WithInitial and WithTerminal

Given a category C, this file constructs two objects:

1. WithTerminal C, the category built from C by formally adjoining a terminal object.
2. WithInitial C, the category built from C by formally adjoining an initial object.

The terminal resp. initial object is WithTerminal.star resp. WithInitial.star, and the proofs that these are terminal resp. initial are in WithTerminal.star_terminal and WithInitial.star_initial.

The inclusion from C into WithTerminal C resp. WithInitial C is denoted WithTerminal.incl resp. WithInitial.incl.

The relevant constructions needed for the universal properties of these constructions are:

1. lift, which lifts F : C ⥤ D to a functor from WithTerminal C resp. WithInitial C in the case where an object Z : D is provided satisfying some additional conditions.
2. inclLift shows that the composition of lift with incl is isomorphic to the functor which was lifted.
3. liftUnique provides the uniqueness property of lift.

In addition to this, we provide WithTerminal.map and WithInitial.map providing the functoriality of these constructions with respect to functors on the base categories.

We define corresponding pseudofunctors WithTerminal.pseudofunctor and WithInitial.pseudofunctor from Cat to Cat.

inductive CategoryTheory.WithTerminal (C : Type u) : Type u

Formally adjoin a terminal object to a category.

• of {C : Type u} : C → WithTerminal C
• star {C : Type u} : WithTerminal C

instance CategoryTheory.instInhabitedWithTerminal {a✝ : Type u_1} : Inhabited (WithTerminal a✝)

Equations

• CategoryTheory.instInhabitedWithTerminal = { default := CategoryTheory.WithTerminal.star }

inductive CategoryTheory.WithInitial (C : Type u) : Type u

Formally adjoin an initial object to a category.

• of {C : Type u} : C → WithInitial C
• star {C : Type u} : WithInitial C

instance CategoryTheory.instInhabitedWithInitial {a✝ : Type u_1} : Inhabited (WithInitial a✝)

Equations

• CategoryTheory.instInhabitedWithInitial = { default := CategoryTheory.WithInitial.star }

def CategoryTheory.WithTerminal.Hom {C : Type u} [Category.{v, u} C] : WithTerminal C → WithTerminal C → Type v

Morphisms for WithTerminal C.

Equations

• (CategoryTheory.WithTerminal.of X).Hom (CategoryTheory.WithTerminal.of Y) = (X ⟶ Y)
• CategoryTheory.WithTerminal.star.Hom (CategoryTheory.WithTerminal.of a) = PEmpty.{?u.28 + 1}
• x✝.Hom CategoryTheory.WithTerminal.star = PUnit.{?u.40 + 1}

def CategoryTheory.WithTerminal.id {C : Type u} [Category.{v, u} C] (X : WithTerminal C) : X.Hom X

Identity morphisms for WithTerminal C.

Equations

• (CategoryTheory.WithTerminal.of a).id = CategoryTheory.CategoryStruct.id a
• CategoryTheory.WithTerminal.star.id = PUnit.unit

def CategoryTheory.WithTerminal.comp {C : Type u} [Category.{v, u} C] {X Y Z : WithTerminal C} : X.Hom Y → Y.Hom Z → X.Hom Z

Composition of morphisms for WithTerminal C.

Equations

• One or more equations did not get rendered due to their size.
• CategoryTheory.WithTerminal.comp = fun (_f : (CategoryTheory.WithTerminal.of _X).Hom x✝) (_g : x✝.Hom CategoryTheory.WithTerminal.star) => PUnit.unit
• CategoryTheory.WithTerminal.comp = fun (f : CategoryTheory.WithTerminal.star.Hom (CategoryTheory.WithTerminal.of _X)) (_g : (CategoryTheory.WithTerminal.of _X).Hom x✝) => PEmpty.elim f
• CategoryTheory.WithTerminal.comp = fun (_f : x✝.Hom CategoryTheory.WithTerminal.star) (g : CategoryTheory.WithTerminal.star.Hom (CategoryTheory.WithTerminal.of _Y)) => PEmpty.elim g
• CategoryTheory.WithTerminal.comp = fun (x x : CategoryTheory.WithTerminal.star.Hom CategoryTheory.WithTerminal.star) => PUnit.unit

instance CategoryTheory.WithTerminal.instCategory {C : Type u} [Category.{v, u} C] : Category.{v, u} (WithTerminal C)

Equations

• CategoryTheory.WithTerminal.instCategory = CategoryTheory.Category.mk ⋯ ⋯ ⋯

def CategoryTheory.WithTerminal.down {C : Type u} [Category.{v, u} C] {X Y : C} (f : of X ⟶ of Y) : X ⟶ Y

Helper function for typechecking.

