Theorems · Definition · category theory

CategoryTheory.ShortComplex.homologyIsKernel

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
      (S : CategoryTheory.ShortComplex C) →
        [inst_2 : S.HasHomology] → CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι S.homologyι ⋯)

The homology S.homology of a short complex is the kernel of the morphism S.fromOpcycles : S.opcycles ⟶ S.X₃.

Defined in: Mathlib.Algebra.Homology.ShortComplex.Homology
Cited by: 4 results in Mathlib
Foundations: Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.ShortComplex.HasHomology

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites19

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 · cited by 32,603
CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.ShortComplex.X₃statement · cited by 876
CategoryTheory.Limits.WalkingParallelPairstatement · cited by 781
CategoryTheory.Limits.parallelPairstatement · cited by 766
CategoryTheory.Limits.IsLimitstatement · cited by 664
CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
CategoryTheory.ShortComplex.homologystatement · cited by 216
CategoryTheory.ShortComplex.opcyclesstatement · cited by 192
CategoryTheory.Limits.KernelFork.ofιstatement · cited by 70

Cited by8

Results whose statement or proof uses this declaration.

CategoryTheory.ShortComplex.exact_iff_monoproof · cited by 8
CategoryTheory.ShortComplex.RightHomologyData.canonicalproof · cited by 7
CategoryTheory.ShortComplex.HomologyData.ofEpiMonoFactorisation.rightHomologyDataproof · cited by 5
CategoryTheory.ShortComplex.liftHomologyproof · cited by 3
HomologicalComplex.homologyIsKernelproof · cited by 1
CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesE_exactproof · cited by 1
CategoryTheory.ShortComplex.liftHomology_ιproof · cited by 1
CategoryTheory.ShortComplex.isIso_homologyMap_of_isIso_opcyclesMap_of_monoproof · cited by 0