Theorems · Definition · category theory

CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
      {S₁ S₂ : CategoryTheory.ShortComplex C} →
        (φ : S₁ ⟶ S₂) →
          S₁.LeftHomologyData →
            [CategoryTheory.Epi φ.τ₁] → [CategoryTheory.IsIso φ.τ₂] → [CategoryTheory.Mono φ.τ₃] → S₂.LeftHomologyData

If φ : S₁ ⟶ S₂ is a morphism of short complexes such that φ.τ₁ is epi, φ.τ₂ is an iso and φ.τ₃ is mono, then a left homology data for S₁ induces a left homology data for S₂ with the same K and H fields. The inverse construction is ofEpiOfIsIsoOfMono'.

Defined in

Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Cited by

9 results in Mathlib

Foundations

Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.EpiCategoryTheory.IsIsoCategoryTheory.Mono