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 CompleteSemiNormedGroup and promote the Completion functor below to a functor landing in this category.
  2. Prove that the functor Completion : SemiNormedGroup ⥤ CompleteSemiNormedGroup is left adjoint to the forgetful functor.

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

Instances For
    theorem SemiNormedGroupCat.completion.incl_apply {V : SemiNormedGroupCat} (v : V) :
    SemiNormedGroupCat.completion.incl v = V v

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

    Instances For
      theorem SemiNormedGroupCat.completion.norm_incl_eq {V : SemiNormedGroupCat} {v : V} :
      SemiNormedGroupCat.completion.incl v = v

      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.

      Instances For

        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.

        Instances For