{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T06:41:49Z","timestamp":1763620909124,"version":"3.37.3"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,2,6]],"date-time":"2021-02-06T00:00:00Z","timestamp":1612569600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,2,6]],"date-time":"2021-02-06T00:00:00Z","timestamp":1612569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N015452\/1","EP\/N016505\/1"],"award-info":[{"award-number":["EP\/N015452\/1","EP\/N016505\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study Maurer\u2013Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several different contexts, Schlessinger\u2013Stasheff type theorems comparing the notions of homotopy and gauge equivalence for Maurer\u2013Cartan elements as well as their categorified versions. As an application, we re-prove and generalize Block\u2013Smith\u2019s higher Riemann\u2013Hilbert correspondence, and develop its analogue for simplicial complexes and topological spaces.<\/jats:p>","DOI":"10.1007\/s10485-021-09631-3","type":"journal-article","created":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T00:15:36Z","timestamp":1612829736000},"page":"685-728","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Maurer\u2013Cartan Moduli and Theorems of Riemann\u2013Hilbert Type"],"prefix":"10.1007","volume":"29","author":[{"given":"Joseph","family":"Chuang","sequence":"first","affiliation":[]},{"given":"Julian","family":"Holstein","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9938-7174","authenticated-orcid":false,"given":"Andrey","family":"Lazarev","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,2,6]]},"reference":[{"key":"9631_CR1","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1073\/pnas.42.7.409","volume":"42","author":"J Adams","year":"1956","unstructured":"Adams, J.: On the cobar construction. Proc. Natl. Acad. Sci. U.S.A. 42, 409\u2013412 (1956)","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"9631_CR2","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1090\/S0894-0347-1989-0991015-7","volume":"2","author":"DJ Anick","year":"1989","unstructured":"Anick, D.J.: Hopf algebras up to homotopy. J. Am. Math. Soc. 2, 417\u2013453 (1989)","journal-title":"J. Am. Math. Soc."},{"key":"9631_CR3","doi-asserted-by":"crossref","unstructured":"Araki, H.: Expansional in banach algebras, in Annales scientifiques de l\u2019\u00c9cole Normale Sup\u00e9rieure, vol.\u00a06, Elsevier, pp.\u00a067\u201384 (1973)","DOI":"10.24033\/asens.1243"},{"issue":"4","key":"9631_CR4","first-page":"961","volume":"2018","author":"CA Abad","year":"2016","unstructured":"Abad, C.A., Sch\u00e4tz, F.: Flat $${\\mathbb{Z}}$$-graded connections and loop spaces. Int. Math. Res. Not. IMRN 2018(4), 961\u20131008 (2016)","journal-title":"Int. Math. Res. Not. IMRN"},{"key":"9631_CR5","doi-asserted-by":"crossref","unstructured":"Behrend, K., Getzler, E.: Geometric higher groupoids and categories, in Geometry, analysis and probability, vol.\u00a0310 of Progr. Math., Birkh\u00e4user\/Springer, Cham, pp.\u00a01\u201345 (2017)","DOI":"10.1007\/978-3-319-49638-2_1"},{"key":"9631_CR6","doi-asserted-by":"crossref","unstructured":"Block, J.: Duality and equivalence of module categories in noncommutative geometry, in CRM Proceedings and Lecture Notes, vol.\u00a050 (2010)","DOI":"10.1090\/crmp\/050\/24"},{"key":"9631_CR7","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/j.aim.2013.11.001","volume":"252","author":"J Block","year":"2014","unstructured":"Block, J., Smith, A.M.: The higher Riemann\u2013Hilbert correspondence. Adv. Math. 252, 382\u2013405 (2014)","journal-title":"Adv. Math."},{"key":"9631_CR8","first-page":"209","volume":"86","author":"M B\u00f6kstedt","year":"1993","unstructured":"B\u00f6kstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86, 209\u2013234 (1993)","journal-title":"Compos. Math."},{"key":"9631_CR9","first-page":"669","volume":"181","author":"AI Bondal","year":"1990","unstructured":"Bondal, A.I., Kapranov, M.M.: Framed triangulated categories. Mat. Sb. 181, 669\u2013683 (1990)","journal-title":"Mat. Sb."