precompose_isIso_iff — Mathlib

English

For an isomorphism α, IsVanKampenColimit ((Cocones.precompose α).obj c) is equivalent to IsVanKampenColimit c.

Русский

Для изоморфизма α эквивалентно: IsVanKampenColimit ((Cocones.precompose α).obj c) эквивалентно IsVanKampenColimit c.

LaTeX

$$$\\IsVanKampenColimit((\\mathrm{Cocones.precompose}(\\alpha)).{ }_{{}}\\!\\!\\!\\!\\!\\!.obj c) \\iff \\IsVanKampenColimit(c)$$$

Lean4

theorem precompose_isIso_iff {F G : J ⥤ C} (α : F ⟶ G) [IsIso α] {c : Cocone G} :
    IsVanKampenColimit ((Cocones.precompose α).obj c) ↔ IsVanKampenColimit c :=
  ⟨fun hc ↦
    IsVanKampenColimit.of_iso (IsVanKampenColimit.precompose_isIso (inv α) hc) (Cocones.ext (Iso.refl _) (by simp)),
    IsVanKampenColimit.precompose_isIso α⟩

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