Subdivided interval categories

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In mathematics, more specifically category theory, 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 Δ and is called the simplicial indexing category. A simplicial set is just a contravariant functor X:\Delta^{op}\rightarrow Sets.

[edit] 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}.

[edit] References

MacLane, S. Categories for the working mathematician.