Theorems · Definition · category theory

HomologicalComplex.restrictionOpcyclesIso

{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.IsRelIff] →
                    (i j : ι) →
                      c.prev j = i →
                        {i' j' : ι'} →
                          e.f i = i' →
                            e.f j = j' →
                              c'.prev j' = i' →
                                [inst_3 : K.HasHomology j'] →
                                  [inst_4 : (K.restriction e).HasHomology j] →
                                    (K.restriction e).opcycles j ≅ K.opcycles j'

The isomorphism (K.restriction e).opcycles j ≅ K.opcycles j' when e.f j = j' and the predecessors i and i' of j and j' satisfy e.f i = i'.

Defined in

Mathlib.Algebra.Homology.Embedding.RestrictionHomology

Cited by

11 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsComplexShape.Embedding.IsRelIffHomologicalComplex.HasHomologyHomologicalComplex.HasHomology