Cartesian closed categories

Given a category with chosen finite products, the cartesian monoidal structure is provided by the instance monoidalOfChosenFiniteProducts.

We define exponentiable objects to be closed objects with respect to this monoidal structure, i.e. (X × -) is a left adjoint.

We say a category is cartesian closed if every object is exponentiable (equivalently, that the category equipped with the cartesian monoidal structure is closed monoidal).

Show that exponential forms a difunctor and define the exponential comparison morphisms.

Implementation Details

Cartesian closed categories require a ChosenFiniteProducts instance. If one whishes to state that a category that hasFiniteProducts is cartesian closed, they should first promote the hasFiniteProducts instance to a ChosenFiniteProducts one using CategoryTheory.ChosenFiniteProducts.ofFiniteProducts.

TODO

Some of the results here are true more generally for closed objects and for closed monoidal categories, and these could be generalised.

@[reducible, inline]

abbrev CategoryTheory.Exponentiable {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] (X : C) : Type (max u v)

An object X is exponentiable if (X × -) is a left adjoint. We define this as being Closed in the cartesian monoidal structure.

Equations

• CategoryTheory.Exponentiable X = CategoryTheory.Closed X

@[reducible, inline]

abbrev CategoryTheory.Exponentiable.mk {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] (X : C) (exp : Functor C C) (adj : MonoidalCategory.tensorLeft X ⊣ exp) : Exponentiable X

Constructor for Exponentiable X which takes as an input an adjunction MonoidalCategory.tensorLeft X ⊣ exp for some functor exp : C ⥤ C.

Equations

• CategoryTheory.Exponentiable.mk X exp adj = { rightAdj := exp, adj := adj }

def CategoryTheory.binaryProductExponentiable {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {X Y : C} (hX : Exponentiable X) (hY : Exponentiable Y) : Exponentiable (MonoidalCategoryStruct.tensorObj X Y)

If X and Y are exponentiable then X ⨯ Y is. This isn't an instance because it's not usually how we want to construct exponentials, we'll usually prove all objects are exponential uniformly.

Equations

• CategoryTheory.binaryProductExponentiable hX hY = CategoryTheory.tensorClosed hX hY

def CategoryTheory.terminalExponentiable {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] : Exponentiable (𝟙_ C)

The terminal object is always exponentiable. This isn't an instance because most of the time we'll prove cartesian closed for all objects at once, rather than just for this one.

Equations

• CategoryTheory.terminalExponentiable = CategoryTheory.unitClosed

@[reducible, inline]

abbrev CategoryTheory.CartesianClosed (C : Type u) [Category.{v, u} C] [ChosenFiniteProducts C] : Type (max u v)

A category C is cartesian closed if it has finite products and every object is exponentiable. We define this as monoidal_closed with respect to the cartesian monoidal structure.

Equations

• CategoryTheory.CartesianClosed C = CategoryTheory.MonoidalClosed C

def CategoryTheory.CartesianClosed.mk (C : Type u) [Category.{v, u} C] [ChosenFiniteProducts C] (exp : (X : C) → Exponentiable X) : CartesianClosed C

Constructor for CartesianClosed C.

Equations

• CategoryTheory.CartesianClosed.mk C exp = { closed := fun (X : C) => exp X }

@[reducible, inline]

abbrev CategoryTheory.exp {C : Type u} [Category.{v, u} C] (A : C) [ChosenFiniteProducts C] [Exponentiable A] : Functor C C

This is (-)^A.

Equations

• CategoryTheory.exp A = CategoryTheory.ihom A

@[reducible, inline]

abbrev CategoryTheory.exp.adjunction {C : Type u} [Category.{v, u} C] (A : C) [ChosenFiniteProducts C] [Exponentiable A] : MonoidalCategory.tensorLeft A ⊣ exp A

The adjunction between A ⨯ - and (-)^A.

Equations

• CategoryTheory.exp.adjunction A = CategoryTheory.ihom.adjunction A

@[reducible, inline]

abbrev CategoryTheory.exp.ev {C : Type u} [Category.{v, u} C] (A : C) [ChosenFiniteProducts C] [Exponentiable A] : (exp A).comp (MonoidalCategory.tensorLeft A) ⟶ Functor.id C

The evaluation natural transformation.

