fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[simps!] comp (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : A →ₐc[R] C :=
{ (φ₁ : B →ₗc[R] C).comp (φ₂ : A →ₗc[R] B), (φ₁ : B →ₐ[R] C).comp (φ₂ : A →ₐ[R] B) with }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | comp | Composition of bialgebra homomorphisms. |
coe_comp (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | coe_comp | null |
comp_toCoalgHom (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) :
φ₁.comp φ₂ = (φ₁ : B →ₗc[R] C).comp (φ₂ : A →ₗc[R] B) :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | comp_toCoalgHom | null |
comp_toAlgHom (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) :
φ₁.comp φ₂ = (φ₁ : B →ₐ[R] C).comp (φ₂ : A →ₐ[R] B) :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | comp_toAlgHom | null |
comp_id : φ.comp (BialgHom.id R A) = φ :=
ext fun _x => rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | comp_id | null |
id_comp : (BialgHom.id R B).comp φ = φ :=
ext fun _x => rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | id_comp | null |
comp_assoc (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) :=
ext fun _x => rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | comp_assoc | null |
map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')
(x : A) : φ (r • x) = r • φ x :=
φ.toLinearMap.map_smul_of_tower r x
@[simps -isSimp toSemigroup_toMul_mul toOne_one] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | map_smul_of_tower | null |
End : Monoid (A →ₐc[R] A) where
mul := comp
mul_assoc _ _ _ := rfl
one := BialgHom.id R A
one_mul _ := ext fun _ => rfl
mul_one _ := ext fun _ => rfl
@[simp] | instance | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | End | null |
one_apply (x : A) : (1 : A →ₐc[R] A) x = x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | one_apply | null |
mul_apply (φ ψ : A →ₐc[R] A) (x : A) : (φ * ψ) x = φ (ψ x) :=
rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | mul_apply | null |
@[simps!]
ofAlgHom (f : A →ₐ[R] B) (counit_comp : (counitAlgHom R B).comp f = counitAlgHom R A)
(map_comp_comul :
(Algebra.TensorProduct.map f f).comp (comulAlgHom _ _) = (comulAlgHom _ _).comp f) :
A →ₐc[R] B where
__ := f
map_smul' := map_smul f
counit_comp := congr(($counit_comp).toLinearMap)
map_comp_comul := congr(($map_comp_comul).toLinearMap) | def | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | ofAlgHom | Construct a bialgebra hom from an algebra hom respecting counit and comultiplication. |
noncomputable unitBialgHom : R →ₐc[R] A :=
.ofAlgHom (Algebra.ofId R A) (by ext) (by ext)
variable (R A) in | def | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | unitBialgHom | The unit of a bialgebra as a `BialgHom`. |
noncomputable counitBialgHom : A →ₐc[R] R :=
{ Coalgebra.counitCoalgHom R A, counitAlgHom R A with }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | counitBialgHom | The counit of a bialgebra as a `BialgHom`. |
counitBialgHom_apply (x : A) :
counitBialgHom R A x = Coalgebra.counit x := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | counitBialgHom_apply | null |
counitBialgHom_toCoalgHom :
counitBialgHom R A = Coalgebra.counitCoalgHom R A := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | counitBialgHom_toCoalgHom | null |
subsingleton_to_ring : Subsingleton (A →ₐc[R] R) :=
⟨fun _ _ => BialgHom.coe_coalgHom_injective (Subsingleton.elim _ _)⟩
@[ext high] | instance | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | subsingleton_to_ring | null |
ext_to_ring (f g : A →ₐc[R] R) : f = g := Subsingleton.elim _ _ | theorem | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom",
"Mathlib.RingTheory.Bialgebra.Basic"
] | Mathlib/RingTheory/Bialgebra/Hom.lean | ext_to_ring | null |
instBialgebra : Bialgebra R (MonoidAlgebra A M) where
counit_one := by simp only [one_def, counit_single, Bialgebra.counit_one]
mul_compr₂_counit := by ext; simp
comul_one := by
simp only [one_def, comul_single, Bialgebra.comul_one, Algebra.TensorProduct.one_def,
TensorProduct.map_tmul, lsingle_apply]
mul_compr₂_comul := by
ext a b c d
simp only [Function.comp_apply, LinearMap.coe_comp, LinearMap.compr₂_apply,
LinearMap.mul_apply', single_mul_single, comul_single, Bialgebra.comul_mul,
← (Coalgebra.Repr.arbitrary R b).eq, ← (Coalgebra.Repr.arbitrary R d).eq, Finset.sum_mul_sum,
Algebra.TensorProduct.tmul_mul_tmul, map_sum, TensorProduct.map_tmul, lsingle_apply,
LinearMap.compl₁₂_apply, LinearMap.coeFn_sum, Finset.sum_apply,
Finset.sum_comm (s := (Coalgebra.Repr.arbitrary R b).index)]
variable (R) in | instance | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | instBialgebra | null |
@[simps!]
