Homomorphisms of R-bialgebras

This file defines bundled homomorphisms of R-bialgebras. We simply mimic Mathlib/Algebra/Algebra/Hom.lean.

Main definitions

• BialgHom R A B: the type of R-bialgebra morphisms from A to B.
• Bialgebra.counitBialgHom R A : A →ₐc[R] R: the counit of a bialgebra as a bialgebra homomorphism.

Notations

• A →ₐc[R] B : R-bialgebra homomorphism from A to B.

structure BialgHom (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] extends CoalgHom : Type (max u_2 u_3)

Given R-algebras A, B with comultiplication maps Δ_A, Δ_B and counit maps ε_A, ε_B, an R-bialgebra homomorphism A →ₐc[R] B is an R-algebra map f such that ε_B ∘ f = ε_A and (f ⊗ f) ∘ Δ_A = Δ_B ∘ f.

• toFun : A → B
• map_add' : ∀ (x y : A), self.toFun (x + y) = self.toFun x + self.toFun y
• map_smul' : ∀ (m : R) (x : A), self.toFun (m • x) = (RingHom.id R) m • self.toFun x
• counit_comp : Coalgebra.counit ∘ₗ self.toLinearMap = Coalgebra.counit
• map_comp_comul : TensorProduct.map self.toLinearMap self.toLinearMap ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ self.toLinearMap
• map_one' : self.toFun 1 = 1

  The proposition that the function preserves 1

• map_mul' : ∀ (x y : A), self.toFun (x * y) = self.toFun x * self.toFun y

  The proposition that the function preserves multiplication

@[reducible]

abbrev BialgHom.toMonoidHom {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (self : A →ₐc[R] B) : A →* B

Reinterpret a BialgHom as a MonoidHom

Equations

• self.toMonoidHom = { toFun := self.toFun, map_one' := ⋯, map_mul' := ⋯ }

def «term_→ₐc_» : Lean.TrailingParserDescr

Given R-algebras A, B with comultiplication maps Δ_A, Δ_B and counit maps ε_A, ε_B, an R-bialgebra homomorphism A →ₐc[R] B is an R-algebra map f such that ε_B ∘ f = ε_A and (f ⊗ f) ∘ Δ_A = Δ_B ∘ f.

Equations

• «term_→ₐc_» = Lean.ParserDescr.trailingNode `term_→ₐc_ 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →ₐc ") (Lean.ParserDescr.cat `term 25))

def «term_→ₐc[_]_» : Lean.TrailingParserDescr

Given R-algebras A, B with comultiplication maps Δ_A, Δ_B and counit maps ε_A, ε_B, an R-bialgebra homomorphism A →ₐc[R] B is an R-algebra map f such that ε_B ∘ f = ε_A and (f ⊗ f) ∘ Δ_A = Δ_B ∘ f.

Equations

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

class BialgHomClass (F : Type u_1) (R : outParam (Type u_2)) (A : outParam (Type u_3)) (B : outParam (Type u_4)) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] extends CoalgHomClass , MonoidHomClass : Prop

BialgHomClass F R A B asserts F is a type of bundled bialgebra homomorphisms from A to B.

@[instance 100]

instance BialgHomClass.toAlgHomClass {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [BialgHomClass F R A B] : AlgHomClass F R A B

Equations

• ⋯ = ⋯

def BialgHomClass.toBialgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [BialgHomClass F R A B] (f : F) : A →ₐc[R] B

Turn an element of a type F satisfying BialgHomClass F R A B into an actual BialgHom. This is declared as the default coercion from F to A →ₐc[R] B.

Equations

• ↑f = { toFun := ⇑f, map_add' := ⋯, map_smul' := ⋯, counit_comp := ⋯, map_comp_comul := ⋯, map_one' := ⋯, map_mul' := ⋯ }

instance BialgHomClass.instCoeToBialgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [BialgHomClass F R A B] : CoeHead F (A →ₐc[R] B)

Equations

• BialgHomClass.instCoeToBialgHom = { coe := BialgHomClass.toBialgHom }

@[simp]

theorem BialgHomClass.counitAlgHom_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Bialgebra R A] [Semiring B] [Bialgebra R B] [FunLike F A B] [BialgHomClass F R A B] (f : F) : (Bialgebra.counitAlgHom R B).comp ↑f = Bialgebra.counitAlgHom R A

