Loogle!

Result

Found 192 declarations mentioning Coalgebra.

Coalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] : Type (max u v)
CommSemiring.toCoalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) [CommSemiring R] : Coalgebra R R
Coalgebra.IsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Prop
Coalgebra.toCoalgebraStruct 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} [self : Coalgebra R A] : CoalgebraStruct R A
Coalgebra.Repr.ι 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (_repr : Coalgebra.Repr R a ι) : Type u_1
Finsupp.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Coalgebra R (ι →₀ A)
Pi.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u_1} {n : Type u_2} [CommSemiring R] [Fintype n] [DecidableEq n] {A : n → Type u_3} [(i : n) → AddCommMonoid (A i)] [(i : n) → Module R (A i)] [(i : n) → Coalgebra R (A i)] : Coalgebra R ((i : n) → A i)
Prod.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : Coalgebra R (A × B)
Prod.instCoalgebraStruct 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct R (A × B)
Finsupp.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (ι : Type v) (A : Type w) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] [Coalgebra.IsCocomm R A] : Coalgebra.IsCocomm R (ι →₀ A)
DFinsupp.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (ι : Type v) (A : ι → Type w) [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [(i : ι) → Coalgebra R (A i)] : Coalgebra R (Π₀ (i : ι), A i)
Equiv.coalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u_1) {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [Coalgebra R B] (e : A ≃ₗ[R] B) : Coalgebra R A
LinearEquiv.coalgebra 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u_1) {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [Coalgebra R B] (e : A ≃ₗ[R] B) : Coalgebra R A
Pi.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u_1} {n : Type u_2} [CommSemiring R] [Fintype n] [DecidableEq n] {A : n → Type u_3} [(i : n) → AddCommMonoid (A i)] [(i : n) → Module R (A i)] [(i : n) → Coalgebra R (A i)] [∀ (i : n), Coalgebra.IsCocomm R (A i)] : Coalgebra.IsCocomm R ((i : n) → A i)
Prod.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] [Coalgebra.IsCocomm R A] [Coalgebra.IsCocomm R B] : Coalgebra.IsCocomm R (A × B)
DFinsupp.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (ι : Type v) (A : ι → Type w) [DecidableEq ι] [CommSemiring R] [(i : ι) → AddCommMonoid (A i)] [(i : ι) → Module R (A i)] [(i : ι) → Coalgebra R (A i)] [∀ (i : ι), Coalgebra.IsCocomm R (A i)] : Coalgebra.IsCocomm R (Π₀ (i : ι), A i)
Equiv.coalgebraIsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u_1) {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [Coalgebra R B] [Coalgebra.IsCocomm R B] (e : A ≃ₗ[R] B) : Coalgebra.IsCocomm R A
LinearEquiv.coalgebraIsCocomm 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u_1) {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B] [Coalgebra R B] [Coalgebra.IsCocomm R B] (e : A ≃ₗ[R] B) : Coalgebra.IsCocomm R A
Prod.counit_comp_inl 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct.counit ∘ₗ LinearMap.inl R A B = CoalgebraStruct.counit
Prod.counit_comp_inr 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct.counit ∘ₗ LinearMap.inr R A B = CoalgebraStruct.counit
Coalgebra.sum_counit_smul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (𝓡 : Coalgebra.Repr R a ι) : ∑ x ∈ 𝓡.index, CoalgebraStruct.counit (𝓡.left x) • 𝓡.right x = a
Coalgebra.sum_counit_tmul_eq 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (repr : Coalgebra.Repr R a ι) : ∑ i ∈ repr.index, CoalgebraStruct.counit (repr.left i) ⊗ₜ[R] repr.right i = 1 ⊗ₜ[R] a
Coalgebra.sum_tmul_counit_eq 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (repr : Coalgebra.Repr R a ι) : ∑ i ∈ repr.index, repr.left i ⊗ₜ[R] CoalgebraStruct.counit (repr.right i) = a ⊗ₜ[R] 1
Coalgebra.comm_comp_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] [Coalgebra.IsCocomm R A] : ↑(TensorProduct.comm R A A) ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul
Coalgebra.IsCocomm.comm_comp_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} {inst✝³ : Coalgebra R A} [self : Coalgebra.IsCocomm R A] : ↑(TensorProduct.comm R A A) ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul
Coalgebra.IsCocomm.mk 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (comm_comp_comul : ↑(TensorProduct.comm R A A) ∘ₗ CoalgebraStruct.comul = CoalgebraStruct.comul) : Coalgebra.IsCocomm R A
Coalgebra.sum_counit_tmul_map_eq 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {B : Type u_4} [AddCommMonoid B] [Module R B] {F : Type u_5} [FunLike F A B] [LinearMapClass F R A B] (f : F) (a : A) {repr : Coalgebra.Repr R a ι} : ∑ i ∈ repr.index, CoalgebraStruct.counit (repr.left i) ⊗ₜ[R] f (repr.right i) = 1 ⊗ₜ[R] f a
Coalgebra.sum_map_tmul_counit_eq 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {B : Type u_4} [AddCommMonoid B] [Module R B] {F : Type u_5} [FunLike F A B] [LinearMapClass F R A B] (f : F) (a : A) {repr : Coalgebra.Repr R a ι} : ∑ i ∈ repr.index, f (repr.left i) ⊗ₜ[R] CoalgebraStruct.counit (repr.right i) = f a ⊗ₜ[R] 1
Coalgebra.lift_lsmul_comp_counit_comp_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : TensorProduct.lift (LinearMap.lsmul R A ∘ₗ CoalgebraStruct.counit) ∘ₗ CoalgebraStruct.comul = LinearMap.id
Prod.counit_apply 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] (r : A × B) : CoalgebraStruct.counit r = CoalgebraStruct.counit r.1 + CoalgebraStruct.counit r.2
