The category of bialgebras over a commutative ring

We introduce the bundled category BialgebraCat of bialgebras over a fixed commutative ring R along with the forgetful functors to CoalgebraCat and AlgebraCat.

This file mimics Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.

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

The category of R-bialgebras.

• carrier : Type v

  The underlying type.

• instRing : Ring self.carrier
• instBialgebra : Bialgebra R self.carrier

instance BialgebraCat.instCoeSortType {R : Type u} [CommRing R] : CoeSort (BialgebraCat R) (Type v)

Equations

• BialgebraCat.instCoeSortType = { coe := fun (x : BialgebraCat R) => x.carrier }

def BialgebraCat.of (R : Type u) [CommRing R] (X : Type v) [Ring X] [Bialgebra R X] : BialgebraCat R

The object in the category of R-bialgebras associated to an R-bialgebra.

Equations

• BialgebraCat.of R X = BialgebraCat.mk X

@[simp]

theorem BialgebraCat.of_instRing (R : Type u) [CommRing R] (X : Type v) [Ring X] [Bialgebra R X] : (of R X).instRing = inst✝

@[simp]

theorem BialgebraCat.of_instBialgebra (R : Type u) [CommRing R] (X : Type v) [Ring X] [Bialgebra R X] : (of R X).instBialgebra = inst✝

@[simp]

theorem BialgebraCat.of_carrier (R : Type u) [CommRing R] (X : Type v) [Ring X] [Bialgebra R X] : (of R X).carrier = X

@[simp]

theorem BialgebraCat.of_comul {R : Type u} [CommRing R] {X : Type v} [Ring X] [Bialgebra R X] : CoalgebraStruct.comul = CoalgebraStruct.comul

@[simp]

theorem BialgebraCat.of_counit {R : Type u} [CommRing R] {X : Type v} [Ring X] [Bialgebra R X] : CoalgebraStruct.counit = CoalgebraStruct.counit

structure BialgebraCat.Hom {R : Type u} [CommRing R] (V W : BialgebraCat R) : Type v

A type alias for BialgHom to avoid confusion between the categorical and algebraic spellings of composition.

• toBialgHom' : V.carrier →ₐc[R] W.carrier

  The underlying BialgHom

theorem BialgebraCat.Hom.ext {R : Type u} {inst✝ : CommRing R} {V W : BialgebraCat R} {x y : V.Hom W} (toBialgHom' : x.toBialgHom' = y.toBialgHom') : x = y

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

Equations

• BialgebraCat.category = CategoryTheory.Category.mk ⋯ ⋯ ⋯

instance BialgebraCat.concreteCategory {R : Type u} [CommRing R] : CategoryTheory.ConcreteCategory (BialgebraCat R) fun (x1 x2 : BialgebraCat R) => x1.carrier →ₐc[R] x2.carrier

Equations

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

@[reducible, inline]

abbrev BialgebraCat.Hom.toBialgHom {R : Type u} [CommRing R] {X Y : BialgebraCat R} (f : X.Hom Y) : X.carrier →ₐc[R] Y.carrier

Turn a morphism in BialgebraCat back into a BialgHom.

Equations

• f.toBialgHom = CategoryTheory.ConcreteCategory.hom f

@[reducible, inline]

abbrev BialgebraCat.ofHom {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [Bialgebra R X] [Bialgebra R Y] (f : X →ₐc[R] Y) : of R X ⟶ of R Y

Typecheck a BialgHom as a morphism in BialgebraCat R.

Equations

• BialgebraCat.ofHom f = CategoryTheory.ConcreteCategory.ofHom f

theorem BialgebraCat.Hom.toBialgHom_injective {R : Type u} [CommRing R] (V W : BialgebraCat R) : Function.Injective toBialgHom

theorem BialgebraCat.hom_ext {R : Type u} [CommRing R] {X Y : BialgebraCat R} (f g : X ⟶ Y) (h : Hom.toBialgHom f = Hom.toBialgHom g) : f = g

@[simp]

theorem BialgebraCat.toBialgHom_comp {R : Type u} [CommRing R] {X Y Z : BialgebraCat R} (f : X ⟶ Y) (g : Y ⟶ Z) : Hom.toBialgHom (CategoryTheory.CategoryStruct.comp f g) = (Hom.toBialgHom g).comp (Hom.toBialgHom f)

@[simp]

theorem BialgebraCat.toBialgHom_id {R : Type u} [CommRing R] {M : BialgebraCat R} : Hom.toBialgHom (CategoryTheory.CategoryStruct.id M) = BialgHom.id R M.carrier

instance BialgebraCat.hasForget {R : Type u} [CommRing R] : CategoryTheory.HasForget (BialgebraCat R)

Equations

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

instance BialgebraCat.hasForgetToAlgebra {R : Type u} [CommRing R] : CategoryTheory.HasForget₂ (BialgebraCat R) (AlgebraCat R)

