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] : ∏ 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 }

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)).toPrefunctor.obj (ModuleCat.productCone Z).pt) : (CategoryTheory.Limits.limit.π (CategoryTheory.Discrete.functor Z) { as := i }) ((CategoryTheory.Limits.limit.isoLimitCone { cone := ModuleCat.productCone Z, isLimit := ModuleCat.productConeIsLimit Z }).inv x) = ((ModuleCat.productCone Z).π.app { as := i }) x

@[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_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)).toPrefunctor.obj (CategoryTheory.Limits.limit.cone (CategoryTheory.Discrete.functor Z)).pt) : ((ModuleCat.productCone Z).π.app { as := i }) ((CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso (ModuleCat.productConeIsLimit Z) (CategoryTheory.Limits.limit.isLimit (CategoryTheory.Discrete.functor Z))).inv x) = (CategoryTheory.Limits.limit.π (CategoryTheory.Discrete.functor Z) { as := 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