Exponential ideals

An exponential ideal of a cartesian closed category C is a subcategory D ⊆ C such that for any B : D and A : C, the exponential A ⟹ B is in D: resembling ring theoretic ideals. We define the notion here for inclusion functors i : D ⥤ C rather than explicit subcategories to preserve the principle of equivalence.

We additionally show that if C is cartesian closed and i : D ⥤ C is a reflective functor, the following are equivalent.

• The left adjoint to i preserves binary (equivalently, finite) products.
• i is an exponential ideal.

class CategoryTheory.ExponentialIdeal {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.CartesianClosed C] : Prop

• exp_closed : ∀ {B : C}, B ∈ CategoryTheory.Functor.essImage i → ∀ (A : C), (A ⟹ B) ∈ CategoryTheory.Functor.essImage i

The subcategory D of C expressed as an inclusion functor is an exponential ideal if B ∈ D implies A ⟹ B ∈ D for all A.

theorem CategoryTheory.ExponentialIdeal.mk' {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.CartesianClosed C] (h : ∀ (B : D) (A : C), (A ⟹ i.obj B) ∈ CategoryTheory.Functor.essImage i) : CategoryTheory.ExponentialIdeal i

To show i is an exponential ideal it suffices to show that A ⟹ iB is "in" D for any A in C and B in D.

instance CategoryTheory.instExponentialIdealId {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.CartesianClosed C] : CategoryTheory.ExponentialIdeal (CategoryTheory.Functor.id C)

The entire category viewed as a subcategory is an exponential ideal.

instance CategoryTheory.instExponentialIdealSubterminalsInstCategorySubterminalsSubterminalInclusion {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.CartesianClosed C] : CategoryTheory.ExponentialIdeal (CategoryTheory.subterminalInclusion C)

The subcategory of subterminal objects is an exponential ideal.

def CategoryTheory.exponentialIdealReflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.CartesianClosed C] (A : C) [CategoryTheory.Reflective i] [CategoryTheory.ExponentialIdeal i] : CategoryTheory.Functor.comp i (CategoryTheory.Functor.comp (CategoryTheory.exp A) (CategoryTheory.Functor.comp (CategoryTheory.leftAdjoint i) i)) ≅ CategoryTheory.Functor.comp i (CategoryTheory.exp A)

If D is a reflective subcategory, the property of being an exponential ideal is equivalent to the presence of a natural isomorphism i ⋙ exp A ⋙ leftAdjoint i ⋙ i ≅ i ⋙ exp A, that is: (A ⟹ iB) ≅ i L (A ⟹ iB), naturally in B. The converse is given in ExponentialIdeal.mk_of_iso.

theorem CategoryTheory.ExponentialIdeal.mk_of_iso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.CartesianClosed C] [CategoryTheory.Reflective i] (h : (A : C) → CategoryTheory.Functor.comp i (CategoryTheory.Functor.comp (CategoryTheory.exp A) (CategoryTheory.Functor.comp (CategoryTheory.leftAdjoint i) i)) ≅ CategoryTheory.Functor.comp i (CategoryTheory.exp A)) : CategoryTheory.ExponentialIdeal i

Given a natural isomorphism i ⋙ exp A ⋙ leftAdjoint i ⋙ i ≅ i ⋙ exp A, we can show i is an exponential ideal.

theorem CategoryTheory.reflective_products {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] : CategoryTheory.Limits.HasFiniteProducts D

instance CategoryTheory.exponentialIdeal_of_preservesBinaryProducts {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) (CategoryTheory.leftAdjoint i)] : CategoryTheory.ExponentialIdeal i

If the reflector preserves binary products, the subcategory is an exponential ideal. This is the converse of preservesBinaryProductsOfExponentialIdeal.

def CategoryTheory.cartesianClosedOfReflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] : CategoryTheory.CartesianClosed D

If i witnesses that D is a reflective subcategory and an exponential ideal, then D is itself cartesian closed.

noncomputable def CategoryTheory.bijection {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] (A : C) (B : C) (X : D) : ((CategoryTheory.leftAdjoint i).obj (A ⨯ B) ⟶ X) ≃ ((CategoryTheory.leftAdjoint i).obj A ⨯ (CategoryTheory.leftAdjoint i).obj B ⟶ X)

We construct a bijection between morphisms L(A ⨯ B) ⟶ X and morphisms LA ⨯ LB ⟶ X. This bijection has two key properties:

• It is natural in X: See bijection_natural.
• When X = LA ⨯ LB, then the backwards direction sends the identity morphism to the product comparison morphism: See bijection_symm_apply_id.

Together these help show that L preserves binary products. This should be considered internal implementation towards preservesBinaryProductsOfExponentialIdeal.

theorem CategoryTheory.bijection_symm_apply_id {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] (A : C) (B : C) : ↑(CategoryTheory.bijection i A B ((CategoryTheory.leftAdjoint i).obj A ⨯ (CategoryTheory.leftAdjoint i).obj B)).symm (CategoryTheory.CategoryStruct.id ((CategoryTheory.leftAdjoint i).obj A ⨯ (CategoryTheory.leftAdjoint i).obj B)) = CategoryTheory.Limits.prodComparison (CategoryTheory.leftAdjoint i) A B

theorem CategoryTheory.bijection_natural {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] (A : C) (B : C) (X : D) (X' : D) (f : (CategoryTheory.leftAdjoint i).obj (A ⨯ B) ⟶ X) (g : X ⟶ X') : ↑(CategoryTheory.bijection i A B X') (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (↑(CategoryTheory.bijection i A B X) f) g

theorem CategoryTheory.prodComparison_iso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] (A : C) (B : C) : CategoryTheory.IsIso (CategoryTheory.Limits.prodComparison (CategoryTheory.leftAdjoint i) A B)

The bijection allows us to show that prodComparison L A B is an isomorphism, where the inverse is the forward map of the identity morphism.

noncomputable def CategoryTheory.preservesBinaryProductsOfExponentialIdeal {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] : CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) (CategoryTheory.leftAdjoint i)

If a reflective subcategory is an exponential ideal, then the reflector preserves binary products. This is the converse of exponentialIdeal_of_preserves_binary_products.

noncomputable def CategoryTheory.preservesFiniteProductsOfExponentialIdeal {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₁, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Reflective i] [CategoryTheory.CartesianClosed C] [CategoryTheory.Limits.HasFiniteProducts D] [CategoryTheory.ExponentialIdeal i] (J : Type) [Fintype J] : CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete J) (CategoryTheory.leftAdjoint i)

If a reflective subcategory is an exponential ideal, then the reflector preserves finite products.