noncomputable mapDomainBialgHom (f : M →* N) : MonoidAlgebra R M →ₐc[R] MonoidAlgebra R N :=
.ofAlgHom (mapDomainAlgHom R R f) (by ext; simp) (by ext; simp)
@[simp] lemma mapDomainBialgHom_id : mapDomainBialgHom R (.id M) = .id _ _ := by ext; simp
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | mapDomainBialgHom | If `f : M → N` is a monoid hom, then `MonoidAlgebra.mapDomain f` is a bialgebra hom between
their monoid algebras. |
mapDomainBialgHom_comp (f : N →* O) (g : M →* N) :
mapDomainBialgHom R (f.comp g) = (mapDomainBialgHom R f).comp (mapDomainBialgHom R g) := by
ext; simp [Finsupp.mapDomain_comp] | lemma | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | mapDomainBialgHom_comp | null |
mapDomainBialgHom_mapDomainBialgHom (f : N →* O) (g : M →* N) (x : MonoidAlgebra R M) :
mapDomainBialgHom R f (mapDomainBialgHom R g x) = mapDomainBialgHom R (f.comp g) x := by
ext; simp | lemma | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | mapDomainBialgHom_mapDomainBialgHom | null |
instBialgebra : Bialgebra R A[M] where
counit_one := by simp only [one_def, counit_single, Bialgebra.counit_one]
mul_compr₂_counit := by ext; simp [single_mul_single]
comul_one := by
simp only [one_def, comul_single, Bialgebra.comul_one, Algebra.TensorProduct.one_def,
TensorProduct.map_tmul, lsingle_apply]
mul_compr₂_comul := by
ext a b c d
simp only [Function.comp_apply, LinearMap.coe_comp, LinearMap.compr₂_apply,
LinearMap.mul_apply', single_mul_single, comul_single, Bialgebra.comul_mul,
← (Coalgebra.Repr.arbitrary R b).eq, ← (Coalgebra.Repr.arbitrary R d).eq, Finset.sum_mul_sum,
Algebra.TensorProduct.tmul_mul_tmul, map_sum, TensorProduct.map_tmul, lsingle_apply,
LinearMap.compl₁₂_apply, LinearMap.coeFn_sum, Finset.sum_apply,
Finset.sum_comm (s := (Coalgebra.Repr.arbitrary R b).index)]
variable (R) in | instance | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | instBialgebra | null |
@[simps]
noncomputable mapDomainBialgHom (f : M →+ N) : R[M] →ₐc[R] R[N] where
__ := mapDomainAlgHom R R f
map_smul' m x := by simp
counit_comp := by ext; simp
map_comp_comul := by ext; simp
@[simp] lemma mapDomainBialgHom_id : mapDomainBialgHom R (.id M) = .id _ _ := by ext; simp
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | mapDomainBialgHom | If `f : M → N` is a monoid hom, then `AddMonoidAlgebra.mapDomain f` is a bialgebra hom between
their monoid algebras. |
mapDomainBialgHom_comp (f : N →+ O) (g : M →+ N) :
mapDomainBialgHom R (f.comp g) = (mapDomainBialgHom R f).comp (mapDomainBialgHom R g) := by
ext; simp | lemma | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | mapDomainBialgHom_comp | null |
mapDomainBialgHom_mapDomainBialgHom (f : N →+ O) (g : M →+ N) (x : R[M]) :
mapDomainBialgHom R f (mapDomainBialgHom R g x) = mapDomainBialgHom R (f.comp g) x := by
ext; simp | lemma | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | mapDomainBialgHom_mapDomainBialgHom | null |
instBialgebra : Bialgebra R A[T;T⁻¹] :=
inferInstanceAs <| Bialgebra R A[ℤ]
@[simp] | instance | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | instBialgebra | null |
comul_T (n : ℤ) :
Coalgebra.comul (R := R) (T n : A[T;T⁻¹]) = T n ⊗ₜ[R] T n := by
simp [T, -single_eq_C_mul_T, Algebra.TensorProduct.one_def]
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | comul_T | null |
counit_T (n : ℤ) :
Coalgebra.counit (R := R) (T n : A[T;T⁻¹]) = 1 := by
simp [T, -single_eq_C_mul_T] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Hom",
"Mathlib.RingTheory.Coalgebra.MonoidAlgebra"
] | Mathlib/RingTheory/Bialgebra/MonoidAlgebra.lean | counit_T | null |
on a tensor product of bialgebras, and the tensor product of two `BialgHom`s as a
`BialgHom`. This is done by combining the corresponding API for coalgebras and algebras.
