Subdivided interval categories

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."


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