Category

Category is a labelled directed graph with associative composition and identity.

  • identity
    • ${f_{a\to b} \implies f\circ id_a = f = id_b \circ f }$
  • composition
    • ${g_{b→c},\ f_{a→b} \implies (g \circ f)_{a→c} }$
  • associative
    • ${h_{c→d},\ g_{b→c},\ f_{a→b} \implies h \circ (g \circ f) = (h \circ g) \circ f = (h \circ g \circ f)_{a→d}}$

Functor

Functor is a categorical strucure-preserving mapping between categoryies.

  • mapping for every dot and arrow
    • ${x \mapsto F(x)}$
    • ${f \mapsto F(f)}$
  • identity preserving
    • ${F(id_x) = id_{F(a)}}$
  • composition preserving
    • ${F(g) \circ F(f) = F(g \circ f)}$