标题： 多项式函子在某些元素范畴上的应用 显示英文标题

标题： Polynomial functors on some categories of elements

摘要： 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 $\mathbb{k}$. 在满足某些诺特性条件时，$S$的范畴有一个方便的描述$\mathfrak{S}_S$。 在此情况下，我们可以在$\mathfrak{S}_S$上定义多项式函子的概念。 并且，如同在有限维向量空间范畴到任意维数向量空间范畴的通常函子设置中一样，我们可以描述商范畴 $\mathcal{P}\mathrm{ol}_{n}(\mathfrak{S}_S,\mathcal{V})/\mathcal{P}\mathrm{ol}_{n-1}(\mathfrak{S}_S,\mathcal{V})$，其中$\mathcal{P}\mathrm{ol}_{n}(\mathfrak{S}_S,\mathcal{V})$表示$\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$中度数小于或等于$n$的多项式函子的满子范畴。 最后，如果对于某个素数 $p$，$\mathbb{k}=\mathbb{F}_p$ 成立且 $S$ 满足所需的 Noether 性条件，我们可以计算出 $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$ 中简约对象的同构类集合。 显示英文摘要

摘要： 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 $\mathcal{F}(\mathfrak{S}_S,\mathcal{V})$.