terminal object


Category theory

Limits and colimits



A terminal object in a category CC is an object 11 of CC satisfying the following universal property:

for every object xx of CC, there exists a unique morphism !:x1!:x\to 1. The terminal object of any category, if it exists, is unique up to unique isomorphism. If the terminal object is also initial, it is called a zero object.


  • Less usual synonyms are final object and terminator.

  • A terminal object is often written 11, since in Set it is a 1-element set, and also because it is the unit for the cartesian product. Other notations for a terminal object include ** and ptpt.

  • A terminal object may also be viewed as a limit over the empty diagram. Conversely, a limit over a diagram is a terminal cone over that diagram.

  • For any object xx in a category with terminal object 11, the categorical product x×1x\times 1 and the exponential object x 1x^1 both exist and are canonically isomorphic to xx.


Some examples of terminal objects in notable categories follow:

Last revised on June 17, 2017 at 16:13:54. See the history of this page for a list of all contributions to it.