Homology of infinity-operads

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id="center"> <div id="article-div" class="article-div"> <div > <div class="col-12"> <div style="font-size:18px;margin-bottom:10px"> <span class="common-title article-title"> Homology of infinity-operads </span> <br> <span class="article-trans-title">[Homologie des infini-opérades]</span> </div> <div class="article-author" style="display:inline-block"> <div style="margin-bottom:5px; font-size: 16px"> <a class="author_link" href="/search?"Hoffbeck, Eric"-c">Hoffbeck, Eric<span href=#aff_1></span></a><a class="author_link" href="#affiliation" data-bs-toggle="collapse" aria-controls="affiliation"><sup><span href=#aff_1>1</span></sup></a> ; <a class="author_link" href="/search?"Moerdijk, Ieke"-c">Moerdijk, Ieke<span href=#aff_2></span></a><a class="author_link" href="#affiliation" data-bs-toggle="collapse" aria-controls="affiliation"><sup><span href=#aff_2>2</span></sup></a> <a data-bs-toggle="collapse" href="#affiliation" aria-controls="affiliation"> <button id='house-affiliation' type="button" class="btn btn-link btn-xs" title="Affiliations des auteurs"> <span class="fa fa-home"></span><span id='house-caret' class="caret-icon fa fa-caret-down" style="margin-left:3px"></span> </button> </a> <div class="collapse" id="affiliation"> <div class="card card-body"> <div class="affiliation-authors"> <span id=aff_1> <sup>1</sup> Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse (France) </span> <br> <span id=aff_2> <sup>2</sup> Department of Mathematics, Utrecht University, PO BOX 80.010, 3508 TA Utrecht (The Netherlands) </span> <br> </div> </div> </div> </div> </div> <div style="margin-bottom:10px"> Annales de l'Institut Fourier, Online first, 37 p. </div> <div class="binary-file-icon" style="margin-top:20px;margin-bottom:10px"> <div class="wrapper-float"> <div class="d-flex flex-nowrap" style="margin-top:0px"> <a href="/item/10.5802/aif.3653.pdf" target="_blank"> <img src="/static/ptf/img/pdf.gif"> </a> </div> </div> </div> </div> </div> <div> <div class="col-12"> <div style="margin-top:20px;margin-bottom:30px"> <ul class="nav nav-tabs article-tab"> <li class="nav-item active"> <a class="nav-link active" lang="fr" data-bs-toggle="tab" data-bs-target="#abstract-fr" href="#abstract-fr"><span title='Français '>Résumé</span></a> </li> <li class="nav-item"> <a class="nav-link" lang="en" data-bs-toggle="tab" data-bs-target="#abstract-en" href="#abstract-en"> <span title='Anglais '>Abstract</span> </a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div lang="fr" id="abstract-fr" class="tab-pane fade show active"> <p>Dans une première partie, nous introduisons une théorie de l’homologie pour les infini-opérades et les espaces dendroïdaux, qui étend l’homologie usuelle des opérades différentielles graduées définie au moyen de la construction bar, et nous prouvons certaines de ses propriétés de base. Dans une seconde partie, nous définissons des constructions bar et cobar générales. Ces constructions envoient les infini-opérades sur des infini-coopérades et vice versa, et définissent une dualité bar-cobar (ou « de Koszul »). De façon assez surprenante, cette dualité est encore vraie pour des préfaisceaux et copréfaisceaux quelconques sur la catégorie des arbres définissant les infini-opérades. Nous insistons sur le caractère élémentaire et explicite de nos méthodes.</p> </div> <div lang="en" id="abstract-en" class="tab-pane fade"> <p>In a first part of this paper, we introduce a homology theory for infinity-operads and for dendroidal spaces which extends the usual homology of differential graded operads defined in terms of the bar construction, and we prove some of its basic properties. In a second part, we define general bar and cobar constructions. These constructions send infinity-operads to infinity-cooperads and vice versa, and define a bar-cobar (or “Koszul”) duality. Somewhat surprisingly, this duality is shown to hold much more generally between arbitrary presheaves and copresheaves on the category of trees defining infinity-operads. We emphasize that our methods are completely elementary and explicit.