nLab strict terminal object

Idea

The trivial ring, among all unital rings, has two characteristic properties:

1. there is a unique function into the trivial ring, from any other unital ring;

2. there is no function from the trivial ring, except to itself.

The first property generalizes to arbitrary categories as the property of a terminal object.

The corresponding generalization including also the second property is that of a strict terminal object:

Definition

A terminal object $\mathbf{1}$ is called a strict terminal object if every morphism from $\mathbf{1}$ is an isomorphism:

$$\mathbf{1} \overset{\simeq}{\longrightarrow} \X \,.$$

In other words, a strict terminal object is a maximal terminal object.

Examples

• Trivial unital ring

• Trivial Boolean algebra

• Trivial absorption monoid

• In general, for any algebraic theory with two constants $0$ and $1$ and a binary operation for which $0$ is a (left) absorbing element and $1$ is a (left) unit, the trivial model is strictly terminal.

Related concepts

• the formal dual concept: strict initial object
• maximal object

References

• Zhen Lin Low, What do algebraic theories with strictly terminal trivial models look like?, MO question:159234