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Theorems · Theorem · category theory

CategoryTheory.ShortComplex.exact_iff_mono_cokernel_desc

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  (S : CategoryTheory.ShortComplex C) [S.HasHomology] [inst_3 : CategoryTheory.Limits.HasCokernel S.f],
  S.Exact ↔ CategoryTheory.Mono (CategoryTheory.Limits.cokernel.desc S.f S.g ⋯)
Defined in
Mathlib.Algebra.Homology.ShortComplex.Exact
Cited by
1 results in Mathlib
Foundations
Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.ShortComplex.HasHomologyCategoryTheory.Limits.HasCokernel
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CategoryTheory.ShortComplex.Exact.mono_cokernelDesc · cited by 0Exact.mono_cokernelDescCategoryTheory.Category · cited by 32673CategoryTheory.CategoryCategoryTheory.CategoryStruct.comp · cited by 17999CategoryStruct.compCategoryTheory.Preadditive · cited by 3309CategoryTheory.PreadditiveCategoryTheory.Category.comp_id · cited by 2119Category.comp_idCategoryTheory.ShortComplex · cited by 1850CategoryTheory.ShortCompl…CategoryTheory.ShortComplex.X₂ · cited by 1115ShortComplex.X₂CategoryTheory.Iso.symm · cited by 993Iso.symmCategoryTheory.Mono · cited by 893CategoryTheory.MonoCategoryTheory.ShortComplex.X₁ · cited by 889ShortComplex.X₁CategoryTheory.ShortComplex.X₃ · cited by 876ShortComplex.X₃CategoryTheory.Iso.refl · cited by 727Iso.reflCategoryTheory.ShortComplex.g · cited by 658ShortComplex.gCategoryTheory.ShortComplex.f · cited by 653ShortComplex.fCategoryTheory.Arrow.right · cited by 423Arrow.rightCategoryTheory.Arrow.mk · cited by 421Arrow.mkShortComplex.exact_iff_mono_c…CITED BYCITES

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