Theorems · Theorem · category theory
CategoryTheory.ShortComplex.ab_exact_iff
∀ (S : CategoryTheory.ShortComplex Ab),
S.Exact ↔
∀ (x₂ : ↑S.X₂),
(CategoryTheory.ConcreteCategory.hom S.g) x₂ = 0 → ∃ x₁, (CategoryTheory.ConcreteCategory.hom S.f) x₁ = x₂- Defined in
- Mathlib.Algebra.Homology.ShortComplex.Ab
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
- AddMonoidHomstatement · cited by 3,230
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
- CategoryTheory.ShortComplex.X₁statement and proof · cited by 889
- CategoryTheory.ShortComplex.X₃statement · cited by 876
- CategoryTheory.ShortComplex.gstatement and proof · cited by 658
- CategoryTheory.ShortComplex.fstatement and proof · cited by 653
- AddCommGrpCatstatement · cited by 462
- AddCommGrpCat.carrierstatement and proof · cited by 407
- CategoryTheory.ShortComplex.Exactstatement · cited by 292
Cited by10
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.ab_exact_iff_function_exactproof · cited by 8
- CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃proof · cited by 5
- CategoryTheory.Abelian.Ext.covariant_sequence_exact₁proof · cited by 5
- CategoryTheory.Abelian.Ext.covariant_sequence_exact₃proof · cited by 3
- CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁proof · cited by 2
- TopCat.Presheaf.sections_exact_of_exactproof · cited by 1
- CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂proof · cited by 1
- CategoryTheory.ShortComplex.exact_iff_of_hasForgetproof · cited by 1
- CategoryTheory.Abelian.Ext.covariant_sequence_exact₂proof · cited by 1
- CategoryTheory.ShortComplex.ab_exact_iff_ker_le_rangeproof · cited by 1