Documentation

Mathlib.Analysis.Normed.Group.SemiNormedGrp.Completion

Completions of normed groups #

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

Main definitions #

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.
source
@[simp]
theorem SemiNormedGrp.completion_map :
∀ {X Y : SemiNormedGrp} (f : X ⟶ Y), SemiNormedGrp.completion.map f = NormedAddGroupHom.completion f
source
@[simp]
theorem SemiNormedGrp.completion_obj (V : SemiNormedGrp) :
SemiNormedGrp.completion.obj V = SemiNormedGrp.of (UniformSpace.Completion ↑V)
source
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.
Instances For
    source
    instance SemiNormedGrp.completion_completeSpace {V : SemiNormedGrp} :
    CompleteSpace ↑(SemiNormedGrp.completion.obj V)
    Equations
    • ⋯ = ⋯
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    @[simp]
    theorem SemiNormedGrp.completion.incl_apply {V : SemiNormedGrp} (v : ↑V) :
    SemiNormedGrp.completion.incl v = ↑↑V v
    source
    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' := ⋯ }
    Instances For
      source
      theorem SemiNormedGrp.completion.norm_incl_eq {V : SemiNormedGrp} {v : ↑V} :
      ‖SemiNormedGrp.completion.incl v‖ = ‖v‖
      source
      theorem SemiNormedGrp.completion.map_normNoninc {V : SemiNormedGrp} {W : SemiNormedGrp} {f : V ⟶ W} (hf : NormedAddGroupHom.NormNoninc f) :
      NormedAddGroupHom.NormNoninc (SemiNormedGrp.completion.map f)
      source
      def SemiNormedGrp.completion.mapHom (V : SemiNormedGrp) (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
      Instances For
        source
        theorem SemiNormedGrp.completion.map_zero (V : SemiNormedGrp) (W : SemiNormedGrp) :
        SemiNormedGrp.completion.map 0 = 0
        source
        instance SemiNormedGrp.instPreadditive :
        CategoryTheory.Preadditive SemiNormedGrp
        Equations
        • One or more equations did not get rendered due to their size.
        source
        instance SemiNormedGrp.instAdditiveCompletion :
        SemiNormedGrp.completion.Additive
        Equations
        source
        def SemiNormedGrp.completion.lift {V : SemiNormedGrp} {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
        Instances For
          source
          theorem SemiNormedGrp.completion.lift_comp_incl {V : SemiNormedGrp} {W : SemiNormedGrp} [CompleteSpace ↑W] [T0Space ↑W] (f : V ⟶ W) :
          CategoryTheory.CategoryStruct.comp SemiNormedGrp.completion.incl (SemiNormedGrp.completion.lift f) = f
          source
          theorem SemiNormedGrp.completion.lift_unique {V : SemiNormedGrp} {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