whiskerEquivalence — Mathlib

English

If e is an equivalence of index categories J ≌ K, then whiskering a cocone by e preserves van Kampen property: IsVanKampenColimit (c.whisker e.functor).

Русский

Если e — эквивалент индексов J ≌ K, то подшивинг кокона по e сохраняет ван kampen свойство: IsVanKampenColimit(c.whisker e.functor).

LaTeX

$$$\\IsVanKampenColimit(c) \\Rightarrow \\IsVanKampenColimit( c.whisker e.functor)$$$

Lean4

theorem whiskerEquivalence {K : Type*} [Category K] (e : J ≌ K) {F : K ⥤ C} {c : Cocone F} (hc : IsUniversalColimit c) :
    IsUniversalColimit (c.whisker e.functor) := by
  intro F' c' α f e' hα H
  convert
    hc (c'.whisker e.inverse) (whiskerLeft e.inverse α ≫ (e.invFunIdAssoc F).hom) f ?_
      ((hα.whiskerLeft _).comp (NatTrans.equifibered_of_isIso _)) ?_ using
    1
  · exact (IsColimit.whiskerEquivalenceEquiv e.symm).nonempty_congr
  · convert congr_arg (whiskerLeft e.inverse) e'
    ext
    simp
  · intro k
    rw [← Category.comp_id f]
    refine (H (e.inverse.obj k)).paste_vert ?_
    exact IsPullback.of_vert_isIso ⟨by simp⟩

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