The List Object Endofunctor is Polynomial
Samuel Desrochers
TL;DR
The paper generalizes the polynomial nature of the list object functor $L$ from sets to arbitrary cartesian categories with finite limits and parametrized list objects, showing $L$ is represented by a polynomial when the ambient category is extensive. The approach hinges on constructing a polynomial diagram from a natural numbers object $N = L(\mathbbm{1})$ and a finite-cardinal-like object $E = \{(m,n) : m<n\}$ with a projection $\pi_2^E: E \to N$, and then proving that $L$ arises as the associated polynomial functor via a right adjoint structure involving $L_N$ and the functors $\Sigma_E$ and $\Delta_{\pi_2^E}$. The authors develop an internal language for reasoning with equalizers, tails, and nth-element extraction to establish the necessary adjunctions, and they show that two hypotheses (one asserting a coproduct-decomposition via extensivity and another ensuring a coproduct-like behavior for the NNO) suffice to derive the result; these can be weakened to two core hypotheses, still yielding the polynomial representation. The work demonstrates that the list-object construction yields a Cartesian monad without requiring exponentials in the ambient category, thereby broadening the applicability of polynomial functors beyond locally Cartesian closed settings. The results provide a foundational tool for understanding list-like constructions categorically and have potential implications for program semantics and related categorical models.
Abstract
In this paper, we study the list object functor $L : \mathcal{C} \rightarrow \mathcal{C}$ for a general category $\mathcal{C}$ with finite limits and parametrized list objects. We show that $L$ is polynomial as long as $\mathcal{C}$ is extensive.