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.


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: * The terminal object of a poset is its top element, if it exists. * Any one-element set is a terminal object in the category Set. * The trivial group is the terminal object of Grp and, as an abelian group, of Ab. * The terminal object of Ring is the zero ring. (Note however that if rings have unities and ring homomorphisms must preserve them, then the zero ring is not a zero object of Ring.) * Including most of the above, the terminal object of an algebraic category is its trivial algebra. * The terminal object of Cat is the discrete category with just one object, the trivial category. * The terminal object of a slice category C/xC/x is the identity morphism xxx \to x.

Revised on February 6, 2014 12:32:51 by Adeel Khan (