--- _id: '9359' 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" 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." article_processing_charge: No article_type: original author: - first_name: Quoc P full_name: Ho, Quoc P id: 3DD82E3C-F248-11E8-B48F-1D18A9856A87 last_name: Ho citation: 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 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 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. 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. ista: Ho QP. 2021. Homological stability and densities of generalized configuration spaces. Geometry & Topology. 25(2), 813–912. 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. date_created: 2021-05-02T06:59:33Z date_published: 2021-04-27T00:00:00Z date_updated: 2023-08-08T13:28:59Z day: '27' ddc: - '514' - '516' - '512' department: - _id: TaHa doi: 10.2140/gt.2021.25.813 ec_funded: 1 external_id: arxiv: - '1802.07948' isi: - '000682738600005' file: - access_level: open_access checksum: 643a8d2d6f06f0888dcd7503f55d0920 content_type: application/pdf creator: qho date_created: 2021-05-03T06:54:06Z date_updated: 2021-05-03T06:54:06Z file_id: '9366' file_name: densities.pdf file_size: 479268 relation: main_file success: 1 file_date_updated: 2021-05-03T06:54:06Z has_accepted_license: '1' intvolume: ' 25' isi: 1 issue: '2' keyword: - Generalized configuration spaces - homological stability - homological densities - chiral algebras - chiral homology - factorization algebras - Koszul duality - Ran space language: - iso: eng month: '04' oa: 1 oa_version: Submitted Version page: 813-912 project: - _id: 25E549F4-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '320593' name: Arithmetic and physics of Higgs moduli spaces - _id: 26B96266-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02751 name: Algebro-Geometric Applications of Factorization Homology publication: Geometry & Topology publication_identifier: issn: - 1364-0380 publication_status: published publisher: Mathematical Sciences Publishers quality_controlled: '1' status: public title: Homological stability and densities of generalized configuration spaces type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 25 year: '2021' ... --- _id: '10033' abstract: - lang: eng 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]. 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.' article_number: '107992' article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Quoc P full_name: Ho, Quoc P id: 3DD82E3C-F248-11E8-B48F-1D18A9856A87 last_name: Ho orcid: 0000-0001-6889-1418 citation: 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 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 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. ieee: Q. P. Ho, “The Atiyah-Bott formula and connectivity in chiral Koszul duality,” Advances in Mathematics, vol. 392. Elsevier, 2021. ista: Ho QP. 2021. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. 392, 107992. 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). date_created: 2021-09-21T15:58:59Z date_published: 2021-09-21T00:00:00Z date_updated: 2023-08-14T06:54:35Z day: '21' ddc: - '514' department: - _id: TaHa doi: 10.1016/j.aim.2021.107992 external_id: arxiv: - '1610.00212' isi: - '000707040300031' file: - access_level: open_access checksum: f3c0086d41af11db31c00014efb38072 content_type: application/pdf creator: qho date_created: 2021-09-21T15:58:52Z date_updated: 2021-09-21T15:58:52Z file_id: '10034' file_name: 1-s2.0-S000187082100431X-main.pdf file_size: 840635 relation: main_file file_date_updated: 2021-09-21T15:58:52Z has_accepted_license: '1' intvolume: ' 392' isi: 1 keyword: - Chiral algebras - Chiral homology - Factorization algebras - Koszul duality - Ran space language: - iso: eng month: '09' oa: 1 oa_version: Published Version project: - _id: 26B96266-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M02751 name: Algebro-Geometric Applications of Factorization Homology publication: Advances in Mathematics publication_identifier: eissn: - 1090-2082 issn: - 0001-8708 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: The Atiyah-Bott formula and connectivity in chiral Koszul duality tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 392 year: '2021' ...