Initial object

An object $I$ in a category $C$ is **initial** if for every object $X$ in $C$, there exists exactly one arrow $a: I \rightarrow X$ going from $I$ to $X$.

Example: in the category $Set$, the empty set is an initial object. Because: from empty to any other set, there is exactly one function, which is the empty function.

The empty set is also the only initial object in the category Set.

See also:

- Terminal object - the dual construct

category: definition, category-theory