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.

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

Instances For

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

    def CategoryTheory.iso_of_both_ways {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [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.

    Instances For