Documentation

Mathlib.Algebra.Category.MonCat.Yoneda

Yoneda embeddings #

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

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def CommMonCat.coyoneda :
CategoryTheory.Functor CommMonCatᵒᵖ (CategoryTheory.Functor CommMonCat CommMonCat)

The CommMonCat-valued coyoneda embedding.

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      def AddCommMonCat.coyoneda :
      CategoryTheory.Functor AddCommMonCatᵒᵖ (CategoryTheory.Functor AddCommMonCat AddCommMonCat)

      The AddCommMonCat-valued coyoneda embedding.

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          @[simp]
          theorem AddCommMonCat.coyoneda_map_app {X✝ Y✝ : AddCommMonCatᵒᵖ} (f : X✝ ⟶ Y✝) (N : AddCommMonCat) :
          (coyoneda.map f).app N = ofHom (Hom.hom f.unop).compHom'
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          @[simp]
          theorem CommMonCat.coyoneda_obj_map (M : CommMonCatᵒᵖ) {X✝ Y✝ : CommMonCat} (f : X✝ ⟶ Y✝) :
          (coyoneda.obj M).map f = ofHom (MonoidHom.compHom (Hom.hom f))
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          @[simp]
          theorem AddCommMonCat.coyoneda_obj_obj_coe (M : AddCommMonCatᵒᵖ) (N : AddCommMonCat) :
          ↑((coyoneda.obj M).obj N) = (↑(Opposite.unop M) →+ ↑N)
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          @[simp]
          theorem CommMonCat.coyoneda_obj_obj_coe (M : CommMonCatᵒᵖ) (N : CommMonCat) :
          ↑((coyoneda.obj M).obj N) = (↑(Opposite.unop M) →* ↑N)
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          @[simp]
          theorem CommMonCat.coyoneda_map_app {X✝ Y✝ : CommMonCatᵒᵖ} (f : X✝ ⟶ Y✝) (N : CommMonCat) :
          (coyoneda.map f).app N = ofHom (Hom.hom f.unop).compHom'
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          @[simp]
          theorem AddCommMonCat.coyoneda_obj_map (M : AddCommMonCatᵒᵖ) {X✝ Y✝ : AddCommMonCat} (f : X✝ ⟶ Y✝) :
          (coyoneda.obj M).map f = ofHom (AddMonoidHom.compHom (Hom.hom f))
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          def CommMonCat.coyonedaForget :
          coyoneda.comp ((CategoryTheory.Functor.whiskeringRight CommMonCat CommMonCat (Type u_1)).obj (CategoryTheory.forget CommMonCat)) ≅ CategoryTheory.coyoneda

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

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

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

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                  @[simp]
                  theorem CommMonCat.coyonedaForget_hom_app_app_hom (X : CommMonCatᵒᵖ) (X✝ : CommMonCat) (a✝ : ((coyoneda.comp ((CategoryTheory.Functor.whiskeringRight CommMonCat CommMonCat (Type u_1)).obj (CategoryTheory.forget CommMonCat))).obj X).obj X✝) :
                  Hom.hom ((coyonedaForget.hom.app X).app X✝ a✝) = a✝
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                  @[simp]
                  theorem CommMonCat.coyonedaForget_inv_app_app (X : CommMonCatᵒᵖ) (X✝ : CommMonCat) (a✝ : (CategoryTheory.coyoneda.obj X).obj X✝) :
                  (coyonedaForget.inv.app X).app X✝ a✝ = Hom.hom a✝
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                  @[simp]
                  theorem AddCommMonCat.coyonedaForget_inv_app_app (X : AddCommMonCatᵒᵖ) (X✝ : AddCommMonCat) (a✝ : (CategoryTheory.coyoneda.obj X).obj X✝) :
                  (coyonedaForget.inv.app X).app X✝ a✝ = Hom.hom a✝
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                  @[simp]
                  theorem AddCommMonCat.coyonedaForget_hom_app_app_hom (X : AddCommMonCatᵒᵖ) (X✝ : AddCommMonCat) (a✝ : ((coyoneda.comp ((CategoryTheory.Functor.whiskeringRight AddCommMonCat AddCommMonCat (Type u_1)).obj (CategoryTheory.forget AddCommMonCat))).obj X).obj X✝) :
                  Hom.hom ((coyonedaForget.hom.app X).app X✝ a✝) = a✝
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                  def CommMonCat.coyonedaType :
                  CategoryTheory.Functor Type uᵒᵖ (CategoryTheory.Functor CommMonCat CommMonCat)

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

                  This is also the coyoneda embedding of Type into CommMonCat-valued presheaves of commutative monoids.

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

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

                      This is also the coyoneda embedding of Type into AddCommMonCat-valued presheaves of commutative monoids.

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                          @[simp]
                          theorem AddCommMonCat.coyonedaType_map_app {X✝ Y✝ : Type uᵒᵖ} (f : X✝ ⟶ Y✝) (N : AddCommMonCat) :
                          (coyonedaType.map f).app N = ofHom (Pi.addMonoidHom fun (i : Opposite.unop Y✝) => Pi.evalAddMonoidHom (fun (a : Opposite.unop X✝) => ↑N) (f.unop i))
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                          @[simp]
                          theorem CommMonCat.coyonedaType_obj_map (X : Type uᵒᵖ) {X✝ Y✝ : CommMonCat} (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))
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                          @[simp]
                          theorem AddCommMonCat.coyonedaType_obj_obj_coe (X : Type uᵒᵖ) (M : AddCommMonCat) :
                          ↑((coyonedaType.obj X).obj M) = (Opposite.unop X → ↑M)
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                          @[simp]
                          theorem AddCommMonCat.coyonedaType_obj_map (X : Type uᵒᵖ) {X✝ Y✝ : AddCommMonCat} (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))
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                          @[simp]
                          theorem CommMonCat.coyonedaType_obj_obj_coe (X : Type uᵒᵖ) (M : CommMonCat) :
                          ↑((coyonedaType.obj X).obj M) = (Opposite.unop X → ↑M)
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                          @[simp]
                          theorem CommMonCat.coyonedaType_map_app {X✝ Y✝ : Type uᵒᵖ} (f : X✝ ⟶ Y✝) (N : CommMonCat) :
                          (coyonedaType.map f).app N = ofHom (Pi.monoidHom fun (i : Opposite.unop Y✝) => Pi.evalMonoidHom (fun (a : Opposite.unop X✝) => ↑N) (f.unop i))