Equations

• CategoryTheory.WithTerminal.down f = f

@[simp]

theorem CategoryTheory.WithTerminal.down_id {C : Type u} [Category.{v, u} C] {X : C} : down (CategoryStruct.id (of X)) = CategoryStruct.id X

@[simp]

theorem CategoryTheory.WithTerminal.down_comp {C : Type u} [Category.{v, u} C] {X Y Z : C} (f : of X ⟶ of Y) (g : of Y ⟶ of Z) : down (CategoryStruct.comp f g) = CategoryStruct.comp (down f) (down g)

theorem CategoryTheory.WithTerminal.false_of_from_star {C : Type u} [Category.{v, u} C] {X : C} (f : star ⟶ of X) : False

def CategoryTheory.WithTerminal.incl {C : Type u} [Category.{v, u} C] : Functor C (WithTerminal C)

The inclusion from C into WithTerminal C.

Equations

• CategoryTheory.WithTerminal.incl = { obj := CategoryTheory.WithTerminal.of, map := fun {X Y : C} (f : X ⟶ Y) => f, map_id := ⋯, map_comp := ⋯ }

instance CategoryTheory.WithTerminal.instFullIncl {C : Type u} [Category.{v, u} C] : incl.Full

instance CategoryTheory.WithTerminal.instFaithfulIncl {C : Type u} [Category.{v, u} C] : incl.Faithful

def CategoryTheory.WithTerminal.map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) : Functor (WithTerminal C) (WithTerminal D)

Map WithTerminal with respect to a functor F : C ⥤ D.

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.map_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) {X Y : WithTerminal C} (f : X ⟶ Y) : (map F).map f = match X, Y, f with | of a, of a_1, f => F.map (down f) | of a, star, x => PUnit.unit | star, star, x => PUnit.unit

@[simp]

theorem CategoryTheory.WithTerminal.map_obj {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) (X : WithTerminal C) : (map F).obj X = match X with | of x => of (F.obj x) | star => star

def CategoryTheory.WithTerminal.mapId (C : Type u_1) [Category.{u_2, u_1} C] : map (Functor.id C) ≅ Functor.id (WithTerminal C)

A natural isomorphism between the functor map (𝟭 C) and 𝟭 (WithTerminal C).

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.mapId_inv_app (C : Type u_1) [Category.{u_2, u_1} C] (X : WithTerminal C) : (mapId C).inv.app X = (match X with | of a => Iso.refl (of a) | star => Iso.refl star).inv

@[simp]

theorem CategoryTheory.WithTerminal.mapId_hom_app (C : Type u_1) [Category.{u_2, u_1} C] (X : WithTerminal C) : (mapId C).hom.app X = (match X with | of a => Iso.refl (of a) | star => Iso.refl star).hom

def CategoryTheory.WithTerminal.mapComp {C : Type u} [Category.{v, u} C] {D : Type u_1} {E : Type u_2} [Category.{u_3, u_1} D] [Category.{u_4, u_2} E] (F : Functor C D) (G : Functor D E) : map (F.comp G) ≅ (map F).comp (map G)

A natural isomorphism between the functor map (F ⋙ G) and map F ⋙ map G .

