Reflective functors

Basic properties of reflective functors, especially those relating to their essential image.

Note properties of reflective functors relating to limits and colimits are included in CategoryTheory.Monad.Limits.

class CategoryTheory.Reflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (R : CategoryTheory.Functor D C) extends CategoryTheory.Functor.Full , CategoryTheory.Functor.Faithful : Type (max (max (max u₁ u₂) v₁) v₂)

A functor is reflective, or a reflective inclusion, if it is fully faithful and right adjoint.

• map_surjective : ∀ {X Y : D}, Function.Surjective R.map
• map_injective : ∀ {X Y : D}, Function.Injective R.map
• L : CategoryTheory.Functor C D

  a choice of a left adjoint to R

• adj : CategoryTheory.Reflective.L R ⊣ R

  R is a right adjoint

def CategoryTheory.reflector {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Reflective i] : CategoryTheory.Functor C D

The reflector C ⥤ D when R : D ⥤ C is reflective.

Equations

• CategoryTheory.reflector i = CategoryTheory.Reflective.L i

def CategoryTheory.reflectorAdjunction {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Reflective i] : CategoryTheory.reflector i ⊣ i

The adjunction reflector i ⊣ i when i is reflective.

Equations

• CategoryTheory.reflectorAdjunction i = CategoryTheory.Reflective.adj

instance CategoryTheory.instIsRightAdjointOfReflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Reflective i] : i.IsRightAdjoint

Equations

• ⋯ = ⋯

instance CategoryTheory.instIsLeftAdjointReflector {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Reflective i] : (CategoryTheory.reflector i).IsLeftAdjoint

Equations

• ⋯ = ⋯

def CategoryTheory.Functor.fullyFaithfulOfReflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Reflective i] : i.FullyFaithful

A reflective functor is fully faithful.

Equations

• i.fullyFaithfulOfReflective = (CategoryTheory.reflectorAdjunction i).fullyFaithfulROfIsIsoCounit

theorem CategoryTheory.unit_obj_eq_map_unit {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (i : CategoryTheory.Functor D C) [CategoryTheory.Reflective i] (X : C) : (CategoryTheory.reflectorAdjunction i).unit.app (i.obj ((CategoryTheory.reflector i).obj X)) = i.map ((CategoryTheory.reflector i).map ((CategoryTheory.reflectorAdjunction i).unit.app X))

For a reflective functor i (with left adjoint L), with unit η, we have η_iL = iL η.

theorem CategoryTheory.Functor.essImage.unit_isIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] {A : C} (h : A ∈ i.essImage) : CategoryTheory.IsIso ((CategoryTheory.reflectorAdjunction i).unit.app A)

If A is essentially in the image of a reflective functor i, then η_A is an isomorphism. This gives that the "witness" for A being in the essential image can instead be given as the reflection of A, with the isomorphism as η_A.

(For any B in the reflective subcategory, we automatically have that ε_B is an iso.)

theorem CategoryTheory.mem_essImage_of_unit_isSplitMono {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] {A : C} [CategoryTheory.IsSplitMono ((CategoryTheory.reflectorAdjunction i).unit.app A)] : A ∈ i.essImage

If η_A is a split monomorphism, then A is in the reflective subcategory.

instance CategoryTheory.Reflective.comp {C : Type u₁} {D : Type u₂} {E : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) [CategoryTheory.Reflective F] [CategoryTheory.Reflective G] : CategoryTheory.Reflective (F.comp G)

Composition of reflective functors.

Equations

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

def CategoryTheory.unitCompPartialBijectiveAux {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (A : C) (B : D) : (A ⟶ i.obj B) ≃ (i.obj ((CategoryTheory.reflector i).obj A) ⟶ i.obj B)

(Implementation) Auxiliary definition for unitCompPartialBijective.