-/
open scoped TensorProduct | instance | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | on | null |
counit_eq_algHom_toLinearMap :
Coalgebra.counit (R := S) (A := A ⊗[R] B) =
((Algebra.TensorProduct.rid _ _ _).toAlgHom.comp (Algebra.TensorProduct.map
(Bialgebra.counitAlgHom S A) (Bialgebra.counitAlgHom R B))).toLinearMap :=
rfl | lemma | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | counit_eq_algHom_toLinearMap | null |
comul_eq_algHom_toLinearMap :
Coalgebra.comul (R := S) (A := A ⊗[R] B) =
((Algebra.TensorProduct.tensorTensorTensorComm R S R S A A B B).toAlgHom.comp
(Algebra.TensorProduct.map (Bialgebra.comulAlgHom S A)
(Bialgebra.comulAlgHom R B))).toLinearMap :=
rfl | lemma | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | comul_eq_algHom_toLinearMap | null |
noncomputable _root_.TensorProduct.instBialgebra : Bialgebra S (A ⊗[R] B) := by
have hcounit := congr(DFunLike.coe $(counit_eq_algHom_toLinearMap R S A B))
have hcomul := congr(DFunLike.coe $(comul_eq_algHom_toLinearMap R S A B))
refine Bialgebra.mk' S (A ⊗[R] B) ?_ (fun {x y} => ?_) ?_ (fun {x y} => ?_) <;>
simp_all only [AlgHom.toLinearMap_apply] <;>
simp only [map_one, map_mul]
variable {R S A B C D}
variable [Semiring C] [Semiring D] [Bialgebra S C]
[Bialgebra R D] [Algebra R C] [IsScalarTower R S C] | instance | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | _root_.TensorProduct.instBialgebra | null |
noncomputable map (f : A →ₐc[S] C) (g : B →ₐc[R] D) :
A ⊗[R] B →ₐc[S] C ⊗[R] D :=
{ Coalgebra.TensorProduct.map (f : A →ₗc[S] C) (g : B →ₗc[R] D),
Algebra.TensorProduct.map (f : A →ₐ[S] C) (g : B →ₐ[R] D) with }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | map | The tensor product of two bialgebra morphisms as a bialgebra morphism. |
map_tmul (f : A →ₐc[S] C) (g : B →ₐc[R] D) (x : A) (y : B) :
map f g (x ⊗ₜ y) = f x ⊗ₜ g y :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | map_tmul | null |
map_toCoalgHom (f : A →ₐc[S] C) (g : B →ₐc[R] D) :
map f g = Coalgebra.TensorProduct.map (f : A →ₗc[S] C) (g : B →ₗc[R] D) := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | map_toCoalgHom | null |
map_toAlgHom (f : A →ₐc[S] C) (g : B →ₐc[R] D) :
(map f g : A ⊗[R] B →ₐ[S] C ⊗[R] D) =
Algebra.TensorProduct.map (f : A →ₐ[S] C) (g : B →ₐ[R] D) :=
rfl
variable (R S A C D) in | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | map_toAlgHom | null |
protected noncomputable assoc :
(A ⊗[S] C) ⊗[R] D ≃ₐc[S] A ⊗[S] (C ⊗[R] D) :=
{ Coalgebra.TensorProduct.assoc R S A C D, Algebra.TensorProduct.assoc R S A C D with }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | assoc | The associator for tensor products of R-bialgebras, as a bialgebra equivalence. |
assoc_tmul (x : A) (y : C) (z : D) :
Bialgebra.TensorProduct.assoc R S A C D ((x ⊗ₜ y) ⊗ₜ z) = x ⊗ₜ (y ⊗ₜ z) :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | assoc_tmul | null |
assoc_symm_tmul (x : A) (y : C) (z : D) :
(Bialgebra.TensorProduct.assoc R S A C D).symm (x ⊗ₜ (y ⊗ₜ z)) = (x ⊗ₜ y) ⊗ₜ z :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | assoc_symm_tmul | null |
assoc_toCoalgEquiv :
(Bialgebra.TensorProduct.assoc R S A C D : _ ≃ₗc[S] _) =
Coalgebra.TensorProduct.assoc R S A C D := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | assoc_toCoalgEquiv | null |
assoc_toAlgEquiv :
(Bialgebra.TensorProduct.assoc R S A C D : _ ≃ₐ[S] _) =
Algebra.TensorProduct.assoc R S A C D := rfl
variable (R B) in | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | assoc_toAlgEquiv | null |
protected noncomputable lid : R ⊗[R] B ≃ₐc[R] B :=
{ Coalgebra.TensorProduct.lid R B, Algebra.TensorProduct.lid R B with }
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | lid | The base ring is a left identity for the tensor product of bialgebras, up to
bialgebra equivalence. |
lid_toCoalgEquiv :
(Bialgebra.TensorProduct.lid R B : R ⊗[R] B ≃ₗc[R] B) = Coalgebra.TensorProduct.lid R B := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | lid_toCoalgEquiv | null |
lid_toAlgEquiv :
(Bialgebra.TensorProduct.lid R B : R ⊗[R] B ≃ₐ[R] B) = Algebra.TensorProduct.lid R B := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | lid_toAlgEquiv | null |
lid_tmul (r : R) (a : B) : Bialgebra.TensorProduct.lid R B (r ⊗ₜ a) = r • a := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | lid_tmul | null |
