Theorems · Theorem · category theory

CochainComplex.HomComplex.Cocycle.fromSingleMk_mem_coboundaries_iff

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Preadditive C]
  [inst_2 : CategoryTheory.Limits.HasZeroObject C] {X : C} {K : CochainComplex C ℤ} {p q : ℤ} (f : X ⟶ K.X q) {n : ℤ}
  (h : p + n = q) (q' : ℤ) (hq' : q + 1 = q') (hf : CategoryTheory.CategoryStruct.comp f (K.d q q') = 0) (q'' : ℤ),
  q'' + 1 = q →
    (CochainComplex.HomComplex.Cocycle.fromSingleMk f h q' hq' hf ∈
        CochainComplex.HomComplex.coboundaries ((CochainComplex.singleFunctor C p).obj X) K n ↔
      ∃ g, CategoryTheory.CategoryStruct.comp g (K.d q'' q) = f)

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasZeroObject