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

    The condition that the end diagram V F₁ F₂ exists, see enrichedHom.

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      @[reducible, inline]
      noncomputable abbrev CategoryTheory.Enriched.FunctorCategory.enrichedHom (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] :
      V

      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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        @[reducible, inline]
        noncomputable abbrev CategoryTheory.Enriched.FunctorCategory.enrichedHomπ (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] (j : J) :
        Quiver.Hom (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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] {i j : J} (f : Quiver.Hom i j) :
          Eq (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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasEnrichedHom V F₁ F₂] {i j : J} (f : Quiver.Hom i j) {Z : V} (h : Quiver.Hom (EnrichedCategory.Hom (F₁.obj i) (F₂.obj j)) Z) :
          Eq (CategoryStruct.comp (enrichedHomπ V F₁ F₂ i) (CategoryStruct.comp (eHomWhiskerLeft V (F₁.obj i) (F₂.map f)) h)) (CategoryStruct.comp (enrichedHomπ V F₁ F₂ j) (CategoryStruct.comp (eHomWhiskerRight V (F₁.map f) (F₂.obj j)) h))
          noncomputable def CategoryTheory.Enriched.FunctorCategory.homEquiv (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasEnrichedHom 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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasEnrichedHom V F₁ F₂] (τ : Quiver.Hom F₁ F₂) (j : J) :
            theorem CategoryTheory.Enriched.FunctorCategory.homEquiv_apply_π_assoc (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasEnrichedHom V F₁ F₂] (τ : Quiver.Hom F₁ F₂) (j : J) {Z : V} (h : Quiver.Hom (EnrichedCategory.Hom (F₁.obj j) (F₂.obj j)) Z) :

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

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              theorem CategoryTheory.Enriched.FunctorCategory.enrichedId_π (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ : Functor J C) [HasEnrichedHom V F₁ F₁] (j : J) :
              Eq (CategoryStruct.comp (enrichedId V F₁) (Limits.end_.π (diagram V F₁ F₁) j)) (eId V (F₁.obj j))
              theorem CategoryTheory.Enriched.FunctorCategory.enrichedId_π_assoc (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ : Functor J C) [HasEnrichedHom V F₁ F₁] (j : J) {Z : V} (h : Quiver.Hom (((diagram V F₁ F₁).obj { unop := j }).obj j) Z) :
              noncomputable def CategoryTheory.Enriched.FunctorCategory.enrichedComp (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category 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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ : Functor J C) [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₁ F₃] (j : J) :
                Eq (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 { unop := j })) (F₂.obj j) (F₃.obj j)))
                theorem CategoryTheory.Enriched.FunctorCategory.enrichedComp_π_assoc (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category 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 : Quiver.Hom (((diagram V F₁ F₃).obj { unop := j }).obj j) Z) :
                Eq (CategoryStruct.comp (enrichedComp V F₁ F₂ F₃) (CategoryStruct.comp (Limits.end_.π (diagram V F₁ F₃) j) h)) (CategoryStruct.comp (MonoidalCategoryStruct.tensorHom (Limits.end_.π (diagram V F₁ F₂) j) (Limits.end_.π (diagram V F₂ F₃) j)) (CategoryStruct.comp (eComp V (Opposite.unop (F₁.op.obj { unop := j })) (F₂.obj j) (F₃.obj j)) h))
                theorem CategoryTheory.Enriched.FunctorCategory.enriched_assoc (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category 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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category 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 : Quiver.Hom (enrichedHom V F₁ F₄) Z) :
                noncomputable def CategoryTheory.Enriched.FunctorCategory.enrichedOrdinaryCategory (V : Type u₁) [Category V] [MonoidalCategory V] (C : Type u₂) [Category C] (J : Type u₃) [Category J] [EnrichedOrdinaryCategory V C] [∀ (F₁ F₂ : Functor J C), HasEnrichedHom V F₁ F₂] :

                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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] {K : Type u₄} [Category 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 : Iso (G.comp F₁) F₁') (e₂ : Iso (G.comp F₂) F₂') :
                  Quiver.Hom (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 V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] {K : Type u₄} [Category 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)] :
                    Quiver.Hom (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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                        noncomputable def CategoryTheory.Enriched.FunctorCategory.functorEnrichedHom (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] :

