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category_theory.functor.fully_faithful

Full and faithful functors #

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We define typeclasses full and faithful, decorating functors.

Main definitions and results #

See category_theory.equivalence.of_fully_faithful_ess_surj for the fact that a functor is an equivalence if and only if it is fully faithful and essentially surjective.

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@[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)

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 of this typeclass
Instances of other typeclasses for category_theory.full
  • category_theory.full.has_sizeof_inst
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@[simp]
theorem category_theory.full.witness {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F : C ⥤ D} [self : category_theory.full F] {X Y : C} (f : F.obj X ⟶ F.obj Y) :
F.map (category_theory.full.preimage f) = f
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@[class]
structure category_theory.faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) :
Prop

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 of this typeclass
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theorem category_theory.faithful.map_injective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) [self : category_theory.faithful F] {X Y : C} :
function.injective F.map
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theorem category_theory.functor.map_injective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {X Y : C} (F : C ⥤ D) [category_theory.faithful F] :
function.injective F.map
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theorem category_theory.functor.map_iso_injective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {X Y : C} (F : C ⥤ D) [category_theory.faithful F] :
function.injective F.map_iso
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def category_theory.functor.preimage {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {X Y : C} (F : C ⥤ D) [category_theory.full F] (f : F.obj X ⟶ F.obj Y) :
X ⟶ Y

The specified preimage of a morphism under a full functor.

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@[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
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theorem category_theory.functor.map_surjective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {X Y : C} (F : C ⥤ D) [category_theory.full F] :
function.surjective F.map
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noncomputable def category_theory.functor.full_of_exists {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) (h : ∀ (X Y : C) (f : F.obj X ⟶ F.obj Y), ∃ (p : X ⟶ Y), F.map p = f) :
category_theory.full F

Deduce that F is full from the existence of preimages, using choice.

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noncomputable def category_theory.functor.full_of_surjective {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C ⥤ D) (h : ∀ (X Y : C), function.surjective F.map) :
category_theory.full F

Deduce that F is full from surjectivity of F.map, using choice.

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@[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
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@[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
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@[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
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@[simp]
theorem category_theory.functor.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) :
(F.preimage_iso f).hom = F.preimage f.hom
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def category_theory.functor.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.

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@[simp]
theorem category_theory.functor.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) :
(F.preimage_iso f).inv = F.preimage f.inv
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@[simp]
theorem category_theory.functor.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) :
F.preimage_iso (F.map_iso f) = f
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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.

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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.

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@[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
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@[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
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def category_theory.iso_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 iso-sets X ≅ Y and F X ≅ F Y.

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@[simp]
theorem category_theory.iso_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.iso_equiv_of_fully_faithful F).symm) f = F.preimage_iso f
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@[simp]
theorem category_theory.iso_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.iso_equiv_of_fully_faithful F) f = F.map_iso f
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@[simp]
theorem category_theory.nat_trans_of_comp_fully_faithful_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (α : F ⋙ H ⟶ G ⋙ H) (X : C) :
(category_theory.nat_trans_of_comp_fully_faithful H α).app X = ⇑((category_theory.equiv_of_fully_faithful H).symm) (α.app X)
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def category_theory.nat_trans_of_comp_fully_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (α : F ⋙ H ⟶ G ⋙ H) :
F ⟶ G

We can construct a natural transformation between functors by constructing a natural transformation between those functors composed with a fully faithful functor.

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def category_theory.nat_iso_of_comp_fully_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (i : F ⋙ H ≅ G ⋙ H) :
F ≅ G

We can construct a natural isomorphism between functors by constructing a natural isomorphism between those functors composed with a fully faithful functor.