Equations

• CategoryTheory.exp.ev A = CategoryTheory.ihom.ev A

@[reducible, inline]

abbrev CategoryTheory.exp.coev {C : Type u} [Category.{v, u} C] (A : C) [ChosenFiniteProducts C] [Exponentiable A] : Functor.id C ⟶ (MonoidalCategory.tensorLeft A).comp (exp A)

The coevaluation natural transformation.

Equations

• CategoryTheory.exp.coev A = CategoryTheory.ihom.coev A

def CategoryTheory.exp.«term_⟹_» : Lean.TrailingParserDescr

Morphisms obtained using an exponentiable object.

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.exp.delabPrefunctorObjExp : Lean.PrettyPrinter.Delaborator.Delab

Delaborator for Prefunctor.obj

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.exp.«term_^^_» : Lean.TrailingParserDescr

Morphisms from an exponentiable object.

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.exp.ev_coev {C : Type u} [Category.{v, u} C] (A B : C) [ChosenFiniteProducts C] [Exponentiable A] : CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A ((coev A).app B)) ((ev A).app (MonoidalCategoryStruct.tensorObj A B)) = CategoryStruct.id (MonoidalCategoryStruct.tensorObj A B)

theorem CategoryTheory.exp.ev_coev_assoc {C : Type u} [Category.{v, u} C] (A B : C) [ChosenFiniteProducts C] [Exponentiable A] {Z : C} (h : MonoidalCategoryStruct.tensorObj A B ⟶ Z) : CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A ((coev A).app B)) (CategoryStruct.comp ((ev A).app (MonoidalCategoryStruct.tensorObj A B)) h) = h

theorem CategoryTheory.exp.coev_ev {C : Type u} [Category.{v, u} C] (A B : C) [ChosenFiniteProducts C] [Exponentiable A] : CategoryStruct.comp ((coev A).app (A ⟹ B)) ((exp A).map ((ev A).app B)) = CategoryStruct.id (A ⟹ B)

theorem CategoryTheory.exp.coev_ev_assoc {C : Type u} [Category.{v, u} C] (A B : C) [ChosenFiniteProducts C] [Exponentiable A] {Z : C} (h : A ⟹ B ⟶ Z) : CategoryStruct.comp ((coev A).app (A ⟹ B)) (CategoryStruct.comp ((exp A).map ((ev A).app B)) h) = h

instance CategoryTheory.instPreservesColimitsTensorLeft_1 {C : Type u} [Category.{v, u} C] (A : C) [ChosenFiniteProducts C] [Exponentiable A] : Limits.PreservesColimits (MonoidalCategory.tensorLeft A)

def CategoryTheory.CartesianClosed.curry {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] : (MonoidalCategoryStruct.tensorObj A Y ⟶ X) → (Y ⟶ A ⟹ X)

Currying in a cartesian closed category.

Equations

• CategoryTheory.CartesianClosed.curry = ⇑((CategoryTheory.exp.adjunction A).homEquiv Y X)

def CategoryTheory.CartesianClosed.uncurry {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] : (Y ⟶ A ⟹ X) → (MonoidalCategoryStruct.tensorObj A Y ⟶ X)

Uncurrying in a cartesian closed category.

Equations

• CategoryTheory.CartesianClosed.uncurry = ⇑((CategoryTheory.exp.adjunction A).homEquiv Y X).symm

@[simp]

theorem CategoryTheory.CartesianClosed.homEquiv_apply_eq {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : MonoidalCategoryStruct.tensorObj A Y ⟶ X) : ((exp.adjunction A).homEquiv Y X) f = curry f

@[simp]

theorem CategoryTheory.CartesianClosed.homEquiv_symm_apply_eq {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : Y ⟶ A ⟹ X) : ((exp.adjunction A).homEquiv Y X).symm f = uncurry f

theorem CategoryTheory.CartesianClosed.curry_natural_left {C : Type u} [Category.{v, u} C] {A X X' Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ X') (g : MonoidalCategoryStruct.tensorObj A X' ⟶ Y) : curry (CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A f) g) = CategoryStruct.comp f (curry g)

theorem CategoryTheory.CartesianClosed.curry_natural_left_assoc {C : Type u} [Category.{v, u} C] {A X X' Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ X') (g : MonoidalCategoryStruct.tensorObj A X' ⟶ Y) {Z : C} (h : A ⟹ Y ⟶ Z) : CategoryStruct.comp (curry (CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A f) g)) h = CategoryStruct.comp f (CategoryStruct.comp (curry g) h)