lid_symm_apply (a : B) : (Bialgebra.TensorProduct.lid R B).symm a = 1 ⊗ₜ a := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | lid_symm_apply | null |
coalgebra_rid_eq_algebra_rid_apply (x : A ⊗[R] R) :
Coalgebra.TensorProduct.rid R S A x = Algebra.TensorProduct.rid R R A x := rfl
variable (R S A) in | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | coalgebra_rid_eq_algebra_rid_apply | null |
protected noncomputable rid : A ⊗[R] R ≃ₐc[S] A where
toCoalgEquiv := Coalgebra.TensorProduct.rid R S A
map_mul' x y := by
simp only [CoalgEquiv.toCoalgHom_eq_coe, CoalgHom.toLinearMap_eq_coe, AddHom.toFun_eq_coe,
LinearMap.coe_toAddHom, CoalgHom.coe_toLinearMap, CoalgHom.coe_coe,
coalgebra_rid_eq_algebra_rid_apply, map_mul]
@[simp] | def | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | rid | The base ring is a right identity for the tensor product of bialgebras, up to
bialgebra equivalence. |
rid_toCoalgEquiv :
(TensorProduct.rid R S A : A ⊗[R] R ≃ₗc[S] A) = Coalgebra.TensorProduct.rid R S A := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | rid_toCoalgEquiv | null |
rid_toAlgEquiv :
(Bialgebra.TensorProduct.rid R S A : A ⊗[R] R ≃ₐ[S] A) = Algebra.TensorProduct.rid R S A := by
ext x
exact coalgebra_rid_eq_algebra_rid_apply x
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | rid_toAlgEquiv | null |
rid_tmul (r : R) (a : A) : Bialgebra.TensorProduct.rid R S A (a ⊗ₜ r) = r • a := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | rid_tmul | null |
rid_symm_apply (a : A) : (Bialgebra.TensorProduct.rid R S A).symm a = a ⊗ₜ 1 := rfl | theorem | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | rid_symm_apply | null |
noncomputable lTensor (f : B →ₐc[R] C) : A ⊗[R] B →ₐc[R] A ⊗[R] C :=
Bialgebra.TensorProduct.map (BialgHom.id R A) f | abbrev | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | lTensor | `lTensor A f : A ⊗ B →ₐc A ⊗ C` is the natural bialgebra morphism induced by `f : B →ₐc C`. |
noncomputable rTensor (f : B →ₐc[R] C) : B ⊗[R] A →ₐc[R] C ⊗[R] A :=
Bialgebra.TensorProduct.map f (BialgHom.id R A) | abbrev | RingTheory | [
"Mathlib.RingTheory.Bialgebra.Equiv",
"Mathlib.RingTheory.Coalgebra.TensorProduct",
"Mathlib.RingTheory.TensorProduct.Basic"
] | Mathlib/RingTheory/Bialgebra/TensorProduct.lean | rTensor | `rTensor A f : B ⊗ A →ₐc C ⊗ A` is the natural bialgebra morphism induced by `f : B →ₐc C`. |
CoalgebraStruct (R : Type u) (A : Type v)
[CommSemiring R] [AddCommMonoid A] [Module R A] where
/-- The comultiplication of the coalgebra -/
comul : A →ₗ[R] A ⊗[R] A
/-- The counit of the coalgebra -/
counit : A →ₗ[R] R | class | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | CoalgebraStruct | Data fields for `Coalgebra`, to allow API to be constructed before proving `Coalgebra.coassoc`.
See `Coalgebra` for documentation. |
Coalgebra.Repr (R : Type u) {A : Type v}
[CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (a : A) where
/-- the indexing type of a representation of `comul a` -/
{ι : Type*}
/-- the finite indexing set of a representation of `comul a` -/
(index : Finset ι)
/-- the first coordinate of a representation of `comul a` -/
(left : ι → A)
/-- the second coordinate of a representation of `comul a` -/
(right : ι → A)
/-- `comul a` is equal to a finite sum of some pure tensors -/
(eq : ∑ i ∈ index, left i ⊗ₜ[R] right i = CoalgebraStruct.comul a) | structure | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | Coalgebra.Repr | A representation of an element `a` of a coalgebra `A` is a finite sum of pure tensors `∑ xᵢ ⊗ yᵢ`
that is equal to `comul a`. |
noncomputable Coalgebra.Repr.arbitrary (R : Type u) {A : Type v}
[CommSemiring R] [AddCommMonoid A] [Module R A] [CoalgebraStruct R A] (a : A) :
Coalgebra.Repr R a where
left := Prod.fst
right := Prod.snd
index := TensorProduct.exists_finset (R := R) (CoalgebraStruct.comul a) |>.choose
eq := TensorProduct.exists_finset (R := R) (CoalgebraStruct.comul a) |>.choose_spec.symm
@[inherit_doc Coalgebra.Repr.arbitrary]
scoped[Coalgebra] notation "ℛ" => Coalgebra.Repr.arbitrary | def | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | Coalgebra.Repr.arbitrary | An arbitrarily chosen representation. |
Coalgebra (R : Type u) (A : Type v)
[CommSemiring R] [AddCommMonoid A] [Module R A] extends CoalgebraStruct R A where
