F monoids commutative monoids

Example 17.3.7. For \(\Ff = (\Fin ,\Fin ,\Fin )\), the \(\Ff \)-monoids are precisely the commutative monoids from Definition 5.3.4.

For \(\Ff = (\Fin ,\Fin _{\inj },\Fin )\), the \(\infty \)-category \(\Span _{\inj ,\all }(\Fin )\) is equivalent to the (classical) category \(\Fin _*\) of finite pointed sets, and the resulting \(\Ff \)-monoids are precisely the commutative monoids as defined by Lurie (2017). By Proposition 5.3.9, restriction along the inclusion induces an equivalence between these two definitions.