Wide subcategories

A wide subcategory of a category C is a subcategory containing all the objects of C.

Main declarations

Given a category D, a function F : C → D from a type C to the objects of D, and a morphism property P on D which contains identities and is stable under composition, the type class InducedWideCategory D F P is a typeclass synonym for C which comes equipped with a category structure whose morphisms X ⟶ Y are the morphisms in D which have the property P.

The instance WideSubcategory.category provides a category structure on WideSubcategory P whose objects are the objects of C and morphisms are the morphisms in C which have the property P.

def CategoryTheory.InducedWideCategory {C : Type u₁} (D : Type u₂) [CategoryTheory.Category.{v₁, u₂} D] (_F : C → D) (_P : CategoryTheory.MorphismProperty D) [_P.IsMultiplicative] : Type u₁

InducedWideCategory D F P, where F : C → D, is a typeclass synonym for C, which provides a category structure so that the morphisms X ⟶ Y are the morphisms in D from F X to F Y which satisfy a property P : MorphismProperty D that is multiplicative.

Equations

• CategoryTheory.InducedWideCategory D _F _P = C

instance CategoryTheory.InducedWideCategory.hasCoeToSort {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] {α : Sort u_1} [CoeSort D α] : CoeSort (CategoryTheory.InducedWideCategory D F P) α

Equations

• CategoryTheory.InducedWideCategory.hasCoeToSort F P = { coe := fun (c : CategoryTheory.InducedWideCategory D F P) => CoeSort.coe (F c) }

@[simp]

theorem CategoryTheory.InducedWideCategory.category_comp_coe {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] {X : CategoryTheory.InducedWideCategory D F P} {Y : CategoryTheory.InducedWideCategory D F P} {Z : CategoryTheory.InducedWideCategory D F P} (f : X ⟶ Y) (g : Y ⟶ Z) : ↑(CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp ↑f ↑g

@[simp]

theorem CategoryTheory.InducedWideCategory.category_id_coe {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] (X : CategoryTheory.InducedWideCategory D F P) : ↑(CategoryTheory.CategoryStruct.id X) = CategoryTheory.CategoryStruct.id (F X)

instance CategoryTheory.InducedWideCategory.category {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] : CategoryTheory.Category.{v₁, u₁} (CategoryTheory.InducedWideCategory D F P)

Equations

• CategoryTheory.InducedWideCategory.category F P = CategoryTheory.Category.mk ⋯ ⋯ ⋯

@[simp]

theorem CategoryTheory.wideInducedFunctor_obj {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] : ∀ (a : C), (CategoryTheory.wideInducedFunctor F P).obj a = F a

@[simp]

theorem CategoryTheory.wideInducedFunctor_map {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] {X : CategoryTheory.InducedWideCategory D F P} {Y : CategoryTheory.InducedWideCategory D F P} (f : X ⟶ Y) : (CategoryTheory.wideInducedFunctor F P).map f = ↑f

def CategoryTheory.wideInducedFunctor {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] : CategoryTheory.Functor (CategoryTheory.InducedWideCategory D F P) D

The forgetful functor from an induced wide category to the original category.

Equations

• CategoryTheory.wideInducedFunctor F P = { obj := F, map := fun {X Y : CategoryTheory.InducedWideCategory D F P} (f : X ⟶ Y) => ↑f, map_id := ⋯, map_comp := ⋯ }

instance CategoryTheory.InducedWideCategory.faithful {C : Type u₁} {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : C → D) (P : CategoryTheory.MorphismProperty D) [P.IsMultiplicative] : (CategoryTheory.wideInducedFunctor F P).Faithful

The induced functor wideInducedFunctor F P : InducedWideCategory D F P ⥤ D is faithful.

Equations

• ⋯ = ⋯

theorem CategoryTheory.WideSubcategory.ext_iff {C : Type u₁} : ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {_P : CategoryTheory.MorphismProperty C} {inst_1 : _P.IsMultiplicative} {x y : CategoryTheory.WideSubcategory _P}, x = y ↔ x.obj = y.obj

theorem CategoryTheory.WideSubcategory.ext {C : Type u₁} : ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {_P : CategoryTheory.MorphismProperty C} {inst_1 : _P.IsMultiplicative} {x y : CategoryTheory.WideSubcategory _P}, x.obj = y.obj → x = y

structure CategoryTheory.WideSubcategory {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (_P : CategoryTheory.MorphismProperty C) [_P.IsMultiplicative] : Type u₁

Structure for wide subcategories. Objects ignore the morphism property.

• obj : C

  The category of which this is a wide subcategory

instance CategoryTheory.WideSubcategory.category {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] : CategoryTheory.Category.{v₁, u₁} (CategoryTheory.WideSubcategory P)

Equations

• CategoryTheory.WideSubcategory.category P = CategoryTheory.InducedWideCategory.category CategoryTheory.WideSubcategory.obj P

@[simp]

theorem CategoryTheory.WideSubcategory.id_def {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] (X : CategoryTheory.WideSubcategory P) : ↑(CategoryTheory.CategoryStruct.id X) = CategoryTheory.CategoryStruct.id X.obj

@[simp]

theorem CategoryTheory.WideSubcategory.comp_def {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] {X : CategoryTheory.WideSubcategory P} {Y : CategoryTheory.WideSubcategory P} {Z : CategoryTheory.WideSubcategory P} (f : X ⟶ Y) (g : Y ⟶ Z) : ↑(CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp ↑f ↑g

def CategoryTheory.wideSubcategoryInclusion {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] : CategoryTheory.Functor (CategoryTheory.WideSubcategory P) C

The forgetful functor from a wide subcategory into the original category ("forgetting" the condition).

Equations

• CategoryTheory.wideSubcategoryInclusion P = CategoryTheory.wideInducedFunctor CategoryTheory.WideSubcategory.obj P

@[simp]

theorem CategoryTheory.wideSubcategoryInclusion.obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] (X : CategoryTheory.WideSubcategory P) : (CategoryTheory.wideSubcategoryInclusion P).obj X = X.obj

@[simp]

theorem CategoryTheory.wideSubcategoryInclusion.map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] {X : CategoryTheory.WideSubcategory P} {Y : CategoryTheory.WideSubcategory P} {f : X ⟶ Y} : (CategoryTheory.wideSubcategoryInclusion P).map f = ↑f

instance CategoryTheory.wideSubcategory.faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (P : CategoryTheory.MorphismProperty C) [P.IsMultiplicative] : (CategoryTheory.wideSubcategoryInclusion P).Faithful

The inclusion of a wide subcategory is faithful.

Equations

• ⋯ = ⋯