Theorems · Theorem · category theory
CochainComplex.isIso_liftCycles_iff
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Abelian C]
(K : CochainComplex C ℕ) {X : C} (φ : X ⟶ K.X 0) [inst_2 : HomologicalComplex.HasHomology K 0]
(hφ : CategoryTheory.CategoryStruct.comp φ (K.d 0 1) = 0),
CategoryTheory.IsIso (HomologicalComplex.liftCycles K φ 1 CochainComplex.isIso_liftCycles_iff._proof_1 hφ) ↔
{ X₁ := X, X₂ := K.X 0, X₃ := K.X 1, f := φ, g := K.d 0 1, zero := hφ }.Exact ∧ CategoryTheory.Mono φ- Cited by
- 0 results in Mathlib
- Foundations
- Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites29
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
- zero_addproof · cited by 2,366
- HomologicalComplex.Xstatement and proof · cited by 1,839
- CategoryTheory.Abelianstatement and proof · cited by 1,753
- CategoryTheory.IsIsostatement and proof · cited by 1,156
- ComplexShape.upstatement and proof · cited by 1,123
- CochainComplexstatement and proof · cited by 1,016
- CategoryTheory.Monostatement and proof · cited by 893
- HomologicalComplex.dstatement and proof · cited by 598
- ComplexShape.Relproof · cited by 518
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