/-- The comultiplication is coassociative -/
coassoc : TensorProduct.assoc R A A A ∘ₗ comul.rTensor A ∘ₗ comul = comul.lTensor A ∘ₗ comul
/-- The counit satisfies the left counitality law -/
rTensor_counit_comp_comul : counit.rTensor A ∘ₗ comul = TensorProduct.mk R _ _ 1
/-- The counit satisfies the right counitality law -/
lTensor_counit_comp_comul : counit.lTensor A ∘ₗ comul = (TensorProduct.mk R _ _).flip 1 | class | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | Coalgebra | A coalgebra over a commutative (semi)ring `R` is an `R`-module equipped with a coassociative
comultiplication `Δ` and a counit `ε` obeying the left and right counitality laws. |
@[simp]
coassoc_apply (a : A) :
TensorProduct.assoc R A A A (comul.rTensor A (comul a)) = comul.lTensor A (comul a) :=
LinearMap.congr_fun coassoc a
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | coassoc_apply | null |
coassoc_symm_apply (a : A) :
(TensorProduct.assoc R A A A).symm (comul.lTensor A (comul a)) = comul.rTensor A (comul a) := by
rw [(TensorProduct.assoc R A A A).symm_apply_eq, coassoc_apply a]
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | coassoc_symm_apply | null |
coassoc_symm :
(TensorProduct.assoc R A A A).symm ∘ₗ comul.lTensor A ∘ₗ comul =
comul.rTensor A ∘ₗ (comul (R := R)) :=
LinearMap.ext coassoc_symm_apply
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | coassoc_symm | null |
rTensor_counit_comul (a : A) : counit.rTensor A (comul a) = 1 ⊗ₜ[R] a :=
LinearMap.congr_fun rTensor_counit_comp_comul a
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | rTensor_counit_comul | null |
lTensor_counit_comul (a : A) : counit.lTensor A (comul a) = a ⊗ₜ[R] 1 :=
LinearMap.congr_fun lTensor_counit_comp_comul a
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | lTensor_counit_comul | null |
sum_counit_tmul_eq {a : A} (repr : Coalgebra.Repr R a) :
∑ i ∈ repr.index, counit (R := R) (repr.left i) ⊗ₜ (repr.right i) = 1 ⊗ₜ[R] a := by
simpa [← repr.eq, map_sum] using congr($(rTensor_counit_comp_comul (R := R) (A := A)) a)
@[simp] | lemma | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_counit_tmul_eq | null |
sum_tmul_counit_eq {a : A} (repr : Coalgebra.Repr R a) :
∑ i ∈ repr.index, (repr.left i) ⊗ₜ counit (R := R) (repr.right i) = a ⊗ₜ[R] 1 := by
simpa [← repr.eq, map_sum] using congr($(lTensor_counit_comp_comul (R := R) (A := A)) a) | lemma | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_tmul_counit_eq | null |
sum_tmul_tmul_eq {a : A} (repr : Repr R a)
(a₁ : (i : repr.ι) → Repr R (repr.left i)) (a₂ : (i : repr.ι) → Repr R (repr.right i)) :
∑ i ∈ repr.index, ∑ j ∈ (a₁ i).index,
(a₁ i).left j ⊗ₜ[R] ((a₁ i).right j ⊗ₜ[R] repr.right i)
= ∑ i ∈ repr.index, ∑ j ∈ (a₂ i).index,
repr.left i ⊗ₜ[R] ((a₂ i).left j ⊗ₜ[R] (a₂ i).right j) := by
simpa [(a₂ _).eq, ← (a₁ _).eq, ← TensorProduct.tmul_sum,
TensorProduct.sum_tmul, ← repr.eq] using congr($(coassoc (R := R)) a)
@[simp] | lemma | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_tmul_tmul_eq | null |
sum_counit_tmul_map_eq {B : Type*} [AddCommMonoid B] [Module R B]
{F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f : F) (a : A) {repr : Repr R a} :
∑ i ∈ repr.index, counit (R := R) (repr.left i) ⊗ₜ f (repr.right i) = 1 ⊗ₜ[R] f a := by
have := sum_counit_tmul_eq repr
apply_fun LinearMap.lTensor R (f : A →ₗ[R] B) at this
simp_all only [map_sum, LinearMap.lTensor_tmul, LinearMap.coe_coe]
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_counit_tmul_map_eq | null |
sum_map_tmul_counit_eq {B : Type*} [AddCommMonoid B] [Module R B]
{F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f : F) (a : A) {repr : Repr R a} :
∑ i ∈ repr.index, f (repr.left i) ⊗ₜ counit (R := R) (repr.right i) = f a ⊗ₜ[R] 1 := by
have := sum_tmul_counit_eq repr
apply_fun LinearMap.rTensor R (f : A →ₗ[R] B) at this
simp_all only [map_sum, LinearMap.rTensor_tmul, LinearMap.coe_coe] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_map_tmul_counit_eq | null |
sum_map_tmul_tmul_eq {B : Type*} [AddCommMonoid B] [Module R B]
{F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f g h : F) (a : A) {repr : Repr R a}
{a₁ : (i : repr.ι) → Repr R (repr.left i)} {a₂ : (i : repr.ι) → Repr R (repr.right i)} :
∑ i ∈ repr.index, ∑ j ∈ (a₂ i).index,
f (repr.left i) ⊗ₜ (g ((a₂ i).left j) ⊗ₜ h ((a₂ i).right j)) =
