Theorems · Theorem · category theory
CategoryTheory.ShortComplex.exact_of_g_is_cokernel
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
(S : CategoryTheory.ShortComplex C)
(hS : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ S.g ⋯)) [S.HasHomology], S.Exact- Cited by
- 25 results in Mathlib
- Foundations
- Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
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.compproof · cited by 17,999
- CategoryTheory.CategoryStruct.idproof · cited by 6,235
- CategoryTheory.Preadditivestatement and proof · cited by 3,309
- CategoryTheory.Category.comp_idproof · cited by 2,119
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.ShortComplex.X₃statement · cited by 876
- CategoryTheory.Limits.WalkingParallelPairstatement · cited by 781
- CategoryTheory.Limits.IsColimitstatement and proof · cited by 773
Cited by25
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.exact_cokernelproof · cited by 9
- HomologicalComplex.shortComplexTruncLE_shortExactproof · cited by 4
- CategoryTheory.Functor.preservesHomology_of_map_exactproof · cited by 2
- CategoryTheory.ProjectiveResolution.exact₀proof · cited by 2
- CategoryTheory.Abelian.SpectralObject.cokernelSequenceCycles_exactproof · cited by 2
- CategoryTheory.ObjectProperty.isoModSerre_isInvertedBy_iffproof · cited by 1
- CategoryTheory.Functor.preservesHomology_of_preservesMonos_and_cokernelsproof · cited by 1
- CochainComplex.injective_opcyclesproof · cited by 1
- CategoryTheory.ShortComplex.exact_and_epi_g_iff_g_is_cokernelproof · cited by 1
- CategoryTheory.exact_f_dproof · cited by 1