English
If IsWeakFactorizationSystem(cofibrations, trivialFibrations) holds and all f_i are cofibrations, then Limits.coprod.map f is a cofibration.
Русский
Если кофибрации устойчивы и все f_i — кофибрации, то Limits.coprod.map f — кофибрация.
LaTeX
$$$[IsWeakFactorizationSystem(\\mathrm{cofibrations}(C)) (\\mathrm{trivialFibrations}(C))] [\\forall i, \\mathrm{Cofibration}(f_i)] : \\mathrm{Cofibration}(\\mathrm{Limits}.coprod.map f).$$$
Lean4
instance [IsWeakFactorizationSystem (cofibrations C) (trivialFibrations C)] [∀ i, WeakEquivalence (f i)] :
WeakEquivalence (Limits.Pi.map f) := by
rw [weakEquivalence_iff]
exact (MorphismProperty.limMap (W := (trivialFibrations C)) _ (fun ⟨i⟩ ↦ mem_trivialFibrations (f i))).2
