Bicones

Given a category J, a walking Bicone J is a category whose objects are the objects of J and two extra vertices Bicone.left and Bicone.right. The morphisms are the morphisms of J and left ⟶ j, right ⟶ j for each j : J such that (· ⟶ j) and (· ⟶ k) commutes with each f : j ⟶ k.

Given a diagram F : J ⥤ C and two Cone Fs, we can join them into a diagram Bicone J ⥤ C via biconeMk.

This is used in CategoryTheory.Functor.Flat.

inductive CategoryTheory.Bicone (J : Type u₁) : Type u₁

Given a category J, construct a walking Bicone J by adjoining two elements.

• left {J : Type u₁} : Bicone J
• right {J : Type u₁} : Bicone J
• diagram {J : Type u₁} (val : J) : Bicone J

instance CategoryTheory.instDecidableEqBicone {J✝ : Type u_1} [DecidableEq J✝] : DecidableEq (Bicone J✝)

Equations

• CategoryTheory.instDecidableEqBicone = CategoryTheory.decEqBicone✝

instance CategoryTheory.instInhabitedBicone (J : Type u₁) : Inhabited (Bicone J)

Equations

• CategoryTheory.instInhabitedBicone J = { default := CategoryTheory.Bicone.left }

instance CategoryTheory.finBicone (J : Type u₁) [Fintype J] : Fintype (Bicone J)

Equations

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

inductive CategoryTheory.BiconeHom (J : Type u₁) [Category.{v₁, u₁} J] : Bicone J → Bicone J → Type (max u₁ v₁)

The homs for a walking Bicone J.

• left_id {J : Type u₁} [Category.{v₁, u₁} J] : BiconeHom J Bicone.left Bicone.left
• right_id {J : Type u₁} [Category.{v₁, u₁} J] : BiconeHom J Bicone.right Bicone.right
• left {J : Type u₁} [Category.{v₁, u₁} J] (j : J) : BiconeHom J Bicone.left (Bicone.diagram j)
• right {J : Type u₁} [Category.{v₁, u₁} J] (j : J) : BiconeHom J Bicone.right (Bicone.diagram j)
• diagram {J : Type u₁} [Category.{v₁, u₁} J] {j k : J} (f : j ⟶ k) : BiconeHom J (Bicone.diagram j) (Bicone.diagram k)

instance CategoryTheory.instInhabitedBiconeHomLeft (J : Type u₁) [Category.{v₁, u₁} J] : Inhabited (BiconeHom J Bicone.left Bicone.left)

Equations

• CategoryTheory.instInhabitedBiconeHomLeft J = { default := CategoryTheory.BiconeHom.left_id }

instance CategoryTheory.BiconeHom.decidableEq (J : Type u₁) [Category.{v₁, u₁} J] {j k : Bicone J} : DecidableEq (BiconeHom J j k)

Equations

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

instance CategoryTheory.biconeCategoryStruct (J : Type u₁) [Category.{v₁, u₁} J] : CategoryStruct.{max u₁ v₁, u₁} (Bicone J)

Equations

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

@[simp]

theorem CategoryTheory.biconeCategoryStruct_id (J : Type u₁) [Category.{v₁, u₁} J] (j : Bicone J) : CategoryStruct.id j = Bicone.casesOn j BiconeHom.left_id BiconeHom.right_id fun (k : J) => BiconeHom.diagram (CategoryStruct.id k)

@[simp]

