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Mathlib.CategoryTheory.Skeletal

Skeleton of a category #

Define skeletal categories as categories in which any two isomorphic objects are equal.

Construct the skeleton of an arbitrary category by taking isomorphism classes, and show it is a skeleton of the original category.

In addition, construct the skeleton of a thin category as a partial ordering, and (noncomputably) show it is a skeleton of the original category. The advantage of this special case being handled separately is that lemmas and definitions about orderings can be used directly, for example for the subobject lattice. In addition, some of the commutative diagrams about the functors commute definitionally on the nose which is convenient in practice.

A category is skeletal if isomorphic objects are equal.

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IsSkeletonOf C D F says that F : D ⥤ C exhibits D as a skeletal full subcategory of C, in particular F is a (strong) equivalence and D is skeletal.

  • skel : Skeletal D

    The category D has isomorphic objects equal

  • The functor F is an equivalence

theorem CategoryTheory.Functor.eq_of_iso {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F₁ F₂ : Functor D C} [Quiver.IsThin C] (hC : Skeletal C) (hF : F₁ F₂) :
F₁ = F₂

If C is thin and skeletal, then any naturally isomorphic functors to C are equal.

If C is thin and skeletal, D ⥤ C is skeletal. CategoryTheory.functor_thin shows it is thin also.

Construct the skeleton category as the induced category on the isomorphism classes, and derive its category structure.

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The functor from the skeleton of C to C.

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@[simp]
theorem CategoryTheory.fromSkeleton_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : InducedCategory C Quotient.out} (f : X✝ Y✝) :
@[reducible, inline]

The class of an object in the skeleton.

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noncomputable def CategoryTheory.preCounitIso {C : Type u₁} [Category.{v₁, u₁} C] (X : C) :

The isomorphism between ⟦X⟧.out and X.

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An inverse to fromSkeleton C that forms an equivalence with it.

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The equivalence between the skeleton and the category itself.

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The skeleton of C given by choice is a skeleton of C.

noncomputable def CategoryTheory.Skeleton.isoOfEq {C : Type u₁} [Category.{v₁, u₁} C] {X Y : C} (h : toSkeleton X = toSkeleton Y) :
X Y

Provides a (noncomputable) isomorphism X ≅ Y given that toSkeleton X = toSkeleton Y.

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From a functor C ⥤ D, construct a map of skeletons Skeleton C → Skeleton D.

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A natural isomorphism between X ↦ ⟦X⟧ ↦ ⟦FX⟧ and X ↦ FX ↦ ⟦FX⟧. On the level of categories, these are C ⥤ Skeleton C ⥤ Skeleton D and C ⥤ D ⥤ Skeleton D. So this says that the square formed by these 4 objects and 4 functors commutes.

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Two categories which are categorically equivalent have skeletons with equivalent objects.

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Construct the skeleton category by taking the quotient of objects. This construction gives a preorder with nice definitional properties, but is only really appropriate for thin categories. If your original category is not thin, you probably want to be using skeleton instead of this.

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The functor from a category to its thin skeleton.

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@[simp]
theorem CategoryTheory.toThinSkeleton_map (C : Type u₁) [Category.{v₁, u₁} C] {X✝ Y✝ : C} (f : X✝ Y✝) :

The constructions here are intended to be used when the category C is thin, even though some of the statements can be shown without this assumption.

The thin skeleton is thin.

A functor C ⥤ D computably lowers to a functor ThinSkeleton C ⥤ ThinSkeleton D.

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@[simp]
theorem CategoryTheory.ThinSkeleton.map_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : ThinSkeleton C} (a✝ : X Y) :
(map F).map a✝ = Quotient.recOnSubsingleton₂ (motive := fun (x x_1 : ThinSkeleton C) => (x x_1) → (Quotient.map F.obj x Quotient.map F.obj x_1)) X Y (fun (x x_1 : C) (k : x x_1) => homOfLE ) a✝
@[simp]
theorem CategoryTheory.ThinSkeleton.map_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (a✝ : Quotient (isIsomorphicSetoid C)) :
(map F).obj a✝ = Quotient.map F.obj a✝
def CategoryTheory.ThinSkeleton.mapNatTrans {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F₁ F₂ : Functor C D} (k : F₁ F₂) :
map F₁ map F₂

Given a natural transformation F₁ ⟶ F₂, induce a natural transformation map F₁ ⟶ map F₂.

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Given a bifunctor, we descend to a function on objects of ThinSkeleton

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For each x : ThinSkeleton C, we promote map₂ObjMap F x to a functor

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def CategoryTheory.ThinSkeleton.map₂NatTrans {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F : Functor C (Functor D E)) {x₁ x₂ : ThinSkeleton C} :
(x₁ x₂) → (map₂Functor F x₁ map₂Functor F x₂)

This provides natural transformations map₂Functor F x₁ ⟶ map₂Functor F x₂ given x₁ ⟶ x₂

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A functor C ⥤ D ⥤ E computably lowers to a functor ThinSkeleton C ⥤ ThinSkeleton D ⥤ ThinSkeleton E

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@[simp]
theorem CategoryTheory.ThinSkeleton.map₂_obj {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F : Functor C (Functor D E)) (a✝ : ThinSkeleton C) :
(map₂ F).obj a✝ = map₂Functor F a✝
@[simp]
theorem CategoryTheory.ThinSkeleton.map₂_map {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {E : Type u₃} [Category.{v₃, u₃} E] (F : Functor C (Functor D E)) {X✝ Y✝ : ThinSkeleton C} (a✝ : X✝ Y✝) :
(map₂ F).map a✝ = map₂NatTrans F a✝

Use Quotient.out to create a functor out of the thin skeleton.

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@[simp]

The equivalence between the thin skeleton and the category itself.

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theorem CategoryTheory.ThinSkeleton.equiv_of_both_ways {C : Type u₁} [Category.{v₁, u₁} C] [Quiver.IsThin C] {X Y : C} (f : X Y) (g : Y X) :
X Y
theorem CategoryTheory.ThinSkeleton.map_iso_eq {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] [Quiver.IsThin C] {F₁ F₂ : Functor D C} (h : F₁ F₂) :
map F₁ = map F₂

An adjunction between thin categories gives an adjunction between their thin skeletons.

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When e : C ≌ α is a categorical equivalence from a thin category C to some partial order α, the ThinSkeleton C is order isomorphic to α.

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