Theorems · Definition · category theory

HomologicalComplex.iCyclesIso

{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 j : ι) → c.next i = j → K.d i j = 0 → [inst_2 : K.HasHomology i] → K.cycles i ≅ K.X i

The canonical isomorphism K.cycles i ≅ K.X i when the differential from i is zero.

Defined in

Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Cited by

11 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsHomologicalComplex.HasHomology