Theorems · Theorem · category theory

CategoryTheory.Abelian.epiWithInjectiveKernel_iff

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Abelian C] {X Y : C}
  (g : X ⟶ Y),
  CategoryTheory.Abelian.epiWithInjectiveKernel g ↔
    ∃ I,
      ∃ (_ : CategoryTheory.Injective I),
        ∃ f,
          ∃ (w : CategoryTheory.CategoryStruct.comp f g = 0),
            Nonempty { X₁ := I, X₂ := X, X₃ := Y, f := f, g := g, zero := w }.Splitting

A morphism g : X ⟶ Y is epi with an injective kernel iff there exists a morphism f : I ⟶ X with I injective such that f ≫ g = 0 and the short complex I ⟶ X ⟶ Y has a splitting.

Defined in

Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Cited by

3 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Abelian