Theorems · Definition · category theory

HomologicalComplex.restrictionHomologyIso

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

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

Defined in

Mathlib.Algebra.Homology.Embedding.RestrictionHomology

Cited by

11 results in Mathlib

Foundations

Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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