Subdivided interval categories

In mathematics, more specifically category theory, there exists an important collection of categories denoted $[n]$ for natural numbers $ninmathbb\{N\}$. The objects of $[n]$ are the integers $0,1,2,ldots,n$, and the morphism set $Hom(i,j)$ for objects $i,jin\; [n]$ is empty if $j$ and consists of a single element if $ileq\; j$.

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 $Delta$ and is called the simplicial indexing category. A simplicial set is just a contravariant functor $X:Delta^\{op\}\; ightarrow\; Sets$.

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

References

MacLane, S. "Categories for the working mathematician."