If functors are morphisms between categories, and natural transformations are morphisms between functors, what's a morphism between natural transformations? Is there ever a need for such a notion?
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(Small) categories form what's called a 2-category, which is a structure that has objects, morphisms (functors), and morphisms between morphisms (natural transformations). There are also n-categories, which have a deeper morphisms structure. A google search will point you to a lot of references about n-categories. But for ordinary categories, the story ends at natural transformations. |
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The next one is called Modifications between ordinary natural transformations between functors between categories are trivial, but they exist between between 2-functors between 2-categories. The next one after that is called perturbation : a perturbation goes between modifications between pseudonatural transformations between 3-functors between 3-categories. Beyond that, no established terms exists. Instead one starts numbering things and speaks of n-Functors are 0-transfors. Transformations are 1-transfors. Modifications are 2-transfors. Perturbations are 3-transfors. And so on. |
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One advantage of the abstraction of category theory is that one is not constrained to "concrete" objects and morphisms (I mean, made by set with a structure together with functions preserving it), and constructions of new categories from simpler ones are very easily performed. As a result, any further and more general categorical notion can always be read as a particular case of a simpler and more basic one, in a suitable category. Thus in the proper context, a morphism is an object; a natural transformation is a morphism; similarly, a universal arrow is a particular case of an initial object, which of course is a particular case of a universal arrow, and so on. So in a sense, there is no need of the notion of "morphism between natural transformations", just because it is already a particular case of a more basic notion already defined. In practice, several used categories (e.g. algebras; preshaves; chain complexes,...) are themselves categories of functors, where arrows are natural transformations. In this context a morphism between natural transformation naturally arises, even if it will be seen as just an ordinary morphism. |
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This is not really a sophisticated answer as the other ones, but maybe it makes visible why higher structures are needed to get an interesting notion of morphism between natural transformations. Assume $F,G : C \to D$ are functors and $\alpha, \beta$ are natural transformations $F \to G$. What could be a morphism $\alpha \to \beta$? Since $\alpha$ and $\beta$ consist of their components $\alpha(c) : F(c) \to G(c), \beta(c) : F(c) \to G(c)$, the only reasonable way of "connecting" these data in our category $D$ is by means of two morphisms $\gamma(c) : F(c) \to F(c), \delta(c) : G(c) \to G(c)$, so that the resulting diagram becomes commutative. Furthermore, $\gamma$ and $\delta$ should become natural transformations $F \to F, G \to G$. But this comes from a more general concept, namely the arrow category: If $C$ is a category (in the above case, this is a functor category), its arrow category has as objects the morphisms of $C$ and a morphism between two morphisms $x \to y, x' \to y'$ is a pair of morphisms $x \to x', y \to y'$, making the obvious diagram commutative. This may be also described as the functor category $C^{\textbf{2}}$. |
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