Coalgebra.lTensor_counit_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (LinearMap.lTensor A CoalgebraStruct.counit) (CoalgebraStruct.comul a) = a ⊗ₜ[R] 1
Coalgebra.rTensor_counit_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (LinearMap.rTensor A CoalgebraStruct.counit) (CoalgebraStruct.comul a) = 1 ⊗ₜ[R] a
Prod.comul_comp_fst 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct.comul ∘ₗ LinearMap.fst R A B = TensorProduct.map (LinearMap.fst R A B) (LinearMap.fst R A B) ∘ₗ CoalgebraStruct.comul
Prod.comul_comp_snd 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct.comul ∘ₗ LinearMap.snd R A B = TensorProduct.map (LinearMap.snd R A B) (LinearMap.snd R A B) ∘ₗ CoalgebraStruct.comul
Coalgebra.comm_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] [Coalgebra.IsCocomm R A] (a : A) : (TensorProduct.comm R A A) (CoalgebraStruct.comul a) = CoalgebraStruct.comul a
Prod.comul_comp_inl 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct.comul ∘ₗ LinearMap.inl R A B = TensorProduct.map (LinearMap.inl R A B) (LinearMap.inl R A B) ∘ₗ CoalgebraStruct.comul
Prod.comul_comp_inr 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] : CoalgebraStruct.comul ∘ₗ LinearMap.inr R A B = TensorProduct.map (LinearMap.inr R A B) (LinearMap.inr R A B) ∘ₗ CoalgebraStruct.comul
Coalgebra.sum_tmul_tmul_eq 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} {κ : ι → Type u_2} {Λ : ι → Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (repr : Coalgebra.Repr R a ι) (a₁ : (i : ι) → Coalgebra.Repr R (repr.left i) (κ i)) (a₂ : (i : ι) → Coalgebra.Repr R (repr.right i) (Λ 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)
Coalgebra.sum_map_tmul_tmul_eq 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {ι : Type u_1} {κ : ι → Type u_2} {Λ : ι → Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {B : Type u_4} [AddCommMonoid B] [Module R B] {F : Type u_5} [FunLike F A B] [LinearMapClass F R A B] (f g h : F) (a : A) {repr : Coalgebra.Repr R a ι} {a₁ : (i : ι) → Coalgebra.Repr R (repr.left i) (κ i)} {a₂ : (i : ι) → Coalgebra.Repr R (repr.right i) (Λ i)} : ∑ i ∈ repr.index, ∑ j ∈ (a₂ i).index, f (repr.left i) ⊗ₜ[R] (g ((a₂ i).left j) ⊗ₜ[R] 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))
Coalgebra.coassoc 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} [self : Coalgebra R A] : ↑(TensorProduct.assoc R A A A) ∘ₗ LinearMap.rTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = LinearMap.lTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul
Coalgebra.rTensor_counit_comp_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} [self : Coalgebra R A] : LinearMap.rTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R R A) 1
Coalgebra.coassoc_symm 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : ↑(TensorProduct.assoc R A A A).symm ∘ₗ LinearMap.lTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = LinearMap.rTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul
Prod.comul_apply 📋 Mathlib.RingTheory.Coalgebra.Basic
(R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B] [Coalgebra R A] [Coalgebra R B] (r : A × B) : CoalgebraStruct.comul r = (TensorProduct.map (LinearMap.inl R A B) (LinearMap.inl R A B)) (CoalgebraStruct.comul r.1) + (TensorProduct.map (LinearMap.inr R A B) (LinearMap.inr R A B)) (CoalgebraStruct.comul r.2)
Coalgebra.coassoc_apply 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (TensorProduct.assoc R A A A) ((LinearMap.rTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)) = (LinearMap.lTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)
Coalgebra.coassoc_symm_apply 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (TensorProduct.assoc R A A A).symm ((LinearMap.lTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)) = (LinearMap.rTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)
Coalgebra.lTensor_counit_comp_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} [self : Coalgebra R A] : LinearMap.lTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R A R).flip 1
Coalgebra.mk 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [toCoalgebraStruct : CoalgebraStruct R A] (coassoc : ↑(TensorProduct.assoc R A A A) ∘ₗ LinearMap.rTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = LinearMap.lTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul) (rTensor_counit_comp_comul : LinearMap.rTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R R A) 1) (lTensor_counit_comp_comul : LinearMap.lTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R A R).flip 1) : Coalgebra R A
Coalgebra.counitCoalgHom 📋 Mathlib.RingTheory.Coalgebra.Hom
(R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : A →ₗc[R] R
Coalgebra.subsingleton_to_ring 📋 Mathlib.RingTheory.Coalgebra.Hom
{R : Type u} (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Subsingleton (A →ₗc[R] R)
Coalgebra.Repr.induced 📋 Mathlib.RingTheory.Coalgebra.Hom
{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 : Coalgebra.Repr R a ι) {F : Type u_2} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) : Coalgebra.Repr R (φ a) ι
Coalgebra.Repr.induced_index 📋 Mathlib.RingTheory.Coalgebra.Hom
{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 : Coalgebra.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
Coalgebra.ext_to_ring 📋 Mathlib.RingTheory.Coalgebra.Hom
{R : Type u} (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (f g : A →ₗc[R] R) : f = g
Coalgebra.ext_to_ring_iff 📋 Mathlib.RingTheory.Coalgebra.Hom
