Basic API for ULift

This file contains a very basic API for working with the categorical instance on ULift C where C is a type with a category instance.

1. CategoryTheory.ULift.upFunctor is the functorial version of the usual ULift.up.
2. CategoryTheory.ULift.downFunctor is the functorial version of the usual ULift.down.
3. CategoryTheory.ULift.equivalence is the categorical equivalence between C and ULift C.

ULiftHom

Given a type C : Type u, ULiftHom.{w} C is just an alias for C. If we have category.{v} C, then ULiftHom.{w} C is endowed with a category instance whose morphisms are obtained by applying ULift.{w} to the morphisms from C.

This is a category equivalent to C. The forward direction of the equivalence is ULiftHom.up, the backward direction is ULiftHom.down and the equivalence is ULiftHom.equiv.

AsSmall

This file also contains a construction which takes a type C : Type u with a category instance Category.{v} C and makes a small category AsSmall.{w} C : Type (max w v u) equivalent to C.

The forward direction of the equivalence, C ⥤ AsSmall C, is denoted AsSmall.up and the backward direction is AsSmall.down. The equivalence itself is AsSmall.equiv.

def CategoryTheory.ULift.upFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor C (ULift.{u₂, u₁} C)

The functorial version of ULift.up.

Equations

• CategoryTheory.ULift.upFunctor = { obj := ULift.up, map := fun {X Y : C} (f : X ⟶ Y) => f, map_id := ⋯, map_comp := ⋯ }

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theorem CategoryTheory.ULift.upFunctor_obj_down {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (down : C) : (CategoryTheory.ULift.upFunctor.obj down).down = down

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theorem CategoryTheory.ULift.upFunctor_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.ULift.upFunctor.map f = f

def CategoryTheory.ULift.downFunctor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (ULift.{u₂, u₁} C) C

The functorial version of ULift.down.

Equations

• CategoryTheory.ULift.downFunctor = { obj := ULift.down, map := fun {X Y : ULift.{?u.22, ?u.23} C} (f : X ⟶ Y) => f, map_id := ⋯, map_comp := ⋯ }

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theorem CategoryTheory.ULift.downFunctor_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (self : ULift.{u₂, u₁} C) : CategoryTheory.ULift.downFunctor.obj self = self.down

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theorem CategoryTheory.ULift.downFunctor_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : ULift.{u₂, u₁} C} (f : X ⟶ Y), CategoryTheory.ULift.downFunctor.map f = f

def CategoryTheory.ULift.equivalence {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : C ≌ ULift.{u₂, u₁} C

The categorical equivalence between C and ULift C.

Equations

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

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theorem CategoryTheory.ULift.equivalence_functor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.ULift.equivalence.functor = CategoryTheory.ULift.upFunctor

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theorem CategoryTheory.ULift.equivalence_unitIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.ULift.equivalence.unitIso.hom = CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.id C)

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theorem CategoryTheory.ULift.equivalence_unitIso_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.ULift.equivalence.unitIso.inv = CategoryTheory.CategoryStruct.id (CategoryTheory.ULift.upFunctor.comp CategoryTheory.ULift.downFunctor)

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theorem CategoryTheory.ULift.equivalence_counitIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ (x : ULift.{u₂, u₁} C), CategoryTheory.ULift.equivalence.counitIso.hom.app x = CategoryTheory.CategoryStruct.id ((CategoryTheory.ULift.downFunctor.comp CategoryTheory.ULift.upFunctor).obj x)

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theorem CategoryTheory.ULift.equivalence_inverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.ULift.equivalence.inverse = CategoryTheory.ULift.downFunctor

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theorem CategoryTheory.ULift.equivalence_counitIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ (x : ULift.{u₂, u₁} C), CategoryTheory.ULift.equivalence.counitIso.inv.app x = CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id (ULift.{u₂, u₁} C)).obj x)

def CategoryTheory.ULiftHom (C : Type u) : Type u

ULiftHom.{w} C is an alias for C, which is endowed with a category instance whose morphisms are obtained by applying ULift.{w} to the morphisms from C.

Equations

• CategoryTheory.ULiftHom C = C

instance CategoryTheory.instInhabitedULiftHom {C : Type u_1} [Inhabited C] : Inhabited (CategoryTheory.ULiftHom C)

Equations

• CategoryTheory.instInhabitedULiftHom = { default := default }

def CategoryTheory.ULiftHom.objDown {C : Type u_1} (A : CategoryTheory.ULiftHom C) : C

The obvious function ULiftHom C → C.

Equations

• A.objDown = A

def CategoryTheory.ULiftHom.objUp {C : Type u_1} (A : C) : CategoryTheory.ULiftHom C

The obvious function C → ULiftHom C.

Equations

• CategoryTheory.ULiftHom.objUp A = A

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theorem CategoryTheory.objDown_objUp {C : Type u_1} (A : C) : (CategoryTheory.ULiftHom.objUp A).objDown = A

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theorem CategoryTheory.objUp_objDown {C : Type u_1} (A : CategoryTheory.ULiftHom C) : CategoryTheory.ULiftHom.objUp A.objDown = A

instance CategoryTheory.ULiftHom.category {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Category.{max v₂ v₁, u₁} (CategoryTheory.ULiftHom C)

Equations

• CategoryTheory.ULiftHom.category = CategoryTheory.Category.mk ⋯ ⋯ ⋯

def CategoryTheory.ULiftHom.up {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor C (CategoryTheory.ULiftHom C)

One half of the quivalence between C and ULiftHom C.

