标题： 高Koszul对偶性和$n$-仿射性 显示英文标题

标题： Higher Koszul duality and $n$-affineness

摘要： 我们研究形式为$\mathrm{C}_{\bullet}(\Omega^{n}_*X;\Bbbk) \leftrightarrow \mathrm{C}^{\bullet}(X;\Bbbk)$的代数对的$\mathbb{E}_n$-Koszul 对偶性，以及与之密切相关的 Betti 簇的$n$-仿射性问题。人们期望但尚未证实的是，$\mathbb{E}_n$-Koszul 对偶性应在迭代模的范畴之间诱导一种 Morita 等价。 我们通过证明关于$\mathrm{C}_{\bullet}(\Omega_*^{n+1}X;\Bbbk)$的迭代模的$(\infty,n)$类与$\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$上$(\infty,n-1)$类的拟凝聚层的$(\infty,n)$类等价来严格建立这一点，其中$\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$是$\mathrm{C}^{\bullet}(X;\Bbbk)$的余谱。 通过单值等价，这些范畴也等价于范畴$X$，$n\mathbf{LocSysCat}^{n-1}(X;\Bbbk)$上的高阶局部系统。 在经典情形$n=1$中，我们的结果已经是新的，尽管它可以被看作是将$\mathbb{E}_1$-Koszul 对偶性作为模范畴的Morita等价（在适当的$t$-结构完成下）的已知表述的恢复。 我们还研究了Betti堆栈的（高阶）仿射性质。 我们根据其迭代环路空间的$0$-仿射性，对$n$-仿射Betti堆栈给出了完整的刻画。 因此，我们证明了$n$-截断的Betti堆栈是$n$-仿射的；并且$\pi_{n+1}(X)$是$n$-仿射性的障碍。 显示英文摘要

摘要： We study $\mathbb{E}_n$-Koszul duality for pairs of algebras of the form $\mathrm{C}_{\bullet}(\Omega^{n}_*X;\Bbbk) \leftrightarrow \mathrm{C}^{\bullet}(X;\Bbbk)$, and the closely related question of $n$-affineness for Betti stacks. It was expected, but not known, that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the $(\infty,n)$-category of iterated modules over $\mathrm{C}_{\bullet}(\Omega_*^{n+1}X;\Bbbk)$ is equivalent to the $(\infty,n)$-category of quasi-coherent sheaves of $(\infty,n-1)$-categories on $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$, where $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$ is the cospectrum of $\mathrm{C}^{\bullet}(X;\Bbbk)$. By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on $X$, $n\mathbf{LocSysCat}^{n-1}(X;\Bbbk)$. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $\pi_{n+1}(X)$ is an obstruction to $n$-affineness.