knowledge-base

Category Theory

• https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_

Lecture 1

Motivation & Philosophy

• https://www.youtube.com/watch?v=I8LbkfSSR58
• (most?) abstract part of mathematics
• Why is CT important for programming?
  • get to highest possible level to get ideas to translate to concrete programming
  • abstracts all kinds of stuff, also geometry, algebra etc
• OO-Programming
  • divide an conquer & composability
  • abstraction (hide details)
  • reusability
  • ! Concurrency does not fit with OO
    • Objects hide mutation
    • share data
    • => data races
• abstraction hast to be on the right level!
• functional programming
  • immutability => hide data races
• for each new language:
  • what are the most complicate features?
• ! Alonzo Church
• ! Curry-Howard-Lambek Isomorphism

Categories

• https://www.youtube.com/watch?v=p54Hd7AmVFU
• Morphism: structure-preserving map from one structure to another structure of same type
• Major Tools
  • Abstraction
  • Composition
  • Identity
• Category: Composition, Identity
• Category:
  • Objects
  • Morphisms (Arrow)
  • Laws / Properties:
    • Composition: for each a =f=> b =g=> c exists a =g°f=> c
    • Identity: for each a exists a =id_a=> a with id_a°f = f & g°id_a = g
    • Associativity: h ° (g ° f) = (h ° g) ° f
• possibly infinite directed graph (edge duplication possible)

Lecture 6

• https://www.youtube.com/watch?v=i9CU4CuHADQ
• Kleisli Arrow:
  • (>=>) :: (a -> Writer b) -> (b -> Writer c) -> (a -> Writer c)

Lecture 6.1: Functors

• https://www.youtube.com/watch?v=FyoQjkwsy7o
• discrete category: only objects and identity morphism
  • => representation of set
• functor: Mapping between categories C & D (for objects & morphism)
  • F :: a -> D
  • f :: a -> b => F f :: F a -> F b
  • h = g ° f => F h = F g ° f = F g ° F f
  • id_a => id_Fa
  • preserves structure
• Collection of all morphisms from X to Y: hom set Hom(X,Y)
• see also: https://en.wikipedia.org/wiki/Hom_functor
• Faithful functor: injective on hom-sets
• Full functor: surjective on hom-sets
• Endo-functor: maps to the same category
• Identity functor: in category C, written 1_C or id_C, maps an object to itself and a morphism to itself

Lecture 10.1: Monads

• https://www.youtube.com/watch?v=gHiyzctYqZ0&