Documentation

Mathlib.CategoryTheory.Preadditive.Yoneda.Basic

The Yoneda embedding for preadditive categories #

The Yoneda embedding for preadditive categories sends an object Y to the presheaf sending an object X to the group of morphisms X ⟶ Y. At each point, we get an additional End Y-module structure.

We also show that this presheaf is additive and that it is compatible with the normal Yoneda embedding in the expected way and deduce that the preadditive Yoneda embedding is fully faithful.

TODO #

The Yoneda embedding for preadditive categories sends an object Y to the presheaf sending an object X to the End Y-module of morphisms X ⟶ Y.

Equations
  • One or more equations did not get rendered due to their size.
Instances For
    @[simp]
    theorem CategoryTheory.preadditiveYonedaObj_obj_isModule {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (Y : C) (X : Cᵒᵖ) :
    ((CategoryTheory.preadditiveYonedaObj Y).obj X).isModule = CategoryTheory.Preadditive.moduleEndRight

    The Yoneda embedding for preadditive categories sends an object Y to the presheaf sending an object X to the group of morphisms X ⟶ Y. At each point, we get an additional End Y-module structure, see preadditiveYonedaObj.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      @[simp]
      theorem CategoryTheory.preadditiveYoneda_map_app_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {X✝ Y✝ : C} (f : X✝ Y✝) (x✝ : Cᵒᵖ) (g : (((fun (Y : C) => (CategoryTheory.preadditiveYonedaObj Y).comp (CategoryTheory.forget₂ (ModuleCat (CategoryTheory.End Y)) AddCommGrp)) X✝).obj x✝)) :
      ((CategoryTheory.preadditiveYoneda.map f).app x✝) g = CategoryTheory.CategoryStruct.comp g f

      The Yoneda embedding for preadditive categories sends an object X to the copresheaf sending an object Y to the End X-module of morphisms X ⟶ Y.

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

        The Yoneda embedding for preadditive categories sends an object X to the copresheaf sending an object Y to the group of morphisms X ⟶ Y. At each point, we get an additional End X-module structure, see preadditiveCoyonedaObj.

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          @[simp]
          theorem CategoryTheory.preadditiveCoyoneda_map_app_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] {X✝ Y✝ : Cᵒᵖ} (f : X✝ Y✝) (x✝ : C) (g : (((fun (X : Cᵒᵖ) => (CategoryTheory.preadditiveCoyonedaObj X).comp (CategoryTheory.forget₂ (ModuleCat (CategoryTheory.End X)) AddCommGrp)) X✝).obj x✝)) :
          ((CategoryTheory.preadditiveCoyoneda.map f).app x✝) g = CategoryTheory.CategoryStruct.comp f.unop g
          instance CategoryTheory.additive_yonedaObj' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (X : C) :
          (CategoryTheory.preadditiveYoneda.obj X).Additive
          instance CategoryTheory.additive_coyonedaObj' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (X : Cᵒᵖ) :
          (CategoryTheory.preadditiveCoyoneda.obj X).Additive
          @[simp]

          Composing the preadditive yoneda embedding with the forgetful functor yields the regular Yoneda embedding.

          @[simp]

          Composing the preadditive yoneda embedding with the forgetful functor yields the regular Yoneda embedding.

          instance CategoryTheory.faithful_preadditiveCoyoneda {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] :
          CategoryTheory.preadditiveCoyoneda.Faithful