As I understand it there is a correspondence between finitary monads and single-sorted (ordinary) Lawvere theories. My first question is, for a monoid $M$ in Set, when will it be the case that the corresponding action monad is finitary ?
As I understand it, when this action monad is finitary the following categories will be equivalent:
- The category of models of $L$ in Set, where $L$ is the Lawvere theory corresponding to the action monad of $M.$
- The category of algebras of the action monad of $M.$
- The category of functors from $M$ to Set.
So in this case we can find a single-sorted Lawvere theory that has models corresponding to copresheaves of $M$ (presuming I haven't misunderstood).
So my second question is, when can I do this for a more general category, instead of just a monoid ? In particular, given a category $\mathcal{C},$ when can I find a multi-sorted Lawvere theory $L$ such that the models of $L$ correspond to functors from $\mathcal{C}$ to Set ?
For an algebraic theory $L : \mathbb F \to \mathscr L$, the category of models of $L$ is given by $\mathbf{Sind}(\mathscr L)$ the cocompletion of $\mathscr L$ under sifted colimits. The presheaf construction on a small category $\mathscr C$ can be exhibited by first cocompleting it under finite coproducts $\mathbf{Fam}(\mathscr C)$ and then by cocompleting this category under sifted colimits, i.e. $\mathbf{Sind}(\mathbf{Fam}(\mathscr C))$. Therefore, the category of presheaves on a category $\mathscr C$ is always the category of models of an algebraic theory: namely, of the identity functor on $\mathbf{Fam}(\mathscr C)$.