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Theorems · Theorem · category theory

CategoryTheory.ShortComplex.Exact.map_of_mono_of_preservesKernel

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Preadditive C]
  [inst_3 : CategoryTheory.Preadditive D] {S : CategoryTheory.ShortComplex C} [CategoryTheory.Balanced C],
  S.Exact →
    ∀ (F : CategoryTheory.Functor C D) [inst_5 : F.PreservesZeroMorphisms] [(S.map F).HasHomology],
      CategoryTheory.Mono S.f →
        CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair S.g 0) F → (S.map F).Exact
Defined in
Mathlib.Algebra.Homology.ShortComplex.Exact
Cited by
1 results in Mathlib
Foundations
Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.BalancedCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.ShortComplex.HasHomology
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CategoryTheory.Abelian.preadditiveCoyonedaObj_map_surjective · cited by 1Abelian.preadditiveCoyone…CategoryTheory.Category · cited by 32673CategoryTheory.CategoryQuiver.Hom · cited by 32603Quiver.HomCategoryTheory.Functor · cited by 16252CategoryTheory.FunctorCategoryTheory.Preadditive · cited by 3309CategoryTheory.PreadditiveCategoryTheory.ShortComplex · cited by 1850CategoryTheory.ShortCompl…CategoryTheory.ShortComplex.X₂ · cited by 1115ShortComplex.X₂CategoryTheory.Mono · cited by 893CategoryTheory.MonoCategoryTheory.ShortComplex.X₁ · cited by 889ShortComplex.X₁CategoryTheory.ShortComplex.X₃ · cited by 876ShortComplex.X₃CategoryTheory.Limits.WalkingParallelPair · cited by 781Limits.WalkingParallelPairCategoryTheory.Limits.parallelPair · cited by 766Limits.parallelPairCategoryTheory.ShortComplex.g · cited by 658ShortComplex.gCategoryTheory.ShortComplex.f · cited by 653ShortComplex.fCategoryTheory.Functor.PreservesZeroMorphisms · cited by 458Functor.PreservesZeroMorp…CategoryTheory.Limits.PreservesLimit · cited by 293Limits.PreservesLimitExact.map_of_mono_of_preserve…CITED BYCITES

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