Theorems · Theorem · category theory
CategoryTheory.ShortComplex.exact_iff_epi_toCycles
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
(S : CategoryTheory.ShortComplex C) [inst_2 : S.HasHomology], S.Exact ↔ CategoryTheory.Epi S.toCycles- Cited by
- 3 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.
Cites11
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
- CategoryTheory.Preadditivestatement and proof · cited by 3,309
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.Epistatement · cited by 688
- CategoryTheory.ShortComplex.Exactstatement · cited by 292
- CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
- CategoryTheory.ShortComplex.cyclesstatement · cited by 220
- CategoryTheory.ShortComplex.leftHomologyDataproof · cited by 83
- CategoryTheory.ShortComplex.toCyclesstatement · cited by 47
- CategoryTheory.ShortComplex.LeftHomologyData.exact_iff_epi_f'proof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.exact_of_f_is_kernelproof · cited by 22
- CategoryTheory.ShortComplex.exact_iff_exact_up_to_refinementsproof · cited by 14
- CategoryTheory.ShortComplex.exact_iff_epi_kernel_liftproof · cited by 3