# nLab terminal object

Contents

### Context

category theory

#### 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

• Less usual synonyms are final object and terminator.

• 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 $pt$.

## Properties

###### Remark

A terminal object may also be viewed as a limit over the empty diagram.

• Conversely, a limit over any diagram is a terminal cone over that diagram.

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

###### Proposition

Let $\mathcal{C}$ be a category.

1. The following are equivalent:

1. $\mathcal{C}$ has a terminal object;

2. the unique functor $\mathcal{C} \to \ast$ to the terminal category has a right adjoint

$\ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}$

Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.

2. Dually, the following are equivalent:

1. $\mathcal{C}$ has an initial object;

2. the unique functor $\mathcal{C} \to \ast$ to the terminal category has a left adjoint

$\mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast$

Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.

###### Proof

Since the unique hom-set in the terminal category is the singleton, the hom-isomorphism characterizing the adjoint functors is directly the universal property of an initial object in $\mathcal{C}$

$Hom_{\mathcal{C}}( L(\ast) , X ) \;\simeq\; Hom_{\ast}( \ast, R(X) ) = \ast$

or of a terminal object

$Hom_{\mathcal{C}}( X , R(\ast) ) \;\simeq\; Hom_{\ast}( L(X), \ast ) = \ast \,,$

respectively.

## Examples

Some examples of terminal objects in notable categories follow:

## References

Textbook account:

Last revised on May 17, 2023 at 05:46:30. See the history of this page for a list of all contributions to it.