iso — Mathlib

English

An alternative form of the inv_comp_pi identity holds in the finite index setting.

Русский

Альтернативная форма тождества inv_comp_pi сохраняется при конечной индексации.

LaTeX

$$$\\text{AltInvCompPi}$$$

Lean4

/-- Given an object `X : C`, the Čech nerve of the hom to the terminal object `X ⟶ ⊤_ C` is
naturally isomorphic to a simplicial object sending `⦋n⦌` to `Xⁿ⁺¹` (when `C` is `G-Set`, this is
`EG`, the universal cover of the classifying space of `G`. -/
def iso (X : C) : (Arrow.mk (terminal.from X)).cechNerve ≅ cechNerveTerminalFrom X :=
  NatIso.ofComponents (fun _ => wideCospan.limitIsoPi _ _)
    (fun {m n} f => by
      dsimp only [cechNerveTerminalFrom, Arrow.cechNerve]
      ext ⟨j⟩
      simp)

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