Subdivided interval categories

In mathematics, more specifically category theory, there exists an important collection of categories denoted [n] for NATURAL numbers n ∈ ℕ. The objects of [n] are the integers 0, 1, 2, …, n, and the morphism set Hom(i,j) for objects i, j ∈ [n] is empty if j < i and consists of a single element if i ≤ j.

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 Δ and is called the simplicial indexing category. A simplicial set is just a contravariant functor X : Δ^op → Sets.

Examples

The category [0] is the one-object, one-morphism category. It is the terminal object in the category of small categories.

The category [1] has two objects and a single morphism between them. If 𝒞 is any category, then 𝒞^[1] is the category of morphisms and commutative squares in 𝒞.

References

MacLane, S. Categories for the working mathematician.