A Polynomial Construction of Nerves for Higher Categories

We show that the construction due to Leinster and Weber of a generalized Lawvere theory for a familially representable monad on a (co)presheaf category, and the associated ``nerve'' functor from monad algebras to (co)presheaves, have an elegant categorical description in the double category $\mathbb{C}\mathbf{at}^{\#}$ of categories, cofunctors, familial functors, and transformations. In $\mathbb{C}\mathbf{at}^{\#}$, which also arises from comonoids in the category of polynomial functors, both a familial monad and a (co)presheaf it acts on can be modeled as horizontal morphisms; from this perspective, the theory category associated to the monad is built using left Kan extension in the category of endomorphisms, and the nerve functor is modeled by a single composition of horizontal morphisms in $\mathbb{C}\mathbf{at}^{\#}$. For the free category monad $path$ on graphs, this provides a new construction of the simplex category as $\Delta := \lens{path}{path \circ path}$. We also explore the free Eilenberg-Moore completion of $\mathbb{C}\mathbf{at}^{\#}$, in which constructions such as the free symmetric monoidal category monad on $\mathbf{Cat}$ can modeled using the rich language of polynomial functors.