Trivial Group is Initial Object

Theorem

Let $\mathbf{Grp}$ be the category of groups.

Let $1 = \set e$ be the trivial group.

Then $1$ is an initial object of $\mathbf{Grp}$.

Let $\struct {G, \circ}$ be a group with identity $e_G$.

$\map \phi e = e_G$

Let us define the mapping $\phi$ in this way.

By Equality of Mappings, only one such mapping $1 \to G$ can exist, establishing uniqueness.

Now to verify that $\phi$ is actually a group homomorphism.

Since $1$ has only the element $e$, this is verified by:

$\ds \map \phi e \circ \map \phi e$

$=$

$\ds e_G \circ e_G$

Definition of $\phi$

$\ds $

$=$

$\ds e_G$

$\ds $

$=$

$\ds \map \phi e$

$\ds $

$=$

$\ds \map \phi {e * e}$

where we used $*$ to denote the group operation on $1$.

We have thus established that there is a unique morphism $1 \to G$ in $\mathbf{Grp}$ for all $G$.

That is, $1$ is an initial object in $\mathbf{Grp}$.

$\blacksquare$