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.mapComp_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} {E : Type u_2} [Category.{u_3, u_1} D] [Category.{u_4, u_2} E] (F : Functor C D) (G : Functor D E) (X : WithTerminal C) : (mapComp F G).inv.app X = (match X with | of a => Iso.refl (of (G.obj (F.obj a))) | star => Iso.refl star).inv

@[simp]

theorem CategoryTheory.WithTerminal.mapComp_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} {E : Type u_2} [Category.{u_3, u_1} D] [Category.{u_4, u_2} E] (F : Functor C D) (G : Functor D E) (X : WithTerminal C) : (mapComp F G).hom.app X = (match X with | of a => Iso.refl (of (G.obj (F.obj a))) | star => Iso.refl star).hom

def CategoryTheory.WithTerminal.map₂ {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {F G : Functor C D} (η : F ⟶ G) : map F ⟶ map G

From a natural transformation of functors C ⥤ D, the induced natural transformation of functors WithTerminal C ⥤ WithTerminal D.

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.map₂_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {F G : Functor C D} (η : F ⟶ G) (X : WithTerminal C) : (map₂ η).app X = match X with | of x => η.app x | star => CategoryStruct.id star

def CategoryTheory.WithTerminal.prelaxfunctor : PrelaxFunctor Cat Cat

The prelax functor from Cat to Cat defined with WithTerminal.

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.prelaxfunctor_toPrelaxFunctorStruct_map₂ {a✝ b✝ : Cat} {f✝ g✝ : a✝ ⟶ b✝} (η : f✝ ⟶ g✝) : prelaxfunctor.map₂ η = map₂ η

@[simp]

theorem CategoryTheory.WithTerminal.prelaxfunctor_toPrelaxFunctorStruct_toPrefunctor_obj (C : Cat) : prelaxfunctor.obj C = Cat.of (WithTerminal ↑C)

@[simp]

theorem CategoryTheory.WithTerminal.prelaxfunctor_toPrelaxFunctorStruct_toPrefunctor_map {X✝ Y✝ : Cat} (F : Functor ↑X✝ ↑Y✝) : prelaxfunctor.map F = map F

def CategoryTheory.WithTerminal.pseudofunctor : Pseudofunctor Cat Cat

The pseudofunctor from Cat to Cat defined with WithTerminal.

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.pseudofunctor_toPrelaxFunctor : pseudofunctor.toPrelaxFunctor = prelaxfunctor

@[simp]

theorem CategoryTheory.WithTerminal.pseudofunctor_mapComp {a✝ b✝ c✝ : Cat} (F : Functor ↑a✝ ↑b✝) (G : Functor ↑b✝ ↑c✝) : pseudofunctor.mapComp F G = mapComp F G

@[simp]

theorem CategoryTheory.WithTerminal.pseudofunctor_mapId (C : Cat) : pseudofunctor.mapId C = mapId ↑C

instance CategoryTheory.WithTerminal.instUniqueHomStar {C : Type u} [Category.{v, u} C] {X : WithTerminal C} : Unique (X ⟶ star)

Equations

• CategoryTheory.WithTerminal.instUniqueHomStar = { default := PUnit.unit, uniq := ⋯ }
• CategoryTheory.WithTerminal.instUniqueHomStar = { default := PUnit.unit, uniq := ⋯ }

def CategoryTheory.WithTerminal.starTerminal {C : Type u} [Category.{v, u} C] : Limits.IsTerminal star

WithTerminal.star is terminal.

Equations

• CategoryTheory.WithTerminal.starTerminal = CategoryTheory.Limits.IsTerminal.ofUnique CategoryTheory.WithTerminal.star

def CategoryTheory.WithTerminal.lift {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) : Functor (WithTerminal C) D

Lift a functor F : C ⥤ D to WithTerminal C ⥤ D.

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.lift_obj {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) (X : WithTerminal C) : (lift F M hM).obj X = match X with | of x => F.obj x | star => Z

@[simp]

theorem CategoryTheory.WithTerminal.lift_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) {X Y : WithTerminal C} (f : X ⟶ Y) : (lift F M hM).map f = match X, Y, f with | of a, of a_1, f => F.map (down f) | of x, star, x_1 => M x | star, star, x => CategoryStruct.id Z

def CategoryTheory.WithTerminal.inclLift {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) : incl.comp (lift F M hM) ≅ F

The isomorphism between incl ⋙ lift F _ _ with F.

