Theorems · Definition · category theory

CategoryTheory.ShortComplex.ShortExact.gIsCokernel

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [CategoryTheory.Balanced C] →
        {S : CategoryTheory.ShortComplex C} →
          S.ShortExact → CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ S.g ⋯)

If S is a short exact short complex in a balanced category, then S.X₃ is the cokernel of S.f.

Defined in: Mathlib.Algebra.Homology.ShortComplex.ShortExact
Cited by: 1 results in Mathlib
Foundations: Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Balanced

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Cites18

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.Preadditivestatement and proof · cited by 3,309
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.ShortComplex.X₂statement · 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 · cited by 773
CategoryTheory.Limits.parallelPairstatement · cited by 766
CategoryTheory.Epiproof · cited by 688
CategoryTheory.ShortComplex.gstatement and proof · cited by 658

Cited by2

Results whose statement or proof uses this declaration.

CategoryTheory.Functor.preservesFiniteColimits_tfaeproof · cited by 4
CategoryTheory.Abelian.Preradical.isColimitCokernelCoforkObjproof · cited by 0