{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T10:23:11Z","timestamp":1771064591419,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T00:00:00Z","timestamp":1649376000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T00:00:00Z","timestamp":1649376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Max Planck Institute for Mathematics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension\/restriction adjunctions.<\/jats:p>","DOI":"10.1007\/s10485-022-09675-z","type":"journal-article","created":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T07:04:17Z","timestamp":1649401457000},"page":"825-920","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Projective and Reedy Model Category Structures for (Infinitesimal) Bimodules over an Operad"],"prefix":"10.1007","volume":"30","author":[{"given":"Julien","family":"Ducoulombier","sequence":"first","affiliation":[]},{"given":"Benoit","family":"Fresse","sequence":"additional","affiliation":[]},{"given":"Victor","family":"Turchin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,8]]},"reference":[{"issue":"7","key":"9675_CR1","doi-asserted-by":"publisher","first-page":"2323","DOI":"10.1090\/S0002-9939-08-09185-5","volume":"136","author":"V Angeltveit","year":"2008","unstructured":"Angeltveit, V.: Enriched Reedy categories. Proc. Am. Math. Soc. 136(7), 2323\u20132332 (2008)","journal-title":"Proc. Am. Math. Soc."},{"issue":"3","key":"9675_CR2","doi-asserted-by":"publisher","first-page":"1445","DOI":"10.2140\/gt.2016.20.1445","volume":"20","author":"G Arone","year":"2016","unstructured":"Arone, G., Ching, M.: Cross-effects and the classification of Taylor towers. Geom. Topol. 20(3), 1445\u20131537 (2016)","journal-title":"Geom. Topol."},{"issue":"3","key":"9675_CR3","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.2140\/gt.2014.18.1261","volume":"18","author":"G Arone","year":"2014","unstructured":"Arone, G., Turchin, V.: On the rational homology of high dimensional analogues of spaces of long knots. Geom. Topol. 18(3), 1261\u20131322 (2014)","journal-title":"Geom. Topol."},{"key":"9675_CR4","unstructured":"Barwick, C.: On Reedy Model Categories, Unpublished. arXiv:0708.2832 (2007)"},{"key":"9675_CR5","doi-asserted-by":"crossref","unstructured":"Bayeh, M., Hess, K., Karpova, V., Kedziorek, M., Riehl, E., Shipley, B.: Left-induced model structures and diagram categories. In: Women in Topology: Collaborations in Homotopy Theory, Contemporary Mathematics, vol. 641, pp. 49\u201381. American Mathematical Society, Providence (2015)","DOI":"10.1090\/conm\/641\/12859"},{"issue":"4","key":"9675_CR6","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1007\/s00014-003-0772-y","volume":"78","author":"C Berger","year":"2003","unstructured":"Berger, C., Moerdijk, I.: Axiomatic homotopy theory for operads. Comment. Math. Helv. 78(4), 805\u2013831 (2003)","journal-title":"Comment. Math. Helv."},{"key":"9675_CR7","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1016\/j.top.2006.05.001","volume":"45","author":"C Berger","year":"2006","unstructured":"Berger, C., Moerdijk, I.: The Boardman\u2013Vogt resolution of operads in monoidal model categories. Topology 45, 807\u2013849 (2006)","journal-title":"Topology"},{"key":"9675_CR8","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1090\/conm\/431\/08265","volume":"431","author":"C Berger","year":"2007","unstructured":"Berger, C., Moerdijk, I.: Resolution of coloured operads and rectification of homotopy algebras. Contemp. Math. 431, 31\u201358 (2007)","journal-title":"Contemp. Math."},{"issue":"3\u20134","key":"9675_CR9","first-page":"977","volume":"269","author":"C Berger","year":"2008","unstructured":"Berger, C., Moerdijk, I.: On an extension of the notion of Reedy category. Math. Z. 269(3\u20134), 977\u20131004 (2008)","journal-title":"Math. Z."