{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
Coalgebra.Repr.induced_left 📋 Mathlib.RingTheory.Coalgebra.Hom
{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 : Coalgebra.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✝
Coalgebra.Repr.induced_right 📋 Mathlib.RingTheory.Coalgebra.Hom
{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 : Coalgebra.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✝
Coalgebra.counitCoalgHom_apply 📋 Mathlib.RingTheory.Coalgebra.Hom
(R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (x : A) : (Coalgebra.counitCoalgHom R A) x = CoalgebraStruct.counit x
Coalgebra.counitCoalgHom_toLinearMap 📋 Mathlib.RingTheory.Coalgebra.Hom
(R : Type u) (A : Type v) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : ↑(Coalgebra.counitCoalgHom R A) = CoalgebraStruct.counit
CoalgEquiv.toCoalgebra 📋 Mathlib.RingTheory.Coalgebra.Equiv
{R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] [AddCommMonoid B] [Module R B] [CoalgebraStruct R B] (f : A ≃ₗc[R] B) : Coalgebra R B
CoalgCat.of 📋 Mathlib.Algebra.Category.CoalgCat.Basic
(R : Type u) [CommRing R] (X : Type v) [AddCommGroup X] [Module R X] [Coalgebra R X] : CoalgCat R
CoalgCat.mk 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] (toModuleCat : ModuleCat R) (instCoalgebra : Coalgebra R ↑toModuleCat) : CoalgCat R
CoalgCat.instCoalgebra 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] (self : CoalgCat R) : Coalgebra R ↑self.toModuleCat
CoalgEquiv.toCoalgIso_refl 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X : Type v} [AddCommGroup X] [Module R X] [Coalgebra R X] : (CoalgEquiv.refl R X).toCoalgIso = CategoryTheory.Iso.refl (CoalgCat.of R X)
CoalgEquiv.toCoalgIso 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [Coalgebra R X] [Coalgebra R Y] (e : X ≃ₗc[R] Y) : CoalgCat.of R X ≅ CoalgCat.of R Y
CoalgCat.ofHom 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [Coalgebra R X] [Coalgebra R Y] (f : X →ₗc[R] Y) : CoalgCat.of R X ⟶ CoalgCat.of R Y
CoalgCat.of_counit 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X : Type v} [AddCommGroup X] [Module R X] [Coalgebra R X] : CoalgebraStruct.counit = CoalgebraStruct.counit
CoalgEquiv.toCoalgIso_symm 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [Coalgebra R X] [Coalgebra R Y] (e : X ≃ₗc[R] Y) : e.symm.toCoalgIso = e.toCoalgIso.symm
CoalgCat.of_comul 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X : Type v} [AddCommGroup X] [Module R X] [Coalgebra R X] : CoalgebraStruct.comul = CoalgebraStruct.comul
CoalgEquiv.toCoalgIso_hom 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [Coalgebra R X] [Coalgebra R Y] (e : X ≃ₗc[R] Y) : e.toCoalgIso.hom = CoalgCat.ofHom ↑e
CoalgEquiv.toCoalgIso_trans 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X Y Z : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [AddCommGroup Z] [Module R Z] [Coalgebra R X] [Coalgebra R Y] [Coalgebra R Z] (e : X ≃ₗc[R] Y) (f : Y ≃ₗc[R] Z) : (e.trans f).toCoalgIso = e.toCoalgIso ≪≫ f.toCoalgIso
CoalgEquiv.toCoalgIso_inv 📋 Mathlib.Algebra.Category.CoalgCat.Basic
{R : Type u} [CommRing R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y] [Coalgebra R X] [Coalgebra R Y] (e : X ≃ₗc[R] Y) : e.toCoalgIso.inv = CoalgCat.ofHom ↑e.symm
Bialgebra.toCoalgebra 📋 Mathlib.RingTheory.Bialgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} [self : Bialgebra R A] : Coalgebra R A
Bialgebra.mk' 📋 Mathlib.RingTheory.Bialgebra.Basic
(R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] [C : Coalgebra R A] (counit_one : CoalgebraStruct.counit 1 = 1) (counit_mul : ∀ {a b : A}, CoalgebraStruct.counit (a * b) = CoalgebraStruct.counit a * CoalgebraStruct.counit b) (comul_one : CoalgebraStruct.comul 1 = 1) (comul_mul : ∀ {a b : A}, CoalgebraStruct.comul (a * b) = CoalgebraStruct.comul a * CoalgebraStruct.comul b) : Bialgebra R A
Bialgebra.mk 📋 Mathlib.RingTheory.Bialgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [toAlgebra : Algebra R A] [toCoalgebra : Coalgebra R A] (counit_one : CoalgebraStruct.counit 1 = 1) (mul_compr₂_counit : (LinearMap.mul R A).compr₂ CoalgebraStruct.counit = (LinearMap.mul R R).compl₁₂ CoalgebraStruct.counit CoalgebraStruct.counit) (comul_one : CoalgebraStruct.comul 1 = 1) (mul_compr₂_comul : (LinearMap.mul R A).compr₂ CoalgebraStruct.comul = (LinearMap.mul R (TensorProduct R A A)).compl₁₂ CoalgebraStruct.comul CoalgebraStruct.comul) : Bialgebra R A
CoassocSimps.map_counit_comp_comul_left 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') : TensorProduct.map CoalgebraStruct.counit f ∘ₗ CoalgebraStruct.comul = TensorProduct.map LinearMap.id f ∘ₗ ↑(TensorProduct.lid R M).symm
CoassocSimps.map_counit_comp_comul_right 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') : TensorProduct.map f CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = TensorProduct.map f LinearMap.id ∘ₗ ↑(TensorProduct.rid R M).symm
CoassocSimps.map_counit_comp_comul_left_assoc 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {P : Type u_5} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') (g : P →ₗ[R] M) : TensorProduct.map CoalgebraStruct.counit f ∘ₗ CoalgebraStruct.comul ∘ₗ g = TensorProduct.map LinearMap.id f ∘ₗ ↑(TensorProduct.lid R M).symm ∘ₗ g
CoassocSimps.map_counit_comp_comul_right_assoc 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {P : Type u_5} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') (g : P →ₗ[R] M) : TensorProduct.map f CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul ∘ₗ g = TensorProduct.map f LinearMap.id ∘ₗ ↑(TensorProduct.rid R M).symm ∘ₗ g
