Coproduct of Free Monoids

Theorem

Let $\mathbf {Mon}$ be the category of monoids.

Let $\map M A$ and $\map M B$ be free monoids on sets $A$ and $B$, respectively.

Let $A \sqcup B$ be the disjoint union of $A$ and $B$.

Then the free monoid $\map M {A \sqcup B}$ on $A \sqcup B$ is the coproduct of $\map M A$ and $\map M B$ in $\mathbf {Mon}$.

Proof

By Coproduct is Unique, it suffices to verify that $\map M {A \sqcup B}$ is a coproduct for $\map M A$ and $\map M B$.

By the universal mapping property of $\map M A$, $\map M B$ and $\map M {A \sqcup B}$, we have the following commutative diagram:

$\quad\quad \begin {xy} <0em, 5em>*+{N} = "N", <-5em,0em>*+{\map M A} = "MA", <0em,0em>*+{\map M {A \sqcup B} } = "MAB", <5em,0em>*+{\map M B} = "MB", <-5em,-5em>*+{A} = "A", <0em,-5em>*+{A \sqcup B} = "AB", <5em,-5em>*+{B} = "B", "A";"MA" **@{-} ?>*@{>}  ?*!/_.8em/{i_A}, "B";"MB" **@{-} ?>*@{>}  ?*!/^.8em/{i_B}, "AB";"MAB" **@{-} ?>*@{>} ?*!/_.8em/{i_{A \mathop \sqcup B} }, "A";"AB" **@{-} ?>*@{>} ?*!/^.8em/{i_1}, "B";"AB" **@{-} ?>*@{>} ?*!/_.8em/{i_2}, "MA";"MAB" **@{-} ?>*@{>} ?*!/_.8em/{j_1}, "MB";"MAB" **@{-} ?>*@{>} ?*!/^.8em/{j_2}, "MA";"N" **@{-} ?>*@{>}  ?*!/_.8em/{\bar f}, "MB";"N" **@{-} ?>*@{>}  ?*!/^.8em/{\bar g}, "MAB";"N" **@{--} ?>*@{>} ?*!/_.8em/{\bar h}, \end{xy}$

Here (in the notation for free monoids):

$j_1 = \overline {\paren {i_{A \mathop \sqcup B} \circ i_1} }$

$j_2 = \overline {\paren {i_{A \mathop \sqcup B} \circ i_2}}$

and $i_1$, $i_2$ are the injections for the coproduct.

This needs considerable tedious hard slog to complete it. In particular: I don't see how to continue formally with the machinery already developed; probably some preliminary results are needed for rigour's sake To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 3.2$: Example $3.5$