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Result

Found 168 declarations mentioning ContinuousMonoidHom.

ContinuousMonoidHom 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Type (max u_2 u_3)
ContinuousMonoidHom.id 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) [Monoid A] [TopologicalSpace A] : A →ₜ* A
ContinuousMonoidHom.instInhabited 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Inhabited (A →ₜ* B)
ContinuousMonoidHom.instOne 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : One (A →ₜ* B)
ContinuousMonoidHom.instFunLike 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : FunLike (A →ₜ* B) A B
ContinuousMonoidHom.diag 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) [Monoid A] [TopologicalSpace A] : A →ₜ* A × A
ContinuousMonoidHom.toContinuousMap 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A →ₜ* B) : C(A, B)
ContinuousMonoidHom.fst 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : A × B →ₜ* A
ContinuousMonoidHom.inl 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : A →ₜ* A × B
ContinuousMonoidHom.inr 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : B →ₜ* A × B
ContinuousMonoidHom.snd 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : A × B →ₜ* B
ContinuousMonoidHom.instContinuousMapClass 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousMapClass (A →ₜ* B) A B
ContinuousMonoidHom.instCommGroup 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (E : Type u_6) [Monoid A] [TopologicalSpace A] [CommGroup E] [TopologicalSpace E] [IsTopologicalGroup E] : CommGroup (A →ₜ* E)
ContinuousMonoidHom.inv 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(E : Type u_6) [CommGroup E] [TopologicalSpace E] [IsTopologicalGroup E] : E →ₜ* E
ContinuousMonoidHom.toContinuousMap_injective 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Function.Injective ContinuousMonoidHom.toContinuousMap
ContinuousMonoidHom.toMonoidHom 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A →ₜ* B) : A →* B
ContinuousMonoidHom.coe_id 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) [Monoid A] [TopologicalSpace A] : ⇑(ContinuousMonoidHom.id A) = id
ContinuousMonoidHom.id_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) [Monoid A] [TopologicalSpace A] (x : A) : (ContinuousMonoidHom.id A) x = x
ContinuousMonoidHom.instCommMonoid 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {E : Type u_6} [Monoid A] [TopologicalSpace A] [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] : CommMonoid (A →ₜ* E)
ContinuousMonoidHom.comp 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : B →ₜ* C) (f : A →ₜ* B) : A →ₜ* C
ContinuousMonoidHom.instMonoidHomClass 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : MonoidHomClass (A →ₜ* B) A B
ContinuousMonoidHom.swap 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : A × B →ₜ* B × A
ContinuousMonoidHom.toMonoidHom_injective 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Function.Injective ContinuousMonoidHom.toMonoidHom
ContinuousMonoidHom.mul 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(E : Type u_6) [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] : E × E →ₜ* E
ContinuousMonoidHom.ofClass 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (F : Type u_7) [FunLike F A B] [ContinuousMapClass F A B] [MonoidHomClass F A B] (f : F) : A →ₜ* B
ContinuousMonoidHom.toContinuousMonoidHom 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {F : Type u_7} [FunLike F A B] [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) : A →ₜ* B
ContinuousMonoidHom.instCoeOutOfMonoidHomClassOfContinuousMapClass 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {F : Type u_7} [FunLike F A B] [MonoidHomClass F A B] [ContinuousMapClass F A B] : CoeOut F (A →ₜ* B)
ContinuousMonoidHom.prod 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : A →ₜ* B) (g : A →ₜ* C) : A →ₜ* B × C
ContinuousMonoidHom.coe_toContinuousMap 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →ₜ* B) : f.toContinuousMap = ↑f
ContinuousMonoidHom.continuous_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A →ₜ* B) : Continuous (↑self.toMonoidHom).toFun
ContinuousMonoidHom.diag_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) [Monoid A] [TopologicalSpace A] (i : A) : (ContinuousMonoidHom.diag A) i = (i, i)
ContinuousMonoidHom.prodMap 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid A] [Monoid B] [Monoid C] [Monoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : A →ₜ* C) (g : B →ₜ* D) : A × B →ₜ* C × D
ContinuousMonoidHom.coprod 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {E : Type u_6} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] (f : A →ₜ* E) (g : B →ₜ* E) : A × B →ₜ* E
ContinuousMonoidHom.mk 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (toMonoidHom : A →* B) (continuous_toFun : Continuous (↑toMonoidHom).toFun := by fun_prop) : A →ₜ* B
ContinuousMonoidHom.one_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (x✝ : A) : 1 x✝ = 1
ContinuousMonoidHom.fst_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousMonoidHom.fst A B) self = self.1
