initial_fst — Mathlib

English

The functor Prod.fst is initial when the first argument category is cofiltred-or-empty and the second is filtered.

Русский

Функтор Prod.fst инициален, если первый аспект — cofiltred-or-empty, а второй — фильтрован.

LaTeX

$$$[IsCofilteredOrEmpty(C)] [IsFiltered(D)] \Rightarrow (\mathrm{Prod.fst}\, C\, D).Initial$$$

Lean4

instance initial_fst [IsCofilteredOrEmpty C] [IsCofiltered D] : (Prod.fst C D).Initial :=
  by
  apply Functor.initial_of_exists_of_isCofiltered
  · exact fun c => ⟨(c, nonempty.some), ⟨𝟙 c⟩⟩
  · intro c ⟨c', d'⟩ f g
    exact ⟨(eq f g, d'), (eqHom f g, 𝟙 d'), eq_condition _ _⟩

Похожие утверждения