∑ i ∈ repr.index, ∑ j ∈ (a₁ i).index,
f ((a₁ i).left j) ⊗ₜ[R] (g ((a₁ i).right j) ⊗ₜ[R] h (repr.right i)) := by
have := sum_tmul_tmul_eq repr a₁ a₂
apply_fun TensorProduct.map (f : A →ₗ[R] B)
(TensorProduct.map (g : A →ₗ[R] B) (h : A →ₗ[R] B)) at this
simp_all only [map_sum, TensorProduct.map_tmul, LinearMap.coe_coe] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_map_tmul_tmul_eq | null |
sum_counit_smul (𝓡 : Repr R a) :
∑ x ∈ 𝓡.index, counit (R := R) (𝓡.left x) • 𝓡.right x = a := by
simpa only [map_sum, TensorProduct.lift.tmul, LinearMap.lsmul_apply, one_smul]
using congr(TensorProduct.lift (LinearMap.lsmul R A) $(sum_counit_tmul_eq (R := R) 𝓡)) | lemma | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | sum_counit_smul | null |
lift_lsmul_comp_counit_comp_comul :
TensorProduct.lift (.lsmul R A ∘ₗ counit) ∘ₗ comul = .id := by
have := rTensor_counit_comp_comul (R := R) (A := A)
apply_fun (TensorProduct.lift (LinearMap.lsmul R A) ∘ₗ ·) at this
rw [LinearMap.rTensor, ← LinearMap.comp_assoc, TensorProduct.lift_comp_map, LinearMap.compl₂_id]
at this
ext
simp [this]
variable (R A) in | lemma | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | lift_lsmul_comp_counit_comp_comul | null |
IsCocomm where
protected comm_comp_comul : (TensorProduct.comm R A A).comp comul = comul
variable [IsCocomm R A]
variable (R A) in
@[simp] lemma comm_comp_comul : (TensorProduct.comm R A A).comp comul = comul :=
IsCocomm.comm_comp_comul
variable (R) in
@[simp] lemma comm_comul (a : A) : TensorProduct.comm R A A (comul a) = comul a :=
congr($(comm_comp_comul R A) a) | class | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | IsCocomm | A coalgebra `A` is cocommutative if its comultiplication `δ : A → A ⊗ A` commutes with the
swapping `β : A ⊗ A ≃ A ⊗ A` of the factors in the tensor product. |
toCoalgebra : Coalgebra R R where
comul := (TensorProduct.mk R R R) 1
counit := .id
coassoc := rfl
rTensor_counit_comp_comul := by ext; rfl
lTensor_counit_comp_comul := by ext; rfl
@[simp] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | toCoalgebra | Every commutative (semi)ring is a coalgebra over itself, with `Δ r = 1 ⊗ₜ r`. |
comul_apply (r : R) : comul r = 1 ⊗ₜ[R] r := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_apply | null |
counit_apply (r : R) : counit r = r := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | counit_apply | null |
instCoalgebraStruct : CoalgebraStruct R (A × B) where
comul := .coprod
(TensorProduct.map (.inl R A B) (.inl R A B) ∘ₗ comul)
(TensorProduct.map (.inr R A B) (.inr R A B) ∘ₗ comul)
counit := .coprod counit counit
@[simp] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instCoalgebraStruct | null |
comul_apply (r : A × B) :
comul r =
TensorProduct.map (.inl R A B) (.inl R A B) (comul r.1) +
TensorProduct.map (.inr R A B) (.inr R A B) (comul r.2) := rfl
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_apply | null |
counit_apply (r : A × B) : (counit r : R) = counit r.1 + counit r.2 := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | counit_apply | null |
comul_comp_inl :
comul ∘ₗ inl R A B = TensorProduct.map (.inl R A B) (.inl R A B) ∘ₗ comul := by
ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_inl | null |
comul_comp_inr :
comul ∘ₗ inr R A B = TensorProduct.map (.inr R A B) (.inr R A B) ∘ₗ comul := by
ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_inr | null |
comul_comp_fst :
comul ∘ₗ .fst R A B = TensorProduct.map (.fst R A B) (.fst R A B) ∘ₗ comul := by
ext : 1
· rw [comp_assoc, fst_comp_inl, comp_id, comp_assoc, comul_comp_inl, ← comp_assoc,
← TensorProduct.map_comp, fst_comp_inl, TensorProduct.map_id, id_comp]
· rw [comp_assoc, fst_comp_inr, comp_zero, comp_assoc, comul_comp_inr, ← comp_assoc,
← TensorProduct.map_comp, fst_comp_inr, TensorProduct.map_zero_left, zero_comp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_fst | null |
comul_comp_snd :
comul ∘ₗ .snd R A B = TensorProduct.map (.snd R A B) (.snd R A B) ∘ₗ comul := by
ext : 1
· rw [comp_assoc, snd_comp_inl, comp_zero, comp_assoc, comul_comp_inl, ← comp_assoc,
← TensorProduct.map_comp, snd_comp_inl, TensorProduct.map_zero_left, zero_comp]
· rw [comp_assoc, snd_comp_inr, comp_id, comp_assoc, comul_comp_inr, ← comp_assoc,
← TensorProduct.map_comp, snd_comp_inr, TensorProduct.map_id, id_comp]
