Polynomial functors and polynomial monads

TL;DR

This paper explores polynomial functors over locally cartesian closed categories, demonstrating their assembly into a framed bicategory, and establishing that the free monad on a polynomial endofunctor remains polynomial, with connections to operads.

Contribution

It develops the theory of polynomial functors in a categorical setting, showing their structure as a framed bicategory and analyzing the polynomial nature of free monads, linking to operads.

Findings

Polynomial functors form a framed bicategory.

The free monad on a polynomial endofunctor is polynomial.

Connections between polynomial functors and operads are established.

Abstract

We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.