whitehead — Mathlib

English

Let X and Y be cofibrant and fibrant, and f: X → Y a weak equivalence. Then there exists g: Y → X with LeftHomotopyRel (f ∘ g) id_X and LeftHomotopyRel (g ∘ f) id_Y.

Русский

Пусть X и Y кофибрива и фибривант, f: X → Y слабая эквивалентность. Тогда существует g: Y → X так, что левым гомотопическим_REL равенству f∘g и id_X, и g∘f и id_Y.

LaTeX

$$$\exists g:\; Y\to X\;\text{ such that } (f\circ g)\;\text{is left-homotopic to } id_X \text{ and } (g\circ f) \text{ is left-homotopic to } id_Y$$$

Lean4

theorem whitehead [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y] (f : X ⟶ Y) [WeakEquivalence f] :
    ∃ (g : Y ⟶ X), LeftHomotopyRel (f ≫ g) (𝟙 X) ∧ LeftHomotopyRel (g ≫ f) (𝟙 Y) :=
  by
  simp only [leftHomotopyRel_iff_rightHomotopyRel]
  apply RightHomotopyClass.whitehead

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