of_final_functor_from_sifted' — Mathlib

English

Auxiliary variant: given a sifted C and a final F: C ⥤ D, D is sifted.

Русский

Вспомогательный вариант: при наличии ситообразной C и финального F: C ⥤ D, D ситообразна.

LaTeX

$$IsSifted D$$

Lean4

/-- Auxiliary version of `IsSifted.of_final_functor_from_sifted` where everything is a
small category. -/
theorem of_final_functor_from_sifted' {D : Type u} [SmallCategory D] [IsSifted C] (F : C ⥤ D) [Final F] : IsSifted D :=
  by
  have : PreservesFiniteProducts (colim : (D ⥤ Type u) ⥤ _) :=
    ⟨fun n ↦ preservesLimitsOfShape_of_natIso (Final.colimIso F)⟩
  exact of_colim_preservesFiniteProducts D

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