# Maths - Category Theory - Terminal Object

An terminal object has a unique morphism from every other object in the category.

Terminal objects give a category theory version of the concept of 'element' in set theory. 1 -> A allows us to pick out an arbitrary element of the set A.

#### Terminal Object in Set

Although category theory is about the external properties of objects, to get an intuitive understanding, it can be helpful to peek inside them. In set-like categories the terminal object is like the single element set.

In set, maps exist from any n-element to an m-element set (provided m≠0).

So every set has maps to every non-empty set, however only maps to the 1-element set are unique.

## Examples in Various Categories

Terminal Object

Notation 1
generalisation a kind of limit
universal cone over diagram

empty diagram

examples: set:

{1}or {a} ...

set with one element (singleton)

group (null object) trivial group (just identity element)
topological space single point
poset greatest element (if exists)
monoid trivial monoid (consisting of only the identity element)
semigroup singleton semigroup
Rng trivial ring consisting only of a single element 0=1
fields does not have terminal object
Vec zero object
Top one-point space
Grf graph with a single vertex and a single loop
Ω-Alg
algebra with signature Ω

Cat category 1 (with a single object and morphism)