Element (Category Theory)

In category theory, an element of an object $X$ is simply a morphism from a terminal object, $x:1\to X$. In many categories, in which objects represent some kind of set, this simply picks out a single element from $X$.

Note that this only makes sense in certain categories. If the category is pointed such as Vect, then every object has a unique element, making the notion useless.

Examples

In Set, elements of an object are just members of the set. In Cat, elements pick out a single object from a category.