Homomorphisms of R-coalgebras

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

Main definitions

• CoalgHom R A B: the type of R-coalgebra morphisms from A to B.
• Coalgebra.counitCoalgHom R A : A →ₗc[R] R: the counit of a coalgebra as a coalgebra homomorphism.

Notation

• A →ₗc[R] B : R-coalgebra homomorphism from A to B.

structure CoalgHom (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗ[R] B : Type (max u_2 u_3)

Given R-modules A, B with comultiplication maps Δ_A, Δ_B and counit maps ε_A, ε_B, an R-coalgebra homomorphism A →ₗc[R] B is an R-linear 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 : CoalgebraStruct.counit ∘ₗ self.toLinearMap = CoalgebraStruct.counit
• map_comp_comul : TensorProduct.map self.toLinearMap self.toLinearMap ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul ∘ₗ self.toLinearMap

def «term_→ₗc_» : Lean.TrailingParserDescr

Given R-modules A, B with comultiplication maps Δ_A, Δ_B and counit maps ε_A, ε_B, an R-coalgebra homomorphism A →ₗc[R] B is an R-linear map f such that ε_B ∘ f = ε_A and (f ⊗ f) ∘ Δ_A = Δ_B ∘ f.

Equations

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

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

Given R-modules A, B with comultiplication maps Δ_A, Δ_B and counit maps ε_A, ε_B, an R-coalgebra homomorphism A →ₗc[R] B is an R-linear 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 CoalgHomClass (F : Type u_1) (R : outParam (Type u_2)) (A : outParam (Type u_3)) (B : outParam (Type u_4)) [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] extends SemilinearMapClass F (RingHom.id R) A B : Prop

CoalgHomClass F R A B asserts F is a type of bundled coalgebra homomorphisms from A to B.

• map_add (f : F) (x y : A) : f (x + y) = f x + f y
• map_smulₛₗ (f : F) (c : R) (x : A) : f (c • x) = (RingHom.id R) c • f x
• counit_comp (f : F) : CoalgebraStruct.counit ∘ₗ ↑f = CoalgebraStruct.counit
• map_comp_comul (f : F) : TensorProduct.map ↑f ↑f ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul ∘ₗ ↑f

def CoalgHom.ofClass {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] (f : F) : A →ₗc[R] B

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

Equations

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

@[deprecated CoalgHom.ofClass (since := "2026-09-09")]

def CoalgHomClass.toCoalgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] (f : F) : A →ₗc[R] B

Alias of CoalgHom.ofClass.

---

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

Equations

• @CoalgHomClass.toCoalgHom = @CoalgHom.ofClass

@[instance_reducible]

instance CoalgHomClass.instCoeToCoalgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] : CoeHead F (A →ₗc[R] B)

Equations

• CoalgHomClass.instCoeToCoalgHom = { coe := CoalgHom.ofClass }

@[simp]

theorem CoalgHomClass.counit_comp_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] (f : F) (x : A) : CoalgebraStruct.counit (f x) = CoalgebraStruct.counit x

@[simp]

theorem CoalgHomClass.map_comp_comul_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] [CoalgHomClass F R A B] (f : F) (x : A) : (TensorProduct.map ↑f ↑f) (CoalgebraStruct.comul x) = CoalgebraStruct.comul (f x)

@[instance_reducible, macro_inline]

instance CoalgHom.funLike {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : FunLike (A →ₗc[R] B) A B

Equations

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

instance CoalgHom.coalgHomClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : CoalgHomClass (A →ₗc[R] B) R A B

def CoalgHom.Simps.apply {R : Type u_6} {α : Type u_7} {β : Type u_8} [CommSemiring R] [AddCommMonoid α] [Module R α] [AddCommMonoid β] [Module R β] [CoalgebraStruct R α] [CoalgebraStruct R β] (f : α →ₗc[R] β) : α → β

See Note [custom simps projection]

Equations

• CoalgHom.Simps.apply f = ⇑f

@[simp]

theorem CoalgHom.coe_ofClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {F : Type u_6} [FunLike F A B] [CoalgHomClass F R A B] (f : F) : ⇑↑f = ⇑f

@[deprecated CoalgHom.coe_ofClass (since := "2026-09-09")]

theorem CoalgHom.coe_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {F : Type u_6} [FunLike F A B] [CoalgHomClass F R A B] (f : F) : ⇑↑f = ⇑f

Alias of CoalgHom.coe_ofClass.