CoassocSimps.coassoc_left 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') : ↑(TensorProduct.assoc R M M M') ∘ₗ TensorProduct.map CoalgebraStruct.comul f ∘ₗ CoalgebraStruct.comul = TensorProduct.map LinearMap.id (TensorProduct.map LinearMap.id f) ∘ₗ TensorProduct.map LinearMap.id CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul
CoassocSimps.coassoc_left_assoc 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {N : Type u_4} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') (g : N →ₗ[R] M) : ↑(TensorProduct.assoc R M M M') ∘ₗ TensorProduct.map CoalgebraStruct.comul f ∘ₗ CoalgebraStruct.comul ∘ₗ g = TensorProduct.map LinearMap.id (TensorProduct.map LinearMap.id f) ∘ₗ TensorProduct.map LinearMap.id CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul ∘ₗ g
CoassocSimps.coassoc_right 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') : ↑(TensorProduct.assoc R M' M M).symm ∘ₗ TensorProduct.map f CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = TensorProduct.map (TensorProduct.map f LinearMap.id) LinearMap.id ∘ₗ TensorProduct.map CoalgebraStruct.comul LinearMap.id ∘ₗ CoalgebraStruct.comul
CoassocSimps.coassoc_right_assoc 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {N : Type u_4} {M' : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid M'] [Module R M'] [Coalgebra R M] (f : M →ₗ[R] M') (g : N →ₗ[R] M) : ↑(TensorProduct.assoc R M' M M).symm ∘ₗ TensorProduct.map f CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul ∘ₗ g = TensorProduct.map (TensorProduct.map f LinearMap.id) LinearMap.id ∘ₗ TensorProduct.map CoalgebraStruct.comul LinearMap.id ∘ₗ CoalgebraStruct.comul ∘ₗ g
CoassocSimps.assoc_comp_map_comm_comp_comul_comp_comul 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {N : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Coalgebra R M] (f : M →ₗ[R] N) : ↑(TensorProduct.assoc R M M N) ∘ₗ TensorProduct.map (↑(TensorProduct.comm R M M) ∘ₗ CoalgebraStruct.comul) f ∘ₗ CoalgebraStruct.comul = TensorProduct.map LinearMap.id (TensorProduct.map LinearMap.id f ∘ₗ ↑(TensorProduct.comm R M M)) ∘ₗ ↑(TensorProduct.assoc R M M M) ∘ₗ TensorProduct.map CoalgebraStruct.comul LinearMap.id ∘ₗ ↑(TensorProduct.comm R M M) ∘ₗ CoalgebraStruct.comul
CoassocSimps.assoc_comp_map_comm_comp_comul_comp_comul_assoc 📋 Mathlib.RingTheory.Coalgebra.CoassocSimps
{R : Type u_1} {M : Type u_3} {N : Type u_4} {Q : Type u_9} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] [Coalgebra R M] (f : M →ₗ[R] N) (h : Q →ₗ[R] M) : ↑(TensorProduct.assoc R M M N) ∘ₗ TensorProduct.map (↑(TensorProduct.comm R M M) ∘ₗ CoalgebraStruct.comul) f ∘ₗ CoalgebraStruct.comul ∘ₗ h = TensorProduct.map LinearMap.id (TensorProduct.map LinearMap.id f ∘ₗ ↑(TensorProduct.comm R M M)) ∘ₗ ↑(TensorProduct.assoc R M M M) ∘ₗ TensorProduct.map CoalgebraStruct.comul LinearMap.id ∘ₗ ↑(TensorProduct.comm R M M) ∘ₗ CoalgebraStruct.comul ∘ₗ h
Coalgebra.comulCoalgHom 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
(R : Type u_5) (C : Type u_6) [CommSemiring R] [AddCommMonoid C] [Module R C] [Coalgebra R C] [Coalgebra.IsCocomm R C] : C →ₗc[R] TensorProduct R C C
Coalgebra.TensorProduct.lid 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
(R : Type u_5) (P : Type u_9) [CommSemiring R] [AddCommMonoid P] [Module R P] [Coalgebra R P] : TensorProduct R R P ≃ₗc[R] P
TensorProduct.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_1} {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [AddCommMonoid A] [AddCommMonoid B] [Algebra R S] [Module R A] [Module S A] [Module R B] [IsScalarTower R S A] [Coalgebra R B] [Coalgebra S A] : Coalgebra S (TensorProduct R A B)
TensorProduct.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_1} {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [AddCommMonoid A] [AddCommMonoid B] [Algebra R S] [Module R A] [Module S A] [Module R B] [IsScalarTower R S A] [Coalgebra R B] [Coalgebra S A] [Coalgebra.IsCocomm S A] [Coalgebra.IsCocomm R B] : Coalgebra.IsCocomm S (TensorProduct R A B)
Coalgebra.TensorProduct.lid_toLinearEquiv 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {P : Type u_9} [CommSemiring R] [AddCommMonoid P] [Module R P] [Coalgebra R P] : (Coalgebra.TensorProduct.lid R P).toLinearEquiv = TensorProduct.lid R P
Coalgebra.TensorProduct.rid 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
(R : Type u_5) (S : Type u_6) (M : Type u_7) [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [Coalgebra S M] : TensorProduct R M R ≃ₗc[S] M
CoalgHom.lTensor 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} (M : Type u_6) {N : Type u_7} {P : Type u_8} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] (f : N →ₗc[R] P) : TensorProduct R M N →ₗc[R] TensorProduct R M P
CoalgHom.rTensor 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} (M : Type u_6) {N : Type u_7} {P : Type u_8} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] (f : N →ₗc[R] P) : TensorProduct R N M →ₗc[R] TensorProduct R P M
Coalgebra.TensorProduct.lid_tmul 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {P : Type u_9} [CommSemiring R] [AddCommMonoid P] [Module R P] [Coalgebra R P] (r : R) (a : P) : (Coalgebra.TensorProduct.lid R P) (r ⊗ₜ[R] a) = r • a
Coalgebra.TensorProduct.rid_toLinearEquiv 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [Coalgebra S M] : (Coalgebra.TensorProduct.rid R S M).toLinearEquiv = TensorProduct.AlgebraTensorModule.rid R S M