Equations

• CategoryTheory.unitCompPartialBijectiveAux A B = ((CategoryTheory.reflectorAdjunction i).homEquiv A B).symm.trans (CategoryTheory.Functor.FullyFaithful.ofFullyFaithful i).homEquiv

theorem CategoryTheory.unitCompPartialBijectiveAux_symm_apply {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] {A : C} {B : D} (f : i.obj ((CategoryTheory.reflector i).obj A) ⟶ i.obj B) : (CategoryTheory.unitCompPartialBijectiveAux A B).symm f = CategoryTheory.CategoryStruct.comp ((CategoryTheory.reflectorAdjunction i).unit.app A) f

The description of the inverse of the bijection unitCompPartialBijectiveAux.

def CategoryTheory.unitCompPartialBijective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (A : C) {B : C} (hB : B ∈ i.essImage) : (A ⟶ B) ≃ (i.obj ((CategoryTheory.reflector i).obj A) ⟶ B)

If i has a reflector L, then the function (i.obj (L.obj A) ⟶ B) → (A ⟶ B) given by precomposing with η.app A is a bijection provided B is in the essential image of i. That is, the function fun (f : i.obj (L.obj A) ⟶ B) ↦ η.app A ≫ f is bijective, as long as B is in the essential image of i. This definition gives an equivalence: the key property that the inverse can be described nicely is shown in unitCompPartialBijective_symm_apply.

This establishes there is a natural bijection (A ⟶ B) ≃ (i.obj (L.obj A) ⟶ B). In other words, from the point of view of objects in D, A and i.obj (L.obj A) look the same: specifically that η.app A is an isomorphism.

Equations

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

@[simp]

theorem CategoryTheory.unitCompPartialBijective_symm_apply {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (A : C) {B : C} (hB : B ∈ i.essImage) (f : i.obj ((CategoryTheory.reflector i).obj A) ⟶ B) : (CategoryTheory.unitCompPartialBijective A hB).symm f = CategoryTheory.CategoryStruct.comp ((CategoryTheory.reflectorAdjunction i).unit.app A) f

theorem CategoryTheory.unitCompPartialBijective_symm_natural {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (A : C) {B : C} {B' : C} (h : B ⟶ B') (hB : B ∈ i.essImage) (hB' : B' ∈ i.essImage) (f : i.obj ((CategoryTheory.reflector i).obj A) ⟶ B) : (CategoryTheory.unitCompPartialBijective A hB').symm (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.unitCompPartialBijective A hB).symm f) h

theorem CategoryTheory.unitCompPartialBijective_natural {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (A : C) {B : C} {B' : C} (h : B ⟶ B') (hB : B ∈ i.essImage) (hB' : B' ∈ i.essImage) (f : A ⟶ B) : (CategoryTheory.unitCompPartialBijective A hB') (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.unitCompPartialBijective A hB) f) h

instance CategoryTheory.instIsIsoAppUnitReflectorAdjunctionObjEssImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (X : i.EssImageSubcategory) : CategoryTheory.IsIso ((CategoryTheory.reflectorAdjunction i).unit.app X.obj)

Equations

• ⋯ = ⋯

def CategoryTheory.equivEssImageOfReflective_counitIso_app {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (X : i.EssImageSubcategory) : ((i.essImageInclusion.comp (CategoryTheory.reflector i)).comp i.toEssImage).obj X ≅ X

The counit isomorphism of the equivalence D ≌ i.EssImageSubcategory given by equivEssImageOfReflective when the functor i is reflective.

Equations

• CategoryTheory.equivEssImageOfReflective_counitIso_app X = (CategoryTheory.asIso ((CategoryTheory.reflectorAdjunction i).unit.app X.obj)).symm

theorem CategoryTheory.equivEssImageOfReflective_map_counitIso_app_hom {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (X : i.EssImageSubcategory) : i.essImageInclusion.map (CategoryTheory.equivEssImageOfReflective_counitIso_app X).hom = CategoryTheory.inv ((CategoryTheory.reflectorAdjunction i).unit.app X.obj)

theorem CategoryTheory.equivEssImageOfReflective_map_counitIso_app_inv {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] (X : i.EssImageSubcategory) : i.essImageInclusion.map (CategoryTheory.equivEssImageOfReflective_counitIso_app X).inv = (CategoryTheory.reflectorAdjunction i).unit.app X.obj

@[simp]

theorem CategoryTheory.equivEssImageOfReflective_inverse {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] : CategoryTheory.equivEssImageOfReflective.inverse = i.essImageInclusion.comp (CategoryTheory.reflector i)

@[simp]

theorem CategoryTheory.equivEssImageOfReflective_unitIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] : CategoryTheory.equivEssImageOfReflective.unitIso = CategoryTheory.NatIso.ofComponents (fun (X : D) => (CategoryTheory.asIso ((CategoryTheory.reflectorAdjunction i).counit.app X)).symm) ⋯

