Theorems · Theorem · category theory
CategoryTheory.ShortComplex.LeftHomologyData.exact_iff
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S : CategoryTheory.ShortComplex C} [S.HasHomology] (h : S.LeftHomologyData),
S.Exact ↔ CategoryTheory.Limits.IsZero h.H- Cited by
- 8 results in Mathlib
- Foundations
- Depth 41 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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 · cited by 292
- CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
- CategoryTheory.ShortComplex.LeftHomologyData.Hstatement and proof · cited by 236
- CategoryTheory.ShortComplex.LeftHomologyDatastatement and proof · cited by 212
- CategoryTheory.ShortComplex.LeftHomologyData.homologyIsoproof · cited by 34
- CategoryTheory.Iso.isZero_iffproof · cited by 11
- CategoryTheory.ShortComplex.exact_iff_isZero_homologyproof · cited by 6
Cited by8
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.Exact.epi_f'proof · cited by 5
- CategoryTheory.ShortComplex.HomologyData.exact_iffproof · cited by 4
- CategoryTheory.ShortComplex.LeftHomologyData.exact_iff_epi_f'proof · cited by 4
- CategoryTheory.ShortComplex.exact_map_iff_of_faithfulproof · cited by 3
- CategoryTheory.ShortComplex.HomologyData.exact_iff_i_p_zeroproof · cited by 2
- CategoryTheory.ShortComplex.Exact.map_of_preservesLeftHomologyOfproof · cited by 1
- CategoryTheory.ShortComplex.LeftHomologyData.exact_map_iffproof · cited by 1
- CategoryTheory.ShortComplex.exact_iff_isZero_leftHomologyproof · cited by 0