Theorems · Theorem · category theory
CategoryTheory.ShortComplex.exact_iff_of_epi_of_isIso_of_mono
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [CategoryTheory.Epi φ.τ₁] [CategoryTheory.IsIso φ.τ₂]
[CategoryTheory.Mono φ.τ₃], S₁.Exact ↔ S₂.Exact- Cited by
- 11 results in Mathlib
- Foundations
- Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
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 and proof · cited by 32,603
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.IsIsostatement and proof · cited by 1,156
- 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.Epistatement and proof · cited by 688
- CategoryTheory.Limits.IsZeroproof · cited by 306
- CategoryTheory.ShortComplex.Exactstatement and proof · cited by 292
Cited by11
Results whose statement or proof uses this declaration.
- CategoryTheory.Functor.preservesFiniteLimits_tfaeproof · cited by 5
- CategoryTheory.Functor.preservesFiniteColimits_tfaeproof · cited by 4
- CategoryTheory.ShortComplex.exact_iff_exact_coimage_πproof · cited by 2
- CategoryTheory.ShortComplex.exact_iff_exact_image_ιproof · cited by 2
- CategoryTheory.Functor.preservesHomology_of_preservesMonos_and_cokernelsproof · cited by 1
- CategoryTheory.exact_d_fproof · cited by 1
- CategoryTheory.exact_f_dproof · cited by 1
- CategoryTheory.ShortComplex.Exact.shortExactproof · cited by 1
- CategoryTheory.Functor.preservesHomology_of_preservesEpis_and_kernelsproof · cited by 0
- HomologicalComplex.HomologySequence.composableArrows₃_exactproof · cited by 0