Subdivided interval categories

In category theory (mathematics) there exists an important collection of categories denoted  [n] for natural numbers  n\in\mathbb{N}. The objects of  [n] are the integers  0,1,2,\ldots,n, and the morphism set  Hom(i,j) for objects  i,j\in[n] is empty if  j<i and consists of a single element if  i\leq 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}\rightarrow Sets.

Examples

The category 𝟘 is an empty interval, that is, an empty category, having any objects or 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  \mathcal{C} is any category, then  \mathcal{C}^{[1]} is the category of morphisms and commutative squares in \mathcal{C}.

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

References

MacLane, S. Categories for the working mathematician.

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