BM Category Theory 3.x Monoid, Kleisli Category (Monad)… Free Monoid 

Math Online Tom Circle

[Continued from 1.1 to 2.2]

3.1 MonoidM (m, m)

Same meaning in Category as in Set: Only 1 object, Associative, Identity

Thin / Thick Category:

  • “Thin” with only 1 arrow between 2 objects;
  • “Thick” with many arrows between 2 objects.

Arrow : relation between 2 objects. We don’t care what an arrow actually is (may be total / partial order relations like = or $latex leq $, or any relation), just treat arrow abstractly.

Note: Category Theory’s “Abstract Nonsense” is like Buddhism “空即色, 色即空” (Form = Emptiness).

Example ofMonoid: String Concatenation: identity = Null string.

Strong Typing: function f calls function g, both types must match.

Weak Typing: no need to match type. eg. Monoid.

Category induces a Hom-Set: (Set of “Arrows”, aka Homomorphism同态, which preserves structure after the “Arrow”)

  • C (a, b) : a -> b
  • C (a,a) for Monoid…

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