Completions of normed groups

This file contains an API for completions of seminormed groups (basic facts about objects and morphisms).

Main definitions

• SemiNormedGrp.Completion : SemiNormedGrp ⥤ SemiNormedGrp : the completion of a seminormed group (defined as a functor on SemiNormedGrp to itself).
• SemiNormedGrp.Completion.lift (f : V ⟶ W) : (Completion.obj V ⟶ W) : a normed group hom from V to complete W extends ("lifts") to a seminormed group hom from the completion of V to W.

Projects

1. Construct the category of complete seminormed groups, say CompleteSemiNormedGrp and promote the Completion functor below to a functor landing in this category.
2. Prove that the functor Completion : SemiNormedGrp ⥤ CompleteSemiNormedGrp is left adjoint to the forgetful functor.

def SemiNormedGrp.completion : CategoryTheory.Functor SemiNormedGrp SemiNormedGrp

The completion of a seminormed group, as an endofunctor on SemiNormedGrp.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem SemiNormedGrp.completion_obj (V : SemiNormedGrp) : SemiNormedGrp.completion.obj V = SemiNormedGrp.of (UniformSpace.Completion ↑V)

@[simp]

theorem SemiNormedGrp.completion_map {X✝ Y✝ : SemiNormedGrp} (f : X✝ ⟶ Y✝) : SemiNormedGrp.completion.map f = NormedAddGroupHom.completion f

instance SemiNormedGrp.completion_completeSpace {V : SemiNormedGrp} : CompleteSpace ↑(SemiNormedGrp.completion.obj V)

def SemiNormedGrp.completion.incl {V : SemiNormedGrp} : V ⟶ SemiNormedGrp.completion.obj V

The canonical morphism from a seminormed group V to its completion.

Equations

• SemiNormedGrp.completion.incl = { toFun := fun (v : ↑V) => ↑↑V v, map_add' := ⋯, bound' := ⋯ }

@[simp]

theorem SemiNormedGrp.completion.incl_apply {V : SemiNormedGrp} (v : ↑V) : SemiNormedGrp.completion.incl v = ↑↑V v

theorem SemiNormedGrp.completion.norm_incl_eq {V : SemiNormedGrp} {v : ↑V} : ‖SemiNormedGrp.completion.incl v‖ = ‖v‖

theorem SemiNormedGrp.completion.map_normNoninc {V W : SemiNormedGrp} {f : V ⟶ W} (hf : NormedAddGroupHom.NormNoninc f) : NormedAddGroupHom.NormNoninc (SemiNormedGrp.completion.map f)

def SemiNormedGrp.completion.mapHom (V W : SemiNormedGrp) : let_fun this := fun (V W : SemiNormedGrp) => inferInstanceAs (AddGroup (NormedAddGroupHom ↑V ↑W)); (V ⟶ W) →+ (SemiNormedGrp.completion.obj V ⟶ SemiNormedGrp.completion.obj W)

Given a normed group hom V ⟶ W, this defines the associated morphism from the completion of V to the completion of W. The difference from the definition obtained from the functoriality of completion is in that the map sending a morphism f to the associated morphism of completions is itself additive.

Equations

• SemiNormedGrp.completion.mapHom V W = AddMonoidHom.mk' SemiNormedGrp.completion.map ⋯

theorem SemiNormedGrp.completion.map_zero (V W : SemiNormedGrp) : SemiNormedGrp.completion.map 0 = 0

instance SemiNormedGrp.instPreadditive : CategoryTheory.Preadditive SemiNormedGrp

Equations

• One or more equations did not get rendered due to their size.

instance SemiNormedGrp.instAdditiveCompletion : SemiNormedGrp.completion.Additive

def SemiNormedGrp.completion.lift {V W : SemiNormedGrp} [CompleteSpace ↑W] [T0Space ↑W] (f : V ⟶ W) : SemiNormedGrp.completion.obj V ⟶ W

Given a normed group hom f : V → W with W complete, this provides a lift of f to the completion of V. The lemmas lift_unique and lift_comp_incl provide the api for the universal property of the completion.

Equations

• SemiNormedGrp.completion.lift f = { toFun := ⇑(NormedAddGroupHom.extension f), map_add' := ⋯, bound' := ⋯ }

theorem SemiNormedGrp.completion.lift_comp_incl {V W : SemiNormedGrp} [CompleteSpace ↑W] [T0Space ↑W] (f : V ⟶ W) : CategoryTheory.CategoryStruct.comp SemiNormedGrp.completion.incl (SemiNormedGrp.completion.lift f) = f

theorem SemiNormedGrp.completion.lift_unique {V W : SemiNormedGrp} [CompleteSpace ↑W] [T0Space ↑W] (f : V ⟶ W) (g : SemiNormedGrp.completion.obj V ⟶ W) : CategoryTheory.CategoryStruct.comp SemiNormedGrp.completion.incl g = f → g = SemiNormedGrp.completion.lift f