Documentation

Mathlib.CategoryTheory.Functor.Basic

Functors #

Defines a functor between categories, extending a Prefunctor between quivers.

Introduces, in the CategoryTheory scope, notations C ⥤ D for the type of all functors from C to D, 𝟭 for the identity functor and for functor composition.

TODO: Switch to using the arrow.

structure CategoryTheory.Functor (C : Type u₁) [Category.{v₁, u₁} C] (D : Type u₂) [Category.{v₂, u₂} D] extends C ⥤q D :
Type (max v₁ v₂ u₁ u₂)

Functor C D represents a functor between categories C and D.

To apply a functor F to an object use F.obj X, and to a morphism use F.map f.

The axiom map_id expresses preservation of identities, and map_comp expresses functoriality.

Stacks Tag 001B

Notation for a functor between categories.

Equations
theorem CategoryTheory.Functor.map_comp_assoc {C : Type u₁} [Category.{u_1, u₁} C] {D : Type u₂} [Category.{u_2, u₂} D] (F : Functor C D) {X Y Z : C} (f : X Y) (g : Y Z) {W : D} (h : F.obj Z W) :

𝟭 C is the identity functor on a category C.

Equations

Notation for the identity functor on a category.

Equations
@[simp]
@[simp]
theorem CategoryTheory.Functor.id_map {C : Type u₁} [Category.{v₁, u₁} C] {X Y : C} (f : X Y) :
(Functor.id C).map f = f
def CategoryTheory.Functor.comp {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F : Functor C D) (G : Functor D E) :

F ⋙ G is the composition of a functor F and a functor G (F first, then G).

Equations
  • F.comp G = { obj := fun (X : C) => G.obj (F.obj X), map := fun {X Y : C} (f : X Y) => G.map (F.map f), map_id := , map_comp := }
@[simp]
theorem CategoryTheory.Functor.comp_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F : Functor C D) (G : Functor D E) (X : C) :
(F.comp G).obj X = G.obj (F.obj X)

Notation for composition of functors.

Equations
@[simp]
theorem CategoryTheory.Functor.comp_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F : Functor C D) (G : Functor D E) {X Y : C} (f : X Y) :
(F.comp G).map f = G.map (F.map f)
@[simp]
theorem CategoryTheory.Functor.map_dite {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} {P : Prop} [Decidable P] (f : P → (X Y)) (g : ¬P → (X Y)) :
F.map (if h : P then f h else g h) = if h : P then F.map (f h) else F.map (g h)