Theorems · Definition · category theory

HomologicalComplex.descOpcycles

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
      {ι : Type u_2} →
        {c : ComplexShape ι} →
          (K : HomologicalComplex C c) →
            {i : ι} →
              [inst_2 : K.HasHomology i] →
                {A : C} →
                  (k : K.X i ⟶ A) →
                    (j : ι) → c.prev i = j → CategoryTheory.CategoryStruct.comp (K.d j i) k = 0 → (K.opcycles i ⟶ A)

The morphism from K.opcycles i that is induced by an "opcycle", i.e. a morphism from K.X i whose precomposition with the differential is zero.

Defined in

Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Cited by

11 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsHomologicalComplex.HasHomology