TY - JOUR AB - We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types, answering a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest. AU - Ho, Quoc P ID - 9359 IS - 2 JF - Geometry & Topology KW - Generalized configuration spaces KW - homological stability KW - homological densities KW - chiral algebras KW - chiral homology KW - factorization algebras KW - Koszul duality KW - Ran space SN - 1364-0380 TI - Homological stability and densities of generalized configuration spaces VL - 25 ER - TY - JOUR AB - The ⊗*-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]. AU - Ho, Quoc P ID - 10033 JF - Advances in Mathematics KW - Chiral algebras KW - Chiral homology KW - Factorization algebras KW - Koszul duality KW - Ran space SN - 0001-8708 TI - The Atiyah-Bott formula and connectivity in chiral Koszul duality VL - 392 ER -