</p> </div> </div> </div> </div> </div> <div> <div class="col-12"> <div style="margin-top:10px;margin-bottom:10px"> <ul class="nav nav-tabs article-tab"> <li class="nav-item"><a class="nav-link active" data-bs-toggle="tab" data-bs-target="#info-tab" href="#info-tab">Détail</a></li> <li class="nav-item"><a class="nav-link" data-bs-toggle="tab"data-bs-target="#citation-tab" href="#citation-tab">Export</a></li> <li class="nav-item"> <a class="nav-link " data-bs-toggle="tab" data-toggle="tab" data-bs-target="#how-to-cite-tab" href="#how-to-cite-tab">Comment citer</a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div id="info-tab" class="tab-pane fade show active"> Reçu le : <time datetime="2021-08-17">2021-08-17</time><br> Révisé le : <time datetime="2023-03-21">2023-03-21</time><br> Accepté le : <time datetime="2023-08-04">2023-08-04</time><br> Première publication : <time datetime="2024-07-03">2024-07-03</time><br>  <div style="margin-top:5px"> </div> DOI : <a href="https://doi.org/10.5802/aif.3653">10.5802/aif.3653</a> <br> <div style="margin-top:5px"> Classification : <span class="mathjax-formula"> 18N70, 55N35, 18M70 </span> <br/> Keywords: <span class="mathjax-formula"> Koszul duality, infinity-operads, dendroidal sets. </span> <br/> Mots-clés : <span class="mathjax-formula"> Dualité de Koszul, infini-opérades, ensembles dendroidaux. </span> <br/> </div> <div style="margin-top:5px"> </div> <div style="margin-top:5px"> <span id="affiliation-toggle" data-bs-toggle="collapse" href="#affiliation_details" role="button" aria-expanded="false" aria-controls="affiliation_details">Affiliations des auteurs : <span id="details-caret" class="caret-icon fa fa-caret-down" style="margin-left:3px"></span></span> <div class="collapse" id="affiliation_details" style="margin-top:10px;margin-left:10px"> <div class="card card-body"> <p> Hoffbeck, Eric&hairsp;<sup class="d-inline-block">1</sup> ; Moerdijk, Ieke&hairsp;<sup class="d-inline-block">2</sup> </p> <div class="affiliation-authors"> <span id=aff_1> <sup>1</sup> Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse (France) </span> <br> <span id=aff_2> <sup>2</sup> Department of Mathematics, Utrecht University, PO BOX 80.010, 3508 TA Utrecht (The Netherlands) </span> <br> </div> </div> </div> </div> </div> <div id="citation-tab" class="tab-pane fade"> <div class="upper"> <ul class="nav nav-pills"> <li role="presentation" class="nav-item"><a class="nav-link active" data-bs-toggle="tab" data-bs-target="#bibtex" href="#bibtex">BibTeX</a></li> <li role="presentation" class="nav-item"><a class="nav-link" data-bs-toggle="tab" data-bs-target="#ris" href="#ris">RIS</a></li> <li role="presentation" class="nav-item"><a class="nav-link" data-bs-toggle="tab" data-bs-target="#enw" href="#enw">EndNote</a></li> </ul> <div class="tab-content"> <div id="bibtex" class="tab-pane fade show active"> <pre class="upper-right-content" id="bibtex-div">@unpublished{AIF_0__0_0_A116_0, author = {Hoffbeck, Eric and Moerdijk, Ieke}, title = {Homology of infinity-operads}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, year = {2024}, doi = {10.5802/aif.3653}, language = {en}, note = {Online first}, }</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#bibtex-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="Télécharger" href="/export_citation/AIF_0__0_0_A116_0/bib/"></a> </div> <div id="ris" class="tab-pane fade"> <pre class="upper-right-content" id="ris-div">TY - UNPB AU - Hoffbeck, Eric AU - Moerdijk, Ieke TI - Homology of infinity-operads JO - Annales de l'Institut Fourier PY - 2024 PB - Association des Annales de l’institut Fourier N1 - Online first DO - 10.5802/aif.3653 LA - en ID - AIF_0__0_0_A116_0 ER - </pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#ris-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="Télécharger" href="/export_citation/AIF_0__0_0_A116_0/ris/"></a> </div> <div id="enw" class="tab-pane fade"> <pre class="upper-right-content" id="enw-div">%0 Unpublished Work %A Hoffbeck, Eric %A Moerdijk, Ieke %T Homology of infinity-operads %J Annales de l'Institut Fourier %D 2024 %I Association des Annales de l’institut Fourier %Z Online first %R 10.5802/aif.3653 %G en %F AIF_0__0_0_A116_0</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#enw-div" aria-hidden="true" title="Copier dans le presse-papier"></button> <a class="fa fa-download export-button btn btn-primary" title="Télécharger" href="/export_citation/AIF_0__0_0_A116_0/enw/"></a> </div> </div> </div> </div> <div id="how-to-cite-tab" class="tab-pane fade "> <div class="upper-right"> <pre class="upper-right-content" id="how-to-cite-div">Hoffbeck, Eric; Moerdijk, Ieke. Homology of infinity-operads. Annales de l'Institut Fourier, Online first, 37 p.