},{"key":"9631_CR10","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1016\/j.aim.2018.02.004","volume":"328","author":"C Braun","year":"2018","unstructured":"Braun, C., Chuang, J., Lazarev, A.: Derived localisation of algebras and modules. Adv. Math. 328, 555\u2013622 (2018)","journal-title":"Adv. Math."},{"key":"9631_CR11","doi-asserted-by":"publisher","first-page":"223","DOI":"10.2307\/1970101","volume":"69","author":"E Brown Jr","year":"1959","unstructured":"Brown Jr., E.: Twisted tensor products, I. Ann. Math. 69, 223\u2013246 (1959)","journal-title":"Ann. Math."},{"key":"9631_CR12","doi-asserted-by":"publisher","first-page":"3202","DOI":"10.1080\/00927872.2018.1554744","volume":"47","author":"U Choudhury","year":"2019","unstructured":"Choudhury, U., de Souza, M.G.A.: Homotopy theory of dg sheaves. Commun. Algebra 47, 3202\u20133228 (2019)","journal-title":"Commun. Algebra"},{"key":"9631_CR13","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1112\/jlms\/jdp073","volume":"81","author":"J Chuang","year":"2010","unstructured":"Chuang, J., Lazarev, A.: Feynman diagrams and minimal models for operadic algebras. J. Lond. Math. Soc. (2) 81, 317\u2013337 (2010)","journal-title":"J. Lond. Math. Soc. (2)"},{"key":"9631_CR14","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1016\/j.jalgebra.2018.10.032","volume":"519","author":"J Chuang","year":"2019","unstructured":"Chuang, J., Lazarev, A.: On the perturbation algebra. J. Algebra 519, 130\u2013148 (2019)","journal-title":"J. Algebra"},{"key":"9631_CR15","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1016\/j.jalgebra.2003.05.001","volume":"272","author":"V Drinfeld","year":"2004","unstructured":"Drinfeld, V.: DG quotients of DG categories. J. Algebra 272, 643\u2013691 (2004)","journal-title":"J. Algebra"},{"key":"9631_CR16","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1017\/S0305004103007175","volume":"136","author":"D Dugger","year":"2004","unstructured":"Dugger, D., Hollander, S., Isaksen, D.: Hypercovers and simplicial presheaves. Math. Proc. Camb. Phil. Soc. 136, 9\u201351 (2004)","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"9631_CR17","doi-asserted-by":"publisher","first-page":"323","DOI":"10.24033\/bsmf.1583","volume":"90","author":"P Gabriel","year":"1962","unstructured":"Gabriel, P.: Des cat\u00e9gories ab\u00e9liennes. Bull. Soc. Math. France 90, 323\u2013448 (1962)","journal-title":"Bull. Soc. Math. France"},{"key":"9631_CR18","volume-title":"Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics","author":"O Garcia-Prada","year":"2007","unstructured":"Garcia-Prada, O., Wells, R.: Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics. Springer, New York (2007)"},{"key":"9631_CR19","doi-asserted-by":"crossref","unstructured":"Goerss, P., Jardine, J.: Simplicial homotopy theory, Progress in Mathematics Series, Birkh\u00e4user Verlag (1999)","DOI":"10.1007\/978-3-0348-8707-6"},{"key":"9631_CR20","volume-title":"Model Categories and Simplicial Methods, in Contemporary Mathematics","author":"P Goerss","year":"2007","unstructured":"Goerss, P., Schemmerhorn, K.: Model Categories and Simplicial Methods, in Contemporary Mathematics. American Mathematical Society, Providence (2007)"},{"key":"9631_CR21","doi-asserted-by":"publisher","first-page":"1503","DOI":"10.2140\/agt.2009.9.1503","volume":"9","author":"A Hamilton","year":"2009","unstructured":"Hamilton, A., Lazarev, A.: Cohomology theories for homotopy algebras and noncommutative geometry. Algebr. Geom. Topol. 9, 1503\u20131583 (2009)","journal-title":"Algebr. Geom. Topol."},{"key":"9631_CR22","volume-title":"Algebraic Topology","author":"A Hatcher","year":"2002","unstructured":"Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)"},{"key":"9631_CR23","doi-asserted-by":"publisher","first-page":"263","DOI":"10.4310\/HHA.2014.v16.n2.a14","volume":"16","author":"JVS Holstein","year":"2014","unstructured":"Holstein, J.V.S.: Properness and simplicial resolutions for the model category dgCat. Homol. Homotopy Appl. 16, 263\u2013273 (2014)","journal-title":"Homol. Homotopy Appl."