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
and Wood (2019). Our computation of the stable homological densities also
yields rational homotopy types which answer a question posed by Vakil and
Wood in 2015. Our approach hinges on the study of homological stability
of cohomological Chevalley complexes, which is of independent interest.
Keywords
generalized configuration spaces, homological stability,
homological densities, chiral algebras, chiral homology,
factorization algebras, Koszul duality, Ran space
