“Subdivided interval categories”的意思、由来-开放百科全书

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Subdivided interval categories

释义

Examples
References

In category theory (mathematics) there exists an important collection of categories denoted for natural numbers . The objects of are the integers , and the morphism set for objects is empty if and consists of a single element if .

Subdivided interval categories are very useful in defining simplicial sets. The category whose objects are the subdivided interval categories and whose morphisms are functors is often written and is called the simplicial indexing category. A simplicial set is just a contravariant functor .

Examples

The category 𝟘 is an empty interval, that is, an empty category, having no objects nor morphisms. It is an initial object in the category of all categories.

The category [0], also denoted as 𝟙, is a one-object, one-morphism category. It is the terminal object in the category of all categories.

The category [1], also denoted as 𝟚 has two objects and a single (non-identity) morphism between them. If is any category, then is the category of morphisms and commutative squares in

The category [2], also denoted as 𝟛 has three objects and three non-identity morphisms.

References

MacLane, S. Categories for the working mathematician.

1 : Category theory

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