</pre> <button class="fa fa-clipboard copy-button btn btn-primary" data-clipboard-action="copy" data-clipboard-target="#how-to-cite-div" aria-hidden="true" title="Copier dans le presse-papier"></button> </div> </div> </div> </div> </div> </div> <div> <div class="col-12"> <div style="margin-top:30px;margin-bottom:10px"> <ul class="nav nav-tabs article-tab" > <li class="nav-item" id="main-citations-li" > <a class="nav-link active" data-bs-toggle="tab" data-bs-target="#references-tab" href="#references-tab">Bibliographie</a> </li> <li class="nav-item" id="citations-html" style="display:none"> <a class="nav-link" data-bs-toggle="tab" data-bs-target="#citations-tab" href="#citations-tab">Cité par</a> </li> </ul> <div class="tab-content" style="margin-top:10px"> <div id="references-tab" class="tab-pane fade show active"> <div style="margin-top:11px;margin-bottom: 0px"> <p class="bibitemcls " id="BN-spectra"> [1] <span class="citation-author">Bašić, Matija; Nikolaus, Thomas</span><span class="citation-document-title"> Dendroidal sets as models for connective spectra</span><span class="citation-publication-title">, J. <span class="mathjax-formula" data-tex="$$"><math xmlns="http://www.w3.org/1998/Math/MathML" ><mi>K</mi></math></span>-Theory</span><span class="citation-volume">, Volume 14</span> (2014) no. 3, pp. 387-421 | <a href="https://doi.org/10.1017/is014005003jkt265" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3349322" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1323.55009" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="BN-hom"> [2] <span class="citation-author">Bašić, Matija; Nikolaus, Thomas</span><span class="citation-document-title"> Homology of dendroidal sets</span><span class="citation-publication-title">, Homology Homotopy Appl.</span><span class="citation-volume">, Volume 19</span> (2017) no. 1, pp. 111-134 | <a href="https://doi.org/10.4310/HHA.2017.v19.n1.a6" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3633714" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1393.55007" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="C"> [3] <span class="citation-author">Ching, Michael</span><span class="citation-document-title"> Bar constructions for topological operads and the Goodwillie derivatives of the identity</span><span class="citation-publication-title">, Geom. Topol.</span><span class="citation-volume">, Volume 9</span> (2005), pp. 833-933 | <a href="https://doi.org/10.2140/gt.2005.9.833" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2140994" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1153.55006" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="C2"> [4] <span class="citation-author">Ching, Michael</span><span class="citation-document-title"> Bar-cobar duality for operads in stable homotopy theory</span><span class="citation-publication-title">, J. Topol.</span><span class="citation-volume">, Volume 5</span> (2012) no. 1, pp. 39-80 | <a href="https://doi.org/10.1112/jtopol/jtr027" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2897049" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1319.55003" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="CM"> [5] <span class="citation-author">Cisinski, Denis-Charles; Moerdijk, Ieke</span><span class="citation-document-title"> Dendroidal sets and simplicial operads</span><span class="citation-publication-title">, J. Topol.