Theorems · Theorem · category theory

HomologicalComplex.cyclesOpIso_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.cycles i ⟶ Z),
  CategoryTheory.CategoryStruct.comp (HomologicalComplex.opcyclesMap φ i).op
      (CategoryTheory.CategoryStruct.comp (K.cyclesOpIso i).inv h) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.CategoryStruct.comp (L.cyclesOpIso i).inv
        (HomologicalComplex.cyclesMap ((HomologicalComplex.opFunctor V c).map φ.op) i))
      h

Defined in

Mathlib.Algebra.Homology.Opposite

Cited by

0 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsHomologicalComplex.HasHomologyHomologicalComplex.HasHomology