The concrete products in the category of modules are products in the categorical sense.

def ModuleCat.productCone {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) : CategoryTheory.Limits.Fan Z

The product cone induced by the concrete product.

Equations

• ModuleCat.productCone Z = CategoryTheory.Limits.Fan.mk (ModuleCat.of R ((i : ι) → ↑(Z i))) fun (i : ι) => LinearMap.proj i

def ModuleCat.productConeIsLimit {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) : CategoryTheory.Limits.IsLimit (ModuleCat.productCone Z)

The concrete product cone is limiting.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def ModuleCat.piIsoPi {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [CategoryTheory.Limits.HasProduct Z] : ∏ᶜ Z ≅ ModuleCat.of R ((i : ι) → ↑(Z i))

The categorical product of a family of objects in ModuleCat agrees with the usual module-theoretical product.

Equations

• ModuleCat.piIsoPi Z = CategoryTheory.Limits.limit.isoLimitCone { cone := ModuleCat.productCone Z, isLimit := ModuleCat.productConeIsLimit Z }

@[simp]

theorem ModuleCat.piIsoPi_inv_kernel_ι {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [CategoryTheory.Limits.HasProduct Z] (i : ι) : CategoryTheory.CategoryStruct.comp (ModuleCat.piIsoPi Z).inv (CategoryTheory.Limits.Pi.π Z i) = LinearMap.proj i

theorem ModuleCat.piIsoPi_inv_kernel_ι_apply {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [CategoryTheory.Limits.HasProduct Z] (i : ι) (x : (CategoryTheory.forget (ModuleCatMax R)).obj (ModuleCat.of R ((i : ι) → ↑(Z i)))) : (CategoryTheory.Limits.Pi.π Z i) ((ModuleCat.piIsoPi Z).inv x) = (LinearMap.proj i) x

@[simp]

theorem ModuleCat.piIsoPi_hom_ker_subtype {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [CategoryTheory.Limits.HasProduct Z] (i : ι) : CategoryTheory.CategoryStruct.comp (ModuleCat.piIsoPi Z).hom (LinearMap.proj i) = CategoryTheory.Limits.Pi.π Z i

theorem ModuleCat.piIsoPi_hom_ker_subtype_apply {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [CategoryTheory.Limits.HasProduct Z] (i : ι) (x : (CategoryTheory.forget (ModuleCatMax R)).obj (∏ᶜ Z)) : (LinearMap.proj i) ((ModuleCat.piIsoPi Z).hom x) = (CategoryTheory.Limits.Pi.π Z i) x

def ModuleCat.coproductCocone {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] : CategoryTheory.Limits.Cofan Z

The coproduct cone induced by the concrete product.

Equations

• ModuleCat.coproductCocone Z = CategoryTheory.Limits.Cofan.mk (ModuleCat.of R (DirectSum ι fun (i : ι) => ↑(Z i))) fun (i : ι) => DirectSum.lof R ι (fun (i : ι) => ↑(Z i)) i

def ModuleCat.coproductCoconeIsColimit {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] : CategoryTheory.Limits.IsColimit (ModuleCat.coproductCocone Z)

The concrete coproduct cone is limiting.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def ModuleCat.coprodIsoDirectSum {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] [CategoryTheory.Limits.HasCoproduct Z] : ∐ Z ≅ ModuleCat.of R (DirectSum ι fun (i : ι) => ↑(Z i))

The categorical coproduct of a family of objects in ModuleCat agrees with direct sum.

Equations

• ModuleCat.coprodIsoDirectSum Z = CategoryTheory.Limits.colimit.isoColimitCocone { cocone := ModuleCat.coproductCocone Z, isColimit := ModuleCat.coproductCoconeIsColimit Z }

@[simp]

theorem ModuleCat.ι_coprodIsoDirectSum_hom {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] [CategoryTheory.Limits.HasCoproduct Z] (i : ι) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Sigma.ι Z i) (ModuleCat.coprodIsoDirectSum Z).hom = DirectSum.lof R ι (fun (i : ι) => ↑(Z i)) i

theorem ModuleCat.ι_coprodIsoDirectSum_hom_apply {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] [CategoryTheory.Limits.HasCoproduct Z] (i : ι) (x : (CategoryTheory.forget (ModuleCatMax R)).obj (Z i)) : (ModuleCat.coprodIsoDirectSum Z).hom ((CategoryTheory.Limits.Sigma.ι Z i) x) = (DirectSum.lof R ι (fun (i : ι) => ↑(Z i)) i) x

@[simp]

theorem ModuleCat.lof_coprodIsoDirectSum_inv {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] [CategoryTheory.Limits.HasCoproduct Z] (i : ι) : CategoryTheory.CategoryStruct.comp (DirectSum.lof R ι (fun (i : ι) => ↑(Z i)) i) (ModuleCat.coprodIsoDirectSum Z).inv = CategoryTheory.Limits.Sigma.ι Z i

theorem ModuleCat.lof_coprodIsoDirectSum_inv_apply {R : Type u} [Ring R] {ι : Type v} (Z : ι → ModuleCatMax R) [DecidableEq ι] [CategoryTheory.Limits.HasCoproduct Z] (i : ι) (x : (CategoryTheory.forget (ModuleCatMax R)).obj (Z i)) : (ModuleCat.coprodIsoDirectSum Z).inv ((DirectSum.lof R ι (fun (i : ι) => ↑(Z i)) i) x) = (CategoryTheory.Limits.Sigma.ι Z i) x