ContinuousMonoidHom.snd_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousMonoidHom.snd A B) self = self.2
ContinuousMonoidHom.coe_one 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ⇑1 = 1
ContinuousMonoidHom.coe_toMonoidHom 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →ₜ* B) : f.toMonoidHom = ↑f
ContinuousMonoidHom.inv_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(E : Type u_6) [CommGroup E] [TopologicalSpace E] [IsTopologicalGroup E] (a✝ : E) : (ContinuousMonoidHom.inv E) a✝ = a✝⁻¹
ContinuousMonoidHom.ext 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {f g : A →ₜ* B} (h : ∀ (x : A), f x = g x) : f = g
ContinuousMonoidHom.coe_coe 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {F : Type u_7} [FunLike F A B] [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) : ⇑↑f = ⇑f
ContinuousMonoidHom.ext_iff 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {f g : A →ₜ* B} : f = g ↔ ∀ (x : A), f x = g x
ContinuousMonoidHom.inl_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A) : (ContinuousMonoidHom.inl A B) i = (i, 1)
ContinuousMonoidHom.inr_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : B) : (ContinuousMonoidHom.inr A B) i = (1, i)
ContinuousMonoidHom.toContinuousMap_toContinuousMonoidHom 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {F : Type u_7} [FunLike F A B] [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) : ↑↑f = ↑f
ContinuousMonoidHom.swap_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (i : A × B) : (ContinuousMonoidHom.swap A B) i = (i.2, i.1)
ContinuousMonoidHom.comp_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : B →ₜ* C) (f : A →ₜ* B) (x : A) : (g.comp f) x = g (f x)
ContinuousMonoidHom.mul_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
(E : Type u_6) [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] (a✝ : E × E) : (ContinuousMonoidHom.mul E) a✝ = a✝.1 * a✝.2
ContinuousMonoidHom.coe_comp 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : B →ₜ* C) (f : A →ₜ* B) : ⇑(g.comp f) = ⇑g ∘ ⇑f
ContinuousMonoidHom.toMonoidHom_toContinuousMonoidHom 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] {F : Type u_7} [FunLike F A B] [MonoidHomClass F A B] [ContinuousMapClass F A B] (f : F) : ↑↑f = ↑f
ContinuousMonoidHom.prod_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : A →ₜ* B) (g : A →ₜ* C) (i : A) : (f.prod g) i = (f i, g i)
ContinuousMonoidHom.pow_apply 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {E : Type u_6} [Monoid A] [TopologicalSpace A] [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] (f : A →ₜ* E) (n : ℕ) (a : A) : (f ^ n) a = f a ^ n
ContinuousMonoidHom.prodMap_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [Monoid A] [Monoid B] [Monoid C] [Monoid D] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] (f : A →ₜ* C) (g : B →ₜ* D) (i : A × B) : (f.prodMap g) i = (f i.1, g i.2)
ContinuousMonoidHom.coprod_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {B : Type u_3} {E : Type u_6} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] (f : A →ₜ* E) (g : B →ₜ* E) (x : A × B) : (f.coprod g) x = f x.1 * g x.2
ContinuousMonoidHom.mul_apply 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
{A : Type u_2} {E : Type u_6} [Monoid A] [TopologicalSpace A] [CommMonoid E] [TopologicalSpace E] [ContinuousMul E] (f g : A →ₜ* E) (a : A) : (f * g) a = f a * g a
SeparationQuotient.liftContinuousMonoidHom 📋 Mathlib.Topology.Algebra.SeparationQuotient.Hom
{M : Type u_1} {N : Type u_2} [TopologicalSpace M] [TopologicalSpace N] [CommMonoid M] [ContinuousMul M] [CommMonoid N] (f : M →ₜ* N) (hf : ∀ (x y : M), Inseparable x y → f x = f y) : SeparationQuotient M →ₜ* N
SeparationQuotient.liftContinuousCommMonoidHom_mk 📋 Mathlib.Topology.Algebra.SeparationQuotient.Hom
{M : Type u_1} {N : Type u_2} [TopologicalSpace M] [TopologicalSpace N] [CommMonoid M] [ContinuousMul M] [CommMonoid N] (f : M →ₜ* N) (hf : ∀ (x y : M), Inseparable x y → f x = f y) (x : M) : (SeparationQuotient.liftContinuousMonoidHom f hf) (SeparationQuotient.mk x) = f x
ContAction.res 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] (f : G →ₜ* H) : CategoryTheory.Functor (ContAction V H) (ContAction V G)
ContAction.resCongr 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] (f f' : G →ₜ* H) (h : f = f') : ContAction.res V f ≅ ContAction.res V f'
ContAction.res_obj_obj 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] (f : G →ₜ* H) (X : ContAction V H) : ((ContAction.res V f).obj X).obj = (Action.res V ↑f).obj X.obj
ContAction.resComp 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] {K : Type u_6} [Monoid K] [TopologicalSpace K] (f : G →ₜ* H) (h : H →ₜ* K) : ContAction.res V (h.comp f) ≅ (ContAction.res V h).comp (ContAction.res V f)
ContAction.res_map 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] (f : G →ₜ* H) {X✝ Y✝ : ContAction V H} (f✝ : X✝ ⟶ Y✝) : (ContAction.res V f).map f✝ = CategoryTheory.ObjectProperty.homMk ((Action.res V ↑f).map f✝.hom)