theorem CategoryTheory.CartesianClosed.curry_natural_right {C : Type u} [Category.{v, u} C] {A X Y Y' : C} [ChosenFiniteProducts C] [Exponentiable A] (f : MonoidalCategoryStruct.tensorObj A X ⟶ Y) (g : Y ⟶ Y') : curry (CategoryStruct.comp f g) = CategoryStruct.comp (curry f) ((exp A).map g)

theorem CategoryTheory.CartesianClosed.curry_natural_right_assoc {C : Type u} [Category.{v, u} C] {A X Y Y' : C} [ChosenFiniteProducts C] [Exponentiable A] (f : MonoidalCategoryStruct.tensorObj A X ⟶ Y) (g : Y ⟶ Y') {Z : C} (h : A ⟹ Y' ⟶ Z) : CategoryStruct.comp (curry (CategoryStruct.comp f g)) h = CategoryStruct.comp (curry f) (CategoryStruct.comp ((exp A).map g) h)

theorem CategoryTheory.CartesianClosed.uncurry_natural_right {C : Type u} [Category.{v, u} C] {A X Y Y' : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ A ⟹ Y) (g : Y ⟶ Y') : uncurry (CategoryStruct.comp f ((exp A).map g)) = CategoryStruct.comp (uncurry f) g

theorem CategoryTheory.CartesianClosed.uncurry_natural_right_assoc {C : Type u} [Category.{v, u} C] {A X Y Y' : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ A ⟹ Y) (g : Y ⟶ Y') {Z : C} (h : Y' ⟶ Z) : CategoryStruct.comp (uncurry (CategoryStruct.comp f ((exp A).map g))) h = CategoryStruct.comp (uncurry f) (CategoryStruct.comp g h)

theorem CategoryTheory.CartesianClosed.uncurry_natural_left {C : Type u} [Category.{v, u} C] {A X X' Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ X') (g : X' ⟶ A ⟹ Y) : uncurry (CategoryStruct.comp f g) = CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A f) (uncurry g)

theorem CategoryTheory.CartesianClosed.uncurry_natural_left_assoc {C : Type u} [Category.{v, u} C] {A X X' Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ X') (g : X' ⟶ A ⟹ Y) {Z : C} (h : Y ⟶ Z) : CategoryStruct.comp (uncurry (CategoryStruct.comp f g)) h = CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A f) (CategoryStruct.comp (uncurry g) h)

@[simp]

theorem CategoryTheory.CartesianClosed.uncurry_curry {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : MonoidalCategoryStruct.tensorObj A X ⟶ Y) : uncurry (curry f) = f

@[simp]

theorem CategoryTheory.CartesianClosed.curry_uncurry {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : X ⟶ A ⟹ Y) : curry (uncurry f) = f

theorem CategoryTheory.CartesianClosed.curry_eq_iff {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : MonoidalCategoryStruct.tensorObj A Y ⟶ X) (g : Y ⟶ A ⟹ X) : curry f = g ↔ f = uncurry g

theorem CategoryTheory.CartesianClosed.eq_curry_iff {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : MonoidalCategoryStruct.tensorObj A Y ⟶ X) (g : Y ⟶ A ⟹ X) : g = curry f ↔ uncurry g = f

theorem CategoryTheory.CartesianClosed.uncurry_eq {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (g : Y ⟶ A ⟹ X) : uncurry g = CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft A g) ((exp.ev A).app X)

theorem CategoryTheory.CartesianClosed.curry_eq {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] (g : MonoidalCategoryStruct.tensorObj A Y ⟶ X) : curry g = CategoryStruct.comp ((exp.coev A).app Y) ((exp A).map g)

theorem CategoryTheory.CartesianClosed.uncurry_id_eq_ev {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] (A X : C) [Exponentiable A] : uncurry (CategoryStruct.id (A ⟹ X)) = (exp.ev A).app X

theorem CategoryTheory.CartesianClosed.curry_id_eq_coev {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] (A X : C) [Exponentiable A] : curry (CategoryStruct.id (MonoidalCategoryStruct.tensorObj A ((Functor.id C).obj X))) = (exp.coev A).app X

theorem CategoryTheory.CartesianClosed.curry_injective {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] : Function.Injective curry

theorem CategoryTheory.CartesianClosed.uncurry_injective {C : Type u} [Category.{v, u} C] {A X Y : C} [ChosenFiniteProducts C] [Exponentiable A] : Function.Injective uncurry

def CategoryTheory.expUnitNatIso {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] [Exponentiable (𝟙_ C)] : Functor.id C ≅ exp (𝟙_ C)

The exponential with the terminal object is naturally isomorphic to the identity. The typeclass argument is explicit: any instance can be used.

