Lawvere Theories

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Exercises

1. Enumerate all morphisms between $2$ and $3$ in $\mathcal{F}$ (the skeleton of $\mathbf{FinSet}$).
2. Show that the category of models for the Lawvere theory of monoids is equivalent to the category of monad algebras for the list monad.
3. The Lawvere theory of monoids generates the list monad. Show that its binary operations can be generated using the corresponding Kleisli arrows.
4. $\mathbf{FinSet}$ is a subcategory of $\mathbf{Set}$ and there is a functor that embeds it in $\mathbf{Set}$. Any functor on $\mathbf{Set}$ can be restricted to $\mathbf{FinSet}$. Show that a finitary functor is the left Kan extension of its own restriction.