Equations

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

@[simp]

theorem CategoryTheory.WithTerminal.inclLift_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) (x✝ : C) : (inclLift F M hM).inv.app x✝ = CategoryStruct.id (F.obj x✝)

@[simp]

theorem CategoryTheory.WithTerminal.inclLift_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) (x✝ : C) : (inclLift F M hM).hom.app x✝ = CategoryStruct.id (match incl.obj x✝ with | of x => F.obj x | star => Z)

def CategoryTheory.WithTerminal.liftStar {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) : (lift F M hM).obj star ≅ Z

The isomorphism between (lift F _ _).obj WithTerminal.star with Z.

Equations

• CategoryTheory.WithTerminal.liftStar F M hM = CategoryTheory.eqToIso ⋯

@[simp]

theorem CategoryTheory.WithTerminal.liftStar_inv {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) : (liftStar F M hM).inv = CategoryStruct.id Z

@[simp]

theorem CategoryTheory.WithTerminal.liftStar_hom {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) : (liftStar F M hM).hom = CategoryStruct.id Z

theorem CategoryTheory.WithTerminal.lift_map_liftStar {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) (x : C) : CategoryStruct.comp ((lift F M hM).map (starTerminal.from (incl.obj x))) (liftStar F M hM).hom = CategoryStruct.comp ((inclLift F M hM).hom.app x) (M x)

def CategoryTheory.WithTerminal.liftUnique {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → F.obj x ⟶ Z) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (F.map f) (M y) = M x) (G : Functor (WithTerminal C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) (hh : ∀ (x : C), CategoryStruct.comp (G.map (starTerminal.from (incl.obj x))) hG.hom = CategoryStruct.comp (h.hom.app x) (M x)) : G ≅ lift F M hM

The uniqueness of lift.

Equations

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

def CategoryTheory.WithTerminal.liftToTerminal {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) : Functor (WithTerminal C) D

A variant of lift with Z a terminal object.

Equations

• CategoryTheory.WithTerminal.liftToTerminal F hZ = CategoryTheory.WithTerminal.lift F (fun (_x : C) => hZ.from (F.obj _x)) ⋯

@[simp]

theorem CategoryTheory.WithTerminal.liftToTerminal_obj {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) (X : WithTerminal C) : (liftToTerminal F hZ).obj X = match X with | of x => F.obj x | star => Z

@[simp]

theorem CategoryTheory.WithTerminal.liftToTerminal_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) {X Y : WithTerminal C} (f : X ⟶ Y) : (liftToTerminal F hZ).map f = match X, Y, f with | of a, of a_1, f => F.map (down f) | of x, star, x_1 => hZ.from (F.obj x) | star, star, x => CategoryStruct.id Z

def CategoryTheory.WithTerminal.inclLiftToTerminal {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) : incl.comp (liftToTerminal F hZ) ≅ F

A variant of incl_lift with Z a terminal object.

Equations

• CategoryTheory.WithTerminal.inclLiftToTerminal F hZ = CategoryTheory.WithTerminal.inclLift F (fun (_x : C) => hZ.from (F.obj _x)) ⋯

@[simp]

theorem CategoryTheory.WithTerminal.inclLiftToTerminal_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) (x✝ : C) : (inclLiftToTerminal F hZ).inv.app x✝ = CategoryStruct.id (F.obj x✝)

@[simp]

theorem CategoryTheory.WithTerminal.inclLiftToTerminal_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) (x✝ : C) : (inclLiftToTerminal F hZ).hom.app x✝ = CategoryStruct.id (match incl.obj x✝ with | of x => F.obj x | star => Z)

def CategoryTheory.WithTerminal.liftToTerminalUnique {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) (G : Functor (WithTerminal C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) : G ≅ liftToTerminal F hZ

A variant of lift_unique with Z a terminal object.

