Document Zbl 1342.18009

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Math.</a> <a href="/?q=in%3A344125" title="Articles in this Issue">279, 575-716 (2015)</a>. </div> <div class="abstract">The authors set up a general theory of $\infty$-categories enriched in monoidal $\infty$-categories. This $\infty$-category approach can be applied to the derived $\infty$-category of a category of modules, in order to get a homotopical version of the classical notion of a dg-category, and to the $\infty$-category of spectra, in order to get a homotopical version of the notion of a category enriched in spectra. The authors actually prove that Lurie’s stable $\infty$-categories are categories enriched in the $\infty$-category of spectra in their sense.<br class="zbmathjax-paragraph">The notion of an enriched $\infty$-category defined in this paper naturally consists of a space of objects together with hom-objects equipped with composition operations which are unital and associative up to homotopy, just like the tensor products of the monoidal $\infty$-categories in which these hom-objects are defined. In a first step, the authors use a non-symmetric counterpart of Lurie’s theory of $\infty$-operad in order to associate an $\infty$-category of enriched $\infty$-categories $\mathrm{Alg}_{\mathrm{cat}}(\mathcal{V})$ to any monoidal $\infty$-category $\mathcal{V}$. Then they define the homotopy theory of $\mathcal{V}$-enriched $\infty$-categories as a localization of this $\infty$-category $\mathrm{Alg}_{\mathrm{cat}}(\mathcal{V})$ with respect to the class of fully faithful and essentially surjective functors. They prove that this localization is actually identified with the full subcategory of $\mathrm{Alg}_{\mathrm{cat}}(\mathcal{V})$ spanned by the $\mathcal{V}$-enriched $\infty$-categories whose space of objects is equivalent to the classifying space of their equivalences.<br class="zbmathjax-paragraph">This construction gives a notion of enriched $\infty$-category which is functorial with respect to the lax monoidal $\infty$-functors between monoidal $\infty$-categories. Furthermore, the authors prove that their $\infty$-category of enriched $\infty$-categories is equipped with internal hom-objects which give $\infty$-categories of enriched functors between enriched $\infty$-categories with the correct homotopy type.<br class="zbmathjax-paragraph">To give first examples of applications of their theory, the authors also explain a definition of the notion of an $(n,k)$-category in terms of enriched $\infty$-categories, and they give a proof of the Baez-Dolan stabilization in this setting.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11711">Benoît Fresse (Villeneuve d’Ascq)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6435583">Cited in <strong>10</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6435583">Cited in <strong>60</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td class="mono"> 18D05 </td> <td class="space"> Double categories, $2$-categories, bicategories and generalizations (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A18D20" title="MSC2020">18D20</a> </td> <td class="space"> Enriched categories (over closed or monoidal categories) </td> </tr><tr> <td class="mono"> 18D50 </td> <td class="space"> Operads (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahigher+category+theory">higher category theory</a>; <a href="/?q=ut%3Ahigher+operads">higher operads</a>; <a href="/?q=ut%3Aenriched+categories">enriched categories</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1342.18009" data-ciurl="/ci/06435583" data-biburl="/bibtex/06435583.bib" data-amsurl="/amsrefs/06435583.bib" data-xmlurl="/xml/06435583.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06435583.pdf" title="Zbl 1342.18009 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.aim.2015.02.007" aria-label="DOI for “Enriched $\infty$-categories via non-symmetric $\infty$-operads”" title="10.1016/j.aim.2015.02.007">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1312.3178"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bacard, H. 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