@[simp]

theorem BialgHomClass.map_comp_comulAlgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Bialgebra R A] [Semiring B] [Bialgebra R B] [FunLike F A B] [BialgHomClass F R A B] (f : F) : (Algebra.TensorProduct.map ↑f ↑f).comp (Bialgebra.comulAlgHom R A) = (Bialgebra.comulAlgHom R B).comp ↑f

instance BialgHom.funLike {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : FunLike (A →ₐc[R] B) A B

Equations

• BialgHom.funLike = { coe := fun (f : A →ₐc[R] B) => f.toFun, coe_injective' := ⋯ }

instance BialgHom.bialgHomClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : BialgHomClass (A →ₐc[R] B) R A B

Equations

• ⋯ = ⋯

def BialgHom.Simps.apply {R : Type u_6} {α : Type u_7} {β : Type u_8} [CommSemiring R] [Semiring α] [Algebra R α] [Semiring β] [Algebra R β] [CoalgebraStruct R α] [CoalgebraStruct R β] (f : α →ₐc[R] β) : α → β

See Note [custom simps projection]

Equations

• BialgHom.Simps.apply f = ⇑f

@[simp]

theorem BialgHom.coe_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {F : Type u_6} [FunLike F A B] [BialgHomClass F R A B] (f : F) : ⇑↑f = ⇑f

@[simp]

theorem BialgHom.coe_mk {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗc[R] B} (h : f.toFun 1 = 1) (h₁ : ∀ (x y : A), f.toFun (x * y) = f.toFun x * f.toFun y) : ⇑{ toCoalgHom := f, map_one' := h, map_mul' := h₁ } = ⇑f

theorem BialgHom.coe_mks {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A → B} (h₀ : ∀ (x y : A), f (x + y) = f x + f y) (h₁ : ∀ (m : R) (x : A), { toFun := f, map_add' := h₀ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₀ }.toFun x) (h₂ : Coalgebra.counit ∘ₗ { toFun := f, map_add' := h₀, map_smul' := h₁ } = Coalgebra.counit) (h₃ : TensorProduct.map { toFun := f, map_add' := h₀, map_smul' := h₁ } { toFun := f, map_add' := h₀, map_smul' := h₁ } ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ { toFun := f, map_add' := h₀, map_smul' := h₁ }) (h₄ : { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun 1 = 1) (h₅ : ∀ (x y : A), { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun (x * y) = { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun x * { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun y) : ⇑{ toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃, map_one' := h₄, map_mul' := h₅ } = f

@[simp]

theorem BialgHom.coe_coalgHom_mk {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗc[R] B} (h : f.toFun 1 = 1) (h₁ : ∀ (x y : A), f.toFun (x * y) = f.toFun x * f.toFun y) : ↑{ toCoalgHom := f, map_one' := h, map_mul' := h₁ } = f

theorem BialgHom.coe_toCoalgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B) : ⇑↑f = ⇑f

@[simp]

theorem BialgHom.coe_toLinearMap {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B) : ⇑↑f = ⇑f

theorem BialgHom.coe_toAlgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B) : ⇑↑f = ⇑f

theorem BialgHom.toAlgHom_toLinearMap {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B) : (↑↑f).toLinearMap = ↑f

theorem BialgHom.coe_fn_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective DFunLike.coe

theorem BialgHom.coe_fn_inj {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ : A →ₐc[R] B} {φ₂ : A →ₐc[R] B} : ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂

theorem BialgHom.coe_coalgHom_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective CoalgHomClass.toCoalgHom

theorem BialgHom.coe_algHom_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective AlgHomClass.toAlgHom

theorem BialgHom.coe_linearMap_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective fun (x : A →ₐc[R] B) => ↑x

theorem BialgHom.congr_fun {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ : A →ₐc[R] B} {φ₂ : A →ₐc[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x

theorem BialgHom.congr_arg {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B) {x : A} {y : A} (h : x = y) : φ x = φ y

theorem BialgHom.ext_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ : A →ₐc[R] B} {φ₂ : A →ₐc[R] B} : φ₁ = φ₂ ↔ ∀ (x : A), φ₁ x = φ₂ x

theorem BialgHom.ext {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ : A →ₐc[R] B} {φ₂ : A →ₐc[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x) : φ₁ = φ₂

theorem BialgHom.ext_of_ring_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] {f : R →ₐc[R] A} {g : R →ₐc[R] A} : f = g ↔ f 1 = g 1

theorem BialgHom.ext_of_ring {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] {f : R →ₐc[R] A} {g : R →ₐc[R] A} (h : f 1 = g 1) : f = g

