Polynomial functors on some categories of elements

TL;DR

This paper investigates polynomial functors on categories of elements derived from presheaves over finite dimensional vector spaces, providing descriptions of polynomial functor categories and classifying simple objects under certain conditions.

Contribution

It introduces a framework for polynomial functors on categories of elements of presheaves, extending classical functor theory to new categorical contexts with explicit classifications.

Findings

Description of polynomial functor categories and their quotients.

Classification of simple objects over finite fields under noetherianity.

Extension of classical polynomial functor theory to categories of elements.

Abstract

We study the category $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$ of functors from the category $\mathfrak{S}_S$, which is the category of elements of some presheaf $S$ on the category $\mathcal{V}^f$ of finite dimensional vector spaces, to $\mathcal{V}$ the category of vector spaces of any dimension on some field $\mathbb{k}$. In the case where $S$ satisfies some noetherianity condition, we have a convenient description of the category $\mathfrak{S}_S$. In this case, we can define a notion of polynomial functors on $\mathfrak{S}_S$. And, like in the usual setting of functors from the category of finite dimensional vector spaces to the one of vector spaces of any dimension, we can describe the quotient $\mathcal{P}\mathrm{ol}_{n}(\mathfrak{S}_S,\mathcal{V})/\mathcal{P}\mathrm{ol}_{n-1}(\mathfrak{S}_S,\mathcal{V})$, where $\mathcal{P}\mathrm{ol}_{n}(\mathfrak{S}_S,\mathcal{V})$ denote the full subcategory of $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$ of polynomial functors of degree less than or equal to $n$. Finally, if $\mathbb{k}=\mathbb{F}_p$ for some prime $p$ and if $S$ satisfies the required noetherianity condition, we can compute the set of isomorphism classes of simple objects in $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$.