Trivial Group is Terminal Object of Category of Groups
Theorem
Let $\mathbf {Grp}$ be the category of groups.
Let $\set e$ be the trivial group.
Then $\set e$ is a terminal object of $\mathbf {Grp}$.
Proof
Let $\struct {G, \circ}$ be any group.
By Singleton is Terminal Object of Category of Sets, there is precisely one mapping:
- $!: G \to \set e$
defined by:
- $\forall g \in G: ! (g) = e$
By definition, any group homomorphism is also a mapping.
Hence, there is at most one morphism $\struct {G, \circ} \to \set e$ in $\mathbf {Grp}$.
Now to verify that the mapping $!$ is a group homomorphism.
For any $g, h \in G$, we have (using $*$ for the group operation on $\set e$):
\(\ds ! (g) * ! (h)\) | \(=\) | \(\ds e * e\) | Definition of $!$ | |||||||||||
\(\ds \) | \(=\) | \(\ds e\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds ! \paren {g \circ h}\) | Definition of $!$ |
That is, $!$ is a group homomorphism.
Thus for all groups $\struct {G, \circ}$, there is a unique group homomorphism $!: G \to \set e$.
That is, $\set e$ is a terminal object of $\mathbf {Grp}$.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $3$