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Skeleton of a category #

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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.

structure category_theory.is_skeleton_of (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (F : D C) :
Type (max u₁ u₂ v₁ v₂)

is_skeleton_of 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.

Instances for category_theory.is_skeleton_of
theorem category_theory.functor.eq_of_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F₁ F₂ : D C} [quiver.is_thin C] (hC : category_theory.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. category_theory.functor_thin shows it is thin also.

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

Instances for category_theory.skeleton

The equivalence between the skeleton and the category itself.


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


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.

Instances for category_theory.thin_skeleton
@[protected, instance]

The functor from a category to its thin skeleton.

Instances for category_theory.to_thin_skeleton

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.

@[protected, instance]

The thin skeleton is thin.

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


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


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

theorem category_theory.thin_skeleton.equiv_of_both_ways {C : Type u₁} [category_theory.category C] [quiver.is_thin C] {X Y : C} (f : X Y) (g : Y X) :

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