added some category theory into FunctionK document #1636

kailuowang · 2017-04-25T16:30:41Z

I think this will be helpful because it's often referred as Natural ~~Transformer~~ Transformation in other context.

wedens · 2017-04-25T16:32:37Z

I think it's known as "natural transformation", not "natural transformer" :)

kailuowang · 2017-04-25T16:37:52Z

@wedens thanks for pointing out. Corrected.

SystemFw · 2017-04-25T16:59:59Z

docs/src/main/tut/datatypes/functionk.md

+
+## Natural Transformation
+
+In category theory, a [Natural Transformation](https://en.wikipedia.org/wiki/Natural_transformation) provides a morphism between Functors while preserving the internal structures. It's one of the most fundamental notions of category theory.


while preserving the internal structures

A few comments as requested on Gitter: I think the internal structure is slightly better. It refers to the composition of morphisms (similarly to what a functor does)

SystemFw · 2017-04-25T17:03:06Z

docs/src/main/tut/datatypes/functionk.md

+
+We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by its parametric polymorphism.
+
+This is why a parametric polymorphic function `FunctionK[F, G]` is sometime referred as a Natural Transformation.


Pedantry warning (sorry): it should be sometimes, not sometime. :)

SystemFw · 2017-04-25T17:03:55Z

docs/src/main/tut/datatypes/functionk.md

+
+This is why a parametric polymorphic function `FunctionK[F, G]` is sometime referred as a Natural Transformation.
+However, they are two different concepts. `FunctionK` is a stronger and stricter construction than Natural Transformation.
+


I'm not entirely sure this is worth including unless we can give an explanation as to how do they differ

codecov-io · 2017-04-25T17:11:31Z

Codecov Report

Merging #1636 into master will decrease coverage by 5.13%.
The diff coverage is n/a.

@@            Coverage Diff             @@
##           master    #1636      +/-   ##
==========================================
- Coverage   93.43%   88.29%   -5.14%     
==========================================
  Files         241       48     -193     
  Lines        4068      940    -3128     
  Branches      140       86      -54     
==========================================
- Hits         3801      830    -2971     
+ Misses        267      110     -157

Impacted Files	Coverage Δ
.../main/scala/cats/laws/discipline/StrongTests.scala
core/src/main/scala/cats/data/EitherK.scala
core/src/main/scala/cats/instances/vector.scala
laws/src/main/scala/cats/laws/CartesianLaws.scala
...scala/cats/laws/discipline/SerializableTests.scala
core/src/main/scala/cats/instances/function.scala
...re/src/main/scala/cats/functor/Contravariant.scala
.../main/scala/cats/laws/discipline/ChoiceTests.scala
...c/main/scala/cats/laws/discipline/ArrowTests.scala
core/src/main/scala/cats/Cartesian.scala
... and 183 more

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sir-wabbit · 2017-04-25T17:14:57Z

docs/src/main/tut/datatypes/functionk.md

+
+If we have two `Functor`s: `F` and `G`, then, being parametric polymorphic, `FunctionK[F, G]` is automatically a Natural Transformation between them. That is, if we have a `fk: F ~> G`, then for any combination of `A`, `B` and function `f: A => B`, the following two functions are equivalent:
+```Scala
+(fa:F[A]) => fK(F.map(fa)(f))


Minor inconsistency: fK and fk.

kailuowang · 2017-05-12T13:45:17Z

@ceedubs I am interested in hearing your thoughts on this, if you have time.

ChristopherDavenport · 2017-05-12T15:25:27Z

docs/src/main/tut/datatypes/functionk.md

+
+If we have two `Functor`s: `F` and `G`, then, being parametric polymorphic, `FunctionK[F, G]` is automatically a Natural Transformation between them. That is, if we have a `fk: F ~> G`, then for any combination of `A`, `B` and function `f: A => B`, the following two functions are equivalent:
+```Scala
+(fa:F[A]) => fK(F.map(fa)(f))


This has fK where is is defined above as fk

ceedubs · 2017-05-13T15:52:43Z

docs/src/main/tut/datatypes/functionk.md

+
+We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by its parametric polymorphism.
+
+This is why a parametric polymorphic function `FunctionK[F, G]` is sometimes referred as a Natural Transformation. However, they are two different concepts.


Minor: is there a particular reason that you capitalized "Natural Transformation" here?

