Documentation

Mathlib.RingTheory.Bialgebra.Hom

Homomorphisms of R-bialgebras #

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

Main definitions #

Notations #

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 A →ₗc[R] B, A →* B :
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.

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

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 F R A B, MonoidHomClass F A B :

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

Instances
    @[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
    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) :

    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] :
    Equations
    @[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) :
    @[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) :
    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
    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] :
    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) :
    αβ

    See Note [custom simps projection]

    Equations
    @[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 : AB} (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 : CoalgebraStruct.counit ∘ₗ { toFun := f, map_add' := h₀, map_smul' := h₁ } = CoalgebraStruct.counit) (h₃ : TensorProduct.map { toFun := f, map_add' := h₀, map_smul' := h₁ } { toFun := f, map_add' := h₀, map_smul' := h₁ } ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.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 = f
    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 φ₂ : A →ₐc[R] B} :
    φ₁ = φ₂ φ₁ = φ₂
    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 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 y : A} (h : x = y) :
    φ x = φ y
    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 φ₂ : A →ₐc[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x) :
    φ₁ = φ₂
    theorem BialgHom.ext_of_ring {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] {f 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 : CoalgebraStruct.counit ∘ₗ { toFun := f, map_add' := h₀, map_smul' := h₁ } = CoalgebraStruct.counit) (h₃ : TensorProduct.map { toFun := f, map_add' := h₀, map_smul' := h₁ } { toFun := f, map_add' := h₀, map_smul' := h₁ } ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.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' : AB) (h : f' = f) :

    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' : AB) (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' : AB) (h : f' = f) :
    f.copy f' h = f
    def BialgHom.id (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] :

    Identity map as a BialgHom.

    Equations
    @[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
    @[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] :
    @[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
    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) :

    Composition of bialgebra homomorphisms.

    Equations
    • φ₁.comp φ₂ = { toCoalgHom := (↑φ₁).comp φ₂, map_one' := , map_mul' := }
    @[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✝)
    @[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
    instance BialgHom.End {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] :
    Equations
    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 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] :
    @[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 ψ : A →ₐc[R] A) (x : A) :
    (φ * ψ) x = φ (ψ x)

    The counit of a bialgebra as a BialgHom.

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