Equations

• CategoryTheory.ULiftHom.up = { obj := CategoryTheory.ULiftHom.objUp, map := fun {X Y : C} (f : X ⟶ Y) => { down := f }, map_id := ⋯, map_comp := ⋯ }

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theorem CategoryTheory.ULiftHom.up_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : C) : CategoryTheory.ULiftHom.up.obj A = CategoryTheory.ULiftHom.objUp A

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theorem CategoryTheory.ULiftHom.up_map_down {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.ULiftHom.up.map f).down = f

def CategoryTheory.ULiftHom.down {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (CategoryTheory.ULiftHom C) C

One half of the quivalence between C and ULiftHom C.

Equations

• CategoryTheory.ULiftHom.down = { obj := CategoryTheory.ULiftHom.objDown, map := fun {X Y : CategoryTheory.ULiftHom C} (f : X ⟶ Y) => f.down, map_id := ⋯, map_comp := ⋯ }

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theorem CategoryTheory.ULiftHom.down_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : CategoryTheory.ULiftHom C) : CategoryTheory.ULiftHom.down.obj A = A.objDown

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theorem CategoryTheory.ULiftHom.down_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : CategoryTheory.ULiftHom C} (f : X ⟶ Y), CategoryTheory.ULiftHom.down.map f = f.down

def CategoryTheory.ULiftHom.equiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : C ≌ CategoryTheory.ULiftHom C

The equivalence between C and ULiftHom C.

Equations

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

def CategoryTheory.AsSmall (D : Type u) [CategoryTheory.Category.{v, u} D] : Type (max u w v)

AsSmall C is a small category equivalent to C. More specifically, if C : Type u is endowed with Category.{v} C, then AsSmall.{w} C : Type (max w v u) is endowed with an instance of a small category.

The objects and morphisms of AsSmall C are defined by applying ULift to the objects and morphisms of C.

Note: We require a category instance for this definition in order to have direct access to the universe level v.

Equations

• CategoryTheory.AsSmall D = ULift.{max ?u.17 ?u.16, ?u.15} D

instance CategoryTheory.instSmallCategoryAsSmall {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.SmallCategory (CategoryTheory.AsSmall C)

Equations

• CategoryTheory.instSmallCategoryAsSmall = CategoryTheory.Category.mk ⋯ ⋯ ⋯

def CategoryTheory.AsSmall.up {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor C (CategoryTheory.AsSmall C)

One half of the equivalence between C and AsSmall C.

Equations

• CategoryTheory.AsSmall.up = { obj := fun (X : C) => { down := X }, map := fun {X Y : C} (f : X ⟶ Y) => { down := f }, map_id := ⋯, map_comp := ⋯ }

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theorem CategoryTheory.AsSmall.up_map_down {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.AsSmall.up.map f).down = f

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theorem CategoryTheory.AsSmall.up_obj_down {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : C) : (CategoryTheory.AsSmall.up.obj X).down = X

def CategoryTheory.AsSmall.down {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.Functor (CategoryTheory.AsSmall C) C

One half of the equivalence between C and AsSmall C.

Equations

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

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theorem CategoryTheory.AsSmall.down_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : ∀ {X Y : CategoryTheory.AsSmall C} (f : X ⟶ Y), CategoryTheory.AsSmall.down.map f = f.down

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theorem CategoryTheory.AsSmall.down_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X : CategoryTheory.AsSmall C) : CategoryTheory.AsSmall.down.obj X = X.down

def CategoryTheory.AsSmall.equiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : C ≌ CategoryTheory.AsSmall C

The equivalence between C and AsSmall C.

Equations

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

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theorem CategoryTheory.AsSmall.equiv_inverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.AsSmall.equiv.inverse = CategoryTheory.AsSmall.down

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theorem CategoryTheory.AsSmall.equiv_unitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.AsSmall.equiv.unitIso = CategoryTheory.NatIso.ofComponents (fun (x : C) => CategoryTheory.eqToIso ⋯) ⋯

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theorem CategoryTheory.AsSmall.equiv_counitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.AsSmall.equiv.counitIso = CategoryTheory.NatIso.ofComponents (fun (x : CategoryTheory.AsSmall C) => CategoryTheory.eqToIso ⋯) ⋯

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theorem CategoryTheory.AsSmall.equiv_functor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] : CategoryTheory.AsSmall.equiv.functor = CategoryTheory.AsSmall.up

instance CategoryTheory.instInhabitedAsSmall {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [Inhabited C] : Inhabited (CategoryTheory.AsSmall C)

Equations

• CategoryTheory.instInhabitedAsSmall = { default := { down := default } }

def CategoryTheory.ULiftHomULiftCategory.equiv (C : Type u) [CategoryTheory.Category.{v, u} C] : C ≌ CategoryTheory.ULiftHom (ULift.{u', u} C)

The equivalence between C and ULiftHom (ULift C).

Equations

• CategoryTheory.ULiftHomULiftCategory.equiv C = CategoryTheory.ULift.equivalence.trans CategoryTheory.ULiftHom.equiv