theorem CategoryTheory.biconeCategoryStruct_comp (J : Type u₁) [Category.{v₁, u₁} J] {X✝ Y✝ Z✝ : Bicone J} (f : X✝ ⟶ Y✝) (g : Y✝ ⟶ Z✝) : CategoryStruct.comp f g = BiconeHom.casesOn (motive := fun (a a_1 : Bicone J) (x : BiconeHom J a a_1) => X✝ = a → Y✝ = a_1 → HEq f x → (X✝ ⟶ Z✝)) f (fun (h : X✝ = Bicone.left) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.left → HEq f BiconeHom.left_id → (X ⟶ Z✝)) (fun (f : Bicone.left ⟶ Y✝) (h : Y✝ = Bicone.left) => Eq.ndrec (motive := fun {Y : Bicone J} => (Y ⟶ Z✝) → (f : Bicone.left ⟶ Y) → HEq f BiconeHom.left_id → (Bicone.left ⟶ Z✝)) (fun (g : Bicone.left ⟶ Z✝) (f : Bicone.left ⟶ Bicone.left) (h : HEq f BiconeHom.left_id) => g) ⋯ g f) ⋯ f) (fun (h : X✝ = Bicone.right) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.right → HEq f BiconeHom.right_id → (X ⟶ Z✝)) (fun (f : Bicone.right ⟶ Y✝) (h : Y✝ = Bicone.right) => Eq.ndrec (motive := fun {Y : Bicone J} => (Y ⟶ Z✝) → (f : Bicone.right ⟶ Y) → HEq f BiconeHom.right_id → (Bicone.right ⟶ Z✝)) (fun (g : Bicone.right ⟶ Z✝) (f : Bicone.right ⟶ Bicone.right) (h : HEq f BiconeHom.right_id) => g) ⋯ g f) ⋯ f) (fun (j : J) (h : X✝ = Bicone.left) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.diagram j → HEq f (BiconeHom.left j) → (X ⟶ Z✝)) (fun (f : Bicone.left ⟶ Y✝) (h : Y✝ = Bicone.diagram j) => Eq.ndrec (motive := fun {Y : Bicone J} => (Y ⟶ Z✝) → (f : Bicone.left ⟶ Y) → HEq f (BiconeHom.left j) → (Bicone.left ⟶ Z✝)) (fun (g : Bicone.diagram j ⟶ Z✝) (f : Bicone.left ⟶ Bicone.diagram j) (h : HEq f (BiconeHom.left j)) => BiconeHom.casesOn (motive := fun (a a_1 : Bicone J) (t : BiconeHom J a a_1) => Bicone.diagram j = a → Z✝ = a_1 → HEq g t → (Bicone.left ⟶ Z✝)) g (fun (h : Bicone.diagram j = Bicone.left) => Bicone.noConfusion h) (fun (h : Bicone.diagram j = Bicone.right) => Bicone.noConfusion h) (fun (j_1 : J) (h : Bicone.diagram j = Bicone.left) => Bicone.noConfusion h) (fun (j_1 : J) (h : Bicone.diagram j = Bicone.right) => Bicone.noConfusion h) (fun {j_1 k : J} (f : j_1 ⟶ k) (h : Bicone.diagram j = Bicone.diagram j_1) => Bicone.noConfusion h fun (val_eq : j = j_1) => Eq.ndrec (motive := fun {j_2 : J} => (f : j_2 ⟶ k) → Z✝ = Bicone.diagram k → HEq g (BiconeHom.diagram f) → (Bicone.left ⟶ Z✝)) (fun (f : j ⟶ k) (h : Z✝ = Bicone.diagram k) => Eq.ndrec (motive := fun {Z : Bicone J} => (g : Bicone.diagram j ⟶ Z) → HEq g (BiconeHom.diagram f) → (Bicone.left ⟶ Z)) (fun (g : Bicone.diagram j ⟶ Bicone.diagram k) (h : HEq g (BiconeHom.diagram f)) => BiconeHom.left k) ⋯ g) val_eq f) ⋯ ⋯ ⋯) ⋯ g f) ⋯ f) (fun (j : J) (h : X✝ = Bicone.right) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.diagram j → HEq f (BiconeHom.right j) → (X ⟶ Z✝)) (fun (f : Bicone.right ⟶ Y✝) (h : Y✝ = Bicone.diagram j) => Eq.ndrec (motive := fun {Y : Bicone J} => (Y ⟶ Z✝) → (f : Bicone.right ⟶ Y) → HEq f (BiconeHom.right j) → (Bicone.right ⟶ Z✝)) (fun (g : Bicone.diagram j ⟶ Z✝) (f : Bicone.right ⟶ Bicone.diagram j) (h : HEq f (BiconeHom.right j)) => BiconeHom.casesOn (motive := fun (a a_1 : Bicone J) (t : BiconeHom J a a_1) => Bicone.diagram j = a → Z✝ = a_1 → HEq g t → (Bicone.right ⟶ Z✝)) g (fun (h : Bicone.diagram j = Bicone.left) => Bicone.noConfusion h) (fun (h : Bicone.diagram j = Bicone.right) => Bicone.noConfusion h) (fun (j_1 : J) (h : Bicone.diagram j = Bicone.left) => Bicone.noConfusion h) (fun (j_1 : J) (h : Bicone.diagram j = Bicone.right) => Bicone.noConfusion h) (fun {j_1 k : J} (f : j_1 ⟶ k) (h : Bicone.diagram j = Bicone.diagram j_1) => Bicone.noConfusion h fun (val_eq : j = j_1) => Eq.ndrec (motive := fun {j_2 : J} => (f : j_2 ⟶ k) → Z✝ = Bicone.diagram k → HEq g (BiconeHom.diagram f) → (Bicone.right ⟶ Z✝)) (fun (f : j ⟶ k) (h : Z✝ = Bicone.diagram k) => Eq.ndrec (motive := fun {Z : Bicone J} => (g : Bicone.diagram j ⟶ Z) → HEq g (BiconeHom.diagram f) → (Bicone.right ⟶ Z)) (fun (g : Bicone.diagram j ⟶ Bicone.diagram k) (h : HEq g (BiconeHom.diagram f)) => BiconeHom.right k) ⋯ g) val_eq f) ⋯ ⋯ ⋯) ⋯ g f) ⋯ f) (fun {j k : J} (f_1 : j ⟶ k) (h : X✝ = Bicone.diagram j) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.diagram k → HEq f (BiconeHom.diagram f_1) → (X ⟶ Z✝)) (fun (f : Bicone.diagram j ⟶ Y✝) (h : Y✝ = Bicone.diagram k) => Eq.ndrec (motive := fun {Y : Bicone J} => (Y ⟶ Z✝) → (f : Bicone.diagram j ⟶ Y) → HEq f (BiconeHom.diagram f_1) → (Bicone.diagram j ⟶ Z✝)) (fun (g : Bicone.diagram k ⟶ Z✝) (f : Bicone.diagram j ⟶ Bicone.diagram k) (h : HEq f (BiconeHom.diagram f_1)) => BiconeHom.casesOn (motive := fun (a a_1 : Bicone J) (x : BiconeHom J a a_1) => Bicone.diagram k = a → Z✝ = a_1 → HEq g x → (Bicone.diagram j ⟶ Z✝)) g (fun (h : Bicone.diagram k = Bicone.left) => Bicone.noConfusion h) (fun (h : Bicone.diagram k = Bicone.right) => Bicone.noConfusion h) (fun (j_1 : J) (h : Bicone.diagram k = Bicone.left) => Bicone.noConfusion h) (fun (j_1 : J) (h : Bicone.diagram k = Bicone.right) => Bicone.noConfusion h) (fun {j_1 k_1 : J} (g_1 : j_1 ⟶ k_1) (h : Bicone.diagram k = Bicone.diagram j_1) => Bicone.noConfusion h fun (val_eq : k = j_1) => Eq.ndrec (motive := fun {j_2 : J} => (g_2 : j_2 ⟶ k_1) → Z✝ = Bicone.diagram k_1 → HEq g (BiconeHom.diagram g_2) → (Bicone.diagram j ⟶ Z✝)) (fun (g_2 : k ⟶ k_1) (h : Z✝ = Bicone.diagram k_1) => Eq.ndrec (motive := fun {Z : Bicone J} => (g : Bicone.diagram k ⟶ Z) → HEq g (BiconeHom.diagram g_2) → (Bicone.diagram j ⟶ Z)) (fun (g : Bicone.diagram k ⟶ Bicone.diagram k_1) (h : HEq g (BiconeHom.diagram g_2)) => BiconeHom.diagram (CategoryStruct.comp f_1 g_2)) ⋯ g) val_eq g_1) ⋯ ⋯ ⋯) ⋯ g f) ⋯ f) ⋯ ⋯ ⋯

