category_theory.fully_faithful

@[class]

structure category_theory.full {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) :

Type (max u₁ v₁ v₂)

preimage : Π {X Y : C}, (F.obj X ⟶ F.obj Y) → (X ⟶ Y)
witness' : (∀ {X Y : C} (f : F.obj X ⟶ F.obj Y), F.map (category_theory.full.preimage f) = f) . "obviously"

A functor F : C ⥤ D is full if for each X Y : C, F.map is surjective. In fact, we use a constructive definition, so the full F typeclass contains data, specifying a particular preimage of each f : F.obj X ⟶ F.obj Y.

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

Instances

source

@[simp]

theorem category_theory.full.witness {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [c : category_theory.full F] {X Y : C} (f : F.obj X ⟶ F.obj Y) :

F.map (category_theory.full.preimage f) = f

source

@[class]

structure category_theory.faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) :

Prop

map_injective' : (∀ {X Y : C}, function.injective F.map) . "obviously"

A functor F : C ⥤ D is faithful if for each X Y : C, F.map is injective.

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

Instances

source

theorem category_theory.faithful.map_injective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [c : category_theory.faithful F] {X Y : C} :

function.injective F.map

source

theorem category_theory.functor.map_injective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.faithful F] {X Y : C} :

function.injective F.map

source

def category_theory.functor.preimage {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.full F] {X Y : C} (f : F.obj X ⟶ F.obj Y) :

X ⟶ Y

The specified preimage of a morphism under a full functor.

Equations

F.preimage f = category_theory.full.preimage f

source

@[simp]

theorem category_theory.functor.image_preimage {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.full F] {X Y : C} (f : F.obj X ⟶ F.obj Y) :

F.map (F.preimage f) = f

source

@[simp]

theorem category_theory.preimage_id {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X : C} :

F.preimage (𝟙 (F.obj X)) = 𝟙 X

source

@[simp]

theorem category_theory.preimage_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X Y Z : C} (f : F.obj X ⟶ F.obj Y) (g : F.obj Y ⟶ F.obj Z) :

F.preimage (f ≫ g) = F.preimage f ≫ F.preimage g

source

@[simp]

theorem category_theory.preimage_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : X ⟶ Y) :

F.preimage (F.map f) = f

source

def category_theory.preimage_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : F.obj X ≅ F.obj Y) :

X ≅ Y

If F : C ⥤ D is fully faithful, every isomorphism F.obj X ≅ F.obj Y has a preimage.

Equations

category_theory.preimage_iso f = {hom := F.preimage f.hom, inv := F.preimage f.inv, hom_inv_id' := _, inv_hom_id' := _}

source

@[simp]

theorem category_theory.preimage_iso_hom {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : F.obj X ≅ F.obj Y) :

(category_theory.preimage_iso f).hom = F.preimage f.hom

source

@[simp]

theorem category_theory.preimage_iso_inv {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : F.obj X ≅ F.obj Y) :

(category_theory.preimage_iso f).inv = F.preimage f.inv

source

@[simp]

theorem category_theory.preimage_iso_map_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : X ≅ Y) :

category_theory.preimage_iso (F.map_iso f) = f

source

theorem category_theory.is_iso_of_fully_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : X ⟶ Y) [category_theory.is_iso (F.map f)] :

category_theory.is_iso f

If the image of a morphism under a fully faithful functor in an isomorphism, then the original morphisms is also an isomorphism.

source

def category_theory.equiv_of_fully_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.full F] [category_theory.faithful F] {X Y : C} :

(X ⟶ Y) ≃ (F.obj X ⟶ F.obj Y)

If F is fully faithful, we have an equivalence of hom-sets X ⟶ Y and F X ⟶ F Y.

Equations

category_theory.equiv_of_fully_faithful F = {to_fun := λ (f : X ⟶ Y), F.map f, inv_fun := λ (f : F.obj X ⟶ F.obj Y), F.preimage f, left_inv := _, right_inv := _}

source

@[simp]

theorem category_theory.equiv_of_fully_faithful_apply {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : X ⟶ Y) :

⇑(category_theory.equiv_of_fully_faithful F) f = F.map f

source

@[simp]

theorem category_theory.equiv_of_fully_faithful_symm_apply {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [category_theory.full F] [category_theory.faithful F] {X Y : C} (f : F.obj X ⟶ F.obj Y) :

⇑((category_theory.equiv_of_fully_faithful F).symm) f = F.preimage f

source

@[protected, instance]

def category_theory.full.id {C : Type u₁} [category_theory.category C] :

category_theory.full (𝟭 C)

