Category instance for algebras over a commutative ring

We introduce the bundled category AlgebraCat of algebras over a fixed commutative ring R along with the forgetful functors to RingCat and ModuleCat. We furthermore show that the functor associating to a type the free R-algebra on that type is left adjoint to the forgetful functor.

structure AlgebraCat (R : Type u) [CommRing R] : Type (max u (v + 1))

The category of R-algebras and their morphisms.

• carrier : Type v
• isRing : Ring ↑self
• isAlgebra : Algebra R ↑self

@[reducible, inline]

abbrev AlgebraCatMax (R : Type u₁) [CommRing R] : Type (max u₁ ((max v₁ v₂) + 1))

An alias for AlgebraCat.{max u₁ u₂}, to deal around unification issues. Since the universe the ring lives in can be inferred, we put that last.

Equations

• AlgebraCatMax R = AlgebraCat R

instance AlgebraCat.instCoeSortType (R : Type u) [CommRing R] : CoeSort (AlgebraCat R) (Type v)

Equations

• AlgebraCat.instCoeSortType R = { coe := AlgebraCat.carrier }

instance AlgebraCat.instCategory (R : Type u) [CommRing R] : CategoryTheory.Category.{v, max (v + 1) u} (AlgebraCat R)

Equations

• AlgebraCat.instCategory R = CategoryTheory.Category.mk ⋯ ⋯ ⋯

instance AlgebraCat.instFunLikeHomCarrier (R : Type u) [CommRing R] {M : AlgebraCat R} {N : AlgebraCat R} : FunLike (M ⟶ N) ↑M ↑N

Equations

• AlgebraCat.instFunLikeHomCarrier R = AlgHom.funLike

instance AlgebraCat.instAlgHomClassHomCarrier (R : Type u) [CommRing R] {M : AlgebraCat R} {N : AlgebraCat R} : AlgHomClass (M ⟶ N) R ↑M ↑N

Equations

• ⋯ = ⋯

instance AlgebraCat.instConcreteCategory (R : Type u) [CommRing R] : CategoryTheory.ConcreteCategory (AlgebraCat R)

Equations

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

instance AlgebraCat.instRingObjForget (R : Type u) [CommRing R] {S : AlgebraCat R} : Ring ((CategoryTheory.forget (AlgebraCat R)).obj S)

Equations

• AlgebraCat.instRingObjForget R = inferInstance

instance AlgebraCat.instAlgebraObjForget (R : Type u) [CommRing R] {S : AlgebraCat R} : Algebra R ((CategoryTheory.forget (AlgebraCat R)).obj S)

Equations

• AlgebraCat.instAlgebraObjForget R = inferInstance

instance AlgebraCat.hasForgetToRing (R : Type u) [CommRing R] : CategoryTheory.HasForget₂ (AlgebraCat R) RingCat

Equations

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

instance AlgebraCat.hasForgetToModule (R : Type u) [CommRing R] : CategoryTheory.HasForget₂ (AlgebraCat R) (ModuleCat R)

Equations

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

@[simp]

theorem AlgebraCat.forget₂_module_obj (R : Type u) [CommRing R] (X : AlgebraCat R) : (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj X = ModuleCat.of R ↑X

@[simp]

theorem AlgebraCat.forget₂_module_map (R : Type u) [CommRing R] {X : AlgebraCat R} {Y : AlgebraCat R} (f : X ⟶ Y) : (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).map f = ModuleCat.ofHom (AlgHom.toLinearMap f)

def AlgebraCat.of (R : Type u) [CommRing R] (X : Type v) [Ring X] [Algebra R X] : AlgebraCat R

The object in the category of R-algebras associated to a type equipped with the appropriate typeclasses.

Equations

• AlgebraCat.of R X = AlgebraCat.mk X

def AlgebraCat.ofHom {R : Type u} [CommRing R] {X : Type v} {Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y] (f : X →ₐ[R] Y) : AlgebraCat.of R X ⟶ AlgebraCat.of R Y

Typecheck a AlgHom as a morphism in AlgebraCat R.