Coalgebra.TensorProduct.map 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} {Q : Type u_10} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] [Coalgebra R Q] (f : M →ₗc[S] N) (g : P →ₗc[R] Q) : TensorProduct R M P →ₗc[S] TensorProduct R N Q
Coalgebra.TensorProduct.lid_symm_apply 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {P : Type u_9} [CommSemiring R] [AddCommMonoid P] [Module R P] [Coalgebra R P] (a : P) : (Coalgebra.TensorProduct.lid R P).symm a = 1 ⊗ₜ[R] a
Coalgebra.TensorProduct.rid_tmul 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [Coalgebra S M] (r : R) (a : M) : (Coalgebra.TensorProduct.rid R S M) (a ⊗ₜ[R] r) = r • a
Coalgebra.TensorProduct.rid_symm_apply 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [Coalgebra S M] (a : M) : (Coalgebra.TensorProduct.rid R S M).symm a = a ⊗ₜ[R] 1
Coalgebra.TensorProduct.map_tmul 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} {Q : Type u_10} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] [Coalgebra R Q] (f : M →ₗc[S] N) (g : P →ₗc[R] Q) (x : M) (y : P) : (Coalgebra.TensorProduct.map f g) (x ⊗ₜ[R] y) = f x ⊗ₜ[R] g y
Coalgebra.TensorProduct.assoc 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
(R : Type u_5) (S : Type u_6) (M : Type u_7) (N : Type u_8) (P : Type u_9) [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] : TensorProduct R (TensorProduct S M N) P ≃ₗc[S] TensorProduct S M (TensorProduct R N P)
Coalgebra.TensorProduct.assoc_toLinearEquiv 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] : (Coalgebra.TensorProduct.assoc R S M N P).toLinearEquiv = TensorProduct.AlgebraTensorModule.assoc R S S M N P
Coalgebra.TensorProduct.map_toLinearMap 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} {Q : Type u_10} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] [Coalgebra R Q] (f : M →ₗc[S] N) (g : P →ₗc[R] Q) : ↑(Coalgebra.TensorProduct.map f g) = TensorProduct.AlgebraTensorModule.map ↑f ↑g
Coalgebra.TensorProduct.assoc_tmul 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] (x : M) (y : N) (z : P) : (Coalgebra.TensorProduct.assoc R S M N P) (x ⊗ₜ[S] y ⊗ₜ[R] z) = x ⊗ₜ[S] (y ⊗ₜ[R] z)
Coalgebra.TensorProduct.assoc_symm_tmul 📋 Mathlib.RingTheory.Coalgebra.TensorProduct
{R : Type u_5} {S : Type u_6} {M : Type u_7} {N : Type u_8} {P : Type u_9} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module S M] [IsScalarTower R S M] [Coalgebra S M] [Module S N] [IsScalarTower R S N] [Coalgebra S N] [Coalgebra R P] (x : M) (y : N) (z : P) : (Coalgebra.TensorProduct.assoc R S M N P).symm (x ⊗ₜ[S] (y ⊗ₜ[R] z)) = x ⊗ₜ[S] y ⊗ₜ[R] z
CoalgCat.tensorUnit_instCoalgebra 📋 Mathlib.Algebra.Category.CoalgCat.Monoidal
(R : Type u) [CommRing R] : (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CoalgCat R)).instCoalgebra = inferInstance
CoalgCat.tensorObj_instCoalgebra 📋 Mathlib.Algebra.Category.CoalgCat.Monoidal
(R : Type u) [CommRing R] (X Y : CoalgCat R) : (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y).instCoalgebra = inferInstance
CoalgCat.MonoidalCategoryAux.counit_tensorObj 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N : Type u} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Coalgebra R M] [Coalgebra R N] : CoalgebraStruct.counit = CoalgebraStruct.counit
CoalgCat.MonoidalCategoryAux.leftUnitor_hom_toLinearMap 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M : Type u} [AddCommGroup M] [Module R M] [Coalgebra R M] : (CategoryTheory.MonoidalCategoryStruct.leftUnitor (CoalgCat.of R M)).hom.toCoalgHom'.toLinearMap = ↑(TensorProduct.lid R M)
CoalgCat.MonoidalCategoryAux.rightUnitor_hom_toLinearMap 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M : Type u} [AddCommGroup M] [Module R M] [Coalgebra R M] : (CategoryTheory.MonoidalCategoryStruct.rightUnitor (CoalgCat.of R M)).hom.toCoalgHom'.toLinearMap = ↑(TensorProduct.rid R M)
CoalgCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_right 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] : CoalgebraStruct.counit = CoalgebraStruct.counit
CoalgCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_left 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] : CoalgebraStruct.counit = CoalgebraStruct.counit
CoalgCat.MonoidalCategoryAux.tensorHom_toLinearMap 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N P Q : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [AddCommGroup Q] [Module R M] [Module R N] [Module R P] [Module R Q] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] [Coalgebra R Q] (f : M →ₗc[R] N) (g : P →ₗc[R] Q) : (CategoryTheory.MonoidalCategoryStruct.tensorHom (CoalgCat.ofHom f) (CoalgCat.ofHom g)).toCoalgHom'.toLinearMap = TensorProduct.map f.toLinearMap g.toLinearMap
CoalgCat.MonoidalCategoryAux.comul_tensorObj 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N : Type u} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Coalgebra R M] [Coalgebra R N] : CoalgebraStruct.comul = CoalgebraStruct.comul
CoalgCat.MonoidalCategoryAux.associator_hom_toLinearMap 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] : (CategoryTheory.MonoidalCategoryStruct.associator (CoalgCat.of R M) (CoalgCat.of R N) (CoalgCat.of R P)).hom.toCoalgHom'.toLinearMap = ↑(TensorProduct.assoc R M N P)
CoalgCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_right 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] : CoalgebraStruct.comul = CoalgebraStruct.comul
CoalgCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_left 📋 Mathlib.Algebra.Category.CoalgCat.ComonEquivalence
