Theorems · Definition · category theory

HomologicalComplex.truncGE.rightHomologyMapData

{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.IsTruncGE] →
                    [inst_3 : ∀ (i' : ι'), K.HasHomology i'] →
                      [inst_4 : CategoryTheory.Limits.HasZeroObject C] →
                        {i j k : ι} →
                          {j' : ι'} →
                            (hj' : e.f j = j') →
                              (hi : c.prev j = i) →
                                (hk : c.next j = k) →
                                  (hj : e.BoundaryGE j) →
                                    CategoryTheory.ShortComplex.RightHomologyMapData
                                      ((HomologicalComplex.shortComplexFunctor C c' j').map (K.πTruncGE e))
                                      (CategoryTheory.ShortComplex.RightHomologyData.canonical (K.sc j'))
                                      (HomologicalComplex.extend.rightHomologyData (K.truncGE' e) e hj' hi ⋯ hk ⋯
                                        (HomologicalComplex.truncGE'.homologyData K e i j k hk hj' hj).right)

The right homology data which allows to show that K.πTruncGE e induces an isomorphism in homology in degrees j' such that e.f j = j' for some j.

Defined in

Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Cited by

3 results in Mathlib

Foundations

Depth 51 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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