更高的对偶性和单生成的Grothendieck范畴
Higher dualizability and singly-generated Grothendieck categories
摘要：
设$k$为字段。我们表明，局部可呈现，$k$ -线性范畴$\mathcal{C}$ dualizable在这个意义上的单位函子可以恢复为$\coprod_i x_i\otimes f_i$的对象$x_i\in \mathcal{C}$和左伴随$f_i$从$\mathcal{C}$到$\mathrm{Vect}_k$是产品的副本$\mathrm{Vect}_k$。这部分证实了Brandenburg、作者和T.约翰逊-佛瑞德受此启发，我们还刻画了Grothendieck范畴包含一个对象$x$的性质，每个对象是一个上幂的$x$：他们正是范畴的非奇异内射右模的简单，正则，右自内射环的类型I或III。
英文摘要：
Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints $f_i$ from $\mathcal{C}$ to $\mathrm{Vect}_k$ are products of copies of $\mathrm{Vect}_k$. This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object $x$ with the property that every object is a copower of $x$: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
