The Combinatorics of Polynomial Functors

TL;DR

This paper introduces a combinatorial framework using finite sets and mazes to describe endofunctors of module categories, unifying polynomial and numerical functors and providing criteria for strict polynomial functors.

Contribution

It presents a new combinatorial approach to understanding polynomial functors, linking numerical and strict polynomial functors through a unified structure.

Findings

A new combinatorial category for endofunctors is proposed.

Polynomial and numerical functors are naturally interpreted within this framework.

The Polynomial Functor Theorem provides an effective criterion for strict polynomiality.

Abstract

We propose a new description of Endofunctors of Module Categories, based upon a combinatorial category comprising finite sets and so-called mazes. Polynomial and numerical functors both find a natural interpretation in this frame-work. Since strict polynomial functors, according to the work of Salomonsson, are encoded by multi-sets, the two strains of functors may be compared and contrasted through juxtaposing the respective combinatorial structures, leading to the Polynomial Functor Theorem, giving an effective criterion for when a numerical (polynomial) functor is strict polynomial.