},{"key":"9631_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0305004114000516","volume":"158","author":"JVS Holstein","year":"2015","unstructured":"Holstein, J.V.S.: Morita cohomology. Math. Proc. Camb. Phil. Soc. 158, 1\u201326 (2015)","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"9631_CR25","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1017\/S0305004114000528","volume":"158","author":"JVS Holstein","year":"2015","unstructured":"Holstein, J.V.S.: Morita cohomology and homotopy locally constant sheaves. Math. Proc. Camb. Phil. Soc. 158, 27\u201335 (2015)","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"9631_CR26","volume-title":"An introduction of Complex Geometry, Universitext","author":"D Huybrechts","year":"2005","unstructured":"Huybrechts, D.: An introduction of Complex Geometry, Universitext. Springer, Berlin (2005)"},{"key":"9631_CR27","unstructured":"Jarchow, H.: Locally Convex Spaces. Vieweg+Teubner Verlag, Mathematische Leitf\u00e4den (2012)"},{"issue":"1","key":"9631_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4310\/HHA.2001.v3.n1.a1","volume":"3","author":"B Keller","year":"2001","unstructured":"Keller, B.: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3(1), 1\u201335 (2001)","journal-title":"Homology Homotopy Appl."},{"key":"9631_CR29","doi-asserted-by":"crossref","unstructured":"Keller, B.: On differential graded categories, in International Congress of Mathematicians, vol. II, pp. 151\u2013190. Z\u00fcrich, Eur. Math. Soc. (2006)","DOI":"10.4171\/022-2\/8"},{"key":"9631_CR30","unstructured":"Kontsevich, M.: The symplectic geometry of homological algebra. http:\/\/www.ihes.fr\/~maxim\/TEXTS\/SymplecticAT2009.pdf"},{"key":"9631_CR31","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1023\/B:MATH.0000027508.00421.bf","volume":"66","author":"M Kontsevich","year":"2003","unstructured":"Kontsevich, M.: Deformation quantization of poisson manifolds. Lett. Math. Phys. 66, 157\u2013216 (2003)","journal-title":"Lett. Math. Phys."},{"key":"9631_CR32","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/053","volume-title":"The Convenient Setting of Global Analysis, vol.\u00a053 of Mathematical Surveys and Monographs","author":"A Kriegl","year":"1997","unstructured":"Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis, vol.\u00a053 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)"},{"key":"9631_CR33","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.aim.2012.11.009","volume":"235","author":"A Lazarev","year":"2013","unstructured":"Lazarev, A.: Maurer\u2013Cartan moduli and models for function spaces. Adv. Math. 235, 296\u2013320 (2013)","journal-title":"Adv. Math."},{"key":"9631_CR34","unstructured":"MacLane, S.: Homology, Springer-Verlag, Berlin-New York, first\u00a0ed., 1967. Die Grundlehren der mathematischen Wissenschaften, Band 114"},{"key":"9631_CR35","volume-title":"Ideals of Differentiable Functions","author":"B Malgrange","year":"1966","unstructured":"Malgrange, B.: Ideals of Differentiable Functions. Oxford University Press, Oxford (1966)"},{"key":"9631_CR36","doi-asserted-by":"publisher","first-page":"5209","DOI":"10.1081\/AGB-100106814","volume":"29","author":"M Markl","year":"2001","unstructured":"Markl, M.: Ideal perturbation lemma. Commun. Algebra 29, 5209\u20135232 (2001)","journal-title":"Commun. Algebra"},{"key":"9631_CR37","unstructured":"May, J.P.: Simplicial objects in algebraic topology. Van Nostrand Mathematical Studies, No. 11 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, p vi+161 (1967)"},{"key":"9631_CR38","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1070\/RM1971v026n04ABEH003983","volume":"26","author":"V Mityagin","year":"1971","unstructured":"Mityagin, V., Khenkin, G.: Linear problems of complex analysis. Russ. Math. Surv. 26, 99\u2013164 (1971)","journal-title":"Russ. Math. Surv."