@[simp]

theorem CoalgHom.coe_mk {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗ[R] B} (h : CoalgebraStruct.counit ∘ₗ f = CoalgebraStruct.counit) (h₁ : TensorProduct.map f f ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul ∘ₗ f) : ⇑{ toLinearMap := f, counit_comp := h, map_comp_comul := h₁ } = ⇑f

theorem CoalgHom.coe_mks {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A → B} (h₁ : ∀ (x y : A), f (x + y) = f x + f y) (h₂ : ∀ (m : R) (x : A), { toFun := f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₁ }.toFun x) (h₃ : CoalgebraStruct.counit ∘ₗ { toFun := f, map_add' := h₁, map_smul' := h₂ } = CoalgebraStruct.counit) (h₄ : TensorProduct.map { toFun := f, map_add' := h₁, map_smul' := h₂ } { toFun := f, map_add' := h₁, map_smul' := h₂ } ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul ∘ₗ { toFun := f, map_add' := h₁, map_smul' := h₂ }) : ⇑{ toFun := f, map_add' := h₁, map_smul' := h₂, counit_comp := h₃, map_comp_comul := h₄ } = f

@[simp]

theorem CoalgHom.coe_linearMap_mk {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗ[R] B} (h : CoalgebraStruct.counit ∘ₗ f = CoalgebraStruct.counit) (h₁ : TensorProduct.map f f ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul ∘ₗ f) : ↑{ toLinearMap := f, counit_comp := h, map_comp_comul := h₁ } = f

@[simp]

theorem CoalgHom.toLinearMap_eq_ofClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) : f.toLinearMap = ↑f

@[deprecated CoalgHom.toLinearMap_eq_ofClass (since := "2026-09-09")]

theorem CoalgHom.toLinearMap_eq_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) : f.toLinearMap = ↑f

Alias of CoalgHom.toLinearMap_eq_ofClass.

@[simp]

theorem CoalgHom.coe_linearMapOfClass {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) : ⇑↑f = ⇑f

@[deprecated CoalgHom.coe_linearMapOfClass (since := "2026-09-09")]

theorem CoalgHom.coe_toLinearMap {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) : ⇑↑f = ⇑f

Alias of CoalgHom.coe_linearMapOfClass.

theorem CoalgHom.coe_toAddMonoidHom {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) : ⇑↑f = ⇑f

theorem CoalgHom.coe_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective DFunLike.coe

theorem CoalgHom.coe_inj {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} : ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂

@[deprecated CoalgHom.coe_injective (since := "2026-09-09")]

theorem CoalgHom.coe_fn_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective DFunLike.coe

Alias of CoalgHom.coe_injective.

@[deprecated CoalgHom.coe_inj (since := "2026-09-09")]

theorem CoalgHom.coe_fn_inj {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} : ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂

Alias of CoalgHom.coe_inj.

theorem CoalgHom.linearMapOfClass_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective fun (x : A →ₗc[R] B) => ↑x

@[deprecated CoalgHom.linearMapOfClass_injective (since := "2026-09-09")]

theorem CoalgHom.coe_linearMap_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective fun (x : A →ₗc[R] B) => ↑x

Alias of CoalgHom.linearMapOfClass_injective.

theorem CoalgHom.coe_addMonoidHom_injective {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] : Function.Injective AddMonoidHomClass.toAddMonoidHom

theorem CoalgHom.congr_fun {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x

theorem CoalgHom.congr_arg {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B) {x y : A} (h : x = y) : φ x = φ y

theorem CoalgHom.ext {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x) : φ₁ = φ₂

theorem CoalgHom.ext_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₗc[R] B} : φ₁ = φ₂ ↔ ∀ (x : A), φ₁ x = φ₂ x

theorem CoalgHom.ext_of_ring {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] {f g : R →ₗc[R] A} (h : f 1 = g 1) : f = g

theorem CoalgHom.ext_of_ring_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] {f g : R →ₗc[R] A} : f = g ↔ f 1 = g 1

@[simp]

theorem CoalgHom.mk_coe {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗc[R] B} (h₁ : ∀ (x y : A), f (x + y) = f x + f y) (h₂ : ∀ (m : R) (x : A), { toFun := ⇑f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₁ }.toFun x) (h₃ : CoalgebraStruct.counit ∘ₗ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } = CoalgebraStruct.counit) (h₄ : TensorProduct.map { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul ∘ₗ { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ }) : { toFun := ⇑f, map_add' := h₁, map_smul' := h₂, counit_comp := h₃, map_comp_comul := h₄ } = f

def CoalgHom.copy {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f) : A →ₗc[R] B

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

Equations

• f.copy f' h = { toLinearMap := (↑f).copy f' h, counit_comp := ⋯, map_comp_comul := ⋯ }

@[simp]

theorem CoalgHom.coe_copy {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module 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 CoalgHom.copy_eq {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f) : f.copy f' h = f

def CoalgHom.id (R : Type u_1) (A : Type u_2) [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] : A →ₗc[R] A

Identity map as a CoalgHom.

