Theorems · Definition · category theory

CategoryTheory.ShortComplex.Splitting.fIsKernel

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {S : CategoryTheory.ShortComplex C} →
        [CategoryTheory.Limits.HasZeroObject C] →
          S.Splitting → CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι S.f ⋯)

If a short complex S is equipped with a splitting, then S.X₁ is the kernel of S.g.

Defined in: Mathlib.Algebra.Homology.ShortComplex.Exact
Cited by: 0 results in Mathlib
Foundations: Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Limits.HasZeroObject

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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.Limits.HasZeroObjectstatement and proof · cited by 1,298
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.parallelPairstatement · cited by 766
CategoryTheory.Limits.IsLimitstatement · cited by 664
CategoryTheory.ShortComplex.gstatement · cited by 658

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