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Mathlib.Algebra.Category.ModuleCat.Kernels

The concrete (co)kernels in the category of modules are (co)kernels in the categorical sense. #

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def ModuleCat.kernelCone {R : Type u} [Ring R] {M N : ModuleCat R} (f : M ⟶ N) :
CategoryTheory.Limits.KernelFork f

The kernel cone induced by the concrete kernel.

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    def ModuleCat.kernelIsLimit {R : Type u} [Ring R] {M N : ModuleCat R} (f : M ⟶ N) :
    CategoryTheory.Limits.IsLimit (kernelCone f)

    The kernel of a linear map is a kernel in the categorical sense.

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      def ModuleCat.cokernelCocone {R : Type u} [Ring R] {M N : ModuleCat R} (f : M ⟶ N) :
      CategoryTheory.Limits.CokernelCofork f

      The cokernel cocone induced by the projection onto the quotient.

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        def ModuleCat.cokernelIsColimit {R : Type u} [Ring R] {M N : ModuleCat R} (f : M ⟶ N) :
        CategoryTheory.Limits.IsColimit (cokernelCocone f)

        The projection onto the quotient is a cokernel in the categorical sense.

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          theorem ModuleCat.hasKernels_moduleCat {R : Type u} [Ring R] :
          CategoryTheory.Limits.HasKernels (ModuleCat R)

          The category of R-modules has kernels, given by the inclusion of the kernel submodule.

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          theorem ModuleCat.hasCokernels_moduleCat {R : Type u} [Ring R] :
          CategoryTheory.Limits.HasCokernels (ModuleCat R)

          The category of R-modules has cokernels, given by the projection onto the quotient.

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          noncomputable def ModuleCat.kernelIsoKer {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) :
          CategoryTheory.Limits.kernel f ≅ of R ↥(LinearMap.ker (Hom.hom f))

          The categorical kernel of a morphism in ModuleCat agrees with the usual module-theoretical kernel.

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            @[simp]
            theorem ModuleCat.kernelIsoKer_inv_kernel_ι {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) :
            CategoryTheory.CategoryStruct.comp (kernelIsoKer f).inv (CategoryTheory.Limits.kernel.ι f) = ofHom (LinearMap.ker (Hom.hom f)).subtype
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            theorem ModuleCat.kernelIsoKer_inv_kernel_ι_apply {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) (x : CategoryTheory.ToType (of R ↥(LinearMap.ker (Hom.hom f)))) :
            (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.kernel.ι f)) ((CategoryTheory.ConcreteCategory.hom (kernelIsoKer f).inv) x) = ↑x
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            @[simp]
            theorem ModuleCat.kernelIsoKer_hom_ker_subtype {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) :
            CategoryTheory.CategoryStruct.comp (kernelIsoKer f).hom (ofHom (LinearMap.ker (Hom.hom f)).subtype) = CategoryTheory.Limits.kernel.ι f
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            theorem ModuleCat.kernelIsoKer_hom_ker_subtype_apply {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) (x : CategoryTheory.ToType (CategoryTheory.Limits.kernel f)) :
            ↑((CategoryTheory.ConcreteCategory.hom (kernelIsoKer f).hom) x) = (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.kernel.ι f)) x
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            noncomputable def ModuleCat.cokernelIsoRangeQuotient {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) :
            CategoryTheory.Limits.cokernel f ≅ of R (↑H ⧸ LinearMap.range (Hom.hom f))

            The categorical cokernel of a morphism in ModuleCat agrees with the usual module-theoretical quotient.

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              theorem ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) :
              CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.π f) (cokernelIsoRangeQuotient f).hom = ofHom (LinearMap.range (Hom.hom f)).mkQ
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              theorem ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom_apply {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) (x : CategoryTheory.ToType H) :
              (CategoryTheory.ConcreteCategory.hom (cokernelIsoRangeQuotient f).hom) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.cokernel.π f)) x) = Submodule.Quotient.mk x
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              theorem ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) :
              CategoryTheory.CategoryStruct.comp (ofHom (LinearMap.range (Hom.hom f)).mkQ) (cokernelIsoRangeQuotient f).inv = CategoryTheory.Limits.cokernel.π f
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              theorem ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv_apply {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) (x : CategoryTheory.ToType (of R ↑H)) :
              (CategoryTheory.ConcreteCategory.hom (cokernelIsoRangeQuotient f).inv) (Submodule.Quotient.mk x) = (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.cokernel.π f)) x
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              theorem ModuleCat.cokernel_π_ext {R : Type u} [Ring R] {M N : ModuleCat R} (f : M ⟶ N) {x y : ↑N} (m : ↑M) (w : x = y + (CategoryTheory.ConcreteCategory.hom f) m) :
              (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.cokernel.π f)) x = (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.cokernel.π f)) y