                        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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                          theorem CategoryTheory.Enriched.FunctorCategory.functorEnrichedHom_map (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] {X✝ Y✝ : J} (f : Quiver.Hom X✝ Y✝) :
                          Eq ((functorEnrichedHom V F₁ F₂).map f) (precompEnrichedHom' V (Under.map f) (Iso.refl ((Under.map f).comp ((Under.forget X✝).comp F₁))) (Iso.refl ((Under.map f).comp ((Under.forget X✝).comp F₂))))
                          theorem CategoryTheory.Enriched.FunctorCategory.functorEnrichedHom_obj (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] (j : J) :
                          Eq ((functorEnrichedHom V F₁ F₂).obj j) (enrichedHom V ((Under.forget j).comp F₁) ((Under.forget j).comp F₂))
                          noncomputable def CategoryTheory.Enriched.FunctorCategory.coneFunctorEnrichedHom (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] :

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

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                            theorem CategoryTheory.Enriched.FunctorCategory.coneFunctorEnrichedHom_π_app (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] (j : J) :
                            Eq ((coneFunctorEnrichedHom V F₁ F₂).π.app j) (precompEnrichedHom V F₁ F₂ (Under.forget j))
                            theorem CategoryTheory.Enriched.FunctorCategory.coneFunctorEnrichedHom_pt (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] :
                            Eq (coneFunctorEnrichedHom V F₁ F₂).pt (enrichedHom V F₁ F₂)
                            noncomputable def CategoryTheory.Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom.lift {V : Type u₁} [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] (s : Limits.Cone (functorEnrichedHom V F₁ F₂)) :
                            Quiver.Hom s.pt (enrichedHom V F₁ F₂)

                            Auxiliary definition for Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom.

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                              theorem CategoryTheory.Enriched.FunctorCategory.isLimitConeFunctorEnrichedHom.fac {V : Type u₁} [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] (s : Limits.Cone (functorEnrichedHom V F₁ F₂)) (j : J) :

                              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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                                  theorem CategoryTheory.Enriched.FunctorCategory.functorEnrichedComp_app (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] (F₁ F₂ F₃ : Functor J C) [HasFunctorEnrichedHom V F₁ F₂] [HasFunctorEnrichedHom V F₂ F₃] [HasFunctorEnrichedHom V F₁ F₃] (j : J) :
                                  Eq ((functorEnrichedComp V F₁ F₂ F₃).app j) (enrichedComp V ((Under.forget j).comp F₁) ((Under.forget j).comp F₂) ((Under.forget j).comp F₃))
                                  noncomputable def CategoryTheory.Enriched.FunctorCategory.functorEnrichedCategory (V : Type u₁) [Category V] [MonoidalCategory V] (C : Type u₂) [Category C] (J : Type u₃) [Category J] [EnrichedOrdinaryCategory V C] [∀ (F₁ F₂ : Functor J C), HasFunctorEnrichedHom V F₁ F₂] :

                                  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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                                    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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                                      theorem CategoryTheory.Enriched.FunctorCategory.functorHomEquiv_apply_app (V : Type u₁) [Category V] [MonoidalCategory V] {C : Type u₂} [Category C] {J : Type u₃} [Category J] [EnrichedOrdinaryCategory V C] {F₁ F₂ : Functor J C} [HasFunctorEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₂] (a✝ : Quiver.Hom F₁ F₂) (X : J) :
                                      noncomputable def CategoryTheory.Enriched.FunctorCategory.functorEnrichedOrdinaryCategory (V : Type u₁) [Category V] [MonoidalCategory V] (C : Type u₂) [Category C] (J : Type u₃) [Category J] [EnrichedOrdinaryCategory V C] [∀ (F₁ F₂ : Functor J C), HasFunctorEnrichedHom V F₁ F₂] [∀ (F₁ F₂ : Functor J C), HasEnrichedHom V F₁ F₂] :

                                      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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