injective_of_map_injective — Mathlib

English

If F : C ⥤ D is a functor with full, faithful, and preserves monos, then Injective (F(X)) holds whenever X is injective.

Русский

Если F: C → D полно и полнота, сохраняет моно morphisms, и X инъективен, то F(X) инъективен.

LaTeX

$$$$\\forall {F : C \\to D}\\;[F\\text{Full}] [F\\text{Faithful}] [F\\text{PreservesMonomorphisms}]\\; {X} \\; (\\text{Injective}(X) \\Rightarrow \\text{Injective}(F(X))).$$$$

Lean4

theorem injective_of_map_injective [F.Full] [F.Faithful] [F.PreservesMonomorphisms] {I : C} (hI : Injective (F.obj I)) :
    Injective I where
  factors g f
    _ := by
    obtain ⟨h, fac⟩ := hI.factors (F.map g) (F.map f)
    exact ⟨F.preimage h, F.map_injective (by simp [fac])⟩

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