πŸ“Category

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Β§ Category Theory

Category:

  • a class of Objects (ob(C))

  • a class of Morphisms (Arrows) (hom(C))

    • Morphisms have source and target, no other properties
  • composition

    • βˆ€f:Aβ†’B,βˆ€g:Bβ†’C,βˆƒh:Aβ†’Cβˆ‹h=g∘f\forall f : A \to B, \forall g : B \to C, \exists h : A \to C \ni h = g \circ f
    • βˆ€f:Aβ†’B,βˆ€g:Bβ†’C,βˆ€h:Cβ†’D,(f∘g)∘h=f∘(g∘h)\forall f : A \to B, \forall g : B \to C, \forall h : C \to D, (f \circ g) \circ h = f \circ (g \circ h) (associativity)
    • identity morphism

      • βˆ€A,βˆƒidA:Aβ†’Aβˆ‹βˆ€f:Aβ†’B,idA∘f=f\forall A, \exists id_A : A \to A \ni \forall f : A \to B, id_A \circ f = f (left identity)
      • βˆ€B,βˆƒidB:Bβ†’Bβˆ‹βˆ€f:Aβ†’B,f∘idB=f\forall B, \exists id_B : B \to B \ni \forall f : A \to B, f \circ id_B = f (right identity)

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