Theorems · Definition · category theory

CategoryTheory.ShortComplex.abelianImageToKernelIsKernel

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

Abelian.image S.f is the kernel of kernel.ι S.g ≫ cokernel.π S.f

Defined in: Mathlib.Algebra.Homology.ShortComplex.Abelian
Cited by: 0 results in Mathlib
Foundations: Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Abelian

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Cites23

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 and proof · cited by 32,603
CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.Abelianstatement and proof · cited by 1,753
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 and proof · cited by 658

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.ShortComplex.LeftHomologyData.ofAbelianproof · cited by 4