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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 $x\times 1$ and the exponential object $x^1$ both exist and are canonically isomorphic to $x$.

Examples

Some examples of terminal objects in notable categories follow:

• 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 $C/x$ is the identity morphism $x\to x$.

Related concepts

• initial object

• unit type

• terminal object in a quasi-category.