From Category Theory for Programmers - Series 1

Lecture 1

• A category is a bunch of objects
• Morphisms (or an arrow) is something that goes between two objects
• Object is the primitive, take it on faith, can be anything (think object class in C#)
• Or consider, an object is simply used to mark the beginning and end of arrows
• You can have zero or more arrows going between objects
• Can be infinite/uncountable number of arrows
• Doesn't have to be different objects, can be arrows back to itself
• Composition is a property
• For a → b → c
• If a→b is called 𝑓, and b→c is called 𝑔
• There's a composition 𝑔 ∘ 𝑓, which is identical a→b→c, which must exist
• Different composition tables will give different categories
• For every object (a) there is an identity arrow id_a
• For a→b (call it 𝑓) and b→b
• id_b ∘ 𝑓 = 𝑓
• 𝑓 followed by id_b = 𝑓
• Left identity
• For (different) a,b where a→a and a→b (call it 𝑔)
• 𝑔 ∘ id_a = 𝑔
• Id_a followed by 𝑔 = 𝑔
• Right identity
• Left and right identity are not necessarily the same
• Axiom of category
• Aka associativity
• ℎ ∘ (g ∘ 𝑓) = (ℎ ∘ 𝑔) ∘ 𝑓
• (side note for higher math) If objects form a set, it's a small category; if they don't form a set, it's a large category
• Haskell note: "bottom" is base lower type, for example, describes what gets "returned" if the function enters an infinite loop