category_theory.md

Category Theory

Data

• Objects
• Arrows
  • Source object
  • Target object
• Rules
  • Identity
    • Arrow where source = target
  • Composition
    • if f: X -> Y and g: Y -> Z, then h: X -> Z = g . f
  • Associativity
    • if f: A -> B and g: B -> C and h: C -> D, then (h . g) . f = h . (g . f)

Monoid

Category with a single object.

trait Monoid[A] {
  def identity: A
  def compose(x: A, y: A): A
}

class Monoid a where
    mempty  :: a
    mappend :: a -> a -> a

Functors

Objects that implement map.

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