Yoneda embeddings

This file defines a few Yoneda embeddings for the category of commutative groups.

def CommGrpCat.coyoneda : CategoryTheory.Functor CommGrpCatᵒᵖ (CategoryTheory.Functor CommGrpCat CommGrpCat)

The CommGrpCat-valued coyoneda embedding.

def AddCommGrpCat.coyoneda : CategoryTheory.Functor AddCommGrpCatᵒᵖ (CategoryTheory.Functor AddCommGrpCat AddCommGrpCat)

The AddCommGrpCat-valued coyoneda embedding.

@[simp]

theorem AddCommGrpCat.coyoneda_obj_map (M : AddCommGrpCatᵒᵖ) {X✝ Y✝ : AddCommGrpCat} (f : X✝ ⟶ Y✝) : (coyoneda.obj M).map f = ofHom (AddMonoidHom.compHom (Hom.hom f))

@[simp]

theorem CommGrpCat.coyoneda_obj_obj_coe (M : CommGrpCatᵒᵖ) (N : CommGrpCat) : ↑((coyoneda.obj M).obj N) = (↑(Opposite.unop M) →* ↑N)

@[simp]

theorem AddCommGrpCat.coyoneda_obj_obj_coe (M : AddCommGrpCatᵒᵖ) (N : AddCommGrpCat) : ↑((coyoneda.obj M).obj N) = (↑(Opposite.unop M) →+ ↑N)

@[simp]

theorem CommGrpCat.coyoneda_obj_map (M : CommGrpCatᵒᵖ) {X✝ Y✝ : CommGrpCat} (f : X✝ ⟶ Y✝) : (coyoneda.obj M).map f = ofHom (MonoidHom.compHom (Hom.hom f))

@[simp]

theorem AddCommGrpCat.coyoneda_map_app {X✝ Y✝ : AddCommGrpCatᵒᵖ} (f : X✝ ⟶ Y✝) (N : AddCommGrpCat) : (coyoneda.map f).app N = ofHom (Hom.hom f.unop).compHom'

@[simp]

theorem CommGrpCat.coyoneda_map_app {X✝ Y✝ : CommGrpCatᵒᵖ} (f : X✝ ⟶ Y✝) (N : CommGrpCat) : (coyoneda.map f).app N = ofHom (Hom.hom f.unop).compHom'

def CommGrpCat.coyonedaForget : coyoneda.comp ((CategoryTheory.Functor.whiskeringRight CommGrpCat CommGrpCat (Type u_1)).obj (CategoryTheory.forget CommGrpCat)) ≅ CategoryTheory.coyoneda

The CommGrpCat-valued coyoneda embedding composed with the forgetful functor is the usual coyoneda embedding.

def AddCommGrpCat.coyonedaForget : coyoneda.comp ((CategoryTheory.Functor.whiskeringRight AddCommGrpCat AddCommGrpCat (Type u_1)).obj (CategoryTheory.forget AddCommGrpCat)) ≅ CategoryTheory.coyoneda

The AddCommGrpCat-valued coyoneda embedding composed with the forgetful functor is the usual coyoneda embedding.

@[simp]

theorem AddCommGrpCat.coyonedaForget_inv_app_app (X : AddCommGrpCatᵒᵖ) (X✝ : AddCommGrpCat) (a✝ : (CategoryTheory.coyoneda.obj X).obj X✝) : (coyonedaForget.inv.app X).app X✝ a✝ = Hom.hom a✝

@[simp]

theorem CommGrpCat.coyonedaForget_inv_app_app (X : CommGrpCatᵒᵖ) (X✝ : CommGrpCat) (a✝ : (CategoryTheory.coyoneda.obj X).obj X✝) : (coyonedaForget.inv.app X).app X✝ a✝ = Hom.hom a✝

