Thin categories

A thin category (also known as a sparse category) is a category with at most one morphism between each pair of objects. Examples include posets, but also some indexing categories (diagrams) for special shapes of (co)limits. To construct a category instance one only needs to specify the category_struct part, as the axioms hold for free. If C is thin, then the category of functors to C is also thin. Further, to show two objects are isomorphic in a thin category, it suffices only to give a morphism in each direction.

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def CategoryTheory.thin_category {C : Type u₁} [inst : CategoryTheory.CategoryStruct C] [inst : Quiver.IsThin C] : CategoryTheory.Category C

Construct a category instance from a category_struct, using the fact that hom spaces are subsingletons to prove the axioms.

Equations

• CategoryTheory.thin_category = CategoryTheory.Category.mk

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instance CategoryTheory.functor_thin {C : Type u₁} [inst : CategoryTheory.Category C] {D : Type u₂} [inst : CategoryTheory.Category D] [inst : Quiver.IsThin C] : Quiver.IsThin (D ⥤ C)

If C is a thin category, then D ⥤ C is a thin category.

Equations

• @CategoryTheory.functor_thin = @CategoryTheory.functor_thin.proof_1

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def CategoryTheory.iso_of_both_ways {C : Type u₁} [inst : CategoryTheory.Category C] [inst : Quiver.IsThin C] {X : C} {Y : C} (f : X ⟶ Y) (g : Y ⟶ X) : X ≅ Y

To show X ≅ Y in a thin category, it suffices to just give any morphism in each direction.

Equations

• CategoryTheory.iso_of_both_ways f g = CategoryTheory.Iso.mk f g

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instance CategoryTheory.subsingleton_iso {C : Type u₁} [inst : CategoryTheory.Category C] [inst : Quiver.IsThin C] {X : C} {Y : C} : Subsingleton (X ≅ Y)

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• @CategoryTheory.subsingleton_iso = @CategoryTheory.subsingleton_iso.proof_1