@[simp] theorem counit_comp_inr : counit ∘ₗ inr R A B = counit := by ext; simp
@[simp] theorem counit_comp_inl : counit ∘ₗ inl R A B = counit := by ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_snd | null |
instCoalgebra : Coalgebra R (A × B) where
rTensor_counit_comp_comul := by
ext : 1
· rw [comp_assoc, comul_comp_inl, ← comp_assoc, rTensor_comp_map, counit_comp_inl,
← lTensor_comp_rTensor, comp_assoc, rTensor_counit_comp_comul, lTensor_comp_mk]
· rw [comp_assoc, comul_comp_inr, ← comp_assoc, rTensor_comp_map, counit_comp_inr,
← lTensor_comp_rTensor, comp_assoc, rTensor_counit_comp_comul, lTensor_comp_mk]
lTensor_counit_comp_comul := by
ext : 1
· rw [comp_assoc, comul_comp_inl, ← comp_assoc, lTensor_comp_map, counit_comp_inl,
← rTensor_comp_lTensor, comp_assoc, lTensor_counit_comp_comul, rTensor_comp_flip_mk]
· rw [comp_assoc, comul_comp_inr, ← comp_assoc, lTensor_comp_map, counit_comp_inr,
← rTensor_comp_lTensor, comp_assoc, lTensor_counit_comp_comul, rTensor_comp_flip_mk]
coassoc := by
dsimp only [instCoalgebraStruct]
ext x : 2 <;> dsimp only [comp_apply, LinearEquiv.coe_coe, coe_inl, coe_inr, coprod_apply]
· simp only [map_zero, add_zero]
simp_rw [← comp_apply, ← comp_assoc, rTensor_comp_map, lTensor_comp_map, coprod_inl,
← map_comp_rTensor, ← map_comp_lTensor, comp_assoc, ← coassoc, ← comp_assoc,
TensorProduct.map_map_comp_assoc_eq, comp_apply, LinearEquiv.coe_coe]
· simp only [map_zero, zero_add]
simp_rw [← comp_apply, ← comp_assoc, rTensor_comp_map, lTensor_comp_map, coprod_inr,
← map_comp_rTensor, ← map_comp_lTensor, comp_assoc, ← coassoc, ← comp_assoc,
TensorProduct.map_map_comp_assoc_eq, comp_apply, LinearEquiv.coe_coe] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instCoalgebra | null |
instCoalgebraStruct : CoalgebraStruct R (Π₀ i, A i) where
comul := DFinsupp.lsum R fun i =>
TensorProduct.map (DFinsupp.lsingle i) (DFinsupp.lsingle i) ∘ₗ comul
counit := DFinsupp.lsum R fun _ => counit
@[simp] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instCoalgebraStruct | null |
comul_single (i : ι) (a : A i) :
comul (R := R) (DFinsupp.single i a) =
(TensorProduct.map (DFinsupp.lsingle i) (DFinsupp.lsingle i) : _ →ₗ[R] _) (comul a) :=
lsum_single _ _ _ _
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_single | null |
counit_single (i : ι) (a : A i) : counit (DFinsupp.single i a) = counit (R := R) a :=
lsum_single _ _ _ _ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | counit_single | null |
comul_comp_lsingle (i : ι) :
comul ∘ₗ (lsingle i : A i →ₗ[R] _) = TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul := by
ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_lsingle | null |
comul_comp_lapply (i : ι) :
comul ∘ₗ (lapply i : _ →ₗ[R] A i) = TensorProduct.map (lapply i) (lapply i) ∘ₗ comul := by
ext j : 1
conv_rhs => rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, ← TensorProduct.map_comp]
obtain rfl | hij := eq_or_ne i j
· rw [comp_assoc, lapply_comp_lsingle_same, comp_id, TensorProduct.map_id, id_comp]
· rw [comp_assoc, lapply_comp_lsingle_of_ne _ _ hij, comp_zero, TensorProduct.map_zero_left,
zero_comp]
@[simp] theorem counit_comp_lsingle (i : ι) : counit ∘ₗ (lsingle i : A i →ₗ[R] _) = counit := by
ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_lapply | null |
instCoalgebra : Coalgebra R (Π₀ i, A i) where
rTensor_counit_comp_comul := by
ext : 1
rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, rTensor_comp_map, counit_comp_lsingle,
← lTensor_comp_rTensor, comp_assoc, rTensor_counit_comp_comul, lTensor_comp_mk]
lTensor_counit_comp_comul := by
ext : 1
rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, lTensor_comp_map, counit_comp_lsingle,
← rTensor_comp_lTensor, comp_assoc, lTensor_counit_comp_comul, rTensor_comp_flip_mk]
coassoc := by
ext i : 1
simp_rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, lTensor_comp_map, comul_comp_lsingle,
comp_assoc, ← comp_assoc comul, rTensor_comp_map, comul_comp_lsingle, ← map_comp_rTensor,
← map_comp_lTensor, comp_assoc, ← coassoc, ← comp_assoc comul, ← comp_assoc,
TensorProduct.map_map_comp_assoc_eq] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instCoalgebra | The `R`-module whose elements are dependent functions `(i : ι) → A i` which are zero on all but
finitely many elements of `ι` has a coalgebra structure.
