Programming in Scala”

Chapter notes 34
def map[R,A,B](f: A => B): Reader[R, A] => Reader[R, B] =
Reader(r => f compose r.run)
The meaning of join is to pass the same argument as both parameters to a binary function:
def join[R,A](x: Reader[R, Reader[R, A]]): Reader[R, A] =
Reader(r => x.run(r).run(r))
And the meaning of unit is to ignore the argument:
def unit[R,A](a: A): Reader[R, A] = Reader(_ => a)
The reader monad subsumes (and is simpler than) dependency injection¹⁴⁴. See Rúnar’s talks on
dependency injection with the reader monad:
“Dead-simple dependency inection”, from the 2012 Northeast Scala Symposium in Boston¹⁴⁵
“Lambda: the ultimate dependency injection framework”, from the 2012 YOW! Summer Conference
in Brisbane¹⁴⁶
See also Tony Morris’s talk “Dependency injection without the gymnastics”¹⁴⁷ from ETE 2012.
Eilenberg-Moore categories
Another categorical view of monads is through Eilenberg-Moore categories¹⁴⁸. The EM category of
a monad is the category of its algebras¹⁴⁹.
For example, the algebras for the List monad are Scala Monoids. The EM category of the List monad
is the category with monoids as its objects and monoid morphisms¹⁵⁰ (see chapter 10) as its arrows.
In general, the EM category for a monad can be found by the following method (source: Theory
Lunch¹⁵¹):
1. An F-algebra for the monad F is a type A together with a function a: F[A] => A such that
a(unit(x)) == x and a(join(x)) == a(map(x)(a)).
2. A morphism of F-algebras from an F-algebra a: F[A] => A to an F-algebra b: F[B] => B is
a function f: A => B such that b(map(x)(f)) == f(a(x)).
¹⁴⁴http://en.wikipedia.org/wiki/Dependency_injection
¹⁴⁵https://www.youtube.com/watch?v=ZasXwtTRkio
¹⁴⁶http://yow.eventer.com/yow-2012-1012/lambda-the-ultimate-dependency-injection-framework-by-runar-bjarnason-1277
¹⁴⁷http://vimeo.com/44502327
¹⁴⁸http://ncatlab.org/nlab/show/Eilenberg-Moore+category
¹⁴⁹https://www.fpcomplete.com/user/bartosz/understanding-algebras
¹⁵⁰http://en.wikipedia.org/wiki/Monoid#Monoid_homomorphisms
¹⁵¹http://theorylunch.wordpress.com/2013/06/06/an-initial-solution-to-the-monad-problem-and-then-some-more/#more-885