Theorems · Theorem · category theory
CategoryTheory.ShortComplex.toCycles_i
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
(S : CategoryTheory.ShortComplex C) [inst_2 : S.HasLeftHomology],
CategoryTheory.CategoryStruct.comp S.toCycles S.iCycles = S.f- Cited by
- 16 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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.CategoryStruct.compstatement · cited by 17,999
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.ShortComplex.X₂statement · cited by 1,115
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.ShortComplex.fstatement · cited by 653
- CategoryTheory.ShortComplex.cyclesstatement · cited by 220
- CategoryTheory.ShortComplex.HasLeftHomologystatement and proof · cited by 132
- CategoryTheory.ShortComplex.iCyclesstatement · cited by 100
- CategoryTheory.ShortComplex.leftHomologyDataproof · cited by 83
Cited by16
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.mono_gproof · cited by 10
- CategoryTheory.ShortComplex.exact_iff_epiproof · cited by 8
- CategoryTheory.ShortComplex.Exact.liftFromProjective_compproof · cited by 5
- CategoryTheory.ShortComplex.exact_iff_epi_kernel_liftproof · cited by 3
- CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_invproof · cited by 2
- CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_homproof · cited by 2
- CategoryTheory.ShortComplex.comp_homologyπ_eq_zero_iff_up_to_refinementsproof · cited by 1
- HomologicalComplex.toCycles_cyclesIsoSc'_homproof · cited by 1
- CategoryTheory.Abelian.SpectralObject.δToCycles_cyclesIso_invproof · cited by 1