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Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers

Normal mono categories with finite products and kernels have all equalizers. #

This, and the dual result, are used in the development of abelian categories.

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@[irreducible]
def CategoryTheory.NormalMonoCategory.pullback_of_mono {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.NormalMonoCategory C] {X Y Z : C} (a : X ⟶ Z) (b : Y ⟶ Z) [CategoryTheory.Mono a] [CategoryTheory.Mono b] :
CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.cospan a b)

The pullback of two monomorphisms exists.

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  • ⋯ = ⋯
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    @[irreducible]
    def CategoryTheory.NormalMonoCategory.hasLimit_parallelPair {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.NormalMonoCategory C] {X Y : C} (f g : X ⟶ Y) :
    CategoryTheory.Limits.HasLimit (CategoryTheory.Limits.parallelPair f g)

    The equalizer of f and g exists.

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    • ⋯ = ⋯
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      @[instance 100]
      instance CategoryTheory.NormalMonoCategory.hasEqualizers {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.NormalMonoCategory C] :
      CategoryTheory.Limits.HasEqualizers C

      A NormalMonoCategory category with finite products and kernels has all equalizers.

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      • ⋯ = ⋯
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      theorem CategoryTheory.NormalMonoCategory.epi_of_zero_cokernel {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.NormalMonoCategory C] {X Y : C} (f : X ⟶ Y) (Z : C) (l : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ 0 ⋯)) :
      CategoryTheory.Epi f

      If a zero morphism is a cokernel of f, then f is an epimorphism.

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      theorem CategoryTheory.NormalMonoCategory.epi_of_zero_cancel {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.NormalMonoCategory C] [CategoryTheory.Limits.HasZeroObject C] {X Y : C} (f : X ⟶ Y) (hf : ∀ (Z : C) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f g = 0 → g = 0) :
      CategoryTheory.Epi f

      If f ≫ g = 0 implies g = 0 for all g, then g is a monomorphism.

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      @[irreducible]
      def CategoryTheory.NormalEpiCategory.pushout_of_epi {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.NormalEpiCategory C] {X Y Z : C} (a : X ⟶ Y) (b : X ⟶ Z) [CategoryTheory.Epi a] [CategoryTheory.Epi b] :
      CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.span a b)

      The pushout of two epimorphisms exists.

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      • ⋯ = ⋯
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        @[irreducible]
        def CategoryTheory.NormalEpiCategory.hasColimit_parallelPair {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.NormalEpiCategory C] {X Y : C} (f g : X ⟶ Y) :
        CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.parallelPair f g)

        The coequalizer of f and g exists.

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        • ⋯ = ⋯
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          @[instance 100]
          instance CategoryTheory.NormalEpiCategory.hasCoequalizers {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.NormalEpiCategory C] :
          CategoryTheory.Limits.HasCoequalizers C

          A NormalEpiCategory category with finite coproducts and cokernels has all coequalizers.

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          • ⋯ = ⋯
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          theorem CategoryTheory.NormalEpiCategory.mono_of_zero_kernel {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.NormalEpiCategory C] {X Y : C} (f : X ⟶ Y) (Z : C) (l : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι 0 ⋯)) :
          CategoryTheory.Mono f

          If a zero morphism is a kernel of f, then f is a monomorphism.

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          theorem CategoryTheory.NormalEpiCategory.mono_of_cancel_zero {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasFiniteCoproducts C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.NormalEpiCategory C] [CategoryTheory.Limits.HasZeroObject C] {X Y : C} (f : X ⟶ Y) (hf : ∀ (Z : C) (g : Z ⟶ X), CategoryTheory.CategoryStruct.comp g f = 0 → g = 0) :
          CategoryTheory.Mono f

          If g ≫ f = 0 implies g = 0 for all g, then f is a monomorphism.