</span><span class="citation-volume">, Volume 6</span> (2013) no. 3, pp. 705-756 | <a href="https://doi.org/10.1112/jtopol/jtt006" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3100888" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1291.55005" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="F"> [6] <span class="citation-author">Fresse, Benoit</span><span class="citation-document-title"> Koszul duality of operads and homology of partition posets</span><span class="citation-publication-title">, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic <span class="mathjax-formula" data-tex="${K}$"><math xmlns="http://www.w3.org/1998/Math/MathML" ><mi>K</mi></math></span>-theory</span><span class="citation-series"> (Contemporary Mathematics)</span><span class="citation-volume">, Volume 346</span>, American Mathematical Society, 2004, pp. 115-215 | <a href="https://doi.org/10.1090/conm/346/06287" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2066499" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1077.18007" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="FG"> [7] <span class="citation-author">Fresse, Benoit; Guerra, Lorenzo</span><span class="citation-document-title"> On a notion of homotopy Segal <span class="mathjax-formula" data-tex="${E}_\infty $"><math xmlns="http://www.w3.org/1998/Math/MathML" ><msub><mi>E</mi> <mi>∞</mi> </msub></math></span>-Hopf cooperad</span> (2020) (<a href="https://arxiv.org/abs/2011.11333" target="_blank">https://arxiv.org/abs/2011.11333</a>) </p> <p class="bibitemcls " id="GJ"> [8] <span class="citation-author">Getzler, Ezra; Jones, J. D. S.</span><span class="citation-document-title"> Operads, homotopy algebra and iterated integrals for double loop spaces</span> (1994) (<a href="https://arxiv.org/abs/hep-th/9403055" target="_blank">https://arxiv.org/abs/hep-th/9403055</a>) </p> <p class="bibitemcls " id="GK"> [9] <span class="citation-author">Ginzburg, Victor; Kapranov, Mikhail</span><span class="citation-document-title"> Koszul duality for operads</span><span class="citation-publication-title">, Duke Math. J.</span><span class="citation-volume">, Volume 76</span> (1994) no. 1, pp. 203-272 | <a href="https://doi.org/10.1215/S0012-7094-94-07608-4" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1301191" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:0855.18006" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="HM"> [10] <span class="citation-author">Heuts, Gijs; Moerdijk, Ieke</span><span class="citation-document-title"> Simplicial and dendroidal homotopy theory</span><span class="citation-series">, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge</span><span class="citation-volume">, 3</span>, Springer, 2022 | <a href="https://doi.org/10.1007/978-3-031-10447-3" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=4485749" class="badge bg-secondary rounded-pill ref-badge"> MR </a> </p> <p class="bibitemcls " id="KS"> [11] <span class="citation-author">Kontsevich, Maxim; Soibelman, Yan</span><span class="citation-document-title"> Deformations of algebras over operads and the Deligne conjecture</span><span class="citation-publication-title">, Conférence Moshé Flato 1999, Vol. I (Dijon)</span><span class="citation-series"> (Mathematical Physics Studies)</span><span class="citation-volume">, Volume 21</span>, Kluwer Academic Publishers, 2000, pp. 255-307 | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1805894" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:0972.18005" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="LV"> [12] <span class="citation-author">Loday, Jean-Louis; Vallette, Bruno</span><span class="citation-document-title"> Algebraic operads</span><span class="citation-series">, Grundlehren der Mathematischen Wissenschaften</span><span class="citation-volume">, 346</span>, Springer, 2012, xxiv+634 pages | <a href="https://doi.org/10.1007/978-3-642-30362-3" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2954392" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1260.18001" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="MW1"> [13] <span class="citation-author">Moerdijk, Ieke; Weiss, Ittay</span><span class="citation-document-title"> Dendroidal sets</span><span class="citation-publication-title">, Algebr. Geom. Topol.