weakEquivalence_of_precomp — Mathlib

English

If f: X → Y is a weak equivalence with X,Y cofibrant and X,Y fibrant, there exists g: Y → X with LeftHomotopyRel (f ≫ g) id_X and LeftHomotopyRel (g ≫ f) id_Y.

Русский

Если f: X→Y — слабая эквивалентность, X,Y ко-фибрива и фибривант, существует g: Y→X с LeftHomotopyRel (f ≫ g) id_X и LeftHomotopyRel (g ≫ f) id_Y.

LaTeX

$$$\exists g:\; Y\to X\;\text{ such that } (f\circ g)\sim_{\mathrm{L}} id_X \wedge (g\circ f)\sim_{\mathrm{L}} id_Y$$$

Lean4

theorem weakEquivalence_of_precomp [hf : WeakEquivalence f] [hfg : WeakEquivalence (f ≫ g)] : WeakEquivalence g :=
  by
  rw [weakEquivalence_iff] at hf hfg ⊢
  exact of_precomp _ _ _ hf hfg

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