Equations

• CoalgHom.id R A = { toLinearMap := LinearMap.id, counit_comp := ⋯, map_comp_comul := ⋯ }

@[simp]

theorem CoalgHom.id_apply (R : Type u_1) (A : Type u_2) [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (x : A) : (CoalgHom.id R A) x = x

@[simp]

theorem CoalgHom.coe_id {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] : ⇑(CoalgHom.id R A) = id

@[simp]

theorem CoalgHom.id_toLinearMap {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] : ↑(CoalgHom.id R A) = LinearMap.id

def CoalgHom.comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) : A →ₗc[R] C

Composition of coalgebra homomorphisms.

Equations

• φ₁.comp φ₂ = { toLinearMap := ↑φ₁ ∘ₗ ↑φ₂, counit_comp := ⋯, map_comp_comul := ⋯ }

@[simp]

theorem CoalgHom.comp_apply {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) (x : A) : (φ₁.comp φ₂) x = φ₁ (φ₂ x)

@[simp]

theorem CoalgHom.coe_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) : ⇑(φ₁.comp φ₂) = ⇑φ₁ ∘ ⇑φ₂

@[simp]

theorem CoalgHom.comp_toLinearMap {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₗc[R] C) (φ₂ : A →ₗc[R] B) : ↑(φ₁.comp φ₂) = ↑φ₁ ∘ₗ ↑φ₂

@[simp]

theorem CoalgHom.comp_id {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B) : φ.comp (CoalgHom.id R A) = φ

@[simp]

theorem CoalgHom.id_comp {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[R] B) : (CoalgHom.id R B).comp φ = φ

theorem CoalgHom.comp_assoc {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [AddCommMonoid C] [Module R C] [AddCommMonoid D] [Module R D] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] [CoalgebraStruct R D] (φ₁ : C →ₗc[R] D) (φ₂ : B →ₗc[R] C) (φ₃ : A →ₗc[R] B) : (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃)

theorem CoalgHom.map_smul_of_tower {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₗc[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_reducible]

instance CoalgHom.End {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] : Monoid (A →ₗc[R] A)

Equations

• CoalgHom.End = { one := CoalgHom.id R A, mul := CoalgHom.comp, mul_assoc := ⋯, one_mul := ⋯, mul_one := ⋯, npow_zero := ⋯, npow_succ := ⋯ }

theorem CoalgHom.End_toOne_one {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] : 1 = CoalgHom.id R A

theorem CoalgHom.End_toMul_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (φ₁ φ₂ : A →ₗc[R] A) : φ₁ * φ₂ = φ₁.comp φ₂

@[simp]

theorem CoalgHom.one_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (x : A) : 1 x = x

@[simp]

theorem CoalgHom.mul_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (φ ψ : A →ₗc[R] A) (x : A) : (φ * ψ) x = φ (ψ x)

noncomputable def Coalgebra.counitCoalgHom (R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : A →ₗc[R] R

The counit of a coalgebra as a CoalgHom.

Equations

• Coalgebra.counitCoalgHom R A = { toLinearMap := CoalgebraStruct.counit, counit_comp := ⋯, map_comp_comul := ⋯ }

@[simp]

theorem Coalgebra.counitCoalgHom_apply (R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (x : A) : (counitCoalgHom R A) x = counit x

@[simp]

theorem Coalgebra.counitCoalgHom_toLinearMap (R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : ↑(counitCoalgHom R A) = counit

instance Coalgebra.subsingleton_to_ring {R : Type u} (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Subsingleton (A →ₗc[R] R)

theorem Coalgebra.ext_to_ring {R : Type u} (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (f g : A →ₗc[R] R) : f = g

theorem Coalgebra.ext_to_ring_iff {R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {f g : A →ₗc[R] R} : f = g ↔ True

def Coalgebra.Repr.induced {R : Type u} {A : Type v} {B : Type w} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] {a : A} (repr : Repr R a ι) {F : Type u_2} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) : Repr R (φ a) ι

If φ : A → B is a coalgebra map and a = ∑ xᵢ ⊗ yᵢ, then φ a = ∑ φ xᵢ ⊗ φ yᵢ

Equations

• repr.induced φ = { index := repr.index, left := ⇑φ ∘ repr.left, right := ⇑φ ∘ repr.right, eq := ⋯ }

@[simp]

theorem Coalgebra.Repr.induced_index {R : Type u} {A : Type v} {B : Type w} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] {a : A} (repr : Repr R a ι) {F : Type u_2} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) : (repr.induced φ).index = repr.index

@[simp]

theorem Coalgebra.Repr.induced_left {R : Type u} {A : Type v} {B : Type w} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] {a : A} (repr : Repr R a ι) {F : Type u_2} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) (a✝ : ι) : (repr.induced φ).left a✝ = (⇑φ ∘ repr.left) a✝

@[simp]

theorem Coalgebra.Repr.induced_right {R : Type u} {A : Type v} {B : Type w} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] {a : A} (repr : Repr R a ι) {F : Type u_2} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) (a✝ : ι) : (repr.induced φ).right a✝ = (⇑φ ∘ repr.right) a✝