ContAction.resComp_hom 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] {K : Type u_6} [Monoid K] [TopologicalSpace K] (f : G →ₜ* H) (h : H →ₜ* K) : (ContAction.resComp V f h).hom = { app := fun X => CategoryTheory.CategoryStruct.id ((ContAction.res V (h.comp f)).obj X), naturality := ⋯ }
ContAction.resComp_inv 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] {K : Type u_6} [Monoid K] [TopologicalSpace K] (f : G →ₜ* H) (h : H →ₜ* K) : (ContAction.resComp V f h).inv = { app := fun X => CategoryTheory.CategoryStruct.id ((ContAction.res V (h.comp f)).obj X), naturality := ⋯ }
ContAction.resCongr_hom 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] (f f' : G →ₜ* H) (h : f = f') : (ContAction.resCongr V f f' h).hom = { app := fun X => CategoryTheory.ObjectProperty.homMk (Action.mkIso (CategoryTheory.Iso.refl X.obj.V) ⋯).hom, naturality := ⋯ }
ContAction.resCongr_inv 📋 Mathlib.CategoryTheory.Action.Continuous
(V : Type u_1) [CategoryTheory.Category.{v_1, u_1} V] {FV : V → V → Type u_2} {CV : V → Type u_3} [(X Y : V) → FunLike (FV X Y) (CV X) (CV Y)] [CategoryTheory.ConcreteCategory V FV] [CategoryTheory.HasForget₂ V TopCat] {G : Type u_4} [Monoid G] [TopologicalSpace G] {H : Type u_5} [Monoid H] [TopologicalSpace H] (f f' : G →ₜ* H) (h : f = f') : (ContAction.resCongr V f f' h).inv = { app := fun X => CategoryTheory.ObjectProperty.homMk (Action.mkIso (CategoryTheory.Iso.refl X.obj.V) ⋯).inv, naturality := ⋯ }
Field.absoluteGaloisGroup.map 📋 Mathlib.FieldTheory.AbsoluteGaloisGroup
{K : Type u_1} {L : Type u_2} [Field K] [Field L] (f : K →+* L) : Field.absoluteGaloisGroup L →ₜ* Field.absoluteGaloisGroup K
Field.absoluteGaloisGroup.mapOfAlgebra 📋 Mathlib.FieldTheory.AbsoluteGaloisGroup
(K : Type u_1) (L : Type u_2) [Field K] [Field L] [Algebra K L] [Algebra (AlgebraicClosure K) (AlgebraicClosure L)] [IsScalarTower K (AlgebraicClosure K) (AlgebraicClosure L)] : Field.absoluteGaloisGroup L →ₜ* Field.absoluteGaloisGroup K
Field.absoluteGaloisGroup.mapOfAlgebra_toFun_apply 📋 Mathlib.FieldTheory.AbsoluteGaloisGroup
(K : Type u_1) (L : Type u_2) [Field K] [Field L] [Algebra K L] [Algebra (AlgebraicClosure K) (AlgebraicClosure L)] [IsScalarTower K (AlgebraicClosure K) (AlgebraicClosure L)] (x : Field.absoluteGaloisGroup L) (a : AlgebraicClosure K) : ((Field.absoluteGaloisGroup.mapOfAlgebra K L) x) a = ((↑((AlgEquiv.restrictScalarsHom K) x)).restrictNormal (AlgebraicClosure K)) a
Field.absoluteGaloisGroup.map_toFun_apply 📋 Mathlib.FieldTheory.AbsoluteGaloisGroup
{K : Type u_1} {L : Type u_2} [Field K] [Field L] (f : K →+* L) (x : Field.absoluteGaloisGroup L) (a : AlgebraicClosure K) : ((Field.absoluteGaloisGroup.map f) x) a = ((↑((AlgEquiv.restrictScalarsHom K) x)).restrictNormal (AlgebraicClosure K)) a
Field.absoluteGaloisGroup.mapOfAlgebra_toFun_symm_apply 📋 Mathlib.FieldTheory.AbsoluteGaloisGroup
(K : Type u_1) (L : Type u_2) [Field K] [Field L] [Algebra K L] [Algebra (AlgebraicClosure K) (AlgebraicClosure L)] [IsScalarTower K (AlgebraicClosure K) (AlgebraicClosure L)] (x : Field.absoluteGaloisGroup L) (b : AlgebraicClosure K) : (AlgEquiv.symm ((Field.absoluteGaloisGroup.mapOfAlgebra K L) x)) b = Function.surjInv ⋯ b
Field.absoluteGaloisGroup.map_toFun_symm_apply 📋 Mathlib.FieldTheory.AbsoluteGaloisGroup
{K : Type u_1} {L : Type u_2} [Field K] [Field L] (f : K →+* L) (x : Field.absoluteGaloisGroup L) (b : AlgebraicClosure K) : (AlgEquiv.symm ((Field.absoluteGaloisGroup.map f) x)) b = Function.surjInv ⋯ b
ProfiniteGrp.ofHom 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{X Y : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X] [CompactSpace X] [TotallyDisconnectedSpace X] [Group Y] [TopologicalSpace Y] [IsTopologicalGroup Y] [CompactSpace Y] [TotallyDisconnectedSpace Y] (f : X →ₜ* Y) : ProfiniteGrp.of X ⟶ ProfiniteGrp.of Y
ProfiniteGrp.Hom.hom 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{M N : ProfiniteGrp.{u}} (f : M.Hom N) : ↑M.toProfinite.toTop →ₜ* ↑N.toProfinite.toTop
ProfiniteGrp.Hom.hom' 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} (self : A.Hom B) : ↑A.toProfinite.toTop →ₜ* ↑B.toProfinite.toTop
ProfiniteGrp.Hom.ext 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} {x y : A.Hom B} (hom' : x.hom' = y.hom') : x = y
ProfiniteGrp.Hom.ext_iff 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} {x y : A.Hom B} : x = y ↔ x.hom' = y.hom'
ProfiniteGrp.hom_ext 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} {f g : A ⟶ B} (hf : ProfiniteGrp.Hom.hom f = ProfiniteGrp.Hom.hom g) : f = g
ProfiniteGrp.hom_ext_iff 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} {f g : A ⟶ B} : f = g ↔ ProfiniteGrp.Hom.hom f = ProfiniteGrp.Hom.hom g
ProfiniteGrp.hom_id 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A : ProfiniteGrp.{u}} : ProfiniteGrp.Hom.hom (CategoryTheory.CategoryStruct.id A) = ContinuousMonoidHom.id ↑A.toProfinite.toTop
ProfiniteGrp.ofHom_comp 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{X Y Z : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X] [CompactSpace X] [TotallyDisconnectedSpace X] [Group Y] [TopologicalSpace Y] [IsTopologicalGroup Y] [CompactSpace Y] [TotallyDisconnectedSpace Y] [Group Z] [TopologicalSpace Z] [IsTopologicalGroup Z] [CompactSpace Z] [TotallyDisconnectedSpace Z] (f : X →ₜ* Y) (g : Y →ₜ* Z) : ProfiniteGrp.ofHom (g.comp f) = CategoryTheory.CategoryStruct.comp (ProfiniteGrp.ofHom f) (ProfiniteGrp.ofHom g)
instConcreteCategoryProfiniteGrpContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfinite 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: CategoryTheory.ConcreteCategory ProfiniteGrp.{u_1} fun X Y => ↑X.toProfinite.toTop →ₜ* ↑Y.toProfinite.toTop
ProfiniteGrp.instReflectsIsomorphismsForgetContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfinite 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: (CategoryTheory.forget ProfiniteGrp.{u}).ReflectsIsomorphisms
ProfiniteGrp.hom_ofHom 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{X Y : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X] [CompactSpace X] [TotallyDisconnectedSpace X] [Group Y] [TopologicalSpace Y] [IsTopologicalGroup Y] [CompactSpace Y] [TotallyDisconnectedSpace Y] (f : X →ₜ* Y) : ProfiniteGrp.Hom.hom (ProfiniteGrp.ofHom f) = f
ProfiniteGrp.id_apply 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
(A : ProfiniteGrp.{u}) (a : ↑A.toProfinite.toTop) : (ProfiniteGrp.Hom.hom (CategoryTheory.CategoryStruct.id A)) a = a
ProfiniteGrp.coe_id 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
(X : ProfiniteGrp.{u_1}) : ⇑(ProfiniteGrp.Hom.hom (CategoryTheory.CategoryStruct.id X)) = id
ProfiniteGrp.hom_comp 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B C : ProfiniteGrp.{u}} (f : A ⟶ B) (g : B ⟶ C) : ProfiniteGrp.Hom.hom (CategoryTheory.CategoryStruct.comp f g) = (ProfiniteGrp.Hom.hom g).comp (ProfiniteGrp.Hom.hom f)
ProfiniteGrp.instHasForget₂ContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfiniteGrpCatMonoidHomCarrier 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: CategoryTheory.HasForget₂ ProfiniteGrp.{u_1} GrpCat
ProfiniteGrp.instHasForget₂FiniteGrpMonoidHomCarrierToGrpContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfinite 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: CategoryTheory.HasForget₂ FiniteGrp.{u_1} ProfiniteGrp.{u_1}
ProfiniteGrp.instHasForget₂ContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfiniteProfiniteContinuousMap 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: CategoryTheory.HasForget₂ ProfiniteGrp.{u_1} Profinite
ProfiniteGrp.instFaithfulProfiniteForget₂ContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfiniteContinuousMap 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: (CategoryTheory.forget₂ ProfiniteGrp.{u_1} Profinite).Faithful
ProfiniteGrp.instPreservesLimitsProfiniteForget₂ContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfiniteContinuousMap 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget₂ ProfiniteGrp.{u_1} Profinite)
ProfiniteGrp.instReflectsIsomorphismsProfiniteForget₂ContinuousMonoidHomCarrierToTopTotallyDisconnectedSpaceToProfiniteContinuousMap 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
: (CategoryTheory.forget₂ ProfiniteGrp.{u_1} Profinite).ReflectsIsomorphisms
ProfiniteGrp.hom_inv_apply 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} (e : A ≅ B) (x : ↑B.toProfinite.toTop) : (ProfiniteGrp.Hom.hom e.hom) ((ProfiniteGrp.Hom.hom e.inv) x) = x
ProfiniteGrp.inv_hom_apply 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B : ProfiniteGrp.{u}} (e : A ≅ B) (x : ↑A.toProfinite.toTop) : (ProfiniteGrp.Hom.hom e.inv) ((ProfiniteGrp.Hom.hom e.hom) x) = x
ProfiniteGrp.ofHom_apply 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{X Y : Type u} [Group X] [TopologicalSpace X] [IsTopologicalGroup X] [CompactSpace X] [TotallyDisconnectedSpace X] [Group Y] [TopologicalSpace Y] [IsTopologicalGroup Y] [CompactSpace Y] [TotallyDisconnectedSpace Y] (f : X →ₜ* Y) (x : X) : (ProfiniteGrp.Hom.hom (ProfiniteGrp.ofHom f)) x = f x
ProfiniteGrp.comp_apply 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{A B C : ProfiniteGrp.{u}} (f : A ⟶ B) (g : B ⟶ C) (a : ↑A.toProfinite.toTop) : (ProfiniteGrp.Hom.hom (CategoryTheory.CategoryStruct.comp f g)) a = (ProfiniteGrp.Hom.hom g) ((ProfiniteGrp.Hom.hom f) a)
ProfiniteGrp.coe_comp 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{X Y Z : ProfiniteGrp.{u_1}} (f : X ⟶ Y) (g : Y ⟶ Z) : ⇑(ProfiniteGrp.Hom.hom (CategoryTheory.CategoryStruct.comp f g)) = ⇑(ProfiniteGrp.Hom.hom g) ∘ ⇑(ProfiniteGrp.Hom.hom f)
ProfiniteGrp.instGroupCarrierToTopTotallyDisconnectedSpacePtProfiniteLimitConeCompForget₂ContinuousMonoidHomToProfiniteContinuousMap 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J ProfiniteGrp.{max v u}) : Group ↑(Profinite.limitCone (F.comp (CategoryTheory.forget₂ ProfiniteGrp.{max u v} Profinite))).pt.toTop
ProfiniteGrp.instIsTopologicalGroupCarrierToTopTotallyDisconnectedSpacePtProfiniteLimitConeCompForget₂ContinuousMonoidHomToProfiniteContinuousMap 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
{J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J ProfiniteGrp.{max v u}) : IsTopologicalGroup ↑(Profinite.limitCone (F.comp (CategoryTheory.forget₂ ProfiniteGrp.{max u v} Profinite))).pt.toTop