@[simp]

theorem BialgHom.mk_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₐc[R] B} (h₀ : ∀ (x y : A), f (x + y) = f x + f y) (h₁ : ∀ (m : R) (x : A), { toFun := ⇑f, map_add' := h₀ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₀ }.toFun x) (h₂ : Coalgebra.counit ∘ₗ { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ } = Coalgebra.counit) (h₃ : TensorProduct.map { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ } { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ } ∘ₗ Coalgebra.comul = Coalgebra.comul ∘ₗ { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ }) (h₄ : { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun 1 = 1) (h₅ : ∀ (x y : A), { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun (x * y) = { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun x * { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun y) : { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃, map_one' := h₄, map_mul' := h₅ } = f

def BialgHom.copy {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B) (f' : A → B) (h : f' = ⇑f) : A →ₐc[R] B

Copy of a BialgHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• f.copy f' h = { toCoalgHom := (↑f).copy f' h, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem BialgHom.coe_copy {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem BialgHom.copy_eq {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f) : f.copy f' h = f

@[simp]

theorem BialgHom.id_apply (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (a : A) : (BialgHom.id R A) a = a

def BialgHom.id (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] : A →ₐc[R] A

Identity map as a BialgHom.

Equations

• BialgHom.id R A = { toCoalgHom := CoalgHom.id R A, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem BialgHom.coe_id {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] : ⇑(BialgHom.id R A) = id

@[simp]

theorem BialgHom.id_toCoalgHom {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] : ↑(BialgHom.id R A) = CoalgHom.id R A

@[simp]

theorem BialgHom.id_toAlgHom {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] : ↑(BialgHom.id R A) = AlgHom.id R A

@[simp]

theorem BialgHom.comp_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : ∀ (a : A), (φ₁.comp φ₂) a = φ₁ (φ₂ a)

def BialgHom.comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : A →ₐc[R] C

Composition of bialgebra homomorphisms.

Equations

• φ₁.comp φ₂ = { toCoalgHom := (↑φ₁).comp ↑φ₂, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem BialgHom.coe_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : ⇑(φ₁.comp φ₂) = ⇑φ₁ ∘ ⇑φ₂

@[simp]

theorem BialgHom.comp_toCoalgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂

@[simp]

theorem BialgHom.comp_toAlgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂

@[simp]

theorem BialgHom.comp_id {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B) : φ.comp (BialgHom.id R A) = φ

@[simp]

theorem BialgHom.id_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B) : (BialgHom.id R B).comp φ = φ

theorem BialgHom.comp_assoc {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] [CoalgebraStruct R D] (φ₁ : C →ₐc[R] D) (φ₂ : B →ₐc[R] C) (φ₃ : A →ₐc[R] B) : (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃)

theorem BialgHom.map_smul_of_tower {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B) {R' : Type u_6} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R') (x : A) : φ (r • x) = r • φ x

theorem BialgHom.End_toSemigroup_toMul_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (φ₁ : A →ₐc[R] A) (φ₂ : A →ₐc[R] A) : φ₁ * φ₂ = φ₁.comp φ₂

theorem BialgHom.End_toOne_one {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] : 1 = BialgHom.id R A

instance BialgHom.End {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] : Monoid (A →ₐc[R] A)

Equations

• BialgHom.End = Monoid.mk ⋯ ⋯ npowRecAuto ⋯ ⋯

@[simp]

theorem BialgHom.one_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (x : A) : 1 x = x

@[simp]

theorem BialgHom.mul_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (φ : A →ₐc[R] A) (ψ : A →ₐc[R] A) (x : A) : (φ * ψ) x = φ (ψ x)

def Bialgebra.counitBialgHom (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] : A →ₐc[R] R

The counit of a bialgebra as a BialgHom.

Equations

• Bialgebra.counitBialgHom R A = { toCoalgHom := Coalgebra.counitCoalgHom R A, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem Bialgebra.counitBialgHom_apply (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] (x : A) : (Bialgebra.counitBialgHom R A) x = Coalgebra.counit x

@[simp]

theorem Bialgebra.counitBialgHom_toCoalgHom (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] : ↑(Bialgebra.counitBialgHom R A) = Coalgebra.counitCoalgHom R A

instance Bialgebra.subsingleton_to_ring {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] : Subsingleton (A →ₐc[R] R)

Equations

• ⋯ = ⋯

theorem Bialgebra.ext_to_ring_iff {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] {f : A →ₐc[R] R} {g : A →ₐc[R] R} : f = g ↔ True

theorem Bialgebra.ext_to_ring {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] (f : A →ₐc[R] R) (g : A →ₐc[R] R) : f = g