Daryoush Mehrtash wrote:
Thanks this was helpful.
In many of Conal Elliot's writings I see that he shows that his semantic function is a natural transformation. Is that just basically showing the polymorphic nature of his semantic functions, or are there other benifits you get by showing a particular function is a natural transformation?
Daryoush
Natural transformations give you a lot of very regular structure to work with. This is helpful down the road when trying to prove other laws, e.g. about how your thing (that happens to be a NT) interacts with other things. For example, the catamorphism fusion law: eps : F :~> G ------------- cata_G psi . cata_F (in_G . eps) = cata_F (psi . eps) Because eps is a natural transformation we know that it can't be doing any of the bad things that would violate the equation. Often times when trying to prove something non-trivial you find yourself wishing "if only I had a very well-behaved morphism to go here". The components of NTs are frequently the exact thing you want there. -- Live well, ~wren