nLab terminal object

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:

• 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 $x \to x$.

Related concepts

• strict terminal object

• initial object

• unit type

• terminal object in a 2-category

• terminal object in an $(\infty,1)$-category

References

Textbook account:

• Francis Borceux, Section 2.3 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)