Equations

• CategoryTheory.expUnitNatIso = CategoryTheory.MonoidalClosed.unitNatIso

def CategoryTheory.expUnitIsoSelf {C : Type u} [Category.{v, u} C] {X : C} [ChosenFiniteProducts C] [Exponentiable (𝟙_ C)] : 𝟙_ C ⟹ X ≅ X

The exponential of any object with the terminal object is isomorphic to itself, i.e. X^1 ≅ X. The typeclass argument is explicit: any instance can be used.

Equations

• CategoryTheory.expUnitIsoSelf = (CategoryTheory.expUnitNatIso.app X).symm

def CategoryTheory.internalizeHom {C : Type u} [Category.{v, u} C] {A Y : C} [ChosenFiniteProducts C] [Exponentiable A] (f : A ⟶ Y) : ⊤_ C ⟶ A ⟹ Y

The internal element which points at the given morphism.

Equations

• CategoryTheory.internalizeHom f = CategoryTheory.CartesianClosed.curry (CategoryTheory.CategoryStruct.comp (CategoryTheory.ChosenFiniteProducts.fst A (⊤_ C)) f)

def CategoryTheory.pre {C : Type u} [Category.{v, u} C] {A B : C} [ChosenFiniteProducts C] [Exponentiable A] (f : B ⟶ A) [Exponentiable B] : exp A ⟶ exp B

Pre-compose an internal hom with an external hom.

Equations

• CategoryTheory.pre f = (CategoryTheory.conjugateEquiv (CategoryTheory.exp.adjunction A) (CategoryTheory.exp.adjunction B)) ((CategoryTheory.MonoidalCategory.tensoringLeft C).map f)

theorem CategoryTheory.prod_map_pre_app_comp_ev {C : Type u} [Category.{v, u} C] {A B : C} [ChosenFiniteProducts C] [Exponentiable A] (f : B ⟶ A) [Exponentiable B] (X : C) : CategoryStruct.comp (MonoidalCategoryStruct.whiskerLeft B ((pre f).app X)) ((exp.ev B).app X) = CategoryStruct.comp (MonoidalCategoryStruct.whiskerRight f (A ⟹ X)) ((exp.ev A).app X)

theorem CategoryTheory.uncurry_pre {C : Type u} [Category.{v, u} C] {A B : C} [ChosenFiniteProducts C] [Exponentiable A] (f : B ⟶ A) [Exponentiable B] (X : C) : CartesianClosed.uncurry ((pre f).app X) = CategoryStruct.comp (MonoidalCategoryStruct.whiskerRight f (A ⟹ X)) ((exp.ev A).app X)

theorem CategoryTheory.coev_app_comp_pre_app {C : Type u} [Category.{v, u} C] {A B X : C} [ChosenFiniteProducts C] [Exponentiable A] (f : B ⟶ A) [Exponentiable B] : CategoryStruct.comp ((exp.coev A).app X) ((pre f).app (MonoidalCategoryStruct.tensorObj A X)) = CategoryStruct.comp ((exp.coev B).app X) ((exp B).map (MonoidalCategoryStruct.tensorHom f (CategoryStruct.id X)))

@[simp]

theorem CategoryTheory.pre_id {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] (A : C) [Exponentiable A] : pre (CategoryStruct.id A) = CategoryStruct.id (exp A)

@[simp]

theorem CategoryTheory.pre_map {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {A₁ A₂ A₃ : C} [Exponentiable A₁] [Exponentiable A₂] [Exponentiable A₃] (f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) : pre (CategoryStruct.comp f g) = CategoryStruct.comp (pre g) (pre f)

def CategoryTheory.internalHom {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] [CartesianClosed C] : Functor Cᵒᵖ (Functor C C)

The internal hom functor given by the cartesian closed structure.

