Empty Set is Initial Object

Theorem

Let $\mathbf{Set}$ be the category of sets.

Then the empty set $\O$ is an initial object of $\mathbf{Set}$.

Proof

Follows from Empty Mapping is Unique and the definition of initial object.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $1$