y(x)= x + y$] purely as sliding the [real] line with the following rule: You slide until the point corresponding to zero ends up where the point corresponding to the adder itself started.

When you apply two adders, the effect will be the same as applying some other adder. This is how we define their sum.

Hmm... That sounds like there's a category here somewhere.

He goes on to describe multiplication similarly:

Now, the rule is to fix 0 in place, and bring the point corresponding with 1 to where the point corresponding with the multiplier itself started off, keeping everything evenly spaced as you do so.

We can now redefine multiplication as the successive application of two different actions.

Then he segues into a discussion of $e^x$:

The life's ambition of $e^x$ is to transform adders into multipliers, and to do so as naturally as possible.

Wait a minute... "transform ... as naturally as possible"... Lo and behold, the next screen shows none other than a "naturality diagram" for $e^x$!

This "diagram" (I put it in quotes because we don't have a formally defined category) happens to express exactly the equation $e^{x+y}=e^x e^y$.

The video continues:

Multiple functions satisfy this property, but ... one stands out as being the most natural, and we express it with this infinite sum:

$$ e^x := \sum {1\over n!} x^n$$

By the way, the number $e$ is just defined to be the value of this function at 1.

3Blue then proceeds to develop complex arithmetic, eventually deriving Euler's identity, using the same intuitions as before.

So now it seems we have not only a "natural transformation", but also some "universal properties" on our hands as well. Indeed, by definition all three of these operations are completely determined by their actions on the respective identity elements. I think I sense Yoneda in here somewhere.

So, my actual question is:

Question: Am I actually onto something here, or is this just nonsense?

  • Is there a precise sense in which $e^x$ is a natural transformation?

  • Is there a precise sense in which real or complex addition, multiplication, and/or exponentiation are universal in the sense of category theory?

If so, what are the categories/functors involved?

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    $\begingroup$ You may be roughly "on to something", but I don't consider it really on-topic for MO. That composition of two adders is another adder is a facet of associativity, which is connected with Cayley's theorem on groups and with the Yoneda lemma. As for the significance of the series for $\exp$ and its taking addition to multiplication: there is a categorified interpretation described here which considers $\exp$ as reflecting a functor: mathoverflow.net/a/43584 But this response is more free association than it is a coherent answer to these questions, which are somewhat loosely worded. $\endgroup$
    – Todd Trimble
    Commented Apr 16, 2016 at 0:24
  • $\begingroup$ If it is off topic, I apologize. This is my first post on MO, though I have posted on MSE before. $\endgroup$
    – BenW
    Commented Apr 16, 2016 at 3:00
  • $\begingroup$ Actually, the composition of two "adders" or "multipliers" (those are the terms used in the video) simply reminded me of composition of arrows. The series for $\exp$ isn't really that important to my line of thought, it was just part of the quote; rather what struck me was the "value at 1" bit and the possible relation to Yoneda. $\endgroup$
    – BenW
    Commented Apr 16, 2016 at 3:02
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    $\begingroup$ It's very good that you are learning category theory; it sounds like you've made good progress. If $M$ is a monoid (a one-object category) and left multiplication by $a \in M$ is denoted $\lambda_a: M \to M$, then $\lambda_a \circ \lambda_b = \lambda_{ab}$ by associativity, and the assignment $a \mapsto \lambda_a$ is the Cayley-Yoneda embedding $M \to Set^{M^{op}}$ in disguise; the unique object of $M$ is sent to a representation with $M$ acting on itself (on the right), with the $\lambda_a$ serving as natural transformations, again by associativity. (continued next comment) $\endgroup$
    – Todd Trimble
    Commented Apr 16, 2016 at 10:25
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    $\begingroup$ As for $\exp$: in order to assert that an exponential map $E: (\mathbb{R}, +) \to (\mathbb{R}, \cdot)$ is determined by $E(1)$, it's necessary to have some extra assumption such as continuity, just as in order to have an abelian group map $f: \mathbb{R} \to \mathbb{R}$ determined by $f(1)$, you need some such assumption. So this is not quite Yoneda-like, and the word "natural" in that video is not in the technical sense of Eilenberg-Mac Lane, but just in an informal sense. These are delightful things to discuss, but unfortunately MO is not a discussion forum and I think it's a bit off-topic. $\endgroup$
    – Todd Trimble
    Commented Apr 16, 2016 at 10:34

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In category theory exponentiation is not a natural transformation. Rather, it is a functor, or a family of functors, equipped with a natural transformation.

If we have a category $C$ with finite products, for each object we have a functor $Y \times \mbox{-}$. We can then ask for an adjoint: $$ C(X \times Y, Z) \cong C(X, Z^Y). $$

To define an adjoint we need a functor $\mbox{-}^Y$ and a natural transformation as defined above.

In the category of sets, $Z^Y$ is the set of functions $Y \rightarrow Z$, and the natural transformation from $\theta : \mathbf{Set}(X \times Y, Z) \rightarrow \mathbf{Set}(X, Z^Y)$ is defined as $$ \theta(f)(x)(y) = f(x,y). $$ This operation is often called "currying", after Haskell Curry, because it was invented by Moses Schönfinkel.

Other examples of this can be found in logic in the relationship between conjunction and implication (in both Boolean and intuitionistic logic): $$ A \land B \vdash C \Leftrightarrow A \vdash B \rightarrow C $$

However, the set example really only relates to how exponentiation works on integers, or, if pushed, cardinals. For how this relates to real numbers, I cannot fully answer, but:

  1. $e$ occurs as the "groupoid cardinality" of the groupoid whose objects are finite sets and whose maps are bijections. For more information on this see page 15 of: https://arxiv.org/abs/math/0004133
  2. There is this interesting paper about using facts about polynomials that are true over the complex numbers to deduce isomorphisms in categories: http://arxiv.org/abs/math/0212377

I think Tom Leinster is a member here, he would undoubtedly give a much better answer.

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  • $\begingroup$ I know a little bit about exponential objects and cartesian closed categories, and indeed we can indeed view exponentiation in $\mathbb{N}$ this way, but I don't think this works for the extension to $\mathbb{R}$, at least not directly. $\endgroup$
    – BenW
    Commented Apr 16, 2016 at 2:36
  • $\begingroup$ A big stumbling block is that the obvious way of defining "additive inverses", $A + B \cong 0$ in a category with coproducts, doesn't work, as it forces $A$ and $B$ to be initial. The dual result holds for products. So it's not entirely obvious how to make sense of subtraction and division in terms of objects in a category, if it's even possible. There goes the obvious way of categorifying $\mathbb{Z}$, let alone $\mathbb{R}$ or $\mathbb{C}$. $\endgroup$
    – BenW
    Commented Apr 16, 2016 at 2:44

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