Theorems · Definition · category theory
CategoryTheory.ShortComplex.leftHomologyIso
{C : Type u} →
[inst : CategoryTheory.Category.{v, u} C] →
[inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
(S : CategoryTheory.ShortComplex C) → [inst_2 : S.HasHomology] → S.leftHomology ≅ S.homologyWhen a short complex has homology, this is the canonical isomorphism
S.leftHomology ≅ S.homology.
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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.Isostatement · cited by 3,963
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.Iso.reflproof · cited by 727
- CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
- CategoryTheory.ShortComplex.homologystatement · cited by 216
- CategoryTheory.ShortComplex.HomologyData.leftproof · cited by 130
- CategoryTheory.ShortComplex.leftHomologyDataproof · cited by 83
- CategoryTheory.ShortComplex.leftHomologystatement · cited by 66
- CategoryTheory.ShortComplex.homologyDataproof · cited by 33
- CategoryTheory.ShortComplex.leftHomologyMapIso'proof · cited by 6
Cited by36
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.homologyπproof · cited by 71
- CategoryTheory.ShortComplex.LeftHomologyData.homologyIsoproof · cited by 34
- CategoryTheory.ShortComplex.toCycles_comp_homologyπproof · cited by 12
- CategoryTheory.ShortComplex.homologyπ_naturalityproof · cited by 10
- groupHomology.H1Isoproof · cited by 7
- groupHomology.H2Isoproof · cited by 6
- CategoryTheory.ShortComplex.homologyIsCokernelproof · cited by 5
- CategoryTheory.ShortComplex.homologyOpIsoproof · cited by 5
- CategoryTheory.ShortComplex.homology_π_ιproof · cited by 4
- CategoryTheory.ShortComplex.homologyπ_comp_leftHomologyIso_inv_assocstatement and proof · cited by 4
- groupHomology.H1π_eq_zero_iffproof · cited by 4