Theorems · Definition · category theory

HomologicalComplex.truncLEXIsoCycles

{ι : Type u_1} →
  {ι' : Type u_2} →
    {c : ComplexShape ι} →
      {c' : ComplexShape ι'} →
        {C : Type u_3} →
          [inst : CategoryTheory.Category.{v_1, u_3} C] →
            [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
              (K : HomologicalComplex C c') →
                (e : c.Embedding c') →
                  [inst_2 : e.IsTruncLE] →
                    [inst_3 : ∀ (i' : ι'), K.HasHomology i'] →
                      [inst_4 : CategoryTheory.Limits.HasZeroObject C] →
                        {i : ι} → {i' : ι'} → e.f i = i' → e.BoundaryLE i → ((K.truncLE e).X i' ≅ K.cycles i')

The isomorphism (K.truncLE e).X i' ≅ K.cycles i' when e.f i = i' and e.BoundaryLE i holds.

Defined in

Mathlib.Algebra.Homology.Embedding.TruncLE

Cited by

0 results in Mathlib

Foundations

Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsComplexShape.Embedding.IsTruncLEHomologicalComplex.HasHomologyCategoryTheory.Limits.HasZeroObject