Theorems · Theorem · category theory
CategoryTheory.ShortComplex.exact_iff_isZero_homology
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) [inst_2 : S.HasHomology], S.Exact ↔ CategoryTheory.Limits.IsZero S.homology- Cited by
- 6 results in Mathlib
- Foundations
- Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.Limits.IsZerostatement and proof · cited by 306
- CategoryTheory.ShortComplex.Exactstatement and proof · cited by 292
- CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
- CategoryTheory.ShortComplex.LeftHomologyData.Hproof · cited by 236
- CategoryTheory.ShortComplex.homologystatement and proof · cited by 216
- CategoryTheory.ShortComplex.HomologyData.leftproof · cited by 130
- CategoryTheory.ShortComplex.HomologyDataproof · cited by 102
- CategoryTheory.Limits.IsZero.of_isoproof · cited by 35
- CategoryTheory.ShortComplex.LeftHomologyData.homologyIsoproof · cited by 34
Cited by6
Results whose statement or proof uses this declaration.
- HomologicalComplex.exactAt_iff_isZero_homologyproof · cited by 13
- CategoryTheory.ShortComplex.LeftHomologyData.exact_iffproof · cited by 8
- CategoryTheory.ShortComplex.RightHomologyData.exact_iffproof · cited by 6
- CategoryTheory.Abelian.SpectralObject.isZero_E_of_isZero_Hproof · cited by 1
- TopCat.Sheaf.exact_iff_stalkFunctor_map_exactproof · cited by 0
- CategoryTheory.ShortComplex.exact_iff_homology_iso_zeroproof · cited by 0