---
_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
arxiv: 1
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. Homological stability and densities of generalized configuration spaces.
    <i>Geometry &#38; Topology</i>. 2021;25(2):813-912. doi:<a href="https://doi.org/10.2140/gt.2021.25.813">10.2140/gt.2021.25.813</a>
  apa: Ho, Q. P. (2021). Homological stability and densities of generalized configuration
    spaces. <i>Geometry &#38; Topology</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/gt.2021.25.813">https://doi.org/10.2140/gt.2021.25.813</a>
  chicago: Ho, Quoc P. “Homological Stability and Densities of Generalized Configuration
    Spaces.” <i>Geometry &#38; Topology</i>. Mathematical Sciences Publishers, 2021.
    <a href="https://doi.org/10.2140/gt.2021.25.813">https://doi.org/10.2140/gt.2021.25.813</a>.
  ieee: Q. P. Ho, “Homological stability and densities of generalized configuration
    spaces,” <i>Geometry &#38; Topology</i>, vol. 25, no. 2. Mathematical Sciences
    Publishers, pp. 813–912, 2021.
  ista: Ho QP. 2021. Homological stability and densities of generalized configuration
    spaces. Geometry &#38; Topology. 25(2), 813–912.
  mla: Ho, Quoc P. “Homological Stability and Densities of Generalized Configuration
    Spaces.” <i>Geometry &#38; Topology</i>, vol. 25, no. 2, Mathematical Sciences
    Publishers, 2021, pp. 813–912, doi:<a href="https://doi.org/10.2140/gt.2021.25.813">10.2140/gt.2021.25.813</a>.
  short: Q.P. Ho, Geometry &#38; Topology 25 (2021) 813–912.
corr_author: '1'
date_created: 2021-05-02T06:59:33Z
date_published: 2021-04-27T00:00:00Z
date_updated: 2025-04-14T09:09:36Z
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'
scopus_import: '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'
...