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.
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 $pt$.
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 $x$.
Let $\mathcal{C}$ be a category.
The following are equivalent:
$\mathcal{C}$ has a terminal object;
the unique functor $\mathcal{C} \to \ast$ to the terminal category has a right adjoint
Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.
Dually, the following are equivalent:
$\mathcal{C}$ has an initial object;
the unique functor $\mathcal{C} \to \ast$ to the terminal category has a left adjoint
Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.
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 $\mathcal{C}$
or of a terminal object
respectively.
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 terminal object of Top is the point space.
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 $x \to x$.
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.