Essential image of a functor

The essential image essImage of a functor consists of the objects in the target category which are isomorphic to an object in the image of the object function. This, for instance, allows us to talk about objects belonging to a subcategory expressed as a functor rather than a subtype, preserving the principle of equivalence. For example this lets us define exponential ideals.

The essential image can also be seen as a subcategory of the target category, and witnesses that a functor decomposes into an essentially surjective functor and a fully faithful functor. (TODO: show that this decomposition forms an orthogonal factorisation system).

def CategoryTheory.Functor.essImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : Set D

The essential image of a functor F consists of those objects in the target category which are isomorphic to an object in the image of the function F.obj. In other words, this is the closure under isomorphism of the function F.obj. This is the "non-evil" way of describing the image of a functor.

Equations

• CategoryTheory.Functor.essImage F Y = ∃ X, Nonempty (F.obj X ≅ Y)

def CategoryTheory.Functor.essImage.witness {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {Y : D} (h : Y ∈ CategoryTheory.Functor.essImage F) : C

Get the witnessing object that Y is in the subcategory given by F.

Equations

• CategoryTheory.Functor.essImage.witness h = Exists.choose h

def CategoryTheory.Functor.essImage.getIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {Y : D} (h : Y ∈ CategoryTheory.Functor.essImage F) : F.obj (CategoryTheory.Functor.essImage.witness h) ≅ Y

Extract the isomorphism between F.obj h.witness and Y itself.

Equations

• CategoryTheory.Functor.essImage.getIso h = Classical.choice (_ : Nonempty (F.obj (Exists.choose h) ≅ Y))

theorem CategoryTheory.Functor.essImage.ofIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {Y : D} {Y' : D} (h : Y ≅ Y') (hY : Y ∈ CategoryTheory.Functor.essImage F) : Y' ∈ CategoryTheory.Functor.essImage F

Being in the essential image is a "hygienic" property: it is preserved under isomorphism.

theorem CategoryTheory.Functor.essImage.ofNatIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} (h : F ≅ F') {Y : D} (hY : Y ∈ CategoryTheory.Functor.essImage F) : Y ∈ CategoryTheory.Functor.essImage F'

If Y is in the essential image of F then it is in the essential image of F' as long as F ≅ F'.

theorem CategoryTheory.Functor.essImage_eq_of_natIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {F' : CategoryTheory.Functor C D} (h : F ≅ F') : CategoryTheory.Functor.essImage F = CategoryTheory.Functor.essImage F'

Isomorphic functors have equal essential images.

theorem CategoryTheory.Functor.obj_mem_essImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor D C) (Y : D) : F.obj Y ∈ CategoryTheory.Functor.essImage F

An object in the image is in the essential image.

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

The essential image of a functor, interpreted as a full subcategory of the target category.

Equations

• CategoryTheory.Functor.EssImageSubcategory F = CategoryTheory.FullSubcategory (CategoryTheory.Functor.essImage F)

instance CategoryTheory.Functor.instCategoryEssImageSubcategory {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} : CategoryTheory.Category.{v₂, u₂} (CategoryTheory.Functor.EssImageSubcategory F)

Equations

• CategoryTheory.Functor.instCategoryEssImageSubcategory = inferInstance

@[simp]

theorem CategoryTheory.Functor.essImageInclusion_map {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : ∀ {X Y : CategoryTheory.InducedCategory D CategoryTheory.FullSubcategory.obj} (f : X ⟶ Y), (CategoryTheory.Functor.essImageInclusion F).map f = f

@[simp]

theorem CategoryTheory.Functor.essImageInclusion_obj {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (self : CategoryTheory.FullSubcategory (CategoryTheory.Functor.essImage F)) : (CategoryTheory.Functor.essImageInclusion F).obj self = self.obj

def CategoryTheory.Functor.essImageInclusion {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : CategoryTheory.Functor (CategoryTheory.Functor.EssImageSubcategory F) D

The essential image as a subcategory has a fully faithful inclusion into the target category.

