Theorems · Theorem · category theory

HomologicalComplex.opcyclesOpIso_inv_naturality_assoc

∀ {ι : Type u_1} {V : Type u_2} [inst : CategoryTheory.Category.{v_1, u_2} V] {c : ComplexShape ι}
  [inst_1 : CategoryTheory.Limits.HasZeroMorphisms V] {K L : HomologicalComplex V c} (φ : K ⟶ L) (i : ι)
  [inst_2 : K.HasHomology i] [inst_3 : L.HasHomology i] {Z : Vᵒᵖ} (h : K.op.opcycles i ⟶ Z),
  CategoryTheory.CategoryStruct.comp (HomologicalComplex.cyclesMap φ i).op
      (CategoryTheory.CategoryStruct.comp (K.opcyclesOpIso i).inv h) =
    CategoryTheory.CategoryStruct.comp (L.opcyclesOpIso i).inv
      (CategoryTheory.CategoryStruct.comp
        (HomologicalComplex.opcyclesMap ((HomologicalComplex.opFunctor V c).map φ.op) i) h)

Defined in

Mathlib.Algebra.Homology.Opposite

Cited by

1 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsHomologicalComplex.HasHomologyHomologicalComplex.HasHomology