Theorems · Definition · category theory

CategoryTheory.ShortComplex.opcyclesIsCokernel

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] →
      (S : CategoryTheory.ShortComplex C) →
        [inst_2 : S.HasRightHomology] →
          CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ S.pOpcycles ⋯)

Via S.pOpcycles : S.X₂ ⟶ S.opcycles, the object S.opcycles identifies to the cokernel of S.f : S.X₁ ⟶ S.X₂.

Defined in: Mathlib.Algebra.Homology.ShortComplex.RightHomology
Cited by: 5 results in Mathlib
Foundations: Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms CategoryTheory.ShortComplex.HasRightHomology

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Around this declaration

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

Cites17

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.X₂statement · cited by 1,115
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.fstatement · cited by 653
CategoryTheory.ShortComplex.opcyclesstatement · cited by 192
CategoryTheory.ShortComplex.HasRightHomologystatement and proof · cited by 125

Cited by11

Results whose statement or proof uses this declaration.

CategoryTheory.ShortComplex.isoOpcyclesOfIsColimitproof · cited by 12
CategoryTheory.ShortComplex.Exact.gIsCokernelproof · cited by 11
CategoryTheory.ShortComplex.RightHomologyData.canonicalproof · cited by 7
CategoryTheory.ComposableArrows.Exact.opcyclesIsoCyclesproof · cited by 3
CategoryTheory.ShortComplex.π_isoOpcyclesOfIsColimit_homproof · cited by 3
CategoryTheory.ShortComplex.pOpcycles_π_isoOpcyclesOfIsColimit_invproof · cited by 2
HomologicalComplex.opcyclesIsCokernelproof · cited by 2
CategoryTheory.ComposableArrows.IsComplex.opcyclesToCyclesproof · cited by 2
CategoryTheory.ComposableArrows.IsComplex.opcyclesToCycles_facproof · cited by 2
CategoryTheory.ShortComplex.comp_pOpcycles_eq_zero_iff_up_to_refinementsproof · cited by 1
CategoryTheory.ShortComplex.quasiIso_iff_isIso_descOpcyclesproof · cited by 1