"Alternatives" to Natural Transformations

Question

I would like someone to either (1) point out the mistake in what follows or (2) confirm what is said is correct.

This would be accomplished by addressing the part in yellow only. The rest of the question is context, optional for the reader to consider.

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Natural transformations are often explained as being "morphisms between functors".

morphism between functors - consider a category $\mathscr{F}$ with objects that are functors (for some other category $\mathcal{C}$), and the morphisms are whatever as long as they satisfy the category axioms. The "morphisms between functors" are the morphisms of the category $\mathscr{F}$.

However, I feel like this characterization is insufficient, because although natural transformations are the best way to define morphisms between functors, they are not the only way.

As far as I can tell, there are other conditions imposed on natural transformations in their definition which are not strictly necessary to be a morphism between functors.

Is this true?

Is the only thing which differentiates natural transformations between arbitrary morphisms between functors is that natural transformations satsify the interchange law?

The question is essentially whether or not the morphisms of such a category $\mathscr{F}$ must also be natural transformations for the category $\mathcal{C}$. I do not think this is the case, and I have given a purported counterexample below using an arrow category, but I am not sure if it correct, since I have not seen something like this discussed before in any textbook I have looked at.

Counterexample??? If we form the arrow category $\mathscr{D}^{\to}$ of some category $\mathscr{D}$ whose objects are categories $\mathcal{C} \in \mathcal{Ob}(\mathscr{D})$, then the objects of this arrow category $\mathscr{D}^{\to}=\mathscr{F}$ are functors and the morphisms of this arrow category $\mathscr{D}^{\to}=\mathscr{F}$ are therefore morphisms between functors, but not necessarily natural transformations.

(I think, because these can be defined even between functors which do not have the same source and target categories, whereas natural transformations always map between functors with the same source and target categories, hence one reason why they are an appropriate choice of the morphisms of functor categories whose objects are functors between two fixed categories.)

As far as I can tell, natural transformations can be used to generalize the notion of homotopy (and perhaps were even invented for this purpose), hence the homotopy hypothesis and homotopy type theory. Since homotopies satisfy a (weak version of?) the interchange law, any attempt to generalize them must be a 2-morphism, and not an arbitrary morphism between functors.

Hence the "proper", or "more natural", way to think about natural transformations is as the 2-morphisms of the 2-category $Cat$ of locally small categories, since thinking of them only as morphisms between functors seems to suggest less structure than they actually have.

Answer 1

There's nothing preventing you from taking any class, and then making any category $\mathcal{C}$ you want for which $\mathrm{ob}(\mathcal{C})$ is that class, as long as $\mathcal{C}$ satisfies the axioms for a category. If you want the class to be the class of all functors $\mathcal{A}\to\mathcal{B}$ for some other categories $\mathcal{A}$ and $\mathcal{B}$, that's fine.

Analogously, you could take abelian groups $A$ and $B$, form the set of all group homomorphisms $A\to B$, and then make that set into an abelian group $C$ in any way you want.

Of course, the usefulness of doing these things in any way but the usual way is a separate issue.

Answer 2

If you have some experience with algebraic topology, here's some motivation as to why natural transformations are a useful notion of morphism between functors. One can rephrase the definition of natural transformation as follows.

Let $I$ be the category with two objects, $0$ and $1$, and one nontrivial morphism $0 \to 1$. For any category $\mathcal{C}$, we have two "inclusion" functors $i_0, i_1 : \mathcal{C} \to \mathcal{C} \times I$. Then a natural transformation between functors $F, G: \mathcal{C} \to \mathcal{D}$ gives exactly the same data as a functor $H : \mathcal{C} \times I \to \mathcal{D}$ such that, the compositions $H \circ i_0$ and $H\circ i_1: \mathcal{C} \to \mathcal{C} \times I \to \mathcal{D}$ are $F$ and $G$, respectively.

This is exactly analogous to the definition of a homotopy between continuous maps, i.e. a morphism between morphisms. In fact, higher category theory draws a lot of inspiration from algebraic topology.

Hope this helps!

Answer 3

Here is, in my opinion, a good explanation of why natural transformations are the most natural notion of morphism between functors, based on the algebra of $\mathbf{Cat}$.

Suppose $\mathcal{C}$ is a category.

The objects of $\mathcal{C}$ are precisely the functors $\mathbf{1} \to \mathcal{C}$, where $\mathbf{1}$ is the terminal category. (one object, no nonidentity arrows)

The arrows of $\mathcal{C}$ are precisely the functors $\mathbf{2} \to \mathcal{C}$, whre $\mathbf{2}$ is the arrow category. (two objects, one nonidentity arrow from one to the other)

Now, suppose there was a good notion of a functor category: a category $\mathcal{D}^\mathcal{C}$ whose objects are functors $\mathcal{C} \to \mathcal{D}$. If we insist the usual relationship bewteen products and exponentials holds, then the following notions are equivalent:

• A functor $\mathbf{2} \to \mathcal{D}^\mathcal{C}$
• A functor $\mathbf{2} \times \mathcal{C} \to \mathcal{D}$

Objects of the first type tell us what morphisms between functors should be. Objects of the second type we can handle explicitly, and it's not hard to show they give the usual definition of natural transformation.