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Mathlib.CategoryTheory.Enriched.FunctorCategory

Functor categories are enriched #

If C is a V-enriched ordinary category, then J ⥤ C is also both a V-enriched ordinary category and a J ⥤ V-enriched ordinary category, provided C has suitable limits.

We first define the V-enriched structure on J ⥤ C by saying that if F₁ and F₂ are in J ⥤ C, then enrichedHom V F₁ F₂ : V is a suitable limit involving F₁.obj j ⟶[V] F₂.obj j for all j : C. The J ⥤ V object of morphisms functorEnrichedHom V F₁ F₂ : J ⥤ V is defined by sending j : J to the previously defined enrichedHom for the "restriction" of F₁ and F₂ to the category Under j. The definition isLimitConeFunctorEnrichedHom shows that enriched V F₁ F₂ is the limit of the functor functorEnrichedHom V F₁ F₂.

Given two functors F₁ and F₂ from a category J to a V-enriched ordinary category C, this is the diagram Jᵒᵖ ⥤ J ⥤ V whose end shall be the V-morphisms in J ⥤ V from F₁ to F₂.

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    theorem CategoryTheory.Enriched.FunctorCategory.diagram_map_app (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) {X✝ Y✝ : Jᵒᵖ} (f : X✝ Y✝) (X : J) :
    ((diagram V F₁ F₂).map f).app X = eHomWhiskerRight V (F₁.map f.unop) (F₂.obj X)
    @[simp]
    theorem CategoryTheory.Enriched.FunctorCategory.diagram_obj_obj (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) (X : Jᵒᵖ) (X✝ : J) :
    ((diagram V F₁ F₂).obj X).obj X✝ = EnrichedCategory.Hom (F₁.obj (Opposite.unop X)) (F₂.obj X✝)
    @[simp]
    theorem CategoryTheory.Enriched.FunctorCategory.diagram_obj_map (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) (X : Jᵒᵖ) {X✝ Y✝ : J} (f : X✝ Y✝) :
    ((diagram V F₁ F₂).obj X).map f = eHomWhiskerLeft V (F₁.obj (Opposite.unop X)) (F₂.map f)
    @[reducible, inline]

    The V-enriched hom from F₁ to F₂ when F₁ and F₂ are functors J ⥤ C and C is a V-enriched category.

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      noncomputable abbrev CategoryTheory.Enriched.FunctorCategory.enrichedHomπ (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] (j : J) :
      enrichedHom V F₁ F₂ EnrichedCategory.Hom (F₁.obj j) (F₂.obj j)

      The projection enrichedHom V F₁ F₂ ⟶ F₁.obj j ⟶[V] F₂.obj j in the category V for any j : J when F₁ and F₂ are functors J ⥤ C and C is a V-enriched category.

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        theorem CategoryTheory.Enriched.FunctorCategory.enrichedHom_condition (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] {i j : J} (f : i j) :
        CategoryStruct.comp (enrichedHomπ V F₁ F₂ i) (eHomWhiskerLeft V (F₁.obj i) (F₂.map f)) = CategoryStruct.comp (enrichedHomπ V F₁ F₂ j) (eHomWhiskerRight V (F₁.map f) (F₂.obj j))
        theorem CategoryTheory.Enriched.FunctorCategory.enrichedHom_condition_assoc (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] {i j : J} (f : i j) {Z : V} (h : EnrichedCategory.Hom (F₁.obj i) (F₂.obj j) Z) :
        noncomputable def CategoryTheory.Enriched.FunctorCategory.homEquiv (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasEnrichedHom V F₁ F₂] :
        (F₁ F₂) (𝟙_ V enrichedHom V F₁ F₂)

        Given functors F₁ and F₂ in J ⥤ C, where C is a V-enriched ordinary category, this is the bijection (F₁ ⟶ F₂) ≃ (𝟙_ V ⟶ enrichedHom V F₁ F₂).

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          theorem CategoryTheory.Enriched.FunctorCategory.homEquiv_apply_π (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasEnrichedHom V F₁ F₂] (τ : F₁ F₂) (j : J) :
          CategoryStruct.comp ((homEquiv V) τ) (enrichedHomπ V F₁ F₂ j) = (eHomEquiv V) (τ.app j)
          @[simp]
          theorem CategoryTheory.Enriched.FunctorCategory.homEquiv_apply_π_assoc (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasEnrichedHom V F₁ F₂] (τ : F₁ F₂) (j : J) {Z : V} (h : EnrichedCategory.Hom (F₁.obj j) (F₂.obj j) Z) :

          The identity for the V-enrichment of the category J ⥤ C.

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            noncomputable def CategoryTheory.Enriched.FunctorCategory.enrichedComp (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ : Functor J C) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₁ F₃] :

            The composition for the V-enrichment of the category J ⥤ C.