ContRepresentation.coindV 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) (π : ContRepresentation R G V) : Submodule R C(H, V)
ContRepresentation.coe_coindV 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) (π : ContRepresentation R G V) : ↑(ContRepresentation.coindV φ π) = {f | ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)}
ContRepresentation.coind₁Res 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] [IsTopologicalGroup G] [IsTopologicalGroup H] (φ : H →ₜ* G) (π : ContRepresentation R G V) : ContIntertwiningMap (π.coind₁.restrict ↑φ) (π.restrict ↑φ).coind₁
ContRepresentation.mem_coindV 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) (π : ContRepresentation R G V) (f : C(H, V)) : f ∈ ContRepresentation.coindV φ π ↔ ∀ (g : G) (h : H), f (φ g * h) = (π g) (f h)
ContRepresentation.coind₁ResMap 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} {W : Type u_4} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] [AddCommGroup W] [TopologicalSpace W] [IsTopologicalAddGroup W] [Module R W] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [ContinuousSMul R W] [Group H] [TopologicalSpace H] {π : ContRepresentation R G V} [IsTopologicalGroup G] [IsTopologicalGroup H] {π' : ContRepresentation R H W} (φ : H →ₜ* G) (f : ContIntertwiningMap (π.restrict ↑φ) π') : ContIntertwiningMap (π.coind₁.restrict ↑φ) π'.coind₁
ContRepresentation.coind 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) [IsTopologicalGroup H] (π : ContRepresentation R G V) : ContRepresentation R H ↥(ContRepresentation.coindV φ π)
ContRepresentation.coind₁Res_apply 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] [IsTopologicalGroup G] [IsTopologicalGroup H] (φ : H →ₜ* G) (π : ContRepresentation R G V) (F : C(G, V)) (x : H) : ((ContRepresentation.coind₁Res φ π) F) x = F (φ x)
ContRepresentation.instContinuousSMulSubtypeContinuousMapMemSubmoduleCoindV 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) (π : ContRepresentation R G V) : ContinuousSMul R ↥(ContRepresentation.coindV φ π)
ContRepresentation.coind₁ResMap_apply 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} {W : Type u_4} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] [AddCommGroup W] [TopologicalSpace W] [IsTopologicalAddGroup W] [Module R W] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [ContinuousSMul R W] [Group H] [TopologicalSpace H] {π : ContRepresentation R G V} [IsTopologicalGroup G] [IsTopologicalGroup H] {π' : ContRepresentation R H W} (φ : H →ₜ* G) (f : ContIntertwiningMap (π.restrict ↑φ) π') (F : C(G, V)) (x : H) : ((ContRepresentation.coind₁ResMap φ f) F) x = f (F (φ x))
ContRepresentation.coind₁ResMap_comp_coind₁ι_restrict 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} {W : Type u_4} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] [AddCommGroup W] [TopologicalSpace W] [IsTopologicalAddGroup W] [Module R W] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [ContinuousSMul R W] [Group H] [TopologicalSpace H] {π : ContRepresentation R G V} [IsTopologicalGroup G] [IsTopologicalGroup H] {π' : ContRepresentation R H W} (φ : H →ₜ* G) (f : ContIntertwiningMap (π.restrict ↑φ) π') : (ContRepresentation.coind₁ResMap φ f).comp (ContIntertwiningMap.restrict (↑φ) π.coind₁ι) = π'.coind₁ι.comp f
ContRepresentation.coind₁Map_comp_coind₁ResMap 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} {W : Type u_4} {U : Type u_5} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] [AddCommGroup W] [TopologicalSpace W] [IsTopologicalAddGroup W] [Module R W] [AddCommGroup U] [Module R U] [TopologicalSpace U] [IsTopologicalAddGroup U] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [ContinuousSMul R W] [Group H] [TopologicalSpace H] {π : ContRepresentation R G V} [IsTopologicalGroup G] [IsTopologicalGroup H] [ContinuousSMul R U] {π' : ContRepresentation R H W} (φ : H →ₜ* G) {σ : ContRepresentation R H U} (f : ContIntertwiningMap (π.restrict ↑φ) π') (g : ContIntertwiningMap π' σ) : (ContRepresentation.coind₁Map g).comp (ContRepresentation.coind₁ResMap φ f) = ContRepresentation.coind₁ResMap φ (g.comp f)
ContRepresentation.coind₁ResMap_comp_coind₁Map_restrict 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} {W : Type u_4} {U : Type u_5} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] [AddCommGroup W] [TopologicalSpace W] [IsTopologicalAddGroup W] [Module R W] [AddCommGroup U] [Module R U] [TopologicalSpace U] [IsTopologicalAddGroup U] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [ContinuousSMul R W] [Group H] [TopologicalSpace H] {π : ContRepresentation R G V} [IsTopologicalGroup G] [IsTopologicalGroup H] [ContinuousSMul R U] {π' : ContRepresentation R H W} (φ : H →ₜ* G) {ρ : ContRepresentation R G U} (g : ContIntertwiningMap ρ π) (f : ContIntertwiningMap (π.restrict ↑φ) π') : (ContRepresentation.coind₁ResMap φ f).comp (ContIntertwiningMap.restrict (↑φ) (ContRepresentation.coind₁Map g)) = ContRepresentation.coind₁ResMap φ (f.comp (ContIntertwiningMap.restrict (↑φ) g))
