After a review of the concept of “monad with arities” we show that the category of algebras
for such a monad has a canonical dense generator. This is used to extend the
correspondence between finitary monads on sets and Lawvere's algebraic theories to a
general correspondence between monads and theories for a given category with arities. As
an application we determine arities for the free groupoid monad on involutive graphs and
recover the symmetric simplicial nerve characterisation of groupoids.