Theorems · Theorem · category theory

HomologicalComplex.extendCyclesIso_hom_naturality_assoc

∀ {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 L : HomologicalComplex C c} (φ : K ⟶ L) (e : c.Embedding c')
  {j : ι} {j' : ι'} (hj' : e.f j = j') [inst_3 : K.HasHomology j] [inst_4 : L.HasHomology j]
  [inst_5 : (K.extend e).HasHomology j'] [inst_6 : (L.extend e).HasHomology j'] {Z : C} (h : L.cycles j ⟶ Z),
  CategoryTheory.CategoryStruct.comp (HomologicalComplex.cyclesMap (HomologicalComplex.extendMap φ e) j')
      (CategoryTheory.CategoryStruct.comp (L.extendCyclesIso e hj').hom h) =
    CategoryTheory.CategoryStruct.comp (K.extendCyclesIso e hj').hom
      (CategoryTheory.CategoryStruct.comp (HomologicalComplex.cyclesMap φ j) h)

Defined in

Mathlib.Algebra.Homology.Embedding.ExtendHomology

Cited by

1 results in Mathlib

Foundations

Depth 49 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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