Picosecond ultrasonics

2013-04-26T11:59:00Z

132.166.21.231: /* Applications and future challenges */

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

Subdivided interval categories are very useful in defining [[simplicial set]]s. The category whose objects are the subdivided interval categories and whose morphisms are functors is often written <math>\Delta</math> and is called the [[simplicial category|simplicial indexing category]]. A simplicial set is just a contravariant functor <math> X:\Delta^{op}\rightarrow Sets</math>.

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

[[Category:Category theory]]

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Picosecond ultrasonics