[{"issue":"2","abstract":[{"lang":"eng","text":"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.\r\n"}],"type":"journal_article","oa_version":"Submitted Version","file":[{"content_type":"application/pdf","file_size":479268,"creator":"qho","access_level":"open_access","file_name":"densities.pdf","checksum":"643a8d2d6f06f0888dcd7503f55d0920","success":1,"date_updated":"2021-05-03T06:54:06Z","date_created":"2021-05-03T06:54:06Z","relation":"main_file","file_id":"9366"}],"_id":"9359","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","intvolume":" 25","status":"public","ddc":["514","516","512"],"title":"Homological stability and densities of generalized configuration spaces","article_processing_charge":"No","has_accepted_license":"1","day":"27","keyword":["Generalized configuration spaces","homological stability","homological densities","chiral algebras","chiral homology","factorization algebras","Koszul duality","Ran space"],"date_published":"2021-04-27T00:00:00Z","citation":{"mla":"Ho, Quoc P. “Homological Stability and Densities of Generalized Configuration Spaces.” Geometry & Topology, vol. 25, no. 2, Mathematical Sciences Publishers, 2021, pp. 813–912, doi:10.2140/gt.2021.25.813.","short":"Q.P. Ho, Geometry & Topology 25 (2021) 813–912.","chicago":"Ho, Quoc P. “Homological Stability and Densities of Generalized Configuration Spaces.” Geometry & Topology. Mathematical Sciences Publishers, 2021. https://doi.org/10.2140/gt.2021.25.813.","ama":"Ho QP. Homological stability and densities of generalized configuration spaces. Geometry & Topology. 2021;25(2):813-912. doi:10.2140/gt.2021.25.813","ista":"Ho QP. 2021. Homological stability and densities of generalized configuration spaces. Geometry & Topology. 25(2), 813–912.","apa":"Ho, Q. P. (2021). Homological stability and densities of generalized configuration spaces. Geometry & Topology. Mathematical Sciences Publishers. https://doi.org/10.2140/gt.2021.25.813","ieee":"Q. P. Ho, “Homological stability and densities of generalized configuration spaces,” Geometry & Topology, vol. 25, no. 2. Mathematical Sciences Publishers, pp. 813–912, 2021."},"publication":"Geometry & Topology","page":"813-912","article_type":"original","ec_funded":1,"file_date_updated":"2021-05-03T06:54:06Z","author":[{"last_name":"Ho","first_name":"Quoc P","id":"3DD82E3C-F248-11E8-B48F-1D18A9856A87","full_name":"Ho, Quoc P"}],"volume":25,"date_updated":"2023-08-08T13:28:59Z","date_created":"2021-05-02T06:59:33Z","acknowledgement":"This paper owes an obvious intellectual debt to the illuminating treatments of factorization homology by J.\r\nFrancis, D. Gaitsgory, and J. Lurie in [GL,G1, FG]. The author would like to thank B. Farb and J. Wolfson for\r\nbringing the question of explaining coincidences in homological densities to his attention. Moreover, the author\r\nthanks J. Wolfson for many helpful conversations on the subject, O. Randal-Williams for many comments which\r\ngreatly help improve the exposition, and G. C. Drummond-Cole for many useful conversations on L∞-algebras.\r\nFinally, the author is grateful to the anonymous referee for carefully reading the manuscript and for providing\r\nnumerous comments which greatly helped improve the clarity and precision of the exposition.\r\nThis work is supported by the Advanced Grant “Arithmetic and Physics of Higgs moduli spaces” No. 320593 of\r\nthe European Research Council and the Lise Meitner fellowship “Algebro-Geometric Applications of Factorization\r\nHomology,” Austrian Science Fund (FWF): M 2751.","year":"2021","department":[{"_id":"TaHa"}],"publisher":"Mathematical Sciences Publishers","publication_status":"published","publication_identifier":{"issn":["1364-0380"]},"month":"04","doi":"10.2140/gt.2021.25.813","language":[{"iso":"eng"}],"oa":1,"external_id":{"isi":["000682738600005"],"arxiv":["1802.07948"]},"project":[{"call_identifier":"FP7","name":"Arithmetic and physics of Higgs moduli spaces","grant_number":"320593","_id":"25E549F4-B435-11E9-9278-68D0E5697425"},{"_id":"26B96266-B435-11E9-9278-68D0E5697425","grant_number":"M02751","call_identifier":"FWF","name":"Algebro-Geometric Applications of Factorization Homology"}],"isi":1,"quality_controlled":"1"},{"project":[{"_id":"26B96266-B435-11E9-9278-68D0E5697425","grant_number":"M02751","name":"Algebro-Geometric Applications of Factorization Homology","call_identifier":"FWF"}],"quality_controlled":"1","isi":1,"oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"arxiv":["1610.00212"],"isi":["000707040300031"]},"language":[{"iso":"eng"}],"doi":"10.1016/j.aim.2021.107992","publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]},"month":"09","publisher":"Elsevier","department":[{"_id":"TaHa"}],"publication_status":"published","acknowledgement":"The author would like to express his gratitude to D. Gaitsgory, without whose tireless guidance and encouragement in pursuing this problem, this work would not have been possible. The author is grateful to his advisor B.C. Ngô for many years of patient guidance and support. This paper is revised while the author is a postdoc in Hausel group at IST Austria. We thank him and the group for providing a wonderful research environment. The author also gratefully acknowledges the support of the Lise Meitner fellowship “Algebro-Geometric Applications of Factorization Homology,” Austrian Science Fund (FWF): M 2751.","year":"2021","volume":392,"date_created":"2021-09-21T15:58:59Z","date_updated":"2023-08-14T06:54:35Z","author":[{"last_name":"Ho","first_name":"Quoc P","orcid":"0000-0001-6889-1418","id":"3DD82E3C-F248-11E8-B48F-1D18A9856A87","full_name":"Ho, Quoc P"}],"article_number":"107992","file_date_updated":"2021-09-21T15:58:52Z","article_type":"original","citation":{"mla":"Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.” Advances in Mathematics, vol. 392, 107992, Elsevier, 2021, doi:10.1016/j.aim.2021.107992.","short":"Q.P. Ho, Advances in Mathematics 392 (2021).","chicago":"Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.” Advances in Mathematics. Elsevier, 2021. https://doi.org/10.1016/j.aim.2021.107992.","ama":"Ho QP. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. 2021;392. doi:10.1016/j.aim.2021.107992","ista":"Ho QP. 2021. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. 392, 107992.","ieee":"Q. P. Ho, “The Atiyah-Bott formula and connectivity in chiral Koszul duality,” Advances in Mathematics, vol. 392. Elsevier, 2021.","apa":"Ho, Q. P. (2021). The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2021.107992"},"publication":"Advances in Mathematics","date_published":"2021-09-21T00:00:00Z","keyword":["Chiral algebras","Chiral homology","Factorization algebras","Koszul duality","Ran space"],"scopus_import":"1","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","day":"21","intvolume":" 392","ddc":["514"],"status":"public","title":"The Atiyah-Bott formula and connectivity in chiral Koszul duality","_id":"10033","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa_version":"Published Version","file":[{"file_name":"1-s2.0-S000187082100431X-main.pdf","access_level":"open_access","creator":"qho","content_type":"application/pdf","file_size":840635,"file_id":"10034","relation":"main_file","date_created":"2021-09-21T15:58:52Z","date_updated":"2021-09-21T15:58:52Z","checksum":"f3c0086d41af11db31c00014efb38072"}],"type":"journal_article","abstract":[{"text":"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].","lang":"eng"}]}]