Adjunctions involving evaluation

We show that evaluation of functors have adjoints, given the existence of (co)products.

def CategoryTheory.evaluationLeftAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) : CategoryTheory.Functor D (CategoryTheory.Functor C D)

The left adjoint of evaluation.

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theorem CategoryTheory.evaluationLeftAdjoint_obj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) (d : D) : ∀ {X Y : C} (f : X ⟶ Y), ((CategoryTheory.evaluationLeftAdjoint D c).obj d).map f = CategoryTheory.Limits.Sigma.desc fun (g : c ⟶ X) => CategoryTheory.Limits.Sigma.ι (fun (x : c ⟶ Y) => d) (CategoryTheory.CategoryStruct.comp g f)

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theorem CategoryTheory.evaluationLeftAdjoint_map_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) : ∀ {x d₂ : D} (f : x ⟶ d₂) (x_1 : C), ((CategoryTheory.evaluationLeftAdjoint D c).map f).app x_1 = CategoryTheory.Limits.Sigma.desc fun (h : c ⟶ x_1) => CategoryTheory.CategoryStruct.comp f (CategoryTheory.Limits.Sigma.ι (fun (x : c ⟶ x_1) => d₂) h)

@[simp]

theorem CategoryTheory.evaluationLeftAdjoint_obj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) (d : D) (t : C) : ((CategoryTheory.evaluationLeftAdjoint D c).obj d).obj t = ∐ fun (x : c ⟶ t) => d

def CategoryTheory.evaluationAdjunctionRight {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) : CategoryTheory.evaluationLeftAdjoint D c ⊣ (CategoryTheory.evaluation C D).obj c

The adjunction showing that evaluation is a right adjoint.

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theorem CategoryTheory.evaluationAdjunctionRight_counit_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) (Y : CategoryTheory.Functor C D) : ∀ (x : C), ((CategoryTheory.evaluationAdjunctionRight D c).counit.app Y).app x = CategoryTheory.Limits.Sigma.desc fun (h : c ⟶ x) => Y.map h

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theorem CategoryTheory.evaluationAdjunctionRight_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) (X : D) : (CategoryTheory.evaluationAdjunctionRight D c).unit.app X = CategoryTheory.Limits.Sigma.ι (fun (x : c ⟶ c) => X) (CategoryTheory.CategoryStruct.id c)

instance CategoryTheory.evaluationIsRightAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] (c : C) : ((CategoryTheory.evaluation C D).obj c).IsRightAdjoint

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• ⋯ = ⋯

theorem CategoryTheory.NatTrans.mono_iff_mono_app' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (η : F ⟶ G) : CategoryTheory.Mono η ↔ ∀ (c : C), CategoryTheory.Mono (η.app c)

See also the file CategoryTheory.Limits.FunctorCategory.EpiMono for a similar result under a HasPullbacks assumption.

def CategoryTheory.evaluationRightAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) : CategoryTheory.Functor D (CategoryTheory.Functor C D)

The right adjoint of evaluation.

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theorem CategoryTheory.evaluationRightAdjoint_map_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) : ∀ {X Y : D} (f : X ⟶ Y) (x : C), ((CategoryTheory.evaluationRightAdjoint D c).map f).app x = CategoryTheory.Limits.Pi.lift fun (g : x ⟶ c) => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Pi.π (fun (x : x ⟶ c) => X) g) f

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theorem CategoryTheory.evaluationRightAdjoint_obj_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) (d : D) (t : C) : ((CategoryTheory.evaluationRightAdjoint D c).obj d).obj t = ∏ᶜ fun (x : t ⟶ c) => d

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theorem CategoryTheory.evaluationRightAdjoint_obj_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) (d : D) : ∀ {X Y : C} (f : X ⟶ Y), ((CategoryTheory.evaluationRightAdjoint D c).obj d).map f = CategoryTheory.Limits.Pi.lift fun (g : Y ⟶ c) => CategoryTheory.Limits.Pi.π (fun (x : X ⟶ c) => d) (CategoryTheory.CategoryStruct.comp f g)

def CategoryTheory.evaluationAdjunctionLeft {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) : (CategoryTheory.evaluation C D).obj c ⊣ CategoryTheory.evaluationRightAdjoint D c

The adjunction showing that evaluation is a left adjoint.

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theorem CategoryTheory.evaluationAdjunctionLeft_unit_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) (X : CategoryTheory.Functor C D) : ∀ (x : C), ((CategoryTheory.evaluationAdjunctionLeft D c).unit.app X).app x = CategoryTheory.Limits.Pi.lift fun (g : x ⟶ c) => X.map g

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theorem CategoryTheory.evaluationAdjunctionLeft_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) (Y : D) : (CategoryTheory.evaluationAdjunctionLeft D c).counit.app Y = CategoryTheory.Limits.Pi.π (fun (x : c ⟶ c) => Y) (CategoryTheory.CategoryStruct.id c)

instance CategoryTheory.evaluationIsLeftAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] (c : C) : ((CategoryTheory.evaluation C D).obj c).IsLeftAdjoint

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• ⋯ = ⋯

theorem CategoryTheory.NatTrans.epi_iff_epi_app' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (η : F ⟶ G) : CategoryTheory.Epi η ↔ ∀ (c : C), CategoryTheory.Epi (η.app c)

See also the file CategoryTheory.Limits.FunctorCategory.EpiMono for a similar result under a HasPushouts assumption.