Equations

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

@[simp]

theorem BialgebraCat.forget₂_algebra_obj {R : Type u} [CommRing R] (X : BialgebraCat R) : (CategoryTheory.forget₂ (BialgebraCat R) (AlgebraCat R)).obj X = AlgebraCat.of R X.carrier

@[simp]

theorem BialgebraCat.forget₂_algebra_map {R : Type u} [CommRing R] (X Y : BialgebraCat R) (f : X ⟶ Y) : (CategoryTheory.forget₂ (BialgebraCat R) (AlgebraCat R)).map f = AlgebraCat.ofHom ↑(Hom.toBialgHom f)

instance BialgebraCat.hasForgetToCoalgebra {R : Type u} [CommRing R] : CategoryTheory.HasForget₂ (BialgebraCat R) (CoalgebraCat R)

Equations

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

@[simp]

theorem BialgebraCat.forget₂_coalgebra_obj {R : Type u} [CommRing R] (X : BialgebraCat R) : (CategoryTheory.forget₂ (BialgebraCat R) (CoalgebraCat R)).obj X = CoalgebraCat.of R X.carrier

@[simp]

theorem BialgebraCat.forget₂_coalgebra_map {R : Type u} [CommRing R] (X Y : BialgebraCat R) (f : X ⟶ Y) : (CategoryTheory.forget₂ (BialgebraCat R) (CoalgebraCat R)).map f = CoalgebraCat.ofHom ↑(Hom.toBialgHom f)

def BialgEquiv.toBialgebraCatIso {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [Bialgebra R X] [Bialgebra R Y] (e : X ≃ₐc[R] Y) : BialgebraCat.of R X ≅ BialgebraCat.of R Y

Build an isomorphism in the category BialgebraCat R from a BialgEquiv.

Equations

• e.toBialgebraCatIso = { hom := BialgebraCat.ofHom ↑e, inv := BialgebraCat.ofHom ↑e.symm, hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem BialgEquiv.toBialgebraCatIso_inv {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [Bialgebra R X] [Bialgebra R Y] (e : X ≃ₐc[R] Y) : e.toBialgebraCatIso.inv = BialgebraCat.ofHom ↑e.symm

@[simp]

theorem BialgEquiv.toBialgebraCatIso_hom {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [Bialgebra R X] [Bialgebra R Y] (e : X ≃ₐc[R] Y) : e.toBialgebraCatIso.hom = BialgebraCat.ofHom ↑e

@[simp]

theorem BialgEquiv.toBialgebraCatIso_refl {R : Type u} [CommRing R] {X : Type v} [Ring X] [Bialgebra R X] : (refl R X).toBialgebraCatIso = CategoryTheory.Iso.refl (BialgebraCat.of R X)

@[simp]

theorem BialgEquiv.toBialgebraCatIso_symm {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Ring Y] [Bialgebra R X] [Bialgebra R Y] (e : X ≃ₐc[R] Y) : e.symm.toBialgebraCatIso = e.toBialgebraCatIso.symm

@[simp]

theorem BialgEquiv.toBialgebraCatIso_trans {R : Type u} [CommRing R] {X Y Z : Type v} [Ring X] [Ring Y] [Ring Z] [Bialgebra R X] [Bialgebra R Y] [Bialgebra R Z] (e : X ≃ₐc[R] Y) (f : Y ≃ₐc[R] Z) : (e.trans f).toBialgebraCatIso = e.toBialgebraCatIso ≪≫ f.toBialgebraCatIso

def CategoryTheory.Iso.toBialgEquiv {R : Type u} [CommRing R] {X Y : BialgebraCat R} (i : X ≅ Y) : X.carrier ≃ₐc[R] Y.carrier

Build a BialgEquiv from an isomorphism in the category BialgebraCat R.

Equations

• i.toBialgEquiv = { toCoalgHom := (BialgebraCat.Hom.toBialgHom i.hom).toCoalgHom, invFun := ⇑(BialgebraCat.Hom.toBialgHom i.inv), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }

@[simp]

theorem CategoryTheory.Iso.toBialgEquiv_toBialgHom {R : Type u} [CommRing R] {X Y : BialgebraCat R} (i : X ≅ Y) : ↑i.toBialgEquiv = i.hom.toBialgHom'

@[simp]

theorem CategoryTheory.Iso.toBialgEquiv_refl {R : Type u} [CommRing R] {X : BialgebraCat R} : (refl X).toBialgEquiv = BialgEquiv.refl R X.carrier

@[simp]

theorem CategoryTheory.Iso.toBialgEquiv_symm {R : Type u} [CommRing R] {X Y : BialgebraCat R} (e : X ≅ Y) : e.symm.toBialgEquiv = e.toBialgEquiv.symm

@[simp]

theorem CategoryTheory.Iso.toBialgEquiv_trans {R : Type u} [CommRing R] {X Y Z : BialgebraCat R} (e : X ≅ Y) (f : Y ≅ Z) : (e ≪≫ f).toBialgEquiv = e.toBialgEquiv.trans f.toBialgEquiv

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