Theorems · Theorem · category theory

CategoryTheory.ShortComplex.Exact.comp_eq_zero

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  {S : CategoryTheory.ShortComplex C},
  S.Exact →
    ∀ {X Y : C} {a : X ⟶ S.X₂},
      CategoryTheory.CategoryStruct.comp a S.g = 0 →
        ∀ {b : S.X₂ ⟶ Y}, CategoryTheory.CategoryStruct.comp S.f b = 0 → CategoryTheory.CategoryStruct.comp a b = 0

Defined in: Mathlib.Algebra.Homology.ShortComplex.Exact
Cited by: 1 results in Mathlib
Foundations: Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Limits.HasZeroMorphisms

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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.Category.assocproof · cited by 6,433
CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
CategoryTheory.ShortComplexstatement and proof · cited by 1,850
CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
CategoryTheory.ShortComplex.X₁statement · cited by 889
CategoryTheory.ShortComplex.X₃statement · cited by 876
CategoryTheory.ShortComplex.gstatement and proof · cited by 658
CategoryTheory.ShortComplex.fstatement and proof · cited by 653
CategoryTheory.Limits.comp_zeroproof · cited by 365

Cited by1

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CategoryTheory.ShortComplex.Exact.comp_eq_zero_assocproof · cited by 0