Equations

• CategoryTheory.internalHom = { obj := fun (X : Cᵒᵖ) => CategoryTheory.exp (Opposite.unop X), map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) => CategoryTheory.pre f.unop, map_id := ⋯, map_comp := ⋯ }

def CategoryTheory.zeroMul {C : Type u} [Category.{v, u} C] {A : C} [ChosenFiniteProducts C] [Exponentiable A] {I : C} (t : Limits.IsInitial I) : MonoidalCategoryStruct.tensorObj A I ≅ I

If an initial object I exists in a CCC, then A ⨯ I ≅ I.

Equations

• CategoryTheory.zeroMul t = { hom := CategoryTheory.ChosenFiniteProducts.snd A I, inv := t.to (CategoryTheory.MonoidalCategoryStruct.tensorObj A I), hom_inv_id := ⋯, inv_hom_id := ⋯ }

@[simp]

theorem CategoryTheory.zeroMul_hom {C : Type u} [Category.{v, u} C] {A : C} [ChosenFiniteProducts C] [Exponentiable A] {I : C} (t : Limits.IsInitial I) : (zeroMul t).hom = ChosenFiniteProducts.snd A I

@[simp]

theorem CategoryTheory.zeroMul_inv {C : Type u} [Category.{v, u} C] {A : C} [ChosenFiniteProducts C] [Exponentiable A] {I : C} (t : Limits.IsInitial I) : (zeroMul t).inv = t.to (MonoidalCategoryStruct.tensorObj A I)

def CategoryTheory.mulZero {C : Type u} [Category.{v, u} C] {A : C} [ChosenFiniteProducts C] [Exponentiable A] {I : C} (t : Limits.IsInitial I) : MonoidalCategoryStruct.tensorObj I A ≅ I

If an initial object 0 exists in a CCC, then 0 ⨯ A ≅ 0.

Equations

• CategoryTheory.mulZero t = β_ I A ≪≫ CategoryTheory.zeroMul t

def CategoryTheory.powZero {C : Type u} [Category.{v, u} C] (B : C) [ChosenFiniteProducts C] {I : C} (t : Limits.IsInitial I) [CartesianClosed C] : I ⟹ B ≅ ⊤_ C

If an initial object 0 exists in a CCC then 0^B ≅ 1 for any B.

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.prodCoprodDistrib {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] [Limits.HasBinaryCoproducts C] [CartesianClosed C] (X Y Z : C) : MonoidalCategoryStruct.tensorObj Z X ⨿ MonoidalCategoryStruct.tensorObj Z Y ≅ MonoidalCategoryStruct.tensorObj Z (X ⨿ Y)

In a CCC with binary coproducts, the distribution morphism is an isomorphism.

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.strict_initial {C : Type u} [Category.{v, u} C] {A : C} [ChosenFiniteProducts C] [Exponentiable A] {I : C} (t : Limits.IsInitial I) (f : A ⟶ I) : IsIso f

If an initial object I exists in a CCC then it is a strict initial object, i.e. any morphism to I is an iso. This actually shows a slightly stronger version: any morphism to an initial object from an exponentiable object is an isomorphism.

instance CategoryTheory.to_initial_isIso {C : Type u} [Category.{v, u} C] {A : C} [ChosenFiniteProducts C] [Exponentiable A] [Limits.HasInitial C] (f : A ⟶ ⊥_ C) : IsIso f

theorem CategoryTheory.initial_mono {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {I : C} (B : C) (t : Limits.IsInitial I) [CartesianClosed C] : Mono (t.to B)

If an initial object 0 exists in a CCC then every morphism from it is monic.

instance CategoryTheory.Initial.mono_to {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] [Limits.HasInitial C] (B : C) [CartesianClosed C] : Mono (Limits.initial.to B)

def CategoryTheory.cartesianClosedOfEquiv {C : Type u} [Category.{v, u} C] [ChosenFiniteProducts C] {D : Type u₂} [Category.{v₂, u₂} D] [ChosenFiniteProducts D] (e : C ≌ D) [CartesianClosed C] : CartesianClosed D

Transport the property of being cartesian closed across an equivalence of categories.

Note we didn't require any coherence between the choice of finite products here, since we transport along the prodComparison isomorphism.

Equations

• CategoryTheory.cartesianClosedOfEquiv e = CategoryTheory.MonoidalClosed.ofEquiv e.inverse e.symm.toAdjunction