English
If IsWeakFactorizationSystem(cofibrations, trivialFibrations) and all f_i are weak equivalences, then Limits.Pi.map f is a weak equivalence.
Русский
Если есть слабая факторизационная система кофибраций и тривиальных фибраций, и все f_i являются слабые эквивалентности, то Limits.Pi.map f — слабая эквивалентность.
LaTeX
$$$[IsWeakFactorizationSystem(\\mathrm{cofibrations}(C)) (\\mathrm{trivialFibrations}(C))] [\\forall i, \\mathrm{WeakEquivalence}(f_i)] : \\mathrm{WeakEquivalence}(\\mathrm{Limits}.\\Pi.map f).$$$
Lean4
instance [IsWeakFactorizationSystem (trivialCofibrations C) (fibrations C)] : Fibration (Limits.Pi.map f) :=
by
simp only [fibration_iff] at h ⊢
exact MorphismProperty.limMap _ (fun ⟨i⟩ ↦ h i)
