Functors

Part 1 of the Functors, Monoids, and Monads presentation here: http://www.meetup.com/dynamic/events/222501291/

Transcript

1. A monad is a monoid in the category of endofunctors,
   what's the problem?
   

   View Slide

2. View Slide

3. The magic of functional composition
   

   View Slide

4. add2(1)
   add2(x) = x + 2
   3
   

   View Slide

5. sub1(add2(1))
   sub1(x) = x - 1
   2
   

   View Slide

6. crunch(5)
   crunch(x) = sub1(add2(x))
   6
   

   View Slide

7. crunch(5)
   crunch = sub1 . add2
   6
   

   View Slide

8. crunch([1,2,3])
   WTF?????
   

   View Slide

9. FUNCTORS
   

   View Slide

10. fcrunch([1,2,3])
    fcrunch = fmap crunch
    [2,3,4]
    

    View Slide

11. class Functor f where
    fmap :: (a -> b) -> f a -> f b
    

    View Slide

12. fmap transforms a “normal” function (g :: a -> b) into
    one which operates over containers/contexts (fmap
    g :: f a -> f b). This transformation is often referred to
    as a lift; fmap “lifts” a function from the “normal world”
    into the “f world”.
    

    View Slide

13. *> :t add2
    add2 :: Num a => a -> a
    *> :t crunch
    crunch :: Num a => a -> a
    *> :t fcrunch
    fcrunch :: (Functor f, Num b) => f b -> f b
    

    View Slide

14. instance Functor [] where
    fmap _ [] = []
    fmap g (x:xs) = g x : fmap g xs
    -- or we could just say fmap = map
    

    View Slide

15. instance Functor Maybe where
    fmap _ Nothing = Nothing
    fmap g (Just a) = Just (g a)
    

    View Slide

16. •[]
    •Tree
    •Map
    •Sequence
    •Maybe
    •IO
    

    View Slide

17. MATH TIME
    

    View Slide

18. –Wikipedia
    “In mathematics, a functor is a type of mapping
    between categories, which is applied in category
    theory. Functors can be thought of as
    homomorphisms between categories”
    

    View Slide

19. Categories are bags of stuff that hold:
    • Objects
    • Morphisms
    Math
    

    View Slide

20. Categories are bags of stuff that hold:
    • Types
    • Functions
    Code
    

    View Slide

21. add2
    sub1
    Number
    f add2
    f sub1
    f Number
    Functor
    

    View Slide

22. add2
    sub1
    10
    fmap add2
    fmap sub1
    [10]
    []
    LIFT
    

    View Slide

23. FUNCTOR RULES
    Math
    

    View Slide

24. FUNCTOR RULES
    •fmap id x = x
    •fmap(g.h) = (fmap g).(fmap h)
    Code
    

    View Slide

25. BROKEN FUNCTOR
    instance Functor [] where
    fmap _ [] = []
    fmap g (x:xs) = g x : g x : fmap g xs
    

    View Slide