},{"key":"9675_CR10","unstructured":"Brown, R.: Topology and groupoids. Third edition of Elements of modern topology [McGraw-Hill, New York, 1968]. BookSurge, LLC, Charleston, SC, 2006. xxvi+512 pp"},{"key":"9675_CR11","doi-asserted-by":"crossref","first-page":"409","DOI":"10.4171\/dm\/147","volume":"8","author":"B Dundas","year":"2003","unstructured":"Dundas, B., R\u00f6ndigs, O., \u00d8stv\u00e6r, P.A.: Enriched functors and stable homotopy theory. Doc. Math. 8, 409\u2013488 (2003)","journal-title":"Doc. Math."},{"issue":"3","key":"9675_CR12","doi-asserted-by":"publisher","first-page":"1683","DOI":"10.2140\/agt.2016.16.1683","volume":"16","author":"J Ducoulombier","year":"2016","unstructured":"Ducoulombier, J.: Swiss-Cheese action on the totalization of action-operads. Algebr. Geom. Topol. 16(3), 1683\u20131726 (2016)","journal-title":"Algebr. Geom. Topol."},{"issue":"2","key":"9675_CR13","doi-asserted-by":"publisher","first-page":"661","DOI":"10.5802\/aif.3175","volume":"68","author":"J Ducoulombier","year":"2018","unstructured":"Ducoulombier, J.: From maps between coloured operads to Swiss-Cheese algebras. Ann. Inst. Fourier (Grenoble) 68(2), 661\u2013724 (2018)","journal-title":"Ann. Inst. Fourier (Grenoble)"},{"key":"9675_CR14","unstructured":"Ducoulombier, J.: Delooping of high-dimensional spaces of string links. arXiv:1809.00682 (2018)"},{"key":"9675_CR15","doi-asserted-by":"crossref","unstructured":"Ducoulombier, J., Turchin, V.: Delooping the functor calculus tower. In: The Proceedings of the London Mathematical Society. arXiv:1708.02203 (2022) (to appear)","DOI":"10.1112\/plms.12440"},{"key":"9675_CR16","doi-asserted-by":"publisher","first-page":"7657","DOI":"10.1090\/tran\/8451","volume":"374","author":"J Ducoulombier","year":"2021","unstructured":"Ducoulombier, J., Turchin, V., Willwacher, T.: On the delooping of (framed) embedding spaces. Trans. Am. Math. Soc. 374, 7657\u20137677 (2021)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"9675_CR17","doi-asserted-by":"publisher","first-page":"181","DOI":"10.2307\/2046572","volume":"101","author":"ED Farjoun","year":"1987","unstructured":"Farjoun, E.D.: Homotopy theories for diagrams of spaces. Proc. Am. Math. Soc. 101(1), 181\u2013189 (1987)","journal-title":"Proc. Am. Math. Soc."},{"key":"9675_CR18","doi-asserted-by":"crossref","unstructured":"Fresse, B.: Homotopy of operads and Grothendieck\u2013Teichm\u00fcller groups. Part 1. The algebraic theory and its topological background. In: Mathematical Surveys and Monographs, vol. 217. American Mathematical Society, Providence (2017)","DOI":"10.1090\/surv\/217.1"},{"key":"9675_CR19","doi-asserted-by":"crossref","unstructured":"Fresse, B.: Homotopy of operads and Grothendieck-Teichm\u00fcller groups. Part 2. The applications of (rational) homotopy theory methods. In: Mathematical Surveys and Monographs, vol. 217. American Mathematical Society, Providence (2017)","DOI":"10.1090\/surv\/217.1"},{"key":"9675_CR20","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-89056-0","volume-title":"Modules Over Operads and Functors","author":"B Fresse","year":"2009","unstructured":"Fresse, B.: Modules Over Operads and Functors. Lecture Notes in Mathematics, vol. 1967. Springer, Berlin (2009)"},{"issue":"4","key":"9675_CR21","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s40062-018-0198-2","volume":"13","author":"B Fresse","year":"2018","unstructured":"Fresse, B., Turchin, V., Willwacher, T.: The homotopy theory of operad subcategories. J. Homotopy Relat. Struct. 13(4), 689\u2013702 (2018)","journal-title":"J. Homotopy Relat. Struct."},{"issue":"1","key":"9675_CR22","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1112\/topo.12123","volume":"13","author":"R Garner","year":"2020","unstructured":"Garner, R., Kedziorek, M., Riehl, E.: Lifting accessible model structures. J. Topol. 13(1), 59\u201376 (2020)","journal-title":"J. Topol."