Theorems · Definition · category theory

CategoryTheory.ShortComplex.homologyIsCokernel

{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.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ S.homologyπ ⋯)

The homology S.homology of a short complex is the cokernel of the morphism S.toCycles : S.X₁ ⟶ S.cycles.

Defined in: Mathlib.Algebra.Homology.ShortComplex.Homology
Cited by: 5 results in Mathlib
Foundations: Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.ShortComplex.HasHomology

Mathlib docs Source Loogle LeanSearch

Around this declaration

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.X₁statement · cited by 889
CategoryTheory.Limits.WalkingParallelPairstatement · cited by 781
CategoryTheory.Limits.IsColimitstatement · cited by 773
CategoryTheory.Limits.parallelPairstatement · cited by 766
CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
CategoryTheory.ShortComplex.cyclesstatement · cited by 220
CategoryTheory.ShortComplex.homologystatement · cited by 216
CategoryTheory.Limits.CokernelCofork.ofπstatement · cited by 77

Cited by9

Results whose statement or proof uses this declaration.

CategoryTheory.ShortComplex.HomologyData.ofEpiMonoFactorisation.leftHomologyDataproof · cited by 12
CategoryTheory.ShortComplex.exact_iff_epiproof · cited by 8
CategoryTheory.ShortComplex.LeftHomologyData.canonicalproof · cited by 7
CategoryTheory.ShortComplex.descHomologyproof · cited by 3
CategoryTheory.ShortComplex.liftCycles_comp_homologyπ_eq_zero_iff_up_to_refinementsproof · cited by 3
CategoryTheory.ShortComplex.π_descHomologyproof · cited by 1
HomologicalComplex.homologyIsCokernelproof · cited by 1
CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesE_exactproof · cited by 1
CategoryTheory.ShortComplex.isIso_homologyMap_of_isIso_cyclesMap_of_epiproof · cited by 0