Theorems · Theorem · category theory

HomologicalComplex.pOpcycles_opcyclesToCycles_assoc

∀ {C : Type u_1} {ι : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {c : ComplexShape ι} (K : HomologicalComplex C c) (i j : ι)
  [inst_2 : K.HasHomology i] [inst_3 : K.HasHomology j] {Z : C} (h : K.cycles j ⟶ Z),
  CategoryTheory.CategoryStruct.comp (K.pOpcycles i) (CategoryTheory.CategoryStruct.comp (K.opcyclesToCycles i j) h) =
    CategoryTheory.CategoryStruct.comp (K.toCycles i j) h

Defined in: Mathlib.Algebra.Homology.HomologySequence
Cited by: 2 results in Mathlib
Foundations: Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms HomologicalComplex.HasHomology HomologicalComplex.HasHomology

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Cites15

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.Category.assocproof · cited by 6,433
CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
HomologicalComplex.Xstatement · cited by 1,839
HomologicalComplexstatement and proof · cited by 1,691
ComplexShapestatement and proof · cited by 1,684
HomologicalComplex.HasHomologystatement and proof · cited by 342
HomologicalComplex.cyclesstatement and proof · cited by 164
HomologicalComplex.opcyclesstatement · cited by 153
HomologicalComplex.pOpcyclesstatement and proof · cited by 71

Cited by2

Results whose statement or proof uses this declaration.

CategoryTheory.ShortComplex.ShortExact.δ_applyproof · cited by 2
HomologicalComplex.opcyclesToCycles_homologyπproof · cited by 2