@[simp]

theorem CategoryTheory.biconeCategoryStruct_Hom (J : Type u₁) [Category.{v₁, u₁} J] (a✝ a✝¹ : Bicone J) : (a✝ ⟶ a✝¹) = BiconeHom J a✝ a✝¹

instance CategoryTheory.biconeCategory (J : Type u₁) [Category.{v₁, u₁} J] : Category.{max u₁ v₁, u₁} (Bicone J)

Equations

• CategoryTheory.biconeCategory J = CategoryTheory.Category.mk ⋯ ⋯ ⋯

def CategoryTheory.biconeMk (J : Type v₁) [SmallCategory J] {C : Type u₁} [Category.{v₁, u₁} C] {F : Functor J C} (c₁ c₂ : Limits.Cone F) : Functor (Bicone J) C

Given a diagram F : J ⥤ C and two Cone Fs, we can join them into a diagram Bicone J ⥤ C.

Equations

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

@[simp]

theorem CategoryTheory.biconeMk_map (J : Type v₁) [SmallCategory J] {C : Type u₁} [Category.{v₁, u₁} C] {F : Functor J C} (c₁ c₂ : Limits.Cone F) {X✝ Y✝ : Bicone J} (f : X✝ ⟶ Y✝) : (biconeMk J c₁ c₂).map f = BiconeHom.casesOn (motive := fun (a a_1 : Bicone J) (x : BiconeHom J a a_1) => X✝ = a → Y✝ = a_1 → HEq f x → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) X✝ ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y✝)) f (fun (h : X✝ = Bicone.left) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.left → HEq f BiconeHom.left_id → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) X ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y✝)) (fun (f : Bicone.left ⟶ Y✝) (h : Y✝ = Bicone.left) => Eq.ndrec (motive := fun {Y : Bicone J} => (f : Bicone.left ⟶ Y) → HEq f BiconeHom.left_id → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Bicone.left ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y)) (fun (f : Bicone.left ⟶ Bicone.left) (h : HEq f BiconeHom.left_id) => CategoryStruct.id ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Bicone.left)) ⋯ f) ⋯ f) (fun (h : X✝ = Bicone.right) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.right → HEq f BiconeHom.right_id → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) X ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y✝)) (fun (f : Bicone.right ⟶ Y✝) (h : Y✝ = Bicone.right) => Eq.ndrec (motive := fun {Y : Bicone J} => (f : Bicone.right ⟶ Y) → HEq f BiconeHom.right_id → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Bicone.right ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y)) (fun (f : Bicone.right ⟶ Bicone.right) (h : HEq f BiconeHom.right_id) => CategoryStruct.id ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Bicone.right)) ⋯ f) ⋯ f) (fun (j : J) (h : X✝ = Bicone.left) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.diagram j → HEq f (BiconeHom.left j) → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) X ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y✝)) (fun (f : Bicone.left ⟶ Y✝) (h : Y✝ = Bicone.diagram j) => Eq.ndrec (motive := fun {Y : Bicone J} => (f : Bicone.left ⟶ Y) → HEq f (BiconeHom.left j) → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Bicone.left ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y)) (fun (f : Bicone.left ⟶ Bicone.diagram j) (h : HEq f (BiconeHom.left j)) => c₁.π.app j) ⋯ f) ⋯ f) (fun (j : J) (h : X✝ = Bicone.right) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.diagram j → HEq f (BiconeHom.right j) → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) X ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y✝)) (fun (f : Bicone.right ⟶ Y✝) (h : Y✝ = Bicone.diagram j) => Eq.ndrec (motive := fun {Y : Bicone J} => (f : Bicone.right ⟶ Y) → HEq f (BiconeHom.right j) → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Bicone.right ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y)) (fun (f : Bicone.right ⟶ Bicone.diagram j) (h : HEq f (BiconeHom.right j)) => c₂.π.app j) ⋯ f) ⋯ f) (fun {j k : J} (f_1 : j ⟶ k) (h : X✝ = Bicone.diagram j) => Eq.ndrec (motive := fun {X : Bicone J} => (f : X ⟶ Y✝) → Y✝ = Bicone.diagram k → HEq f (BiconeHom.diagram f_1) → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) X ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y✝)) (fun (f : Bicone.diagram j ⟶ Y✝) (h : Y✝ = Bicone.diagram k) => Eq.ndrec (motive := fun {Y : Bicone J} => (f : Bicone.diagram j ⟶ Y) → HEq f (BiconeHom.diagram f_1) → ((fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) (Bicone.diagram j) ⟶ (fun (X : Bicone J) => Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j) Y)) (fun (f : Bicone.diagram j ⟶ Bicone.diagram k) (h : HEq f (BiconeHom.diagram f_1)) => F.map f_1) ⋯ f) ⋯ f) ⋯ ⋯ ⋯

@[simp]

theorem CategoryTheory.biconeMk_obj (J : Type v₁) [SmallCategory J] {C : Type u₁} [Category.{v₁, u₁} C] {F : Functor J C} (c₁ c₂ : Limits.Cone F) (X : Bicone J) : (biconeMk J c₁ c₂).obj X = Bicone.casesOn X c₁.pt c₂.pt fun (j : J) => F.obj j

instance CategoryTheory.finBiconeHom (J : Type v₁) [SmallCategory J] [FinCategory J] (j k : Bicone J) : Fintype (j ⟶ k)

Equations

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

instance CategoryTheory.biconeSmallCategory (J : Type v₁) [SmallCategory J] : SmallCategory (Bicone J)

Equations

• CategoryTheory.biconeSmallCategory J = CategoryTheory.biconeCategory J

instance CategoryTheory.biconeFinCategory (J : Type v₁) [SmallCategory J] [FinCategory J] : FinCategory (Bicone J)

Equations

• CategoryTheory.biconeFinCategory J = { fintypeObj := inferInstance, fintypeHom := inferInstance }