Equations

category_theory.full.id = {preimage := λ (_x _x_1 : C) (f : (𝟭 C).obj _x ⟶ (𝟭 C).obj _x_1), f, witness' := _}

source

@[protected, instance]

def category_theory.faithful.id {C : Type u₁} [category_theory.category C] :

category_theory.faithful (𝟭 C)

source

@[protected, instance]

def category_theory.faithful.comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) [category_theory.faithful F] [category_theory.faithful G] :

category_theory.faithful (F ⋙ G)

source

theorem category_theory.faithful.of_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) [category_theory.faithful (F ⋙ G)] :

category_theory.faithful F

source

theorem category_theory.faithful.of_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F F' : C ⥤ D} [category_theory.faithful F] (α : F ≅ F') :

category_theory.faithful F'

source

theorem category_theory.faithful.of_comp_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F : C ⥤ D} {G : D ⥤ E} {H : C ⥤ E} [ℋ : category_theory.faithful H] (h : F ⋙ G ≅ H) :

category_theory.faithful F

source

theorem category_theory.iso.faithful_of_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F : C ⥤ D} {G : D ⥤ E} {H : C ⥤ E} [ℋ : category_theory.faithful H] (h : F ⋙ G ≅ H) :

category_theory.faithful F

Alias of faithful.of_comp_iso.

source

theorem category_theory.faithful.of_comp_eq {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F : C ⥤ D} {G : D ⥤ E} {H : C ⥤ E} [ℋ : category_theory.faithful H] (h : F ⋙ G = H) :

category_theory.faithful F

source

theorem eq.faithful_of_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F : C ⥤ D} {G : D ⥤ E} {H : C ⥤ E} [ℋ : category_theory.faithful H] (h : F ⋙ G = H) :

category_theory.faithful F

Alias of faithful.of_comp_eq.

source

@[protected]

def category_theory.faithful.div {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ E) (G : D ⥤ E) [category_theory.faithful G] (obj : C → D) (h_obj : ∀ (X : C), G.obj (obj X) = F.obj X) (map : Π {X Y : C}, (X ⟶ Y) → (obj X ⟶ obj Y)) (h_map : ∀ {X Y : C} {f : X ⟶ Y}, G.map (map f) == F.map f) :

C ⥤ D

“Divide” a functor by a faithful functor.

Equations

category_theory.faithful.div F G obj h_obj map h_map = {obj := obj, map := map, map_id' := _, map_comp' := _}

source

theorem category_theory.faithful.div_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ E) [category_theory.faithful F] (G : D ⥤ E) [category_theory.faithful G] (obj : C → D) (h_obj : ∀ (X : C), G.obj (obj X) = F.obj X) (map : Π {X Y : C}, (X ⟶ Y) → (obj X ⟶ obj Y)) (h_map : ∀ {X Y : C} {f : X ⟶ Y}, G.map (map f) == F.map f) :

category_theory.faithful.div F G obj h_obj map h_map ⋙ G = F

source

theorem category_theory.faithful.div_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ E) [category_theory.faithful F] (G : D ⥤ E) [category_theory.faithful G] (obj : C → D) (h_obj : ∀ (X : C), G.obj (obj X) = F.obj X) (map : Π {X Y : C}, (X ⟶ Y) → (obj X ⟶ obj Y)) (h_map : ∀ {X Y : C} {f : X ⟶ Y}, G.map (map f) == F.map f) :

category_theory.faithful (category_theory.faithful.div F G obj h_obj map h_map)

source

@[protected, instance]

def category_theory.full.comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] (F : C ⥤ D) (G : D ⥤ E) [category_theory.full F] [category_theory.full G] :

category_theory.full (F ⋙ G)

Equations

category_theory.full.comp F G = {preimage := λ (_x _x_1 : C) (f : (F ⋙ G).obj _x ⟶ (F ⋙ G).obj _x_1), F.preimage (G.preimage f), witness' := _}

source

def category_theory.fully_faithful_cancel_right {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (comp_iso : F ⋙ H ≅ G ⋙ H) :

F ≅ G

Given a natural isomorphism between F ⋙ H and G ⋙ H for a fully faithful functor H, we can 'cancel' it to give a natural iso between F and G.

Equations

category_theory.fully_faithful_cancel_right H comp_iso = category_theory.nat_iso.of_components (λ (X : C), category_theory.preimage_iso (comp_iso.app X)) _

source

@[simp]

theorem category_theory.fully_faithful_cancel_right_hom_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F G : C ⥤ D} {H : D ⥤ E} [category_theory.full H] [category_theory.faithful H] (comp_iso : F ⋙ H ≅ G ⋙ H) (X : C) :

(category_theory.fully_faithful_cancel_right H comp_iso).hom.app X = H.preimage (comp_iso.hom.app X)

source

@[simp]

theorem category_theory.fully_faithful_cancel_right_inv_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u₃} [category_theory.category E] {F G : C ⥤ D} {H : D ⥤ E} [category_theory.full H] [category_theory.faithful H] (comp_iso : F ⋙ H ≅ G ⋙ H) (X : C) :

(category_theory.fully_faithful_cancel_right H comp_iso).inv.app X = H.preimage (comp_iso.inv.app X)