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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 #

@[simp]
theorem CategoryTheory.preadditiveYonedaObj_map {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (Y : C) :
∀ {X Y_1 : Cᵒᵖ} (f : X Y_1), (CategoryTheory.preadditiveYonedaObj Y).map f = ModuleCat.ofHom { toFun := fun (g : X.unop Y) => CategoryTheory.CategoryStruct.comp f.unop g, map_add' := , map_smul' := }

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.

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    @[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_1 : Cᵒᵖ) (g : (((fun (Y : C) => (CategoryTheory.preadditiveYonedaObj Y).comp (CategoryTheory.forget₂ (ModuleCat (CategoryTheory.End Y)) AddCommGroupCat)) X).obj X_1)), ((CategoryTheory.preadditiveYoneda.map f).app X_1) g = CategoryTheory.CategoryStruct.comp g f

    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.

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      theorem CategoryTheory.preadditiveCoyonedaObj_map {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (X : Cᵒᵖ) :
      ∀ {X_1 Y : C} (f : X_1 Y), (CategoryTheory.preadditiveCoyonedaObj X).map f = ModuleCat.ofHom { toFun := fun (g : X.unop X_1) => CategoryTheory.CategoryStruct.comp g f, map_add' := , map_smul' := }

      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.

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        theorem CategoryTheory.preadditiveCoyoneda_map_app_apply {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] :
        ∀ {X Y : Cᵒᵖ} (f : X Y) (Y_1 : C) (g : (((fun (X : Cᵒᵖ) => (CategoryTheory.preadditiveCoyonedaObj X).comp (CategoryTheory.forget₂ (ModuleCat (CategoryTheory.End X)) AddCommGroupCat)) X).obj Y_1)), ((CategoryTheory.preadditiveCoyoneda.map f).app Y_1) g = CategoryTheory.CategoryStruct.comp f.unop g

        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.

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          instance CategoryTheory.additive_yonedaObj' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (X : C) :
          (CategoryTheory.preadditiveYoneda.obj X).Additive
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          instance CategoryTheory.additive_coyonedaObj' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] (X : Cᵒᵖ) :
          (CategoryTheory.preadditiveCoyoneda.obj X).Additive
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          @[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.full_preadditiveYoneda {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] :
          CategoryTheory.preadditiveYoneda.Full
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          instance CategoryTheory.full_preadditiveCoyoneda {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] :
          CategoryTheory.preadditiveCoyoneda.Full
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          instance CategoryTheory.faithful_preadditiveYoneda {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] :
          CategoryTheory.preadditiveYoneda.Faithful
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          instance CategoryTheory.faithful_preadditiveCoyoneda {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] :
          CategoryTheory.preadditiveCoyoneda.Faithful
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