ContRepresentation.coind₁Equivcoind 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [IsTopologicalGroup G] : (ContRepresentation.trivial R (↥⊥) V).coind₁.Equiv (ContRepresentation.coind 1 (ContRepresentation.trivial R G V))
ContRepresentation.coind_toMonoidHom_apply_apply_coe 📋 Mathlib.RepresentationTheory.Continuous.Basic
{R : Type u_1} {V : Type u_3} [Ring R] [AddCommGroup V] [TopologicalSpace V] [IsTopologicalAddGroup V] [Module R V] {G : Type u_7} {H : Type u_8} [Group G] [TopologicalSpace G] [TopologicalSpace R] [ContinuousSMul R V] [Group H] [TopologicalSpace H] (φ : G →ₜ* H) [IsTopologicalGroup H] (π : ContRepresentation R G V) (h : H) (x✝ : ↥(ContRepresentation.coindV φ π)) : ↑(((ContRepresentation.coind φ π).toMonoidHom h) x✝) = (↑x✝).comp (ContinuousMap.mulRight h)
ContinuousCohomology.cocyclesMap 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (n : ℕ) : ContinuousCohomology.cocycles X n ⟶ ContinuousCohomology.cocycles Y n
ContinuousCohomology.map 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (n : ℕ) : continuousCohomology n X ⟶ continuousCohomology n Y
ContinuousCohomology.resolutionMap_id 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] (X : TopRep k G) (i : ℕ) : ContinuousCohomology.resolutionMap (ContinuousMonoidHom.id G) (CategoryTheory.CategoryStruct.id X) i = CategoryTheory.CategoryStruct.id (X.resolutionX i)
ContinuousCohomology.resolutionMap 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (i : ℕ) : TopRep.res (↑φ) (X.resolutionX i) ⟶ Y.resolutionX i
ContinuousCohomology.cochainsMap 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) : X.homogeneousCochains ⟶ Y.homogeneousCochains
ContinuousCohomology.resolutionMap_zero 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) : ContinuousCohomology.resolutionMap φ f 0 = f
ContinuousCohomology.cochainsMap_f 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (i : ℕ) : (ContinuousCohomology.cochainsMap φ f).f i = TopRep.invariantsResMap (↑φ) (ContinuousCohomology.resolutionMap φ f (i + 1))
ContinuousCohomology.π_map 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (n : ℕ) : CategoryTheory.CategoryStruct.comp (ContinuousCohomology.π X n) (ContinuousCohomology.map φ f n) = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cocyclesMap φ f n) (ContinuousCohomology.π Y n)
ContinuousCohomology.cocyclesMap_comp 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) (n : ℕ) : ContinuousCohomology.cocyclesMap (φ.comp ψ) (CategoryTheory.CategoryStruct.comp ((TopRep.resFunctor ↑ψ).map f) g) n = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cocyclesMap φ f n) (ContinuousCohomology.cocyclesMap ψ g n)
ContinuousCohomology.map_comp 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) (n : ℕ) : ContinuousCohomology.map (φ.comp ψ) (CategoryTheory.CategoryStruct.comp ((TopRep.resFunctor ↑ψ).map f) g) n = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.map φ f n) (ContinuousCohomology.map ψ g n)
ContinuousCohomology.resolutionMap_comp_d 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (i : ℕ) : CategoryTheory.CategoryStruct.comp (ContinuousCohomology.resolutionMap φ f i) (Y.d i) = CategoryTheory.CategoryStruct.comp ((TopRep.resFunctor ↑φ).map (X.d i)) (ContinuousCohomology.resolutionMap φ f (i + 1))
ContinuousCohomology.π_map_assoc 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (n : ℕ) {Z : TopModuleCat k} (h : continuousCohomology n Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (ContinuousCohomology.π X n) (CategoryTheory.CategoryStruct.comp (ContinuousCohomology.map φ f n) h) = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cocyclesMap φ f n) (CategoryTheory.CategoryStruct.comp (ContinuousCohomology.π Y n) h)
ContinuousCohomology.cochainsMap_comp 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) : ContinuousCohomology.cochainsMap (φ.comp ψ) (CategoryTheory.CategoryStruct.comp ((TopRep.resFunctor ↑ψ).map f) g) = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cochainsMap φ f) (ContinuousCohomology.cochainsMap ψ g)
ContinuousCohomology.resolutionMap_comp 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) (i : ℕ) : ContinuousCohomology.resolutionMap (φ.comp ψ) (CategoryTheory.CategoryStruct.comp ((TopRep.resFunctor ↑ψ).map f) g) i = CategoryTheory.CategoryStruct.comp ((TopRep.resFunctor ↑ψ).map (ContinuousCohomology.resolutionMap φ f i)) (ContinuousCohomology.resolutionMap ψ g i)
ContinuousCohomology.cocyclesMap_comp_assoc 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) (n : ℕ) {Z✝ : TopModuleCat k} (h : ContinuousCohomology.cocycles Z n ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cocyclesMap (φ.comp ψ) (CategoryTheory.CategoryStruct.comp (TopRep.ofHom (ContIntertwiningMap.restrict (↑ψ) (TopRep.Hom.hom f))) g) n) h = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cocyclesMap φ f n) (CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cocyclesMap ψ g n) h)
