Polynomial Functors and Trees

We explore the relationship between polynomial functors and (rooted) trees. In the first part, we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category Ω of Moerdijk and Weiss; its main properties are described in terms of some factorization systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part, we describe polynomial endofunctors and monads as structures built from trees, characterizing the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterized by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterize polynomial endofunctors and monads among (colored) collections and operads.