{R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] : CoalgebraStruct.comul = CoalgebraStruct.comul
LinearMap.convOne 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid C] [Module R C] [Coalgebra R C] : One (WithConv (C →ₗ[R] A))
LinearMap.convSemiring 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid C] [Module R C] [Coalgebra R C] : Semiring (WithConv (C →ₗ[R] A))
LinearMap.convRing 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Ring A] [AddCommMonoid C] [Algebra R A] [Module R C] [Coalgebra R C] : Ring (WithConv (C →ₗ[R] A))
LinearMap.convCommSemiring 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [CommSemiring A] [AddCommMonoid C] [Algebra R A] [Module R C] [Coalgebra R C] [Coalgebra.IsCocomm R C] : CommSemiring (WithConv (C →ₗ[R] A))
LinearMap.convCommRing 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [CommRing A] [AddCommMonoid C] [Algebra R A] [Module R C] [Coalgebra R C] [Coalgebra.IsCocomm R C] : CommRing (WithConv (C →ₗ[R] A))
LinearMap.convAlgebra 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {S : Type u_2} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid C] [Module R C] [Coalgebra R C] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] : Algebra S (WithConv (C →ₗ[R] A))
LinearMap.convNonUnitalSemiring 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [AddCommMonoid C] [Module R C] [Coalgebra R C] : NonUnitalSemiring (WithConv (C →ₗ[R] A))
LinearMap.convOne_def 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid C] [Module R C] [Coalgebra R C] : 1 = WithConv.toConv (Algebra.linearMap R A ∘ₗ CoalgebraStruct.counit)
LinearMap.convNonUnitalRing 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [NonUnitalRing A] [AddCommMonoid C] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [Module R C] [Coalgebra R C] : NonUnitalRing (WithConv (C →ₗ[R] A))
LinearMap.convOne_apply 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid C] [Module R C] [Coalgebra R C] (c : C) : (WithConv.ofConv 1) c = (algebraMap R A) (CoalgebraStruct.counit c)
LinearMap.convAlgebraMap_apply 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {S : Type u_2} {A : Type u_3} {C : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid C] [Module R C] [Coalgebra R C] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] (s : S) (c : C) : ((algebraMap S (WithConv (C →ₗ[R] A))) s).ofConv c = s • (algebraMap R A) (CoalgebraStruct.counit c)
LinearMap.convMul_comp_coalgHom_distrib 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [AddCommMonoid C] [Module R C] [Coalgebra R C] [AddCommMonoid B] [Module R B] [CoalgebraStruct R B] (f g : WithConv (C →ₗ[R] A)) (h : B →ₗc[R] C) : (f * g).ofConv ∘ₗ h.toLinearMap = (WithConv.toConv (f.ofConv ∘ₗ h.toLinearMap) * WithConv.toConv (g.ofConv ∘ₗ h.toLinearMap)).ofConv
LinearMap.nonUnitalAlgHom_comp_convMul_distrib 📋 Mathlib.RingTheory.Coalgebra.Convolution
{R : Type u_1} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [AddCommMonoid C] [Module R C] [Coalgebra R C] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (h : A →ₙₐ[R] B) (f g : WithConv (C →ₗ[R] A)) : ↑h ∘ₗ (f * g).ofConv = (WithConv.toConv (↑h ∘ₗ f.ofConv) * WithConv.toConv (↑h ∘ₗ g.ofConv)).ofConv
LinearMap.convIntrinsicStarRing 📋 Mathlib.Algebra.Star.LinearMap
{R : Type u_5} {A : Type u_6} {C : Type u_7} [CommSemiring R] [StarRing R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [StarRing A] [StarModule R A] [AddCommMonoid C] [Module R C] [StarAddMonoid C] [StarModule R C] [Coalgebra R C] (h : star (WithConv.toConv CoalgebraStruct.comul) = WithConv.toConv (↑(TensorProduct.comm R C C) ∘ₗ CoalgebraStruct.comul)) : StarRing (WithConv (C →ₗ[R] A))
InnerProductSpace.mulOfCoalgebra 📋 Mathlib.Analysis.InnerProductSpace.Coalgebra
{𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [Coalgebra 𝕜 E] : Mul E
InnerProductSpace.ringOfCoalgebra 📋 Mathlib.Analysis.InnerProductSpace.Coalgebra
{𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [Coalgebra 𝕜 E] : Ring E
InnerProductSpace.algebraOfCoalgebra 📋 Mathlib.Analysis.InnerProductSpace.Coalgebra
{𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [Coalgebra 𝕜 E] : Algebra 𝕜 E
InnerProductSpace.coalgebraOfAlgebra 📋 Mathlib.Analysis.InnerProductSpace.Coalgebra
{𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {A : Type u_3} [Ring A] [Module 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (e : E ≃ₗ[𝕜] A) : Coalgebra 𝕜 E
InnerProductSpace.AlgebraOfCoalgebra.mul_def 📋 Mathlib.Analysis.InnerProductSpace.Coalgebra
{𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [Coalgebra 𝕜 E] (x y : E) : x * y = (LinearMap.adjoint CoalgebraStruct.comul) (x ⊗ₜ[𝕜] y)
GroupLike 📋 Mathlib.RingTheory.Coalgebra.GroupLike
(R : Type u_2) (A : Type u_3) [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Type u_3
IsGroupLikeElem 📋 Mathlib.RingTheory.Coalgebra.GroupLike
(R : Type u_2) {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : Prop
GroupLike.val 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (self : GroupLike R A) : A
GroupLike.instCoeOut 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : CoeOut (GroupLike R A) A
GroupLike.mk 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (val : A) (isGroupLikeElem_val : IsGroupLikeElem R val) : GroupLike R A
GroupLike.val_injective 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Function.Injective GroupLike.val
GroupLike.valEquiv 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : GroupLike R A ≃ Subtype (IsGroupLikeElem R)
