cechNerveEquiv — Mathlib

English

Second component of the equivalence proof for the Čech nerve isomorphism.

Русский

Вторая часть доказательства эквивалентности для изоморфности Čech нерва.

LaTeX

$$$ cechNerveEquiv._proof_2 \\;: \\; \\text{Eq } (CategoryTheory.SimplicialObject ...) (CategoryTheory.SimplicialObject ...) $$$

Lean4

/-- A helper function used in defining the Čech adjunction. -/
@[simps]
def cechNerveEquiv (X : SimplicialObject.Augmented C) (F : Arrow C) :
    (Augmented.toArrow.obj X ⟶ F) ≃ (X ⟶ F.augmentedCechNerve)
    where
  toFun := equivalenceLeftToRight _ _
  invFun := equivalenceRightToLeft _ _
  left_inv A := by ext <;> simp
  right_inv := by
    intro A
    ext x : 2
    · refine WidePullback.hom_ext _ _ _ (fun j => ?_) ?_
      · simp
        rfl
      · simpa using congr_app A.w.symm x
    · simp

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