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Pietro Majer
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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. So,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.

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. So, 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.

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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Pietro Majer
  • 60.6k
  • 4
  • 122
  • 269

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. So, 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.