},{"key":"9631_CR39","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1002\/mana.200810263","volume":"286","author":"AS Morye","year":"2013","unstructured":"Morye, A.S.: Note on the Serre-Swan Theorem. Math. Nachr. 286, 272\u2013278 (2013)","journal-title":"Math. Nachr."},{"key":"9631_CR40","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1090\/S0077-1554-08-00169-6","volume":"69","author":"A Pirkovskii","year":"2008","unstructured":"Pirkovskii, A.: Arens\u2013Michael envelopes, homological epimorphisms, and relatively quasi-free algebras. Trans. Moscow Math. Soc. 69, 27\u2013104 (2008)","journal-title":"Trans. Moscow Math. Soc."},{"key":"9631_CR41","first-page":"vi+133","volume":"212","author":"L Positselski","year":"2011","unstructured":"Positselski, L.: Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Am. Math. Soc. 212, vi+133 (2011)","journal-title":"Mem. Am. Math. Soc."},{"key":"9631_CR42","unstructured":"Positselski, L.: Contramodules (2015). arXiv:1503.00991"},{"key":"9631_CR43","first-page":"197","volume":"472","author":"D Quillen","year":"1996","unstructured":"Quillen, D.: $$K_0$$ for nonunital rings and Morita invariance. J. Reine Angew. Math. 472, 197\u2013217 (1996)","journal-title":"J. Reine Angew. Math."},{"key":"9631_CR44","doi-asserted-by":"publisher","first-page":"3789","DOI":"10.2140\/agt.2018.18.3789","volume":"18","author":"M Rivera","year":"2018","unstructured":"Rivera, M., Zeinalian, M.: Cubical rigidification, the cobar construction and the based loop space. Algebr. Geom. Topol. 18, 3789\u20133820 (2018)","journal-title":"Algebr. Geom. Topol."},{"key":"9631_CR45","unstructured":"Schlessinger, M., Stasheff, J.: Deformation theory and rational homotopy type (2012). arXiv:1211.1647"},{"key":"9631_CR46","first-page":"625","volume":"15","author":"RT Seeley","year":"1964","unstructured":"Seeley, R.T.: Extension of $$C^{\\infty }$$ functions defined in a half space. Proc. Am. Math. Soc. 15, 625\u2013626 (1964)","journal-title":"Proc. Am. Math. Soc."},{"key":"9631_CR47","unstructured":"Sella, Y.: Comparison of sheaf cohomology and singular cohomology (2016). arXiv:1602.06674"},{"key":"9631_CR48","doi-asserted-by":"crossref","unstructured":"Tabuada, G.: Invariants additifs de dg-categories. Int. Math. Res. Not., p.\u00a03309 (2005). arXiv:0507227v1","DOI":"10.1155\/IMRN.2005.3309"},{"key":"9631_CR49","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s00222-006-0025-y","volume":"167","author":"B To\u00ebn","year":"2007","unstructured":"To\u00ebn, B.: The homotopy theory of $$dg$$-categories and derived Morita theory. Invent. Math. 167, 615\u2013667 (2007)","journal-title":"Invent. Math."},{"key":"9631_CR50","unstructured":"Treves, F.: Topological Vector Spaces. Dover books on mathematics, Academic Press, Distributions and Kernels (1967)"},{"key":"9631_CR51","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1007\/s00029-014-0166-6","volume":"21","author":"M Van\u00a0den Bergh","year":"2015","unstructured":"Van\u00a0den Bergh, M.: Calabi\u2013Yau algebras and superpotentials. Sel. Math. (N.S.) 21, 555\u2013603 (2015)","journal-title":"Sel. Math. (N.S.)"}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-021-09631-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10485-021-09631-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-021-09631-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T05:07:14Z","timestamp":1625548034000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10485-021-09631-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,6]]},"references-count":51,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,8]]}},"alternative-id":["9631"],"URL":"https:\/\/doi.org\/10.1007\/s10485-021-09631-3","relation":{},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"type":"print","value":"0927-2852"},{"type":"electronic","value":"1572-9095"}],"subject":[],"published":{"date-parts":[[2021,2,6]]},"assertion":[{"value":"21 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 February 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}