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?
3. The magic of functional composition
4. add2(1) add2(x) = x + 2 3
5. sub1(add2(1)) sub1(x) = x - 1 2
6. crunch(5) crunch(x) = sub1(add2(x)) 6
7. crunch(5) crunch = sub1 . add2 6
8. crunch([1,2,3]) WTF?????
9. FUNCTORS
10. fcrunch([1,2,3]) fcrunch = fmap crunch [2,3,4]
11. class Functor f where fmap :: (a -> b) -> f a -> f b
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”.
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
14. instance Functor [] where fmap _ [] = [] fmap g (x:xs) = g x : fmap g xs -- or we could just say fmap = map
15. instance Functor Maybe where fmap _ Nothing = Nothing fmap g (Just a) = Just (g a)
16. •[] •Tree •Map •Sequence •Maybe •IO
17. MATH TIME
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”
19. Categories are bags of stuff that hold: • Objects • Morphisms Math
20. Categories are bags of stuff that hold: • Types • Functions Code
21. add2 sub1 Number f add2 f sub1 f Number Functor
22. add2 sub1 10 fmap add2 fmap sub1 [10] [] LIFT
23. FUNCTOR RULES Math
24. FUNCTOR RULES •fmap id x = x •fmap(g.h) = (fmap g).(fmap h) Code
25. BROKEN FUNCTOR instance Functor [] where fmap _ [] = [] fmap g (x:xs) = g x : g x : fmap g xs