Equations

• AlgebraCat.ofHom f = f

@[simp]

theorem AlgebraCat.ofHom_apply {R : Type u} [CommRing R] {X : Type v} {Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y] (f : X →ₐ[R] Y) (x : X) : (AlgebraCat.ofHom f) x = f x

instance AlgebraCat.instInhabited (R : Type u) [CommRing R] : Inhabited (AlgebraCat R)

Equations

• AlgebraCat.instInhabited R = { default := AlgebraCat.of R R }

@[simp]

theorem AlgebraCat.coe_of (R : Type u) [CommRing R] (X : Type u) [Ring X] [Algebra R X] : ↑(AlgebraCat.of R X) = X

@[simp]

theorem AlgebraCat.ofSelfIso_hom {R : Type u} [CommRing R] (M : AlgebraCat R) : M.ofSelfIso.hom = CategoryTheory.CategoryStruct.id M

@[simp]

theorem AlgebraCat.ofSelfIso_inv {R : Type u} [CommRing R] (M : AlgebraCat R) : M.ofSelfIso.inv = CategoryTheory.CategoryStruct.id M

def AlgebraCat.ofSelfIso {R : Type u} [CommRing R] (M : AlgebraCat R) : AlgebraCat.of R ↑M ≅ M

Forgetting to the underlying type and then building the bundled object returns the original algebra.

Equations

• M.ofSelfIso = { hom := CategoryTheory.CategoryStruct.id M, inv := CategoryTheory.CategoryStruct.id M, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem AlgebraCat.id_apply {R : Type u} [CommRing R] {M : ModuleCat R} (m : ↑M) : (CategoryTheory.CategoryStruct.id M) m = m

@[simp]

theorem AlgebraCat.coe_comp {R : Type u} [CommRing R] {M : ModuleCat R} {N : ModuleCat R} {U : ModuleCat R} (f : M ⟶ N) (g : N ⟶ U) : ⇑(CategoryTheory.CategoryStruct.comp f g) = ⇑g ∘ ⇑f

@[simp]

theorem AlgebraCat.free_map (R : Type u) [CommRing R] : ∀ {X Y : Type u} (f : X ⟶ Y), (AlgebraCat.free R).map f = (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f)

@[simp]

theorem AlgebraCat.free_obj_carrier (R : Type u) [CommRing R] (S : Type u) : ↑((AlgebraCat.free R).obj S) = FreeAlgebra R S

def AlgebraCat.free (R : Type u) [CommRing R] : CategoryTheory.Functor (Type u) (AlgebraCat R)

The "free algebra" functor, sending a type S to the free algebra on S.

Equations

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

def AlgebraCat.adj (R : Type u) [CommRing R] : AlgebraCat.free R ⊣ CategoryTheory.forget (AlgebraCat R)

The free/forget adjunction for R-algebras.

Equations

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

instance AlgebraCat.instIsRightAdjointForget (R : Type u) [CommRing R] : (CategoryTheory.forget (AlgebraCat R)).IsRightAdjoint

Equations

• ⋯ = ⋯

@[simp]

theorem AlgEquiv.toAlgebraIso_hom {R : Type u} [CommRing R] {X₁ : Type u} {X₂ : Type u} {g₁ : Ring X₁} {g₂ : Ring X₂} {m₁ : Algebra R X₁} {m₂ : Algebra R X₂} (e : X₁ ≃ₐ[R] X₂) : e.toAlgebraIso.hom = ↑e

@[simp]

theorem AlgEquiv.toAlgebraIso_inv {R : Type u} [CommRing R] {X₁ : Type u} {X₂ : Type u} {g₁ : Ring X₁} {g₂ : Ring X₂} {m₁ : Algebra R X₁} {m₂ : Algebra R X₂} (e : X₁ ≃ₐ[R] X₂) : e.toAlgebraIso.inv = ↑e.symm

def AlgEquiv.toAlgebraIso {R : Type u} [CommRing R] {X₁ : Type u} {X₂ : Type u} {g₁ : Ring X₁} {g₂ : Ring X₂} {m₁ : Algebra R X₁} {m₂ : Algebra R X₂} (e : X₁ ≃ₐ[R] X₂) : AlgebraCat.of R X₁ ≅ AlgebraCat.of R X₂

Build an isomorphism in the category AlgebraCat R from a AlgEquiv between Algebras.

