Terminal Objects

One is the loneliest number.

Terminal Object

A terminal object S in a category is an object such that each other object in the category has exactly one map to S.

Fun facts:

• Two terminal objects S1 and S2 in a category C are necessarily isomorphic to one another. This is most obvious when we think of the category of sets, where a terminal object is a set with a single element. The sets {potato} and {0} are isomorphic. This isomorphism is unique.

• Two terminal objects in a category C have in common all of the properties that can be expressed by maps in category C.

Examples

• In the category of sets: a singleton set.
• In the category of endomaps: a singleton set with one arrow emanating from and pointing to that single element.
• In the category of irreflexive graphs (arrows, points, source(), target(), and no preferred loop): a single arrow and a single point, where the source and target of the single arrow map to the single point.