Theorems · Theorem · category theory

CategoryTheory.ShortComplex.exact_and_mono_f_iff_f_is_kernel

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  (S : CategoryTheory.ShortComplex C) [CategoryTheory.Balanced C] [S.HasHomology],
  S.Exact ∧ CategoryTheory.Mono S.f ↔
    Nonempty (CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι S.f ⋯))

Defined in: Mathlib.Algebra.Homology.ShortComplex.Exact
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 CategoryTheory.ShortComplex.HasHomology

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Cites21

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.Monostatement and proof · cited by 893
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 and proof · cited by 664
CategoryTheory.ShortComplex.gstatement · cited by 658

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.Functor.preservesFiniteLimits_tfaeproof · cited by 5