Theorems · Definition · category theory

HomologicalComplex.extendOpcyclesIso

{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] →
              [inst_2 : CategoryTheory.Limits.HasZeroObject C] →
                (K : HomologicalComplex C c) →
                  (e : c.Embedding c') →
                    {j : ι} →
                      {j' : ι'} →
                        e.f j = j' →
                          [inst_3 : K.HasHomology j] →
                            [inst_4 : (K.extend e).HasHomology j'] → (K.extend e).opcycles j' ≅ K.opcycles j

The isomorphism (K.extend e).opcycles j' ≅ K.opcycles j when e.f j = j'.

Defined in

Mathlib.Algebra.Homology.Embedding.ExtendHomology

Cited by

9 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroObjectHomologicalComplex.HasHomologyHomologicalComplex.HasHomology