Equations

• CategoryTheory.WithTerminal.liftToTerminalUnique F hZ G h hG = CategoryTheory.WithTerminal.liftUnique F (fun (_z : C) => hZ.from (F.obj _z)) ⋯ G h hG ⋯

@[simp]

theorem CategoryTheory.WithTerminal.liftToTerminalUnique_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) (G : Functor (WithTerminal C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) (X : WithTerminal C) : (liftToTerminalUnique F hZ G h hG).inv.app X = (match X with | of x => h.app x | star => hG).inv

@[simp]

theorem CategoryTheory.WithTerminal.liftToTerminalUnique_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsTerminal Z) (G : Functor (WithTerminal C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) (X : WithTerminal C) : (liftToTerminalUnique F hZ G h hG).hom.app X = (match X with | of x => h.app x | star => hG).hom

def CategoryTheory.WithTerminal.homFrom {C : Type u} [Category.{v, u} C] (X : C) : incl.obj X ⟶ star

Constructs a morphism to star from of X.

Equations

• CategoryTheory.WithTerminal.homFrom X = CategoryTheory.WithTerminal.starTerminal.from (CategoryTheory.WithTerminal.incl.obj X)

instance CategoryTheory.WithTerminal.isIso_of_from_star {C : Type u} [Category.{v, u} C] {X : WithTerminal C} (f : star ⟶ X) : IsIso f

def CategoryTheory.WithInitial.Hom {C : Type u} [Category.{v, u} C] : WithInitial C → WithInitial C → Type v

Morphisms for WithInitial C.

Equations

• (CategoryTheory.WithInitial.of X).Hom (CategoryTheory.WithInitial.of Y) = (X ⟶ Y)
• (CategoryTheory.WithInitial.of a).Hom x✝ = PEmpty.{?u.28 + 1}
• CategoryTheory.WithInitial.star.Hom x✝ = PUnit.{?u.41 + 1}

def CategoryTheory.WithInitial.id {C : Type u} [Category.{v, u} C] (X : WithInitial C) : X.Hom X

Identity morphisms for WithInitial C.

Equations

• (CategoryTheory.WithInitial.of a).id = CategoryTheory.CategoryStruct.id a
• CategoryTheory.WithInitial.star.id = PUnit.unit

def CategoryTheory.WithInitial.comp {C : Type u} [Category.{v, u} C] {X Y Z : WithInitial C} : X.Hom Y → Y.Hom Z → X.Hom Z

Composition of morphisms for WithInitial C.

Equations

• One or more equations did not get rendered due to their size.
• CategoryTheory.WithInitial.comp = fun (_f : CategoryTheory.WithInitial.star.Hom x✝) (_g : x✝.Hom (CategoryTheory.WithInitial.of _X)) => PUnit.unit
• CategoryTheory.WithInitial.comp = fun (_f : x✝.Hom (CategoryTheory.WithInitial.of _X)) (g : (CategoryTheory.WithInitial.of _X).Hom CategoryTheory.WithInitial.star) => PEmpty.elim g
• CategoryTheory.WithInitial.comp = fun (f : (CategoryTheory.WithInitial.of _Y).Hom CategoryTheory.WithInitial.star) (_g : CategoryTheory.WithInitial.star.Hom x✝) => PEmpty.elim f
• CategoryTheory.WithInitial.comp = fun (x x : CategoryTheory.WithInitial.star.Hom CategoryTheory.WithInitial.star) => PUnit.unit

instance CategoryTheory.WithInitial.instCategory {C : Type u} [Category.{v, u} C] : Category.{v, u} (WithInitial C)

Equations

• CategoryTheory.WithInitial.instCategory = CategoryTheory.Category.mk ⋯ ⋯ ⋯

def CategoryTheory.WithInitial.down {C : Type u} [Category.{v, u} C] {X Y : C} (f : of X ⟶ of Y) : X ⟶ Y

Helper function for typechecking.

Equations

• CategoryTheory.WithInitial.down f = f

@[simp]

theorem CategoryTheory.WithInitial.down_id {C : Type u} [Category.{v, u} C] {X : C} : down (CategoryStruct.id (of X)) = CategoryStruct.id X

@[simp]

theorem CategoryTheory.WithInitial.down_comp {C : Type u} [Category.{v, u} C] {X Y Z : C} (f : of X ⟶ of Y) (g : of Y ⟶ of Z) : down (CategoryStruct.comp f g) = CategoryStruct.comp (down f) (down g)

theorem CategoryTheory.WithInitial.false_of_to_star {C : Type u} [Category.{v, u} C] {X : C} (f : of X ⟶ star) : False

def CategoryTheory.WithInitial.incl {C : Type u} [Category.{v, u} C] : Functor C (WithInitial C)

The inclusion of C into WithInitial C.

Equations

• CategoryTheory.WithInitial.incl = { obj := CategoryTheory.WithInitial.of, map := fun {X Y : C} (f : X ⟶ Y) => f, map_id := ⋯, map_comp := ⋯ }

instance CategoryTheory.WithInitial.instFullIncl {C : Type u} [Category.{v, u} C] : incl.Full

instance CategoryTheory.WithInitial.instFaithfulIncl {C : Type u} [Category.{v, u} C] : incl.Faithful

def CategoryTheory.WithInitial.map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) : Functor (WithInitial C) (WithInitial D)

Map WithInitial with respect to a functor F : C ⥤ D.

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.map_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) {X Y : WithInitial C} (f : X ⟶ Y) : (map F).map f = match X, Y, f with | of a, of a_1, f => F.map (down f) | star, of a, x => PUnit.unit | star, star, x => PUnit.unit

@[simp]

theorem CategoryTheory.WithInitial.map_obj {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) (X : WithInitial C) : (map F).obj X = match X with | of x => of (F.obj x) | star => star

def CategoryTheory.WithInitial.mapId (C : Type u_1) [Category.{u_2, u_1} C] : map (Functor.id C) ≅ Functor.id (WithInitial C)

A natural isomorphism between the functor map (𝟭 C) and 𝟭 (WithInitial C).

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.mapId_inv_app (C : Type u_1) [Category.{u_2, u_1} C] (X : WithInitial C) : (mapId C).inv.app X = (match X with | of a => Iso.refl (of a) | star => Iso.refl star).inv

@[simp]

theorem CategoryTheory.WithInitial.mapId_hom_app (C : Type u_1) [Category.{u_2, u_1} C] (X : WithInitial C) : (mapId C).hom.app X = (match X with | of a => Iso.refl (of a) | star => Iso.refl star).hom

def CategoryTheory.WithInitial.mapComp {C : Type u} [Category.{v, u} C] {D : Type u_1} {E : Type u_2} [Category.{u_3, u_1} D] [Category.{u_4, u_2} E] (F : Functor C D) (G : Functor D E) : map (F.comp G) ≅ (map F).comp (map G)

A natural isomorphism between the functor map (F ⋙ G) and map F ⋙ map G .

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.mapComp_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} {E : Type u_2} [Category.{u_3, u_1} D] [Category.{u_4, u_2} E] (F : Functor C D) (G : Functor D E) (X : WithInitial C) : (mapComp F G).inv.app X = (match X with | of a => Iso.refl (of (G.obj (F.obj a))) | star => Iso.refl star).inv

@[simp]

theorem CategoryTheory.WithInitial.mapComp_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} {E : Type u_2} [Category.{u_3, u_1} D] [Category.{u_4, u_2} E] (F : Functor C D) (G : Functor D E) (X : WithInitial C) : (mapComp F G).hom.app X = (match X with | of a => Iso.refl (of (G.obj (F.obj a))) | star => Iso.refl star).hom

def CategoryTheory.WithInitial.map₂ {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {F G : Functor C D} (η : F ⟶ G) : map F ⟶ map G

From a natural transformation of functors C ⥤ D, the induced natural transformation of functors WithInitial C ⥤ WithInitial D.

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.map₂_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {F G : Functor C D} (η : F ⟶ G) (X : WithInitial C) : (map₂ η).app X = match X with | of x => η.app x | star => CategoryStruct.id star

def CategoryTheory.WithInitial.prelaxfunctor : PrelaxFunctor Cat Cat

The prelax functor from Cat to Cat defined with WithInitial.

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.prelaxfunctor_toPrelaxFunctorStruct_map₂ {a✝ b✝ : Cat} {f✝ g✝ : a✝ ⟶ b✝} (η : f✝ ⟶ g✝) : prelaxfunctor.map₂ η = map₂ η

@[simp]

theorem CategoryTheory.WithInitial.prelaxfunctor_toPrelaxFunctorStruct_toPrefunctor_map {X✝ Y✝ : Cat} (F : Functor ↑X✝ ↑Y✝) : prelaxfunctor.map F = map F

@[simp]

theorem CategoryTheory.WithInitial.prelaxfunctor_toPrelaxFunctorStruct_toPrefunctor_obj (C : Cat) : prelaxfunctor.obj C = Cat.of (WithInitial ↑C)

def CategoryTheory.WithInitial.pseudofunctor : Pseudofunctor Cat Cat

The pseudofunctor from Cat to Cat defined with WithInitial.

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.pseudofunctor_mapId (C : Cat) : pseudofunctor.mapId C = mapId ↑C

@[simp]

theorem CategoryTheory.WithInitial.pseudofunctor_mapComp {a✝ b✝ c✝ : Cat} (F : Functor ↑a✝ ↑b✝) (G : Functor ↑b✝ ↑c✝) : pseudofunctor.mapComp F G = mapComp F G

@[simp]

theorem CategoryTheory.WithInitial.pseudofunctor_toPrelaxFunctor : pseudofunctor.toPrelaxFunctor = prelaxfunctor

instance CategoryTheory.WithInitial.instUniqueHomStar {C : Type u} [Category.{v, u} C] {X : WithInitial C} : Unique (star ⟶ X)

Equations

• CategoryTheory.WithInitial.instUniqueHomStar = { default := PUnit.unit, uniq := ⋯ }
• CategoryTheory.WithInitial.instUniqueHomStar = { default := PUnit.unit, uniq := ⋯ }

def CategoryTheory.WithInitial.starInitial {C : Type u} [Category.{v, u} C] : Limits.IsInitial star

WithInitial.star is initial.

Equations

• CategoryTheory.WithInitial.starInitial = CategoryTheory.Limits.IsInitial.ofUnique CategoryTheory.WithInitial.star

def CategoryTheory.WithInitial.lift {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) : Functor (WithInitial C) D

Lift a functor F : C ⥤ D to WithInitial C ⥤ D.

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.lift_obj {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) (X : WithInitial C) : (lift F M hM).obj X = match X with | of x => F.obj x | star => Z

@[simp]

theorem CategoryTheory.WithInitial.lift_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) {X Y : WithInitial C} (f : X ⟶ Y) : (lift F M hM).map f = match X, Y, f with | of a, of a_1, f => F.map (down f) | star, of a, x => M a | star, star, x => CategoryStruct.id ((fun (X : WithInitial C) => match X with | of x => F.obj x | star => Z) star)

def CategoryTheory.WithInitial.inclLift {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) : incl.comp (lift F M hM) ≅ F

The isomorphism between incl ⋙ lift F _ _ with F.

Equations

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

@[simp]

theorem CategoryTheory.WithInitial.inclLift_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) (x✝ : C) : (inclLift F M hM).hom.app x✝ = CategoryStruct.id (match incl.obj x✝ with | of x => F.obj x | star => Z)

@[simp]

theorem CategoryTheory.WithInitial.inclLift_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) (x✝ : C) : (inclLift F M hM).inv.app x✝ = CategoryStruct.id (F.obj x✝)

def CategoryTheory.WithInitial.liftStar {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) : (lift F M hM).obj star ≅ Z

The isomorphism between (lift F _ _).obj WithInitial.star with Z.