@[simp]

theorem AddCommGrpCat.coyonedaForget_hom_app_app_hom (X : AddCommGrpCatᵒᵖ) (X✝ : AddCommGrpCat) (a✝ : ((coyoneda.comp ((CategoryTheory.Functor.whiskeringRight AddCommGrpCat AddCommGrpCat (Type u_1)).obj (CategoryTheory.forget AddCommGrpCat))).obj X).obj X✝) : Hom.hom ((coyonedaForget.hom.app X).app X✝ a✝) = a✝

@[simp]

theorem CommGrpCat.coyonedaForget_hom_app_app_hom (X : CommGrpCatᵒᵖ) (X✝ : CommGrpCat) (a✝ : ((coyoneda.comp ((CategoryTheory.Functor.whiskeringRight CommGrpCat CommGrpCat (Type u_1)).obj (CategoryTheory.forget CommGrpCat))).obj X).obj X✝) : Hom.hom ((coyonedaForget.hom.app X).app X✝ a✝) = a✝

def CommGrpCat.coyonedaType : CategoryTheory.Functor Type uᵒᵖ (CategoryTheory.Functor CommGrpCat CommGrpCat)

The Hom bifunctor sending a type X and a commutative group G to the commutative group X → G with pointwise operations.

This is also the coyoneda embedding of Type into CommGrpCat-valued presheaves of commutative groups.

def AddCommGrpCat.coyonedaType : CategoryTheory.Functor Type uᵒᵖ (CategoryTheory.Functor AddCommGrpCat AddCommGrpCat)

The Hom bifunctor sending a type X and a commutative group G to the commutative group X → G with pointwise operations.

This is also the coyoneda embedding of Type into AddCommGrpCat-valued presheaves of commutative groups.

@[simp]

theorem CommGrpCat.coyonedaType_obj_obj_coe (X : Type uᵒᵖ) (G : CommGrpCat) : ↑((coyonedaType.obj X).obj G) = (Opposite.unop X → ↑G)

@[simp]

theorem AddCommGrpCat.coyonedaType_obj_map (X : Type uᵒᵖ) {X✝ Y✝ : AddCommGrpCat} (f : X✝ ⟶ Y✝) : (coyonedaType.obj X).map f = ofHom (Pi.addMonoidHom fun (i : Opposite.unop X) => (Hom.hom f).comp (Pi.evalAddMonoidHom (fun (a : Opposite.unop X) => ↑X✝) i))

@[simp]

theorem CommGrpCat.coyonedaType_obj_map (X : Type uᵒᵖ) {X✝ Y✝ : CommGrpCat} (f : X✝ ⟶ Y✝) : (coyonedaType.obj X).map f = ofHom (Pi.monoidHom fun (i : Opposite.unop X) => (Hom.hom f).comp (Pi.evalMonoidHom (fun (a : Opposite.unop X) => ↑X✝) i))

@[simp]

theorem AddCommGrpCat.coyonedaType_map_app {X✝ Y✝ : Type uᵒᵖ} (f : X✝ ⟶ Y✝) (G : AddCommGrpCat) : (coyonedaType.map f).app G = ofHom (Pi.addMonoidHom fun (i : Opposite.unop Y✝) => Pi.evalAddMonoidHom (fun (a : Opposite.unop X✝) => ↑G) (f.unop i))

@[simp]

theorem AddCommGrpCat.coyonedaType_obj_obj_coe (X : Type uᵒᵖ) (G : AddCommGrpCat) : ↑((coyonedaType.obj X).obj G) = (Opposite.unop X → ↑G)

@[simp]

theorem CommGrpCat.coyonedaType_map_app {X✝ Y✝ : Type uᵒᵖ} (f : X✝ ⟶ Y✝) (G : CommGrpCat) : (coyonedaType.map f).app G = ofHom (Pi.monoidHom fun (i : Opposite.unop Y✝) => Pi.evalMonoidHom (fun (a : Opposite.unop X✝) => ↑G) (f.unop i))