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

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

The kernel cone induced by the concrete kernel.

Equations

• ModuleCat.kernelCone f = CategoryTheory.Limits.KernelFork.ofι (ModuleCat.asHom (LinearMap.ker f).subtype) ⋯

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

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

Equations

• ModuleCat.kernelIsLimit f = CategoryTheory.Limits.Fork.IsLimit.mk (ModuleCat.kernelCone f) (fun (s : CategoryTheory.Limits.Fork f 0) => LinearMap.codRestrict (LinearMap.ker f) s.ι ⋯) ⋯ ⋯

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

The cokernel cocone induced by the projection onto the quotient.

Equations

• ModuleCat.cokernelCocone f = CategoryTheory.Limits.CokernelCofork.ofπ (ModuleCat.asHom (LinearMap.range f).mkQ) ⋯

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

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

Equations

• ModuleCat.cokernelIsColimit f = CategoryTheory.Limits.Cofork.IsColimit.mk (ModuleCat.cokernelCocone f) (fun (s : CategoryTheory.Limits.Cofork f 0) => (LinearMap.range f).liftQ s.π ⋯) ⋯ ⋯

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.

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.

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

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

Equations

• ModuleCat.kernelIsoKer f = CategoryTheory.Limits.limit.isoLimitCone { cone := ModuleCat.kernelCone f, isLimit := ModuleCat.kernelIsLimit f }

theorem ModuleCat.kernelIsoKer_inv_kernel_ι_apply {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) (x : (CategoryTheory.forget (ModuleCat R)).obj (ModuleCat.of R ↥(LinearMap.ker f))) : (CategoryTheory.Limits.kernel.ι f) ((ModuleCat.kernelIsoKer f).inv x) = (LinearMap.ker f).subtype x

@[simp]

theorem ModuleCat.kernelIsoKer_inv_kernel_ι {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.kernelIsoKer f).inv (CategoryTheory.Limits.kernel.ι f) = (LinearMap.ker f).subtype

theorem ModuleCat.kernelIsoKer_hom_ker_subtype_apply {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) (x : (CategoryTheory.forget (ModuleCat R)).obj (CategoryTheory.Limits.kernel f)) : (LinearMap.ker f).subtype ((ModuleCat.kernelIsoKer f).hom x) = (CategoryTheory.Limits.kernel.ι f) x

@[simp]

theorem ModuleCat.kernelIsoKer_hom_ker_subtype {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.kernelIsoKer f).hom (LinearMap.ker f).subtype = CategoryTheory.Limits.kernel.ι f

noncomputable def ModuleCat.cokernelIsoRangeQuotient {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.Limits.cokernel f ≅ ModuleCat.of R (↑H ⧸ LinearMap.range f)

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

Equations

• ModuleCat.cokernelIsoRangeQuotient f = CategoryTheory.Limits.colimit.isoColimitCocone { cocone := ModuleCat.cokernelCocone f, isColimit := ModuleCat.cokernelIsColimit f }

theorem ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom_apply {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) (x : (CategoryTheory.forget (ModuleCat R)).obj H) : (ModuleCat.cokernelIsoRangeQuotient f).hom ((CategoryTheory.Limits.cokernel.π f) x) = (LinearMap.range f).mkQ x

@[simp]

theorem ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.π f) (ModuleCat.cokernelIsoRangeQuotient f).hom = (LinearMap.range f).mkQ

theorem ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv_apply {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) (x : (CategoryTheory.forget (ModuleCat R)).obj H) : (ModuleCat.cokernelIsoRangeQuotient f).inv ((ModuleCat.asHomLeft (LinearMap.range f).mkQ) x) = (CategoryTheory.Limits.cokernel.π f) x

@[simp]

theorem ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv {R : Type u} [Ring R] {G : ModuleCat R} {H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.asHomLeft (LinearMap.range f).mkQ) (ModuleCat.cokernelIsoRangeQuotient f).inv = CategoryTheory.Limits.cokernel.π f

theorem ModuleCat.cokernel_π_ext {R : Type u} [Ring R] {M : ModuleCat R} {N : ModuleCat R} (f : M ⟶ N) {x : ↑N} {y : ↑N} (m : ↑M) (w : x = y + f m) : (CategoryTheory.Limits.cokernel.π f) x = (CategoryTheory.Limits.cokernel.π f) y