Equations

• CategoryTheory.Functor.essImageInclusion F = CategoryTheory.fullSubcategoryInclusion (CategoryTheory.Functor.essImage F)

instance CategoryTheory.Functor.instFullEssImageSubcategoryInstCategoryEssImageSubcategoryEssImageInclusion {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} : CategoryTheory.Full (CategoryTheory.Functor.essImageInclusion F)

Equations

• CategoryTheory.Functor.instFullEssImageSubcategoryInstCategoryEssImageSubcategoryEssImageInclusion = inferInstance

instance CategoryTheory.Functor.instFaithfulEssImageSubcategoryInstCategoryEssImageSubcategoryEssImageInclusion {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} : CategoryTheory.Faithful (CategoryTheory.Functor.essImageInclusion F)

Equations

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

@[simp]

theorem CategoryTheory.Functor.toEssImage_map {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : ∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.Functor.toEssImage F).map f = F.map f

@[simp]

theorem CategoryTheory.Functor.toEssImage_obj_obj {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C) : ((CategoryTheory.Functor.toEssImage F).obj X).obj = F.obj X

def CategoryTheory.Functor.toEssImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : CategoryTheory.Functor C (CategoryTheory.Functor.EssImageSubcategory F)

Given a functor F : C ⥤ D, we have an (essentially surjective) functor from C to the essential image of F.

Equations

• CategoryTheory.Functor.toEssImage F = CategoryTheory.FullSubcategory.lift (CategoryTheory.Functor.essImage F) F (_ : ∀ (Y : C), F.obj Y ∈ CategoryTheory.Functor.essImage F)

@[simp]

theorem CategoryTheory.Functor.toEssImageCompEssentialImageInclusion_inv_app {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C) : (CategoryTheory.Functor.toEssImageCompEssentialImageInclusion F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)

@[simp]

theorem CategoryTheory.Functor.toEssImageCompEssentialImageInclusion_hom_app {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C) : (CategoryTheory.Functor.toEssImageCompEssentialImageInclusion F).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)

def CategoryTheory.Functor.toEssImageCompEssentialImageInclusion {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : CategoryTheory.Functor.comp (CategoryTheory.Functor.toEssImage F) (CategoryTheory.Functor.essImageInclusion F) ≅ F

The functor F factorises through its essential image, where the first functor is essentially surjective and the second is fully faithful.

Equations

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

class CategoryTheory.EssSurj {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) : Prop

• mem_essImage : ∀ (Y : D), Y ∈ CategoryTheory.Functor.essImage F

  All the objects of the target category are in the essential image.

A functor F : C ⥤ D is essentially surjective if every object of D is in the essential image of F. In other words, for every Y : D, there is some X : C with F.obj X ≅ Y.

See https://stacks.math.columbia.edu/tag/001C.

instance CategoryTheory.EssSurj.toEssImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} : CategoryTheory.EssSurj (CategoryTheory.Functor.toEssImage F)

Equations

• @CategoryTheory.EssSurj.toEssImage = @CategoryTheory.EssSurj.toEssImage.proof_1

def CategoryTheory.Functor.objPreimage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) [CategoryTheory.EssSurj F] (Y : D) : C

Given an essentially surjective functor, we can find a preimage for every object Y in the codomain. Applying the functor to this preimage will yield an object isomorphic to Y, see obj_obj_preimage_iso.

Equations

• CategoryTheory.Functor.objPreimage F Y = CategoryTheory.Functor.essImage.witness (_ : Y ∈ CategoryTheory.Functor.essImage F)

def CategoryTheory.Functor.objObjPreimageIso {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) [CategoryTheory.EssSurj F] (Y : D) : F.obj (CategoryTheory.Functor.objPreimage F Y) ≅ Y

Applying an essentially surjective functor to a preimage of Y yields an object that is isomorphic to Y.

Equations

• CategoryTheory.Functor.objObjPreimageIso F Y = CategoryTheory.Functor.essImage.getIso (_ : Y ∈ CategoryTheory.Functor.essImage F)

instance CategoryTheory.Faithful.toEssImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) [CategoryTheory.Faithful F] : CategoryTheory.Faithful (CategoryTheory.Functor.toEssImage F)

The induced functor of a faithful functor is faithful.

Equations

• @CategoryTheory.Faithful.toEssImage = @CategoryTheory.Faithful.toEssImage.proof_1

instance CategoryTheory.Full.toEssImage {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) [CategoryTheory.Full F] : CategoryTheory.Full (CategoryTheory.Functor.toEssImage F)

The induced functor of a full functor is full.

Equations

• CategoryTheory.Full.toEssImage F = CategoryTheory.Full.ofCompFaithful (CategoryTheory.Functor.toEssImage F) (CategoryTheory.Functor.essImageInclusion F)

instance CategoryTheory.instEssSurjId {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.EssSurj (CategoryTheory.Functor.id C)

Equations

• @CategoryTheory.instEssSurjId = @CategoryTheory.instEssSurjId.proof_1

theorem CategoryTheory.Iso.map_essSurj {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} [CategoryTheory.EssSurj F] (α : F ≅ G) : CategoryTheory.EssSurj G