ContinuousCohomology.map_comp_assoc 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) (n : ℕ) {Z✝ : TopModuleCat k} (h : continuousCohomology n Z ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (ContinuousCohomology.map (φ.comp ψ) (CategoryTheory.CategoryStruct.comp (TopRep.ofHom (ContIntertwiningMap.restrict (↑ψ) (TopRep.Hom.hom f))) g) n) h = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.map φ f n) (CategoryTheory.CategoryStruct.comp (ContinuousCohomology.map ψ g n) h)
ContinuousCohomology.cochainsMap_comp_assoc 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H K : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] [Group K] [TopologicalSpace K] [IsTopologicalGroup K] {X : TopRep k G} {Y : TopRep k H} {Z : TopRep k K} (φ : H →ₜ* G) (ψ : K →ₜ* H) (f : TopRep.res (↑φ) X ⟶ Y) (g : TopRep.res (↑ψ) Y ⟶ Z) {Z✝ : CochainComplex (TopModuleCat k) ℕ} (h : Z.homogeneousCochains ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cochainsMap (φ.comp ψ) (CategoryTheory.CategoryStruct.comp (TopRep.ofHom (ContIntertwiningMap.restrict (↑ψ) (TopRep.Hom.hom f))) g)) h = CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cochainsMap φ f) (CategoryTheory.CategoryStruct.comp (ContinuousCohomology.cochainsMap ψ g) h)
ContinuousCohomology.resolutionMap_succ 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (i : ℕ) : ContinuousCohomology.resolutionMap φ f (i + 1) = TopRep.ofHom (ContRepresentation.coind₁ResMap φ (TopRep.Hom.hom (ContinuousCohomology.resolutionMap φ f i)))
ContinuousCohomology.cochainsMap_f_hom 📋 Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality
{k : Type u} {G H : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {X : TopRep k G} {Y : TopRep k H} (φ : H →ₜ* G) (f : TopRep.res (↑φ) X ⟶ Y) (i : ℕ) : TopModuleCat.Hom.hom ((ContinuousCohomology.cochainsMap φ f).f i) = ContIntertwiningMap.mapInvariantsOfRes (↑φ) (TopRep.Hom.hom (ContinuousCohomology.resolutionMap φ f (i + 1)))
ProfiniteGrp.diagram_obj 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits
(P : ProfiniteGrp.{u}) (X : OpenNormalSubgroup ↑P.toProfinite.toTop) : P.diagram.obj X = (CategoryTheory.forget₂ FiniteGrp.{u} ProfiniteGrp.{u}).obj (P.toFiniteQuotientFunctor.obj X)
ProfiniteGrp.toLimit_injective 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits
(P : ProfiniteGrp.{u}) : Function.Injective ⇑(ProfiniteGrp.Hom.hom P.toLimit)
ProfiniteGrp.toLimit_surjective 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits
(P : ProfiniteGrp.{u}) : Function.Surjective ⇑(ProfiniteGrp.Hom.hom P.toLimit)
ProfiniteGrp.diagram_map 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits
(P : ProfiniteGrp.{u}) {X✝ Y✝ : OpenNormalSubgroup ↑P.toProfinite.toTop} (f : X✝ ⟶ Y✝) : P.diagram.map f = (CategoryTheory.forget₂ FiniteGrp.{u} ProfiniteGrp.{u}).map (P.toFiniteQuotientFunctor.map f)
ProfiniteGrp.denseRange_toLimit 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits
(P : ProfiniteGrp.{u}) : DenseRange ⇑(ProfiniteGrp.Hom.hom P.toLimit)
ProfiniteGrp.ProfiniteCompletion.adjunction 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
: ProfiniteGrp.profiniteCompletion ⊣ CategoryTheory.forget₂ ProfiniteGrp.{u_1} GrpCat
ProfiniteGrp.ProfiniteCompletion.lift_eta 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
{G : GrpCat} {P : ProfiniteGrp.{u}} (f : G ⟶ GrpCat.of ↑P.toProfinite.toTop) : CategoryTheory.CategoryStruct.comp (ProfiniteGrp.ProfiniteCompletion.eta G) ((CategoryTheory.forget₂ ProfiniteGrp.{u} GrpCat).map (ProfiniteGrp.ProfiniteCompletion.lift f)) = f
ProfiniteGrp.ProfiniteCompletion.lift_eta_assoc 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
{G : GrpCat} {P : ProfiniteGrp.{u}} (f : G ⟶ GrpCat.of ↑P.toProfinite.toTop) {Z : GrpCat} (h : (CategoryTheory.forget₂ ProfiniteGrp.{u} GrpCat).obj P ⟶ Z) : CategoryTheory.CategoryStruct.comp (ProfiniteGrp.ProfiniteCompletion.eta G) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.forget₂ ProfiniteGrp.{u} GrpCat).map (ProfiniteGrp.ProfiniteCompletion.lift f)) h) = CategoryTheory.CategoryStruct.comp f h
ProfiniteGrp.ProfiniteCompletion.lift_unique 📋 Mathlib.Topology.Algebra.Category.ProfiniteGrp.Completion
{G : GrpCat} {P : ProfiniteGrp.{u}} (f g : ProfiniteGrp.ProfiniteCompletion.completion G ⟶ P) (h : CategoryTheory.CategoryStruct.comp (ProfiniteGrp.ProfiniteCompletion.eta G) ((CategoryTheory.forget₂ ProfiniteGrp.{u} GrpCat).map f) = CategoryTheory.CategoryStruct.comp (ProfiniteGrp.ProfiniteCompletion.eta G) ((CategoryTheory.forget₂ ProfiniteGrp.{u} GrpCat).map g)) : f = g
ContinuousMonoidHom.instTopologicalSpace 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : TopologicalSpace (A →ₜ* B)
ContinuousMonoidHom.instT2Space 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [T2Space B] : T2Space (A →ₜ* B)
ContinuousMonoidHom.instContinuousEvalConst 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : ContinuousEvalConst (A →ₜ* B) A B
ContinuousMonoidHom.instContinuousEval 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [LocallyCompactPair A B] : ContinuousEval (A →ₜ* B) A B
ContinuousMonoidHom.isEmbedding_toContinuousMap 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Topology.IsEmbedding ContinuousMonoidHom.toContinuousMap
