{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T07:42:27Z","timestamp":1778830947372,"version":"3.51.4"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Praemium Academiae","award":["RVO:67985840"],"award-info":[{"award-number":["RVO:67985840"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s10485-025-09832-0","type":"journal-article","created":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:44:17Z","timestamp":1760175857000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Triple Delooping for Multiplicative Hyperoperads"],"prefix":"10.1007","volume":"33","author":[{"given":"Florian","family":"De Leger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maro\u0161","family":"Grego","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,11]]},"reference":[{"issue":"2","key":"9832_CR1","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1006\/aima.1997.1695","volume":"135","author":"John C Baez","year":"1998","unstructured":"Baez, John C., Dolan, James: Higher-dimensional algebra iii. n-categories and the algebra of opetopes. Adv. Math. 135(2), 145\u2013206 (1998)","journal-title":"Adv. Math."},{"issue":"1","key":"9832_CR2","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1016\/j.aim.2007.06.014","volume":"217","author":"M Batanin","year":"2008","unstructured":"Batanin, M.: The eckmann-hilton argument and higher operads. Adv. Math. 217(1), 334\u2013385 (2008)","journal-title":"Adv. Math."},{"issue":"6","key":"9832_CR3","first-page":"148","volume":"32","author":"M Batanin","year":"2017","unstructured":"Batanin, M., Berger, C.: Homotopy theory for algebras over polynomial monads. Theory Appl. Categ. 32(6), 148\u2013253 (2017)","journal-title":"Theory Appl. Categ."},{"issue":"4","key":"9832_CR4","doi-asserted-by":"publisher","first-page":"1521","DOI":"10.4171\/jncg\/355","volume":"13","author":"M Batanin","year":"2019","unstructured":"Batanin, M., Leger, F.: Polynomial monads and delooping of mapping spaces. J. Noncommutative Geom. 13(4), 1521\u20131576 (2019)","journal-title":"J. Noncommutative Geom."},{"key":"9832_CR5","unstructured":"Batanin, M., De\u00a0Leger, F., White, D.: Model structures on operads and algebras from a global perspective. To be published in the Tunisian Journal of Mathematics (2023). arXiv:2311.07320"},{"key":"9832_CR6","unstructured":"Batanin, M., De\u00a0Leger, F., White, D.: Quasi-tame substitudes and the Grothendieck construction.To be published in Theory and Applications of Categories (2023). arXiv:2311.07322"},{"issue":"2","key":"9832_CR7","first-page":"3569","volume":"357","author":"M Batanin","year":"2022","unstructured":"Batanin, M., White, D.: Homotopy theory of algebras of substitudes and their localisation. Trans. Am. Math. Soc. 357(2), 3569\u20133640 (2022)","journal-title":"Trans. Am. Math. Soc."},{"issue":"39","key":"9832_CR8","first-page":"1485","volume":"38","author":"C Berger","year":"2022","unstructured":"Berger, C.: Moment categories and operads. Theory Appl. Categ. 38(39), 1485\u20131537 (2022)","journal-title":"Theory Appl. Categ."},{"key":"9832_CR9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525872","volume-title":"Handbook of categorical algebra: volume 1, Basic category theory","author":"Francis Borceux","year":"1994","unstructured":"Borceux, Francis: Handbook of categorical algebra: volume 1, Basic category theory, vol. 1. Cambridge University Press, Cambridge (1994)"},{"key":"9832_CR10","unstructured":"Cisinski, D.-C. Les pr\u00e9faisceaux comme mod\u00e8les des types d\u2019homotopie. Soci\u00e9t\u00e9 math\u00e9matique de France, (2006)"},{"key":"9832_CR11","unstructured":"De\u00a0Leger, F.: Cofinal morphism of polynomial monads and double delooping. arXiv preprint arXiv:2205.09149 (2022)"},{"key":"9832_CR12","unstructured":"De\u00a0Leger, F.: The fibration sequences of the lattice path operad. arXiv preprint arXiv:2503.15179 (2025)"},{"key":"9832_CR13","doi-asserted-by":"crossref","unstructured":"De\u00a0Leger, F., Grego, M.: Condensation of the operad for multiplicative hyperoperads. arXiv:2507.10192, (2025)","DOI":"10.1007\/s10485-025-09832-0"},{"issue":"2","key":"9832_CR14","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s40062-018-0217-3","volume":"14","author":"J Ducoulombier","year":"2019","unstructured":"Ducoulombier, J.: Delooping derived mapping spaces of bimodules over an operad. J. Homot. Relat. Struct. 14(2), 411\u2013453 (2019)","journal-title":"J. Homot. Relat. Struct."},{"issue":"6","key":"9832_CR15","doi-asserted-by":"publisher","first-page":"772","DOI":"10.1112\/plms.12440","volume":"124","author":"J Ducoulombier","year":"2022","unstructured":"Ducoulombier, J., Turchin, V.: Delooping the functor calculus tower. Proc. Lond. Math. Soc. 124(6), 772\u2013853 (2022)","journal-title":"Proc. Lond. Math. Soc."