What kind of linearly distributive category do polynomial functors form?

TL;DR

This paper extends the theory of linearly distributive categories by analyzing polynomial functors within a special duoidal category, introducing bi-closed structures and exploring duals, cores, and algebraic objects in this setting.

Contribution

It introduces the notion of bi-closed linearly distributive categories and characterizes dual objects and algebraic structures within polynomial functors.

Findings

Polynomial functors form a bi-closed linearly distributive category.

Linear polynomial functors have right duals that are representable.

Examples of linear monoids, comonoids, and bialgebras are provided in this setting.

Abstract

This paper has two purposes. The first is to extend the theory of linearly distributive categories by considering the structures that emerge in a special case: the normal duoidal category $(\mathsf{Poly} ,\mathcal{y}, \otimes, \triangleleft )$ of polynomial functors under Dirichlet and substitution product. This is an isomix LDC which is neither $*$-autonomous nor fully symmetric. The additional structures of interest here are a closure for $\otimes$ and a co-closure for $\triangleleft$, making $\mathsf{Poly}$ a bi-closed LDC, which is a notion we introduce in this paper. The second purpose is to use $\mathsf{Poly}$ as a source of examples and intuition about various structures that can occur in the setting of LDCs, including duals, cores, linear monoids, and others, as well as how these generalize to the non-symmetric setting. To that end, we characterize the linearly dual objects in $\mathsf{Poly}$: every linear polynomial has a right dual which is a representable. It turns out that the linear and representable polynomials also form the left and right cores of $\mathsf{Poly}$. Finally, we provide examples of linear monoids, linear comonoids, and linear bialgebras in $\mathsf{Poly}$.