# nLab terminal object

category theory

## Applications

#### Limits and colimits

limits and colimits

# Contents

## Definition

A terminal object in a category $C$ is an object $1$ of $C$ satisfying the following universal property:

for every object $x$ of $C$, there exists a unique morphism $!:x\to 1$. 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.

## Remarks

A terminal object is often written $1$, 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 $\mathrm{pt}$.

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 $x$ in a category with terminal object $1$, the categorical product $x×1$ and the exponential object ${x}^{1}$ both exist and are canonically isomorphic to $x$.

## Examples

Some examples of terminal objects in notable categories follow:

Revised on November 20, 2011 23:36:37 by Urs Schreiber (89.204.154.71)