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 $\mathcal{ F}(\mathfrak{ S}_S,\mathcal{ V})$.
Categories: math.CT