Probably don't need to capitalize

This line could use some elaboration.. maybe say something about how FunctionK is more powerful than natural transformations, or natural transformations can be implemented in terms of FunctionK (I prefer the latter)

We had a discussion about this on gitter, and @edmundnoble pointed out that natural transformation could be deemed as more powerful in some ways. My intuition is that FunctionK is more powerful in one category but natural transformation is a concept that could involve two categories. Maybe when we restrict natural transformation to endofunctors we can say something about it. In conclusion we feel it's too complex a thing to include here.

ceedubs · 2017-05-13T15:54:32Z

Thanks @kailuowang. I really like linking this back to the category theory (and probably also providing some insight to people who are used to the NaturalTransformation name in scalaz). I also like that it's added as a separate section at the end so people can get into the theory if they want to but can stop short of it if they prefer. 👍

ceedubs · 2017-05-13T18:38:28Z

@adelbertc as someone who has written a lot of docs and read up on your category theory, does this look good to you?

adelbertc

Minor comments, but otherwise LGTM

adelbertc · 2017-05-13T19:17:37Z

docs/src/main/tut/datatypes/functionk.md

+
+## Natural Transformation
+
+In category theory, a [Natural Transformation](https://en.wikipedia.org/wiki/Natural_transformation) provides a morphism between Functors while preserving the internal structure. It's one of the most fundamental notions of category theory.


ncatlab might be a better reference than Wikipedia for such a subject :-) https://ncatlab.org/nlab/show/natural+transformation

adelbertc · 2017-05-13T19:19:45Z

docs/src/main/tut/datatypes/functionk.md

+
+In category theory, a [Natural Transformation](https://en.wikipedia.org/wiki/Natural_transformation) provides a morphism between Functors while preserving the internal structure. It's one of the most fundamental notions of category theory.
+
+If we have two `Functor`s: `F` and `G`, then, being parametric polymorphic, `FunctionK[F, G]` is automatically a Natural Transformation between them. That is, if we have a `fk: F ~> G`, then for any combination of `A`, `B` and function `f: A => B`, the following two functions are equivalent:


Perhaps drop the colon, and use 'parametricity', e.g.

If we have two Functors F and G, FunctionK[F, G] is a natural transformation via parametricity. That is, given fk: FunctionK[F, G], for all functions A => B the following are equivalent:

adelbertc · 2017-05-13T19:19:54Z

docs/src/main/tut/datatypes/functionk.md

+(fa: F[A]) => G.map(fk(fa))(f)
+````
+
+We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by its parametric polymorphism.


by parametricity

adelbertc · 2017-05-13T19:20:16Z

docs/src/main/tut/datatypes/functionk.md

+
+We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by its parametric polymorphism.
+
+This is why a parametric polymorphic function `FunctionK[F, G]` is sometimes referred as a Natural Transformation. However, they are two different concepts.


Probably don't need to capitalize

adelbertc · 2017-05-13T19:21:11Z

docs/src/main/tut/datatypes/functionk.md

+
+We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by its parametric polymorphism.
+
+This is why a parametric polymorphic function `FunctionK[F, G]` is sometimes referred as a Natural Transformation. However, they are two different concepts.


This line could use some elaboration.. maybe say something about how FunctionK is more powerful than natural transformations, or natural transformations can be implemented in terms of FunctionK (I prefer the latter)

adelbertc · 2017-05-13T19:21:40Z

@ceedubs you flatter me :-)

…to improve-functionK-doc

added some category theory into FunctionK document

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added some category theory into FunctionK document #1636

added some category theory into FunctionK document #1636

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adelbertc commented May 13, 2017


		## Natural Transformation

		In category theory, a [Natural Transformation](https://en.wikipedia.org/wiki/Natural_transformation) provides a morphism between Functors while preserving the internal structures. It's one of the most fundamental notions of category theory.


		We don't need to write a law to test the implementation of the `fk` for the above to be true. It's automatically given by its parametric polymorphism.

		This is why a parametric polymorphic function `FunctionK[F, G]` is sometime referred as a Natural Transformation.


		This is why a parametric polymorphic function `FunctionK[F, G]` is sometime referred as a Natural Transformation.
		However, they are two different concepts. `FunctionK` is a stronger and stricter construction than Natural Transformation.


		In category theory, a [Natural Transformation](https://en.wikipedia.org/wiki/Natural_transformation) provides a morphism between Functors while preserving the internal structure. It's one of the most fundamental notions of category theory.

		If we have two `Functor`s: `F` and `G`, then, being parametric polymorphic, `FunctionK[F, G]` is automatically a Natural Transformation between them. That is, if we have a `fk: F ~> G`, then for any combination of `A`, `B` and function `f: A => B`, the following two functions are equivalent:

added some category theory into FunctionK document #1636

added some category theory into FunctionK document #1636

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