Free Monoid

The free monoid on an H-Set $A$, denoted $A^{*}$, is the set of Lists of $A$, with multiplication given by concatenation.

Properties

• Free monoids are Free Objects for the forgetful functor $U:\mathrm{Mon}{.}\to\mathrm{Sets}{.}$, and in fact assemble into a Left Adjoint.
• Free monoids are always Involutive Monoids, where the involution is given by reversal. This comes from the fact that a Monoid Homomorphism $(-)^{\dagger}:A^{*}\to(A^{*})^{mathrm{op}}$ can be induced by a map $A\to U((A^{*})^{mathrm{op}})$, which in turn is just a map $A\to U(A^{*})$; the unit of the Adjunction fits nicely here. Proving that this is invertible is just the same process.