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              theorem CategoryTheory.Enriched.FunctorCategory.enrichedComp_π (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ : Functor J C) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₁ F₃] (j : J) :
              CategoryStruct.comp (enrichedComp V F₁ F₂ F₃) (Limits.end_.π (diagram V F₁ F₃) j) = CategoryStruct.comp (MonoidalCategoryStruct.tensorHom (Limits.end_.π (diagram V F₁ F₂) j) (Limits.end_.π (diagram V F₂ F₃) j)) (eComp V (Opposite.unop (F₁.op.obj (Opposite.op j))) (F₂.obj j) (F₃.obj j))
              @[simp]
              theorem CategoryTheory.Enriched.FunctorCategory.enrichedComp_π_assoc (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ : Functor J C) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₁ F₃] (j : J) {Z : V} (h : ((diagram V F₁ F₃).obj (Opposite.op j)).obj j Z) :
              theorem CategoryTheory.Enriched.FunctorCategory.enriched_assoc (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ F₄ : Functor J C) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₃] [HasEnrichedHom V F₁ F₄] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₂ F₄] [HasEnrichedHom V F₃ F₄] :
              theorem CategoryTheory.Enriched.FunctorCategory.enriched_assoc_assoc (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ F₄ : Functor J C) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₃] [HasEnrichedHom V F₁ F₄] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₂ F₄] [HasEnrichedHom V F₃ F₄] {Z : V} (h : enrichedHom V F₁ F₄ Z) :

              If C is a V-enriched ordinary category, and C has suitable limits, then J ⥤ C is also a V-enriched ordinary category.

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                @[reducible, inline]
                noncomputable abbrev CategoryTheory.Enriched.FunctorCategory.precompEnrichedHom' (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] {K : Type u₄} [Category.{v₄, u₄} K] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} (G : Functor K J) [HasEnrichedHom V F₁ F₂] {F₁' F₂' : Functor K C} [HasEnrichedHom V F₁' F₂'] (e₁ : G.comp F₁ F₁') (e₂ : G.comp F₂ F₂') :
                enrichedHom V F₁ F₂ enrichedHom V F₁' F₂'

                If F₁ and F₂ are functors J ⥤ C, G : K ⥤ J, and F₁' and F₂' are functors K ⥤ C that are respectively isomorphic to G ⋙ F₁ and G ⋙ F₂, then this is the induced morphism enrichedHom V F₁ F₂ ⟶ enrichedHom V F₁' F₂' in V when C is a category enriched in V.

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                  @[reducible, inline]
                  noncomputable abbrev CategoryTheory.Enriched.FunctorCategory.precompEnrichedHom (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] {K : Type u₄} [Category.{v₄, u₄} K] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) (G : Functor K J) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V (G.comp F₁) (G.comp F₂)] :
                  enrichedHom V F₁ F₂ enrichedHom V (G.comp F₁) (G.comp F₂)

                  If F₁ and F₂ are functors J ⥤ C, and G : K ⥤ J, then this is the induced morphism enrichedHom V F₁ F₂ ⟶ enrichedHom V (G ⋙ F₁) (G ⋙ F₂) in V when C is a category enriched in V.

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                    @[reducible, inline]

                    Given functors F₁ and F₂ in J ⥤ C, where C is a category enriched in V, this condition allows the definition of functorEnrichedHom V F₁ F₂ : J ⥤ V.

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                      Given functors F₁ and F₂ in J ⥤ C, where C is a category enriched in V, this is the enriched hom functor from F₁ to F₂ in J ⥤ V.

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                        @[simp]

                        The (limit) cone expressing that the limit of functorEnrichedHom V F₁ F₂ is enrichedHom V F₁ F₂.

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                          Auxiliary definition for Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom.

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                            The identity for the J ⥤ V-enrichment of the category J ⥤ C.

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                              The composition for the J ⥤ V-enrichment of the category J ⥤ C.

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                                If C is a V-enriched ordinary category, and C has suitable limits, then J ⥤ C is also a J ⥤ V-enriched ordinary category.

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                                  noncomputable def CategoryTheory.Enriched.FunctorCategory.functorHomEquiv (V : Type u₁) [Category.{v₁, u₁} V] [MonoidalCategory V] {C : Type u₂} [Category.{v₂, u₂} C] {J : Type u₃} [Category.{v₃, u₃} J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] :
                                  (F₁ F₂) (𝟙_ (Functor J V) functorEnrichedHom V F₁ F₂)

                                  Given functors F₁ and F₂ in J ⥤ C, where C is a V-enriched ordinary category, this is the bijection (F₁ ⟶ F₂) ≃ (𝟙_ (J ⥤ V) ⟶ functorEnrichedHom V F₁ F₂).

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                                    @[simp]

                                    If C is a V-enriched ordinary category, and C has suitable limits, then J ⥤ C is also a J ⥤ V-enriched ordinary category.

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