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added some category theory into FunctionK document #1636

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merged 7 commits into from
May 17, 2017
Merged

added some category theory into FunctionK document #1636

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f48450d
added some category theory into FunctionK document
kailuowang Apr 25, 2017
64ac09c
correct name
kailuowang Apr 25, 2017
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feedback
kailuowang Apr 25, 2017
c825740
Update functionk.md
kailuowang May 12, 2017
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address feedback
kailuowang May 15, 2017
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Merge branch 'master' into improve-functionK-doc
kailuowang May 17, 2017
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Merge branch 'improve-functionK-doc' of github.com:kailuowang/cats in…
kailuowang May 17, 2017
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22 changes: 22 additions & 0 deletions docs/src/main/tut/datatypes/functionk.md
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Expand Up @@ -121,3 +121,25 @@ type ErrorOr[A] = Either[String, A]
val errorOrFirst: FunctionK[List, ErrorOr] =
λ[FunctionK[List, ErrorOr]](_.headOption.toRight("ERROR: the list was empty!"))
```

## 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.
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@SystemFw SystemFw Apr 25, 2017 •
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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)


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:
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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:

```Scala
(fa:F[A]) => fK(F.map(fa)(f))
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Minor inconsistency: fK and fk.

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This has fK where is is defined above as fk

```
and
```Scala
(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.
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by parametricity


This is why a parametric polymorphic function `FunctionK[F, G]` is sometime referred as a Natural Transformation.
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Pedantry warning (sorry): it should be sometimes, not sometime. :)

However, they are two different concepts. `FunctionK` is a stronger and stricter construction than Natural Transformation.

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

For more details, Bartosz Milewski has written a great blog post titled
["Parametricity: Money for Nothing and Theorems for Free"](https://bartoszmilewski.com/2014/09/22/parametricity-money-for-nothing-and-theorems-for-free/).