ContinuousMonoidHom.isInducing_toContinuousMap 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Topology.IsInducing ContinuousMonoidHom.toContinuousMap
ContinuousMonoidHom.instCompactSpace 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [DiscreteTopology A] [ContinuousMul B] [T2Space B] [CompactSpace B] : CompactSpace (A →ₜ* B)
ContinuousMonoidHom.isClosedEmbedding_toContinuousMap 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [ContinuousMul B] [T2Space B] : Topology.IsClosedEmbedding ContinuousMonoidHom.toContinuousMap
ContinuousMonoidHom.instIsTopologicalGroup 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {E : Type u_4} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [IsTopologicalGroup E] : IsTopologicalGroup (A →ₜ* E)
ContinuousMonoidHom.isClosedEmbedding_coe 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] [DiscreteTopology A] [ContinuousMul B] [T2Space B] : Topology.IsClosedEmbedding DFunLike.coe
ContinuousMonoidHom.continuous_comp_left 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} {C : Type u_3} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : A →ₜ* B) : Continuous fun g => g.comp f
ContinuousMonoidHom.continuous_comp_right 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} {C : Type u_3} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (f : B →ₜ* C) : Continuous fun g => f.comp g
ContinuousMonoidHom.continuous_of_continuous_uncurry 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{B : Type u_2} {C : Type u_3} [Monoid B] [Monoid C] [TopologicalSpace B] [TopologicalSpace C] {A : Type u_5} [TopologicalSpace A] (f : A → B →ₜ* C) (h : Continuous (Function.uncurry fun x y => (f x) y)) : Continuous f
ContinuousMonoidHom.continuous_comp 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} {C : Type u_3} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [LocallyCompactSpace B] : Continuous fun f => f.2.comp f.1
ContinuousMonoidHom.locallyCompactSpace_of_hasBasis 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{X : Type u_5} {Y : Type u_6} [TopologicalSpace X] [Group X] [IsTopologicalGroup X] [UniformSpace Y] [CommGroup Y] [IsUniformGroup Y] [T0Space Y] [CompactSpace Y] [LocallyCompactSpace X] (V : ℕ → Set Y) (hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)) (hVo : (nhds 1).HasBasis (fun x => True) V) : LocallyCompactSpace (X →ₜ* Y)
ContinuousMonoidHom.range_toContinuousMap 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) (B : Type u_2) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : Set.range ContinuousMonoidHom.toContinuousMap = {f | f 1 = 1 ∧ ∀ (x y : A), f (x * y) = f x * f y}
ContinuousMonoidHom.compLeft 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{A : Type u_1} {B : Type u_2} (E : Type u_4) [Monoid A] [Monoid B] [CommGroup E] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace E] [IsTopologicalGroup E] (f : A →ₜ* B) : (B →ₜ* E) →ₜ* A →ₜ* E
ContinuousMonoidHom.compRight 📋 Mathlib.Topology.Algebra.Group.CompactOpen
(A : Type u_1) {E : Type u_4} [Monoid A] [CommGroup E] [TopologicalSpace A] [TopologicalSpace E] [IsTopologicalGroup E] {B : Type u_5} [CommGroup B] [TopologicalSpace B] [IsTopologicalGroup B] (f : B →ₜ* E) : (A →ₜ* B) →ₜ* A →ₜ* E
ContinuousMonoidHom.locallyCompactSpace_of_equicontinuousAt 📋 Mathlib.Topology.Algebra.Group.CompactOpen
{X : Type u_5} {Y : Type u_6} [TopologicalSpace X] [Group X] [IsTopologicalGroup X] [UniformSpace Y] [CommGroup Y] [IsUniformGroup Y] [T0Space Y] [CompactSpace Y] (U : Set X) (V : Set Y) (hU : IsCompact U) (hV : V ∈ nhds 1) (h : EquicontinuousAt (fun f => ⇑↑f) 1) : LocallyCompactSpace (X →ₜ* Y)
PontryaginDual.map 📋 Mathlib.Topology.Algebra.PontryaginDual
{A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →ₜ* B) : PontryaginDual B →ₜ* PontryaginDual A
PontryaginDual.map_apply 📋 Mathlib.Topology.Algebra.PontryaginDual
{A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (f : A →ₜ* B) (x : PontryaginDual B) (y : A) : ((PontryaginDual.map f) x) y = x (f y)
PontryaginDual.map_comp 📋 Mathlib.Topology.Algebra.PontryaginDual
{A : Type u_1} {B : Type u_2} {C : Type u_3} [Monoid A] [Monoid B] [Monoid C] [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] (g : B →ₜ* C) (f : A →ₜ* B) : PontryaginDual.map (g.comp f) = (PontryaginDual.map f).comp (PontryaginDual.map g)
PontryaginDual.map_one 📋 Mathlib.Topology.Algebra.PontryaginDual
{A : Type u_1} {B : Type u_2} [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] : PontryaginDual.map 1 = 1
PontryaginDual.mapHom 📋 Mathlib.Topology.Algebra.PontryaginDual
(A : Type u_1) (G : Type u_4) [Monoid A] [CommGroup G] [TopologicalSpace A] [TopologicalSpace G] [IsTopologicalGroup G] [LocallyCompactSpace G] : (A →ₜ* G) →ₜ* PontryaginDual G →ₜ* PontryaginDual A
PontryaginDual.map_mul 📋 Mathlib.Topology.Algebra.PontryaginDual
{A : Type u_1} {G : Type u_4} [Monoid A] [CommGroup G] [TopologicalSpace A] [TopologicalSpace G] [IsTopologicalGroup G] (f g : A →ₜ* G) : PontryaginDual.map (f * g) = PontryaginDual.map f * PontryaginDual.map g

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 69fae59