Equations

• e.toAlgebraIso = { hom := ↑e, inv := ↑e.symm, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem CategoryTheory.Iso.toAlgEquiv_apply {R : Type u} [CommRing R] {X : AlgebraCat R} {Y : AlgebraCat R} (i : X ≅ Y) (a : ↑X) : i.toAlgEquiv a = i.hom a

@[simp]

theorem CategoryTheory.Iso.toAlgEquiv_symm_apply {R : Type u} [CommRing R] {X : AlgebraCat R} {Y : AlgebraCat R} (i : X ≅ Y) (a : ↑Y) : i.toAlgEquiv.symm a = i.inv a

def CategoryTheory.Iso.toAlgEquiv {R : Type u} [CommRing R] {X : AlgebraCat R} {Y : AlgebraCat R} (i : X ≅ Y) : ↑X ≃ₐ[R] ↑Y

Build a AlgEquiv from an isomorphism in the category AlgebraCat R.

Equations

• i.toAlgEquiv = let __src := i.hom; { toFun := ⇑i.hom, invFun := ⇑i.inv, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem algEquivIsoAlgebraIso_hom {R : Type u} [CommRing R] {X : Type u} {Y : Type u} [Ring X] [Ring Y] [Algebra R X] [Algebra R Y] (e : X ≃ₐ[R] Y) : algEquivIsoAlgebraIso.hom e = e.toAlgebraIso

@[simp]

theorem algEquivIsoAlgebraIso_inv {R : Type u} [CommRing R] {X : Type u} {Y : Type u} [Ring X] [Ring Y] [Algebra R X] [Algebra R Y] (i : AlgebraCat.of R X ≅ AlgebraCat.of R Y) : algEquivIsoAlgebraIso.inv i = i.toAlgEquiv

def algEquivIsoAlgebraIso {R : Type u} [CommRing R] {X : Type u} {Y : Type u} [Ring X] [Ring Y] [Algebra R X] [Algebra R Y] : (X ≃ₐ[R] Y) ≅ AlgebraCat.of R X ≅ AlgebraCat.of R Y

Algebra equivalences between Algebras are the same as (isomorphic to) isomorphisms in AlgebraCat.

Equations

• algEquivIsoAlgebraIso = { hom := fun (e : X ≃ₐ[R] Y) => e.toAlgebraIso, inv := fun (i : AlgebraCat.of R X ≅ AlgebraCat.of R Y) => i.toAlgEquiv, hom_inv_id := ⋯, inv_hom_id := ⋯ }

instance AlgebraCat.forget_reflects_isos {R : Type u} [CommRing R] : (CategoryTheory.forget (AlgebraCat R)).ReflectsIsomorphisms

Equations

• ⋯ = ⋯

@[simp] lemmas for AlgHom.comp and categorical identities.

@[simp]

theorem AlgHom.comp_id_algebraCat {R : Type u_1} [CommRing R] {G : AlgebraCat R} {H : Type u} [Ring H] [Algebra R H] (f : ↑G →ₐ[R] H) : f.comp (CategoryTheory.CategoryStruct.id G) = f

@[simp]

theorem AlgHom.id_algebraCat_comp {R : Type u_1} [CommRing R] {G : Type u} [Ring G] [Algebra R G] {H : AlgebraCat R} (f : G →ₐ[R] ↑H) : AlgHom.comp (CategoryTheory.CategoryStruct.id H) f = f