},{"key":"9675_CR23","doi-asserted-by":"crossref","unstructured":"Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. In: Progress in Mathematics, vol. 174. Birkh\u00e4user Verlag, Basel (1999). xvi+510 pp. ISBN: 3-7643-6064-X","DOI":"10.1007\/978-3-0348-8707-6"},{"key":"9675_CR24","doi-asserted-by":"publisher","first-page":"1407","DOI":"10.1016\/j.jpaa.2009.11.006","volume":"214","author":"JE Harper","year":"2010","unstructured":"Harper, J.E.: Homotopy theory of modules over operads and non-$$\\Sigma $$ operads in monoidal model categories. J. Pure Appl. Algebra 214, 1407\u20131434 (2010)","journal-title":"J. Pure Appl. Algebra"},{"issue":"2","key":"9675_CR25","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1112\/topo.12011","volume":"10","author":"K Hess","year":"2017","unstructured":"Hess, K., Kedziorek, M., Riehl, E., Shipley, B.: A necessary and sufficient condition for induced model structures. J. Topol. 10(2), 324\u2013369 (2017)","journal-title":"J. Topol."},{"key":"9675_CR26","unstructured":"Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs, vol. 99, p.xvi+457 (2003)"},{"issue":"5","key":"9675_CR27","doi-asserted-by":"publisher","first-page":"2691","DOI":"10.2140\/agt.2016.16.2691","volume":"16","author":"P Hackney","year":"2016","unstructured":"Hackney, P., Robertson, M., Yau, D.: Relative left properness of colored operads. Algebr. Geom. Topol. 16(5), 2691\u20132714 (2016)","journal-title":"Algebr. Geom. Topol."},{"key":"9675_CR28","unstructured":"Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)"},{"key":"9675_CR29","unstructured":"Hovey, M.: Monoidal model categories. arXiv:math\/9803002 (1998)"},{"key":"9675_CR30","first-page":"1","volume":"13","author":"GM Kelly","year":"2005","unstructured":"Kelly, G.M.: On the operads of J. P. May. Repr. Theory Appl. Categ. 13, 1\u201313 (2005)","journal-title":"Repr. Theory Appl. Categ."},{"key":"9675_CR31","unstructured":"Lewis, G.: Compactly generated spaces, appendix A of The Stable Category and Generalized Thom Spectra, PhD thesis Chicago (1978)"},{"key":"9675_CR32","series-title":"Annals of Mathematics Studies","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830558","volume-title":"Higher Topos Theory","author":"J Lurie","year":"2009","unstructured":"Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)"},{"issue":"2","key":"9675_CR33","doi-asserted-by":"publisher","first-page":"279","DOI":"10.4310\/HHA.2019.v21.n2.a15","volume":"21","author":"L Moser","year":"2019","unstructured":"Moser, L.: Injective and projective model structures on enriched diagram categories. Homol. Homotopy Appl. 21(2), 279\u2013300 (2019)","journal-title":"Homol. Homotopy Appl."},{"key":"9675_CR34","unstructured":"Reedy, C.: Homotopy theory of model categories, Unpublished. http:\/\/www-math.mit.edu\/~psh\/ (1974)"},{"key":"9675_CR35","unstructured":"Rezk, C.: Spaces of algebra structures and cohomology of operads. PhD thesis, MIT (1996)"},{"issue":"3","key":"9675_CR36","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1090\/S0002-9947-00-02653-2","volume":"353","author":"C Rezk","year":"2001","unstructured":"Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Am. Math. Soc. 353(3), 973\u20131007 (2001)","journal-title":"Trans. Am. Math. Soc."},{"key":"9675_CR37","first-page":"256","volume":"29","author":"E Riehl","year":"2014","unstructured":"Riehl, E., Verity, D.: The theory and practice of Reedy categories. Theory Appl. Categ. 29, 256\u2013301 (2014)","journal-title":"Theory Appl. Categ."}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-022-09675-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10485-022-09675-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-022-09675-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T02:20:02Z","timestamp":1675218002000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10485-022-09675-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,8]]},"references-count":37,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["9675"],"URL":"https:\/\/doi.org\/10.1007\/s10485-022-09675-z","relation":{},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"value":"0927-2852","type":"print"},{"value":"1572-9095","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,8]]},"assertion":[{"value":"13 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}