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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4 Answers
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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.
answered Jul 21, 2010 at 15:28
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13$\begingroup$ Urgh. Is transfor an established term? :( $\endgroup$ Commented Jul 21, 2010 at 15:36
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8$\begingroup$ I prefer to rearrange the terminology, and just have plain functors between n-categories, 2-functors between functors (so, in the n=1 case, 2-functors are the same as natural transformations), 3-functors between 2-functors, and so on. $\endgroup$ Commented Jul 21, 2010 at 16:59
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16$\begingroup$ This sort of reminds me of the naming of higher derivatives of position. We have velocity and acceleration which are common, then after that there are the increasingly obscure 'jerk' for the third derivative, and then 'snap', 'crackle', and 'pop' for the fourth, fifth, and sixth derivatives. $\endgroup$ Commented Jul 21, 2010 at 17:02
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3$\begingroup$ Scott, the term n-functor is already widely established to mean a morphism between n-categories. $\endgroup$ Commented Jul 21, 2010 at 19:15
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3$\begingroup$ "n-transformation" is already used to mean a natural transformation (= 1-transfor) between functors between n-categories, just as "n-functor" means a functor between n-categories. I prefer to drop both prefixes when possible, but I think reusing the same prefixes with a different meaning invites confusion. "Transfor" is due to Sjoerd Crans and is a portmanteau of "transformation" and "functor." One can think of a k-transfor as like a degree-k map of chain complexes: it takes each 0-cell to a k-cell, each 1-cell to a (k+1)-cell, etc. -- this unifies functors with higher transfors. $\endgroup$ Commented Sep 3, 2010 at 6:55
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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.
answered Jul 21, 2010 at 12:19
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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 $C^{\textbf{2}}$.
answered Jul 21, 2010 at 18:04