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<title>Derived stack</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Derived stack</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <b>derived stack</b> is, roughly, a <a href="Stack_(mathematics)" title="Stack (mathematics)">stack</a> together with a <a href="Sheaf_of_spectra" title="Sheaf of spectra">sheaf of commutative ring spectra</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It generalizes a <a href="Derived_scheme" title="Derived scheme">derived scheme</a>. Derived stacks are the "spaces" studied in <a href="Derived_algebraic_geometry" title="Derived algebraic geometry">derived algebraic geometry</a>.<sup id="cite_ref-Vezzosi2011_2-0" class="reference"><a href="#cite_note-Vezzosi2011-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFMathewMeier2013">Mathew & Meier 2013</a>, Definition 2.6.</span>
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<li id="cite_note-Vezzosi2011-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Vezzosi2011_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFVezzosi2011" class="citation journal cs1"><a href="Gabriele_Vezzosi" title="Gabriele Vezzosi">Vezzosi, Gabriele</a> (August 2011). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/201107/rtx110700955p.pdf">"What is ... a Derived Stack?"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>. <b>58</b> (7): <span class="nowrap">955–</span>958<span class="reference-accessdate">. Retrieved <span class="nowrap">4 March</span> 2014</span>.</cite></span>
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFToën2014" class="citation cs2"><a href="Bertrand_To%C3%ABn" title="Bertrand Toën">Toën, Bertrand</a> (2014), <i>Derived Algebraic Geometry</i>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1401.1044">1401.1044</a></span></cite></li>
<li><cite id="CITEREFToën2006" class="citation cs2"><a href="Bertrand_To%C3%ABn" title="Bertrand Toën">Toën, Bertrand</a> (2006), <i>Higher and derived stacks: a global overview</i>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0604504">math/0604504</a></span>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006math......4504T">2006math......4504T</a></cite></li>
<li><cite id="CITEREFLurie2004" class="citation thesis cs1"><a href="Jacob_Lurie" title="Jacob Lurie">Lurie, Jacob</a> (2004). <a rel="nofollow" class="external text" href="https://dspace.mit.edu/handle/1721.1/30144"><i>Derived Algebraic Geometry</i></a> (Thesis). Massachusetts Institute of Technology. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F30144">1721.1/30144</a>.</cite></li>
<li><cite id="CITEREFMathewMeier2013" class="citation journal cs1">Mathew, Akhil; Meier, Lennart (2013). "Affineness and chromatic homotopy theory". <i>Journal of Topology</i>. <b>8</b> (2): <span class="nowrap">476–</span>528. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1311.0514">1311.0514</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjtopol%2Fjtv005">10.1112/jtopol/jtv005</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119713516">119713516</a>.</cite></li></ul>
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