</span><span class="citation-volume">, Volume 7</span> (2007), pp. 1441-1470 | <a href="https://doi.org/10.2140/agt.2007.7.1441" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2366165" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1133.55004" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="RW"> [14] <span class="citation-author">Robinson, Alan; Whitehouse, Sarah</span><span class="citation-document-title"> The tree representation of <span class="mathjax-formula" data-tex="$\Sigma _{n+1}$"><math xmlns="http://www.w3.org/1998/Math/MathML" ><msub><mi>Σ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> </msub></math></span></span><span class="citation-publication-title">, J. Pure Appl. Algebra</span><span class="citation-volume">, Volume 111</span> (1996) no. 1-3, pp. 245-253 | <a href="https://doi.org/10.1016/0022-4049(95)00116-6" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1394355" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:0865.55010" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> <p class="bibitemcls " id="V"> [15] <span class="citation-author">Vallette, Bruno</span><span class="citation-document-title"> Homology of generalized partition posets</span><span class="citation-publication-title">, J. Pure Appl. Algebra</span><span class="citation-volume">, Volume 208</span> (2007) no. 2, pp. 699-725 | <a href="https://doi.org/10.1016/j.jpaa.2006.03.012" class="badge bg-secondary rounded-pill ref-badge"> DOI </a> | <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2277706" class="badge bg-secondary rounded-pill ref-badge"> MR </a> | <a href="https://zbmath.org/?q=an:1109.18002" class="badge bg-secondary rounded-pill ref-badge"> Zbl </a> </p> </div> </div> <div id="citations-tab" class="tab-pane fade show"> <div id="citations-div" data-url="/citedby/10.5802/aif.3653" style="margin-top:11px;margin-bottom: 11px"></div> <p><em>Cité par <span id="citations-number"></span> Sources : <span id="citations-sources"></span></em></p> </div> </div> </div> </div> </div> </div> </div> <div id="right"> </div> </div> </div> </div> </div> <div id="footer"> <div class="logos-footer col-xs-12"> <span class="footer-logo-text">Diffusé par :</span><a class="logo-footer" href="http://www.centre-mersenne.org"><img align="middle" class="logo-footer" src="/static/mersenne_cms/img/logo-mersenne.png"></a> <span class="footer-logo-text">Soutenu par :</span><a class="logo-footer" href="https://www-fourier.ujf-grenoble.fr/"><img align="middle" class="logo-footer" src="/static/aif/img/logo-if-invert.png"></a>  <span class="footer-logo-text">Développé par :</span><a class="logo-footer" href="http://www.mathdoc.fr"><img align="middle" class="logo-footer" src="/static/mersenne_cms/img/logo-mathdoc-footer.png"></a> </div> <div class="row"> <div class="col-xs-12 col-sm-7 col-md-6 footer-text"> ISSN : 0373-0956 - e-ISSN : 1777-5310 <br>  </div> <div class="col-xs-12 col-sm-5 col-md-6"> <nav class="navbar navbar-expand-sm menu-bottom"> <div class="container-fluid"> <ul class="navbar-nav"> <li class="nav-item"> <a class="nav-link" href='/rss/' alt='syndication feed'> <i class="fa fa-rss" aria-hidden="true"></i> Nous suivre </a> </li> <li class="nav-item "><a href="/page/mentions-legales_fr/" class="nav-link">Mentions légales</a></li> <li class="nav-item "><a href="/page/contact_fr/" class="nav-link">Contact</a></li> </ul> </div> </nav> </div> </div> </div> <script type="text/javascript" src="/static/ptf/vendors/jquery/js/jquery-3.5.1.min.js"></script> <script src="/static/ptf/vendors/bootstrap5/js/bootstrap.bundle.min.js"></script> <script src="/static/ptf/vendors/clipboard/js/clipboard.min.js "></script> <script type="text/javascript" src="/static/ptf/js/bs5/ptf.js"></script> <script type="text/javascript" src="/static/ptf/js/bs5/search-bar.js"></script> <script src="/static/ptf/js/mathjax-config.js" defer></script> <script type="text/javascript" id="MathJax-script" defer src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"> </script> </body> </html>

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Homology of infinity-operads