The ${\otimes }^{\star }$-monoidal structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of [3]. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the Ran space and integrating along the Ran space, i.e. taking factorization homology. Based on ideas sketched in [4], we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in [7] and [5].

中文翻译：

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手性 Koszul 对偶中的 Atiyah-Bott 公式和连通性

这 ${\otimes }^{\star }$Ran 空间上滑轮范畴的-幺半群结构在 [3] 的意义上不是幂幂的。然而，在一些连通性假设下，我们证明了 Koszul 对偶性引入了类别的等价性，并且这种等价性在 Ran 空间上的 Verdier 对偶性和沿 Ran 空间的积分方面表现良好，即采用因式分解同调。基于 [4] 中概述的想法，我们表明这些结果还提供了一种更简单的替代方案，可以替代 [7] 和 [5] 中给出的 Atiyah-Bott 公式证明中的两个主要步骤之一。