The coproduct `Δ` is given by `Δ(fᵢ a) = fᵢ a₁ ⊗ fᵢ a₂` where `Δ(a) = a₁ ⊗ a₂` and the counit `ε`
by `ε(fᵢ a) = ε(a)`, where `fᵢ a` is the function sending `i` to `a` and all other elements of `ι`
to zero. |
instIsCocomm [∀ i, IsCocomm R (A i)] : IsCocomm R (Π₀ i, A i) where
comm_comp_comul := by
ext i : 1
simp only [comp_assoc, comul_comp_lsingle]
simp only [← comp_assoc, ← TensorProduct.map_comp_comm_eq]
simp [LinearMap.comp_assoc] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instIsCocomm | null |
noncomputable instCoalgebraStruct : CoalgebraStruct R (ι →₀ A) where
comul := Finsupp.lsum R fun i =>
TensorProduct.map (Finsupp.lsingle i) (Finsupp.lsingle i) ∘ₗ comul
counit := Finsupp.lsum R fun _ => counit
@[simp] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instCoalgebraStruct | null |
comul_single (i : ι) (a : A) :
comul (R := R) (Finsupp.single i a) =
(TensorProduct.map (Finsupp.lsingle i) (Finsupp.lsingle i) : _ →ₗ[R] _) (comul a) :=
lsum_single _ _ _ _
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_single | null |
counit_single (i : ι) (a : A) : counit (Finsupp.single i a) = counit (R := R) a :=
lsum_single _ _ _ _ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | counit_single | null |
comul_comp_lsingle (i : ι) :
comul ∘ₗ (lsingle i : A →ₗ[R] _) = TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul := by
ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_lsingle | null |
comul_comp_lapply (i : ι) :
comul ∘ₗ (lapply i : _ →ₗ[R] A) = TensorProduct.map (lapply i) (lapply i) ∘ₗ comul := by
ext j : 1
conv_rhs => rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, ← TensorProduct.map_comp]
obtain rfl | hij := eq_or_ne i j
· rw [comp_assoc, lapply_comp_lsingle_same, comp_id, TensorProduct.map_id, id_comp]
· rw [comp_assoc, lapply_comp_lsingle_of_ne _ _ hij, comp_zero, TensorProduct.map_zero_left,
zero_comp]
@[simp] theorem counit_comp_lsingle (i : ι) : counit ∘ₗ (lsingle i : A →ₗ[R] _) = counit := by
ext; simp | theorem | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | comul_comp_lapply | null |
noncomputable instCoalgebra : Coalgebra R (ι →₀ A) where
rTensor_counit_comp_comul := by
ext : 1
rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, rTensor_comp_map, counit_comp_lsingle,
← lTensor_comp_rTensor, comp_assoc, rTensor_counit_comp_comul, lTensor_comp_mk]
lTensor_counit_comp_comul := by
ext : 1
rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, lTensor_comp_map, counit_comp_lsingle,
← rTensor_comp_lTensor, comp_assoc, lTensor_counit_comp_comul, rTensor_comp_flip_mk]
coassoc := by
ext i : 1
simp_rw [comp_assoc, comul_comp_lsingle, ← comp_assoc, lTensor_comp_map, comul_comp_lsingle,
comp_assoc, ← comp_assoc comul, rTensor_comp_map, comul_comp_lsingle, ← map_comp_rTensor,
← map_comp_lTensor, comp_assoc, ← coassoc, ← comp_assoc comul, ← comp_assoc,
TensorProduct.map_map_comp_assoc_eq] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instCoalgebra | The `R`-module whose elements are functions `ι → A` which are zero on all but finitely many
elements of `ι` has a coalgebra structure. The coproduct `Δ` is given by `Δ(fᵢ a) = fᵢ a₁ ⊗ fᵢ a₂`
where `Δ(a) = a₁ ⊗ a₂` and the counit `ε` by `ε(fᵢ a) = ε(a)`, where `fᵢ a` is the function sending
`i` to `a` and all other elements of `ι` to zero. |
instIsCocomm [IsCocomm R A] : IsCocomm R (ι →₀ A) where
comm_comp_comul := by
ext i : 1
simp only [comp_assoc, comul_comp_lsingle]
simp only [← comp_assoc, ← TensorProduct.map_comp_comm_eq]
simp [LinearMap.comp_assoc] | instance | RingTheory | [
"Mathlib.LinearAlgebra.TensorProduct.Finiteness",
"Mathlib.LinearAlgebra.TensorProduct.Associator"
] | Mathlib/RingTheory/Coalgebra/Basic.lean | instIsCocomm | null |
CoalgEquiv (R : Type*) [CommSemiring R] (A B : Type*)
[AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B]
[CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗc[R] B, A ≃ₗ[R] B where
attribute [nolint docBlame] CoalgEquiv.toCoalgHom
attribute [nolint docBlame] CoalgEquiv.toLinearEquiv
@[inherit_doc CoalgEquiv]
notation:50 A " ≃ₗc[" R "] " B => CoalgEquiv R A B | structure | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom"
] | Mathlib/RingTheory/Coalgebra/Equiv.lean | CoalgEquiv | An equivalence of coalgebras is an invertible coalgebra homomorphism. |
CoalgEquivClass (F : Type*) (R A B : outParam Type*) [CommSemiring R]
[AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B]
[CoalgebraStruct R A] [CoalgebraStruct R B] [EquivLike F A B] : Prop
extends CoalgHomClass F R A B, SemilinearEquivClass F (RingHom.id R) A B | class | RingTheory | [
"Mathlib.RingTheory.Coalgebra.Hom"
] | Mathlib/RingTheory/Coalgebra/Equiv.lean | CoalgEquivClass | `CoalgEquivClass F R A B` asserts `F` is a type of bundled coalgebra equivalences
from `A` to `B`. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.