Polynomial functors on some 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 $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$.