GroupLike.isGroupLikeElem_val 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (self : GroupLike R A) : IsGroupLikeElem R ↑self
IsGroupLikeElem.ne_zero 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} [Nontrivial R] (ha : IsGroupLikeElem R a) : a ≠ 0
GroupLike.ext 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} {inst✝³ : Coalgebra R A} {x y : GroupLike R A} (val : ↑x = ↑y) : x = y
GroupLike.ext_iff 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} {inst✝³ : Coalgebra R A} {x y : GroupLike R A} : x = y ↔ ↑x = ↑y
GroupLike.val_inj 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a b : GroupLike R A} : ↑a = ↑b ↔ a = b
linearIndepOn_isGroupLikeElem 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommRing R] [IsDomain R] [AddCommGroup A] [Module R A] [Coalgebra R A] [Module.IsTorsionFree R A] : LinearIndepOn R id {a | IsGroupLikeElem R a}
linearIndep_groupLikeVal 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommRing R] [IsDomain R] [AddCommGroup A] [Module R A] [Coalgebra R A] [Module.IsTorsionFree R A] : LinearIndependent R GroupLike.val
IsGroupLikeElem.map 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Coalgebra R A] [Module R B] [Coalgebra R B] {a : A} [FunLike F A B] [CoalgHomClass F R A B] (f : F) (ha : IsGroupLikeElem R a) : IsGroupLikeElem R (f a)
isGroupLikeElem_map_equiv 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Coalgebra R A] [Module R B] [Coalgebra R B] {a : A} [EquivLike F A B] [CoalgEquivClass F R A B] (f : F) : IsGroupLikeElem R (f a) ↔ IsGroupLikeElem R a
IsGroupLikeElem.counit_eq_one 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (self : IsGroupLikeElem R a) : CoalgebraStruct.counit a = 1
GroupLike.valEquiv_apply_coe 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : GroupLike R A) : ↑(GroupLike.valEquiv a) = ↑a
GroupLike.val_valEquiv_symm_apply 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : Subtype (IsGroupLikeElem R)) : ↑(GroupLike.valEquiv.symm a) = ↑a
IsGroupLikeElem.comul_eq_tmul_self 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (self : IsGroupLikeElem R a) : CoalgebraStruct.comul a = a ⊗ₜ[R] a
IsGroupLikeElem.mk 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {a : A} (counit_eq_one : CoalgebraStruct.counit a = 1) (comul_eq_tmul_self : CoalgebraStruct.comul a = a ⊗ₜ[R] a) : IsGroupLikeElem R a
isGroupLikeElem_iff 📋 Mathlib.RingTheory.Coalgebra.GroupLike
(R : Type u_2) {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : IsGroupLikeElem R a ↔ CoalgebraStruct.counit a = 1 ∧ CoalgebraStruct.comul a = a ⊗ₜ[R] a
LaurentPolynomial.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
(R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Module R A] [Coalgebra R A] : Coalgebra R (LaurentPolynomial A)
AddMonoidAlgebra.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
(R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] (X : Type u_3) [Module R A] [Coalgebra R A] : Coalgebra R (AddMonoidAlgebra A X)
MonoidAlgebra.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
(R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] (X : Type u_3) [Module R A] [Coalgebra R A] : Coalgebra R (MonoidAlgebra A X)
LaurentPolynomial.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
(R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Module R A] [Coalgebra R A] [Coalgebra.IsCocomm R A] : Coalgebra.IsCocomm R (LaurentPolynomial A)
AddMonoidAlgebra.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] [Coalgebra.IsCocomm R A] : Coalgebra.IsCocomm R (AddMonoidAlgebra A X)
MonoidAlgebra.instIsCocomm 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] [Coalgebra.IsCocomm R A] : Coalgebra.IsCocomm R (MonoidAlgebra A X)
AddMonoidAlgebra.counit_single 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] (x : X) (a : A) : CoalgebraStruct.counit (AddMonoidAlgebra.single x a) = CoalgebraStruct.counit a
MonoidAlgebra.counit_single 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] (x : X) (a : A) : CoalgebraStruct.counit (MonoidAlgebra.single x a) = CoalgebraStruct.counit a
LaurentPolynomial.counit_C 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Module R A] [Coalgebra R A] (a : A) : CoalgebraStruct.counit (LaurentPolynomial.C a) = CoalgebraStruct.counit a
LaurentPolynomial.counit_C_mul_T 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Module R A] [Coalgebra R A] (a : A) (n : ℤ) : CoalgebraStruct.counit (LaurentPolynomial.C a * LaurentPolynomial.T n) = CoalgebraStruct.counit a
AddMonoidAlgebra.comul_single 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] (x : X) (a : A) : CoalgebraStruct.comul (AddMonoidAlgebra.single x a) = (TensorProduct.map (AddMonoidAlgebra.lsingle x) (AddMonoidAlgebra.lsingle x)) (CoalgebraStruct.comul a)
MonoidAlgebra.comul_single 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] (x : X) (a : A) : CoalgebraStruct.comul (MonoidAlgebra.single x a) = (TensorProduct.map (MonoidAlgebra.lsingle x) (MonoidAlgebra.lsingle x)) (CoalgebraStruct.comul a)
LaurentPolynomial.comul_C 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Module R A] [Coalgebra R A] (a : A) : CoalgebraStruct.comul (LaurentPolynomial.C a) = (TensorProduct.map (AddMonoidAlgebra.lsingle 0) (AddMonoidAlgebra.lsingle 0)) (CoalgebraStruct.comul a)
LaurentPolynomial.comul_C_mul_T 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Module R A] [Coalgebra R A] (a : A) (n : ℤ) : CoalgebraStruct.comul (LaurentPolynomial.C a * LaurentPolynomial.T n) = (TensorProduct.map (AddMonoidAlgebra.lsingle n) (AddMonoidAlgebra.lsingle n)) (CoalgebraStruct.comul a)