@[simp]

theorem CategoryTheory.equivEssImageOfReflective_counitIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] : CategoryTheory.equivEssImageOfReflective.counitIso = CategoryTheory.NatIso.ofComponents CategoryTheory.equivEssImageOfReflective_counitIso_app ⋯

@[simp]

theorem CategoryTheory.equivEssImageOfReflective_functor {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] : CategoryTheory.equivEssImageOfReflective.functor = i.toEssImage

def CategoryTheory.equivEssImageOfReflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {i : CategoryTheory.Functor D C} [CategoryTheory.Reflective i] : D ≌ i.EssImageSubcategory

If i : D ⥤ C is reflective, the inverse functor of i ≌ F.essImage can be explicitly defined by the reflector.

Equations

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

class CategoryTheory.Coreflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (L : CategoryTheory.Functor C D) extends CategoryTheory.Functor.Full , CategoryTheory.Functor.Faithful : Type (max (max (max u₁ u₂) v₁) v₂)

A functor is coreflective, or a coreflective inclusion, if it is fully faithful and left adjoint.

• map_surjective : ∀ {X Y : C}, Function.Surjective L.map
• map_injective : ∀ {X Y : C}, Function.Injective L.map
• R : CategoryTheory.Functor D C

  a choice of a right adjoint to L

• adj : L ⊣ CategoryTheory.Coreflective.R L

  L is a left adjoint

def CategoryTheory.coreflector {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (j : CategoryTheory.Functor C D) [CategoryTheory.Coreflective j] : CategoryTheory.Functor D C

The coreflector D ⥤ C when L : C ⥤ D is coreflective.

Equations

• CategoryTheory.coreflector j = CategoryTheory.Coreflective.R j

def CategoryTheory.coreflectorAdjunction {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (j : CategoryTheory.Functor C D) [CategoryTheory.Coreflective j] : j ⊣ CategoryTheory.coreflector j

The adjunction j ⊣ coreflector j when j is coreflective.

Equations

• CategoryTheory.coreflectorAdjunction j = CategoryTheory.Coreflective.adj

instance CategoryTheory.instIsLeftAdjointOfCoreflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (j : CategoryTheory.Functor C D) [CategoryTheory.Coreflective j] : j.IsLeftAdjoint

Equations

• ⋯ = ⋯

instance CategoryTheory.instIsRightAdjointCoreflector {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (j : CategoryTheory.Functor C D) [CategoryTheory.Coreflective j] : (CategoryTheory.coreflector j).IsRightAdjoint

Equations

• ⋯ = ⋯

def CategoryTheory.Functor.fullyFaithfulOfCoreflective {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (j : CategoryTheory.Functor C D) [CategoryTheory.Coreflective j] : j.FullyFaithful

A coreflective functor is fully faithful.

Equations

• j.fullyFaithfulOfCoreflective = (CategoryTheory.coreflectorAdjunction j).fullyFaithfulLOfIsIsoUnit

theorem CategoryTheory.counit_obj_eq_map_counit {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (j : CategoryTheory.Functor C D) [CategoryTheory.Coreflective j] (X : D) : (CategoryTheory.coreflectorAdjunction j).counit.app (j.obj ((CategoryTheory.coreflector j).obj X)) = j.map ((CategoryTheory.coreflector j).map ((CategoryTheory.coreflectorAdjunction j).counit.app X))

theorem CategoryTheory.Functor.essImage.counit_isIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {j : CategoryTheory.Functor C D} [CategoryTheory.Coreflective j] {A : D} (h : A ∈ j.essImage) : CategoryTheory.IsIso ((CategoryTheory.coreflectorAdjunction j).counit.app A)

theorem CategoryTheory.mem_essImage_of_counit_isSplitEpi {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {j : CategoryTheory.Functor C D} [CategoryTheory.Coreflective j] {A : D} [CategoryTheory.IsSplitEpi ((CategoryTheory.coreflectorAdjunction j).counit.app A)] : A ∈ j.essImage

instance CategoryTheory.Coreflective.comp {C : Type u₁} {D : Type u₂} {E : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) [CategoryTheory.Coreflective F] [CategoryTheory.Coreflective G] : CategoryTheory.Coreflective (F.comp G)

Equations

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