Theorems · Definition · category theory

CategoryTheory.Abelian.Ext.covariantSequence

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      [CategoryTheory.HasExt C] →
        C →
          {S : CategoryTheory.ShortComplex C} →
            S.ShortExact → (n₀ n₁ : ℕ) → n₀ + 1 = n₁ → CategoryTheory.ComposableArrows AddCommGrpCat 5

Given a short exact short complex S in an abelian category C and an object X : C, this is the long exact sequence Ext X S.X₁ n₀ → Ext X S.X₂ n₀ → Ext X S.X₃ n₀ → Ext X S.X₁ n₁ → Ext X S.X₂ n₁ → Ext X S.X₃ n₁ when n₀ + 1 = n₁

Defined in

Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Cited by

1 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasExt