Binary disjoint unions of categories

We define the category instance on C ⊕ D when C and D are categories.

We define:

• inl_ : the functor C ⥤ C ⊕ D
• inr_ : the functor D ⥤ C ⊕ D
• swap : the functor C ⊕ D ⥤ D ⊕ C (and the fact this is an equivalence)

We further define sums of functors and natural transformations, written F.sum G and α.sum β.

instance CategoryTheory.sum (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : CategoryTheory.Category.{v₁, u₁} (C ⊕ D)

sum C D gives the direct sum of two categories.

Equations

• CategoryTheory.sum C D = CategoryTheory.Category.mk ⋯ ⋯ ⋯

theorem CategoryTheory.hom_inl_inr_false (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] {X : C} {Y : D} (f : Sum.inl X ⟶ Sum.inr Y) : False

theorem CategoryTheory.hom_inr_inl_false (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] {X : C} {Y : D} (f : Sum.inr X ⟶ Sum.inl Y) : False

@[simp]

theorem CategoryTheory.sum_comp_inl (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] {P : C} {Q : C} {R : C} (f : Sum.inl P ⟶ Sum.inl Q) (g : Sum.inl Q ⟶ Sum.inl R) : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f g

@[simp]

theorem CategoryTheory.sum_comp_inr (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] {P : D} {Q : D} {R : D} (f : Sum.inr P ⟶ Sum.inr Q) (g : Sum.inr Q ⟶ Sum.inr R) : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp f g

@[simp]

theorem CategoryTheory.Sum.inl__map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] (X : C) (Y : C) (f : X ⟶ Y) : (CategoryTheory.Sum.inl_ C D).map f = f

@[simp]

theorem CategoryTheory.Sum.inl__obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] (X : C) : (CategoryTheory.Sum.inl_ C D).obj X = Sum.inl X

def CategoryTheory.Sum.inl_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : CategoryTheory.Functor C (C ⊕ D)

inl_ is the functor X ↦ inl X.

Equations

• CategoryTheory.Sum.inl_ C D = { toPrefunctor := { obj := fun (X : C) => Sum.inl X, map := fun (X Y : C) (f : X ⟶ Y) => f }, map_id := ⋯, map_comp := ⋯ }

@[simp]

theorem CategoryTheory.Sum.inr__map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] (X : D) (Y : D) (f : X ⟶ Y) : (CategoryTheory.Sum.inr_ C D).map f = f

@[simp]

theorem CategoryTheory.Sum.inr__obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] (X : D) : (CategoryTheory.Sum.inr_ C D).obj X = Sum.inr X

def CategoryTheory.Sum.inr_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : CategoryTheory.Functor D (C ⊕ D)

inr_ is the functor X ↦ inr X.

Equations

• CategoryTheory.Sum.inr_ C D = { toPrefunctor := { obj := fun (X : D) => Sum.inr X, map := fun (X Y : D) (f : X ⟶ Y) => f }, map_id := ⋯, map_comp := ⋯ }

def CategoryTheory.Sum.swap (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : CategoryTheory.Functor (C ⊕ D) (D ⊕ C)

The functor exchanging two direct summand categories.

Equations

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

@[simp]

theorem CategoryTheory.Sum.swap_obj_inl (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] (X : C) : (CategoryTheory.Sum.swap C D).obj (Sum.inl X) = Sum.inr X

@[simp]

theorem CategoryTheory.Sum.swap_obj_inr (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] (X : D) : (CategoryTheory.Sum.swap C D).obj (Sum.inr X) = Sum.inl X

@[simp]

theorem CategoryTheory.Sum.swap_map_inl (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] {X : C} {Y : C} {f : Sum.inl X ⟶ Sum.inl Y} : (CategoryTheory.Sum.swap C D).map f = f

@[simp]

theorem CategoryTheory.Sum.swap_map_inr (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] {X : D} {Y : D} {f : Sum.inr X ⟶ Sum.inr Y} : (CategoryTheory.Sum.swap C D).map f = f

def CategoryTheory.Sum.Swap.equivalence (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : C ⊕ D ≌ D ⊕ C

swap gives an equivalence between C ⊕ D and D ⊕ C.

Equations

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

instance CategoryTheory.Sum.Swap.isEquivalence (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : CategoryTheory.IsEquivalence (CategoryTheory.Sum.swap C D)

Equations

• CategoryTheory.Sum.Swap.isEquivalence C D = inferInstance

def CategoryTheory.Sum.Swap.symmetry (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₁) [CategoryTheory.Category.{v₁, u₁} D] : CategoryTheory.Functor.comp (CategoryTheory.Sum.swap C D) (CategoryTheory.Sum.swap D C) ≅ CategoryTheory.Functor.id (C ⊕ D)

The double swap on C ⊕ D is naturally isomorphic to the identity functor.

Equations

• CategoryTheory.Sum.Swap.symmetry C D = (CategoryTheory.Sum.Swap.equivalence C D).unitIso.symm

def CategoryTheory.Functor.sum {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C D) : CategoryTheory.Functor (A ⊕ C) (B ⊕ D)

The sum of two functors.

Equations

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

@[simp]

theorem CategoryTheory.Functor.sum_obj_inl {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C D) (a : A) : (CategoryTheory.Functor.sum F G).obj (Sum.inl a) = Sum.inl (F.obj a)

@[simp]

theorem CategoryTheory.Functor.sum_obj_inr {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C D) (c : C) : (CategoryTheory.Functor.sum F G).obj (Sum.inr c) = Sum.inr (G.obj c)

@[simp]

theorem CategoryTheory.Functor.sum_map_inl {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C D) {a : A} {a' : A} (f : Sum.inl a ⟶ Sum.inl a') : (CategoryTheory.Functor.sum F G).map f = F.map f

@[simp]

theorem CategoryTheory.Functor.sum_map_inr {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] (F : CategoryTheory.Functor A B) (G : CategoryTheory.Functor C D) {c : C} {c' : C} (f : Sum.inr c ⟶ Sum.inr c') : (CategoryTheory.Functor.sum F G).map f = G.map f

def CategoryTheory.NatTrans.sum {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor A B} {H : CategoryTheory.Functor C D} {I : CategoryTheory.Functor C D} (α : F ⟶ G) (β : H ⟶ I) : CategoryTheory.Functor.sum F H ⟶ CategoryTheory.Functor.sum G I

The sum of two natural transformations.

Equations

• CategoryTheory.NatTrans.sum α β = { app := fun (X : A ⊕ C) => match X with | Sum.inl X => α.app X | Sum.inr X => β.app X, naturality := ⋯ }

@[simp]

theorem CategoryTheory.NatTrans.sum_app_inl {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor A B} {H : CategoryTheory.Functor C D} {I : CategoryTheory.Functor C D} (α : F ⟶ G) (β : H ⟶ I) (a : A) : (CategoryTheory.NatTrans.sum α β).app (Sum.inl a) = α.app a

@[simp]

theorem CategoryTheory.NatTrans.sum_app_inr {A : Type u₁} [CategoryTheory.Category.{v₁, u₁} A] {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₁} [CategoryTheory.Category.{v₁, u₁} D] {F : CategoryTheory.Functor A B} {G : CategoryTheory.Functor A B} {H : CategoryTheory.Functor C D} {I : CategoryTheory.Functor C D} (α : F ⟶ G) (β : H ⟶ I) (c : C) : (CategoryTheory.NatTrans.sum α β).app (Sum.inr c) = β.app c