},{"issue":"2","key":"9832_CR16","doi-asserted-by":"publisher","first-page":"919","DOI":"10.2140\/gt.2012.16.919","volume":"16","author":"W Dwyer","year":"2012","unstructured":"Dwyer, W., Hess, K.: Long knots and maps between operads. Geomet. Topol. 16(2), 919\u2013955 (2012)","journal-title":"Geomet. Topol."},{"issue":"1","key":"9832_CR17","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1017\/S0305004112000394","volume":"154","author":"N Gambino","year":"2013","unstructured":"Gambino, N., Kock, J.: Polynomial functors and polynomial monads. Math. Proc. Cambridge Philos. Soc. 154(1), 153\u2013192 (2013)","journal-title":"Math. Proc. Cambridge Philos. Soc."},{"key":"9832_CR18","doi-asserted-by":"crossref","unstructured":"Hirschhorn, P.: Model categories and their localizations. Number\u00a099. American Mathematical Society, (2009)","DOI":"10.1090\/surv\/099"},{"issue":"3","key":"9832_CR19","first-page":"609","volume":"2011","author":"J Kock","year":"2011","unstructured":"Kock, J.: Polynomial functors and trees. Int. Math. Res. Not. 2011(3), 609\u2013673 (2011)","journal-title":"Int. Math. Res. Not."},{"issue":"6","key":"9832_CR20","doi-asserted-by":"publisher","first-page":"2690","DOI":"10.1016\/j.aim.2010.02.012","volume":"224","author":"J Kock","year":"2010","unstructured":"Kock, J., Joyal, A., Batanin, M., Mascari, J.-F.: Polynomial functors and opetopes. Adv. Math. 224(6), 2690\u20132737 (2010)","journal-title":"Adv. Math."},{"issue":"1","key":"9832_CR21","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1023\/A:1007555725247","volume":"48","author":"M Kontsevich","year":"1999","unstructured":"Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48(1), 35\u201372 (1999)","journal-title":"Lett. Math. Phys."},{"key":"9832_CR22","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/S1570-7954(07)05002-4","volume":"5","author":"M Markl","year":"2008","unstructured":"Markl, M.: Operads and props. Handbook Algebra 5, 87\u2013140 (2008)","journal-title":"Handbook Algebra"},{"key":"9832_CR23","volume-title":"The geometry of iterated loop spaces","author":"JP May","year":"2006","unstructured":"May, J.P.: The geometry of iterated loop spaces, vol. 271. Springer, Berlin (2006)"},{"key":"9832_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0052-5","volume-title":"Simplicial methods for operads and algebraic geometry","author":"Ieke Moerdijk","year":"2010","unstructured":"Moerdijk, Ieke, To\u00ebn, Bertrand: Simplicial methods for operads and algebraic geometry. Springer Science & Business Media, Berlin (2010)"},{"key":"9832_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067053","volume-title":"Higher algebraic K-theory: I","author":"Daniel Quillen","year":"1973","unstructured":"Quillen, Daniel: Higher algebraic K-theory: I. Springer, Berlin (1973)"},{"issue":"2","key":"9832_CR26","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1090\/S0894-0347-05-00510-2","volume":"19","author":"D Sinha","year":"2006","unstructured":"Sinha, D.: Operads and knot spaces. J. Am. Math. Soc. 19(2), 461\u2013486 (2006)","journal-title":"J. Am. Math. Soc."},{"issue":"3","key":"9832_CR27","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1112\/jtopol\/jtq015","volume":"3","author":"V Turchin","year":"2010","unstructured":"Turchin, V.: Hodge-type decomposition in the homology of long knots. J. Topol. 3(3), 487\u2013534 (2010)","journal-title":"J. Topol."},{"issue":"2","key":"9832_CR28","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s40062-013-0032-9","volume":"9","author":"V Turchin","year":"2014","unstructured":"Turchin, V.: Delooping totalization of a multiplicative operad. J. Homot. Relat. Struct. 9(2), 349\u2013418 (2014)","journal-title":"J. Homot. Relat. Struct."}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-025-09832-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10485-025-09832-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-025-09832-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,23]],"date-time":"2025-11-23T06:19:49Z","timestamp":1763878789000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10485-025-09832-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10]]},"references-count":28,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["9832"],"URL":"https:\/\/doi.org\/10.1007\/s10485-025-09832-0","relation":{},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"value":"0927-2852","type":"print"},{"value":"1572-9095","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10]]},"assertion":[{"value":"26 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 October 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"36"}}