Subdivided interval categories

In category theory (mathematics) there exists an important collection of categories denoted [math]\displaystyle{ [n] }[/math] for natural numbers [math]\displaystyle{ n\in\mathbb{N} }[/math]. The objects of [math]\displaystyle{ [n] }[/math] are the integers [math]\displaystyle{ 0,1,2,\ldots,n }[/math], and the morphism set [math]\displaystyle{ Hom(i,j) }[/math] for objects [math]\displaystyle{ i,j\in[n] }[/math] is empty if [math]\displaystyle{ j\lt i }[/math] and consists of a single element if [math]\displaystyle{ i\leq j }[/math]. 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 [math]\displaystyle{ \Delta }[/math] and is called the simplicial indexing category. A simplicial set is just a contravariant functor [math]\displaystyle{ X:\Delta^{op}\rightarrow Sets }[/math].

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 [math]\displaystyle{ \mathcal{C} }[/math] is any category, then [math]\displaystyle{ \mathcal{C}^{[1]} }[/math] is the category of morphisms and commutative squares in [math]\displaystyle{ \mathcal{C}. }[/math]

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

References

MacLane, S. Categories for the working mathematician.