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] extends CategoryTheory.Bundled : Type (max u (v + 1))

The category of R-bialgebras.

• α : Type v
• str : Ring ↑self.toBundled
• instBialgebra : Bialgebra R ↑self.toBundled

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

Equations

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

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 { α := X, str := inferInstance }

@[simp]

theorem BialgebraCat.of_str (R : Type u) [CommRing R] (X : Type v) [Ring X] [Bialgebra R X] : (BialgebraCat.of R X).str = inferInstance

@[simp]

theorem BialgebraCat.of_α (R : Type u) [CommRing R] (X : Type v) [Ring X] [Bialgebra R X] : ↑(BialgebraCat.of R X).toBundled = X

@[simp]

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

@[simp]

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

@[simp]

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

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

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

• toBialgHom : ↑V.toBundled →ₐc[R] ↑W.toBundled

  The underlying BialgHom

theorem BialgebraCat.Hom.ext {R : Type u} : ∀ {inst : CommRing R} {V W : BialgebraCat R} {x y : V.Hom W}, x.toBialgHom = y.toBialgHom → x = y

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

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

Equations

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

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

@[reducible, inline]

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

Typecheck a BialgHom as a morphism in BialgebraCat R.

Equations

• BialgebraCat.ofHom f = { toBialgHom := f }

@[simp]

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

@[simp]

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

instance BialgebraCat.concreteCategory {R : Type u} [CommRing R] : CategoryTheory.ConcreteCategory (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.toBundled

@[simp]

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

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.toBundled

@[simp]

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

def BialgEquiv.toBialgebraCatIso {R : Type u} [CommRing R] {X : Type v} {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_hom_toBialgHom {R : Type u} [CommRing R] {X : Type v} {Y : Type v} [Ring X] [Ring Y] [Bialgebra R X] [Bialgebra R Y] (e : X ≃ₐc[R] Y) : e.toBialgebraCatIso.hom.toBialgHom = ↑e

@[simp]

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

@[simp]

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

@[simp]

theorem BialgEquiv.toBialgebraCatIso_symm {R : Type u} [CommRing R] {X : Type v} {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 : Type v} {Y : Type v} {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 : BialgebraCat R} {Y : BialgebraCat R} (i : X ≅ Y) : ↑X.toBundled ≃ₐc[R] ↑Y.toBundled

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

Equations

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

@[simp]

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

@[simp]

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

@[simp]

theorem CategoryTheory.Iso.toBialgEquiv_trans {R : Type u} [CommRing R] {X : BialgebraCat R} {Y : BialgebraCat R} {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

Equations

• ⋯ = ⋯