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 -