AddMonoidAlgebra.comul_def 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] : CoalgebraStruct.comul = TensorProduct.map ↑(AddMonoidAlgebra.coeffLinearEquiv R).symm ↑(AddMonoidAlgebra.coeffLinearEquiv R).symm ∘ₗ CoalgebraStruct.comul ∘ₗ ↑(AddMonoidAlgebra.coeffLinearEquiv R)
MonoidAlgebra.comul_def 📋 Mathlib.RingTheory.Coalgebra.MonoidAlgebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {X : Type u_3} [Module R A] [Coalgebra R A] : CoalgebraStruct.comul = TensorProduct.map ↑(MonoidAlgebra.coeffLinearEquiv R).symm ↑(MonoidAlgebra.coeffLinearEquiv R).symm ∘ₗ CoalgebraStruct.comul ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv R)
Coalgebra.IsSkewPrimitiveElem 📋 Mathlib.RingTheory.Coalgebra.Primitive
(R : Type u_2) {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (g h a : A) : Prop
Coalgebra.skewPrimitive 📋 Mathlib.RingTheory.Coalgebra.Primitive
(R : Type u_2) {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (g h : A) : Submodule R A
Coalgebra.IsSkewPrimitiveElem.zero 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h : A} : Coalgebra.IsSkewPrimitiveElem R g h 0
Coalgebra.IsSkewPrimitiveElem.neg 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommGroup A] [Module R A] [Coalgebra R A] {g h a : A} (ha : Coalgebra.IsSkewPrimitiveElem R g h a) : Coalgebra.IsSkewPrimitiveElem R g h (-a)
Coalgebra.isSkewPrimitiveElem_neg 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommGroup A] [Module R A] [Coalgebra R A] {g h a : A} : Coalgebra.IsSkewPrimitiveElem R g h (-a) ↔ Coalgebra.IsSkewPrimitiveElem R g h a
Coalgebra.IsSkewPrimitiveElem.add 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a b : A} (ha : Coalgebra.IsSkewPrimitiveElem R g h a) (hb : Coalgebra.IsSkewPrimitiveElem R g h b) : Coalgebra.IsSkewPrimitiveElem R g h (a + b)
Coalgebra.mem_skewPrimitive 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} : a ∈ Coalgebra.skewPrimitive R g h ↔ Coalgebra.IsSkewPrimitiveElem R g h a
Coalgebra.IsSkewPrimitiveElem.sub 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommGroup A] [Module R A] [Coalgebra R A] {g h a b : A} (ha : Coalgebra.IsSkewPrimitiveElem R g h a) (hb : Coalgebra.IsSkewPrimitiveElem R g h b) : Coalgebra.IsSkewPrimitiveElem R g h (a - b)
Coalgebra.IsSkewPrimitiveElem.smul 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} (ha : Coalgebra.IsSkewPrimitiveElem R g h a) (r : R) : Coalgebra.IsSkewPrimitiveElem R g h (r • a)
Coalgebra.IsSkewPrimitiveElem.map 📋 Mathlib.RingTheory.Coalgebra.Primitive
{F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] [AddCommMonoid B] [Module R B] [Coalgebra R B] {g h a : A} [FunLike F A B] [CoalgHomClass F R A B] (f : F) (ha : Coalgebra.IsSkewPrimitiveElem R g h a) : Coalgebra.IsSkewPrimitiveElem R (f g) (f h) (f a)
Coalgebra.isSkewPrimitiveElem_map_equiv 📋 Mathlib.RingTheory.Coalgebra.Primitive
{F : Type u_1} {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] [AddCommMonoid B] [Module R B] [Coalgebra R B] {g h a : A} [EquivLike F A B] [CoalgEquivClass F R A B] (f : F) : Coalgebra.IsSkewPrimitiveElem R (f g) (f h) (f a) ↔ Coalgebra.IsSkewPrimitiveElem R g h a
Coalgebra.IsSkewPrimitiveElem.counit_eq_zero 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} (self : Coalgebra.IsSkewPrimitiveElem R g h a) : CoalgebraStruct.counit a = 0
Coalgebra.IsSkewPrimitiveElem.comul_eq_tmul_add_tmul 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} (self : Coalgebra.IsSkewPrimitiveElem R g h a) : CoalgebraStruct.comul a = g ⊗ₜ[R] a + a ⊗ₜ[R] h
Coalgebra.IsSkewPrimitiveElem.mk 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} (counit_eq_zero : CoalgebraStruct.counit a = 0) (comul_eq_tmul_add_tmul : CoalgebraStruct.comul a = g ⊗ₜ[R] a + a ⊗ₜ[R] h) : Coalgebra.IsSkewPrimitiveElem R g h a
Coalgebra.isSkewPrimitiveElem_iff 📋 Mathlib.RingTheory.Coalgebra.Primitive
(R : Type u_2) {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (g h a : A) : Coalgebra.IsSkewPrimitiveElem R g h a ↔ CoalgebraStruct.counit a = 0 ∧ CoalgebraStruct.comul a = g ⊗ₜ[R] a + a ⊗ₜ[R] h
Coalgebra.IsSkewPrimitiveElem.of_comul_eq_tmul_add_tmul 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} [IsCancelAdd A] (hg : CoalgebraStruct.counit g = 1) (hh : CoalgebraStruct.counit h = 1) (ha : CoalgebraStruct.comul a = g ⊗ₜ[R] a + a ⊗ₜ[R] h) : Coalgebra.IsSkewPrimitiveElem R g h a
Coalgebra.isSkewPrimitiveElem_iff_comul_eq_tmul_add_tmul 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} [IsCancelAdd A] (hg : CoalgebraStruct.counit g = 1) (hh : CoalgebraStruct.counit h = 1) : Coalgebra.IsSkewPrimitiveElem R g h a ↔ CoalgebraStruct.comul a = g ⊗ₜ[R] a + a ⊗ₜ[R] h
Coalgebra.counit_eq_zero_of_comul_eq_tmul_add_tmul 📋 Mathlib.RingTheory.Coalgebra.Primitive
{R : Type u_2} {A : Type u_3} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] {g h a : A} [IsCancelAdd A] (hg : CoalgebraStruct.counit g = 1) (hh : CoalgebraStruct.counit h = 1) (ha : CoalgebraStruct.comul a = g ⊗ₜ[R] a + a ⊗ₜ[R] h) : CoalgebraStruct.counit a = 0
Coalgebra.Quotient.instQuotientSubmodule 📋 Mathlib.RingTheory.Coalgebra.Quotient
{R : Type u_1} {C : Type u_2} [CommRing R] [AddCommGroup C] [Module R C] [Coalgebra R C] (I : Submodule R C) [I.IsCoideal] : Coalgebra R (C ⧸ I)
MulOpposite.instCoalgebra 📋 Mathlib.RingTheory.Coalgebra.MulOpposite
{R : Type u_1} {A : Type u_2} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : Coalgebra R Aᵐᵒᵖ

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c