Theorems · Theorem · category theory
CategoryTheory.ShortComplex.exact_of_f_is_kernel
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
(S : CategoryTheory.ShortComplex C) (hS : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι S.f ⋯))
[S.HasHomology], S.Exact- Cited by
- 22 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.
Cites30
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.Category.assocproof · cited by 6,433
- CategoryTheory.CategoryStruct.idproof · cited by 6,235
- CategoryTheory.Preadditivestatement and proof · cited by 3,309
- CategoryTheory.Category.id_compproof · cited by 1,998
- 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.Limits.WalkingParallelPairstatement · cited by 781
Cited by22
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.exact_kernelproof · cited by 7
- CategoryTheory.Abelian.epiWithInjectiveKernel_iffproof · cited by 3
- CategoryTheory.Functor.preservesHomology_of_map_exactproof · cited by 2
- CategoryTheory.ShortComplex.quasiIso_iff_of_zerosproof · cited by 2
- CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcycles_exactproof · cited by 2
- CategoryTheory.InjectiveResolution.exact₀proof · cited by 2
- CategoryTheory.ObjectProperty.isoModSerre_isInvertedBy_iffproof · cited by 1
- CategoryTheory.exact_d_fproof · cited by 1
- CategoryTheory.ShortComplex.exact_and_mono_f_iff_f_is_kernelproof · cited by 1
- CategoryTheory.IsGrothendieckAbelian.IsPresentable.injectivity₀.epi_fproof · cited by 1
- CategoryTheory.ShortComplex.SnakeInput.exact_C₁_upproof · cited by 1
- CategoryTheory.ShortComplex.Exact.map_of_mono_of_preservesKernelproof · cited by 1