Theorems · Theorem · category theory
CategoryTheory.ShortComplex.exact_iff_mono
∀ {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.f = 0 → (S.Exact ↔ CategoryTheory.Mono S.g)- Cited by
- 8 results in Mathlib
- Foundations
- Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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.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.IsIsoproof · cited by 1,156
- CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
- CategoryTheory.Monostatement and proof · cited by 893
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.ShortComplex.X₃statement and proof · cited by 876
- CategoryTheory.ShortComplex.gstatement and proof · cited by 658
- CategoryTheory.ShortComplex.fstatement and proof · cited by 653
Cited by8
Results whose statement or proof uses this declaration.
- CategoryTheory.Functor.exact_tfaeproof · cited by 3
- CategoryTheory.ShortComplex.SnakeInput.mono_δproof · cited by 3
- CategoryTheory.Abelian.epi_of_mono_of_epi_of_mono'proof · cited by 2
- CategoryTheory.Abelian.mono_of_mono_of_mono_of_monoproof · cited by 1
- CategoryTheory.Abelian.tfae_monoproof · cited by 1
- CategoryTheory.ShortComplex.ShortExact.isIso_g_iffproof · cited by 0
- CategoryTheory.Abelian.isIso_of_isIso_of_monoproof · cited by 0