homEquiv — Mathlib

English

There is a canonical bijection between morphisms K ⟶ L.extend e and the subtype { φ : K.restriction e ⟶ L // e.HasLift φ }.

Русский

Существуют канонические биекция между морфизмами K ⟶ L.extend e и подсодержанием { φ : K.restriction e ⟶ L // e.HasLift φ }.

LaTeX

$$$$ (K \to L.reset e) \;\simeq\; \{ \phi : K.restriction e \to L \; | \; e.HasLift \phi \} $$$$

Lean4

/-- The bijection between `K ⟶ L.extend e` and the subtype of `K.restriction e ⟶ L`
consisting of morphisms `φ` such that `e.HasLift φ`. -/
@[simps]
noncomputable def homEquiv : (K ⟶ L.extend e) ≃ { φ : K.restriction e ⟶ L // e.HasLift φ }
    where
  toFun ψ := ⟨e.homRestrict ψ, e.homRestrict_hasLift ψ⟩
  invFun φ := e.liftExtend φ.1 φ.2
  left_inv ψ := by simp
  right_inv φ := by simp

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