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@[simp]
theorem category_theory.nat_iso_of_comp_fully_faithful_hom_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (i : F ⋙ H ≅ G ⋙ H) (X : C) :
(category_theory.nat_iso_of_comp_fully_faithful H i).hom.app X = H.preimage (i.hom.app X)
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@[simp]
theorem category_theory.nat_iso_of_comp_fully_faithful_inv_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (i : F ⋙ H ≅ G ⋙ H) (X : C) :
(category_theory.nat_iso_of_comp_fully_faithful H i).inv.app X = H.preimage (i.inv.app X)
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theorem category_theory.nat_iso_of_comp_fully_faithful_hom {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (i : F ⋙ H ≅ G ⋙ H) :
(category_theory.nat_iso_of_comp_fully_faithful H i).hom = category_theory.nat_trans_of_comp_fully_faithful H i.hom
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theorem category_theory.nat_iso_of_comp_fully_faithful_inv {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (i : F ⋙ H ≅ G ⋙ H) :
(category_theory.nat_iso_of_comp_fully_faithful H i).inv = category_theory.nat_trans_of_comp_fully_faithful H i.inv
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def category_theory.nat_trans.equiv_of_comp_fully_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] :
(F ⟶ G) ≃ (F ⋙ H ⟶ G ⋙ H)

Horizontal composition with a fully faithful functor induces a bijection on natural transformations.

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@[simp]
theorem category_theory.nat_trans.equiv_of_comp_fully_faithful_apply {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (α : F ⟶ G) :
⇑(category_theory.nat_trans.equiv_of_comp_fully_faithful H) α = α ◫ 𝟙 H
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@[simp]
theorem category_theory.nat_trans.equiv_of_comp_fully_faithful_symm_apply {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (α : F ⋙ H ⟶ G ⋙ H) :
⇑((category_theory.nat_trans.equiv_of_comp_fully_faithful H).symm) α = category_theory.nat_trans_of_comp_fully_faithful H α
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def category_theory.nat_iso.equiv_of_comp_fully_faithful {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] :
(F ≅ G) ≃ (F ⋙ H ≅ G ⋙ H)

Horizontal composition with a fully faithful functor induces a bijection on natural isomorphisms.

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@[simp]
theorem category_theory.nat_iso.equiv_of_comp_fully_faithful_symm_apply {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (i : F ⋙ H ≅ G ⋙ H) :
⇑((category_theory.nat_iso.equiv_of_comp_fully_faithful H).symm) i = category_theory.nat_iso_of_comp_fully_faithful H i
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@[simp]
theorem category_theory.nat_iso.equiv_of_comp_fully_faithful_apply {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {E : Type u_1} [category_theory.category E] {F G : C ⥤ D} (H : D ⥤ E) [category_theory.full H] [category_theory.faithful H] (e : F ≅ G) :
⇑(category_theory.nat_iso.equiv_of_comp_fully_faithful H) e = category_theory.nat_iso.hcomp e (category_theory.iso.refl H)
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@[protected, instance]
def category_theory.full.id {C : Type u₁} [category_theory.category C] :
category_theory.full (𝟭 C)
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@[protected, instance]
def category_theory.faithful.id {C : Type u₁} [category_theory.category C] :
category_theory.faithful (𝟭 C)
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@[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)
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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
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def category_theory.full.of_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F F' : C ⥤ D} [category_theory.full F] (α : F ≅ F') :
category_theory.full F'

If F is full, and naturally isomorphic to some F', then F' is also full.

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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'
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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
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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 category_theory.faithful.of_comp_iso.

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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
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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 category_theory.faithful.of_comp_eq.

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@[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.

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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
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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)
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@[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)
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def category_theory.full.of_comp_faithful {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 ⋙ G)] [category_theory.faithful G] :
category_theory.full F

If F ⋙ G is full and G is faithful, then F is full.

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def category_theory.full.of_comp_faithful_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} [category_theory.full H] [category_theory.faithful G] (h : F ⋙ G ≅ H) :
category_theory.full F

If F ⋙ G is full and G is faithful, then F is full.

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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.

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@[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)
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@[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)