Equations

• CategoryTheory.WithInitial.liftStar F M hM = CategoryTheory.eqToIso ⋯

@[simp]

theorem CategoryTheory.WithInitial.liftStar_inv {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) : (liftStar F M hM).inv = CategoryStruct.id Z

@[simp]

theorem CategoryTheory.WithInitial.liftStar_hom {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) : (liftStar F M hM).hom = CategoryStruct.id Z

theorem CategoryTheory.WithInitial.liftStar_lift_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) (x : C) : CategoryStruct.comp (liftStar F M hM).hom ((lift F M hM).map (starInitial.to (incl.obj x))) = CategoryStruct.comp (M x) ((inclLift F M hM).hom.app x)

def CategoryTheory.WithInitial.liftUnique {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (M : (x : C) → Z ⟶ F.obj x) (hM : ∀ (x y : C) (f : x ⟶ y), CategoryStruct.comp (M x) (F.map f) = M y) (G : Functor (WithInitial C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) (hh : ∀ (x : C), CategoryStruct.comp hG.symm.hom (G.map (starInitial.to (incl.obj x))) = CategoryStruct.comp (M x) (h.symm.hom.app x)) : G ≅ lift F M hM

The uniqueness of lift.

Equations

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

def CategoryTheory.WithInitial.liftToInitial {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) : Functor (WithInitial C) D

A variant of lift with Z an initial object.

Equations

• CategoryTheory.WithInitial.liftToInitial F hZ = CategoryTheory.WithInitial.lift F (fun (_x : C) => hZ.to (F.obj _x)) ⋯

@[simp]

theorem CategoryTheory.WithInitial.liftToInitial_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) {X Y : WithInitial C} (f : X ⟶ Y) : (liftToInitial F hZ).map f = match X, Y, f with | of a, of a_1, f => F.map (down f) | star, of a, x => hZ.to (F.obj a) | star, star, x => CategoryStruct.id Z

@[simp]

theorem CategoryTheory.WithInitial.liftToInitial_obj {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) (X : WithInitial C) : (liftToInitial F hZ).obj X = match X with | of x => F.obj x | star => Z

def CategoryTheory.WithInitial.inclLiftToInitial {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) : incl.comp (liftToInitial F hZ) ≅ F

A variant of incl_lift with Z an initial object.

Equations

• CategoryTheory.WithInitial.inclLiftToInitial F hZ = CategoryTheory.WithInitial.inclLift F (fun (_x : C) => hZ.to (F.obj _x)) ⋯

@[simp]

theorem CategoryTheory.WithInitial.inclLiftToInitial_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) (x✝ : C) : (inclLiftToInitial F hZ).inv.app x✝ = CategoryStruct.id (F.obj x✝)

@[simp]

theorem CategoryTheory.WithInitial.inclLiftToInitial_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) (x✝ : C) : (inclLiftToInitial F hZ).hom.app x✝ = CategoryStruct.id (match incl.obj x✝ with | of x => F.obj x | star => Z)

def CategoryTheory.WithInitial.liftToInitialUnique {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) (G : Functor (WithInitial C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) : G ≅ liftToInitial F hZ

A variant of lift_unique with Z an initial object.

Equations

• CategoryTheory.WithInitial.liftToInitialUnique F hZ G h hG = CategoryTheory.WithInitial.liftUnique F (fun (_z : C) => hZ.to (F.obj _z)) ⋯ G h hG ⋯

@[simp]

theorem CategoryTheory.WithInitial.liftToInitialUnique_inv_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) (G : Functor (WithInitial C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) (X : WithInitial C) : (liftToInitialUnique F hZ G h hG).inv.app X = (match X with | of x => h.app x | star => hG).inv

@[simp]

theorem CategoryTheory.WithInitial.liftToInitialUnique_hom_app {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {Z : D} (F : Functor C D) (hZ : Limits.IsInitial Z) (G : Functor (WithInitial C) D) (h : incl.comp G ≅ F) (hG : G.obj star ≅ Z) (X : WithInitial C) : (liftToInitialUnique F hZ G h hG).hom.app X = (match X with | of x => h.app x | star => hG).hom

def CategoryTheory.WithInitial.homTo {C : Type u} [Category.{v, u} C] (X : C) : star ⟶ incl.obj X

Constructs a morphism from star to of X.

Equations

• CategoryTheory.WithInitial.homTo X = CategoryTheory.WithInitial.starInitial.to (CategoryTheory.WithInitial.incl.obj X)

instance CategoryTheory.WithInitial.isIso_of_to_star {C : Type u} [Category.{v, u} C] {X : WithInitial C} (f : X ⟶ star) : IsIso f