Slide 12
Slide 12 text
Hom functor and Yoneda emedding
Definition.
Category C 〝〒〣 object c ∈ C 〠ର「ɺFunctor
Hom(−, c): Cop → Set ぇɺ࣍〣〽⿸〠ఆٛ『ɻ
1. object d ∈ C 〠ର「ɺHom(d, c) ∈ Set ぇׂ〿〛ɻ
2. f : d → d′ 〠ର「ɺHom(f , c): Hom(d′, c) → Hom(d, c) ぇׂ
〿〛ɻ〔〕「 Hom(f , c) 〤 g : d′ → c ぇ g ◦ f 〠ׂ〿
〛ɻ
〳〔ɺFunctor y : C → SetCop ぇɺc ∈ C 〠 y(c) = Hom(−, c) ぇɺ
f : c → c′ 〠ର「 y(f ): Hom(−, c) ⇒ Hom(−, d) ぇׂ〿⿴〛〷
〣〝『ɻ
〔〕「ɺy(f ) 〤 y(f )d : Hom(d, c) → Hom(d, c′),
Hom(d, c) ∋ g → f ◦ g ∈ Hom(d, c′) 〠〽〘〛ఆ〳〷〣〝『ɻ