mathlib documentation

category_theory.skeletal

Skeleton of a category

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

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 (TODO). In addition, some of the commutative diagrams about the functors commute definitionally on the nose which is convenient in practice.

(TODO): Construct the skeleton of a general category using choice, and show it is a skeleton.

def category_theory.skeletal (C : Type u₁) [category_theory.category C] :
Prop

A category is skeletal if isomorphic objects are equal.

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structure category_theory.is_skeleton_of (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
D CType (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.

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} [∀ (X Y : C), subsingleton (X Y)] :
category_theory.skeletal C(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.

def category_theory.thin_skeleton (C : Type u₁) [category_theory.category C] :
Type u₁

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.

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@[instance]

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The functor from a category to its thin skeleton.

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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.

@[instance]

The thin skeleton is thin.

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

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

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@[simp]

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

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theorem category_theory.thin_skeleton.equiv_of_both_ways {C : Type u₁} [category_theory.category C] [∀ (X Y : C), subsingleton (X Y)] {X Y : C} :
(X Y)(Y X)X Y

theorem category_theory.thin_skeleton.map_iso_eq {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] [∀ (X Y : C), subsingleton (X Y)] {F₁ F₂ : D C} :

from_thin_skeleton C exhibits the thin skeleton as a skeleton.

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