CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Carlos Simpson in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Carlos Simpson </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/17239/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <ul> <li><a href="http://math.unice.fr/~carlos/">website</a></li> </ul> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#selected_writings'>Selected writings</a></li> <li><a href='#related_lab_entries'>Related <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>Lab entries</a></li> </ul> </div> <h2 id="selected_writings">Selected writings</h2> <ul> <li><em>Systems of Hodge bundles and uniformization</em>, Ph.D. Thesis, Harvard Univ., Cambridge, MA, 1987; J. Amer. Math. Soc. 1 (1988), no. 4, 867–918; <a href="http://www.ams.org/mathscinet-getitem?mr=90e:58026">MR90e:58026</a>, <a href="https://doi.org/10.2307/1990994">doi</a></li> <li><em>Some families of local systems over smooth projective varieties</em>, Ann. Math., Sec. Ser. <strong>138</strong>:2 (1993) 337–425 <a href="https://doi.org/10.2307/2946614">doi</a></li> </ul> <p>On a <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> for <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algbraic</a> <a class="existingWikiWord" href="/nlab/show/infinity-stacks"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a> via an <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> realizing the <a class="existingWikiWord" href="/nlab/show/de+Rham+stack">de Rham stack</a>:</p> <ul> <li id="SimpsonTeleman"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>deRham theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em> (1997) [<a href="http://math.berkeley.edu/~teleman/math/simpson.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SimpsonTeleman-deRham.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><em>Homotopy over the complex numbers and generalized de Rham cohomology</em>, in: Moduli of Vector Bundles, edited by Masaki Maruyama, CRC Press 1996</p> </li> <li> <p><em>Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology</em>, arXiv:<a href="https://arxiv.org/abs/alg-geom/9712020">alg-geom/9712020</a></p> </li> </ul> <p>On Deligne’s <a class="existingWikiWord" href="/nlab/show/lambda-connection">lambda-connection</a>s and Hodge moduli space</p> <ul> <li>C. T. Simpson, <em>The Hodge filtration on nonabelian cohomology</em>, in Algebraic geometry-Santa Cruz 1995, 217–281</li> <li>Carlos T. Simpson, <em>A weight two phenomenon for the moduli space of rank one local systems on open varieties</em> <a href="https://arxiv.org/abs/0710.2800">arXiv:0710.2800</a></li> <li>C. T. Simpson, <em>Iterated destability modifications for vector bundles with connection, Contemp. Math. <strong>522</strong> (2010) 183–206</em></li> <li><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>On the notion of lambda-connection</em>, <a href="https://math.univ-cotedazur.fr/~carlos/slides/deligneconf/slideA.html">slides</a></li> <li>C. T. Simpson, <em>Geometricity of the Hodge filtration on the ∞-stack of perfect complexes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>DR</mi></msub></mrow><annotation encoding="application/x-tex">X_{DR}</annotation></semantics></math></em>, Mosc. Math. J. 9 (2009) no. 3, 665–721 arXiv:<a href="https://arxiv.org/abs/math/0510269">math.AG/0510269</a> <a href="http://www.mathjournals.org/mmj/vol9-3-2009/cont9-3-2009.html">journal</a></li> </ul> <p>More on <a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/Higgs+bundles">Higgs bundles</a> and <a class="existingWikiWord" href="/nlab/show/local+systems">local systems</a></em>, Publ. Mathématiques de l’IHÉS <strong>75</strong> (1992), p. 5-95 (<a href="http://www.numdam.org/item?id=PMIHES_1992__75__5_0">numdam:PMIHES_1992__75__5_0</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=94d:32027">MR94d:32027</a>)</li> <li><em>Algebraic aspects of higher nonabelian Hodge theory</em>, arXiv:<a href="https://arxiv.org/abs/math/9902067">math.AG/9902067</a></li> </ul> <p>An early conception of <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-categories">presentable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-categorues</a> and <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-toposes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-toposes</a> (the term seems to originate here) via (what later were called) <a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a>:</p> <ul> <li id="Simpson99"><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>A Giraud-type characterization of the simplicial categories associated to closed model categories as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-pretopoi</em> (<a href="http://arxiv.org/abs/math/9903167">arXiv:math/9903167</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/secondary+characteristic+classes">secondary characteristic classes</a>:</p> <ul> <li>Jaya N. Iyer, <a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>Regulators of canonical extensions are torsion: the smooth divisor case</em> (<a href="https://arxiv.org/abs/0707.0372">arXiv:0707.0372</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Theory+of+Higher+Categories">Homotopy Theory of Higher Categories</a></em>, Cambridge University Press 2011 (ISBN:9780521516952mathematics/geometry-and-topology/homotopy-theory-higher-categories-segal-categories-ini-categories-and-beyond?format=HB)), <a href="https://hal.archives-ouvertes.fr/hal-00449826/document">hal:00449826</a>, <a href="http://arxiv.org/abs/1001.4071">http://arxiv.org/abs/1001.4071</a> )</li> </ul> <p>Lectures on <a class="existingWikiWord" href="/nlab/show/stack">stack</a>s:</p> <ul> <li>Nicole Mestrano, Carlos Simpson, <em>Stacks</em>, 32 pp. <a href="https://hal.science/hal-03347136/document">pdf</a></li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/descent">descent</a>:</p> <p>In January 2009 Carlos Simpson wrote the following short note</p> <ul> <li>C. Simpson, <em>Descent</em> (<a href="http://math.unice.fr/~carlos/slides/ihesAGjan09.pdf">pdf</a>)</li> </ul> <p>The following reproduces the text unabridged, but equipped with hyperlinks on keywords to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>Lab entries as far as existent:</p> <p>“ The notion of <a class="existingWikiWord" href="/nlab/show/descent">descent</a>, piecing together a global picture out of local pieces and glueing data, permeates Grothendieck’s work. The history of this idea dates to the middle ages with mapmakers drawing an ever more precise picture of the world, as modern terminology of ”<a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>es“ and ”<a class="existingWikiWord" href="/nlab/show/chart">chart</a>s“ reminds us. It is crucial to the notion of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, where we first meet higher glueing data.</p> <p><a class="existingWikiWord" href="/nlab/show/descent">Descent</a> comes into Grothendieck’s philosophy and work in a myriad of forms, starting with his papers <em>Technique de descente et théorèmes</em></p> <p>d’existence en géométrie algébrique_ . The technical requirements of this theory incited him to introduce the notion of <a class="existingWikiWord" href="/nlab/show/fibered+category">fibered category</a>. He extended the domain of application of this point of view in a revolutionary way by introducing the notion of “<a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>”, integrating all of <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> and giving us <a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a>. A further transformation occured with the notion of <a class="existingWikiWord" href="/nlab/show/topos">topos</a>. Another incarnation of the idea was cohomological <a class="existingWikiWord" href="/nlab/show/descent">descent</a> used in Deligne’s papers on Hodge theory, where <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s enter in a way which differs significantly from their original occurences in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>.</p> <p>Between the 1960’s and the 1980’s, the notion of descent for objects of a <a class="existingWikiWord" href="/nlab/show/category">category</a>, slowly gave way to a notion of “higher descent” for objects in generalized categorical situations. Examples include Breen’s calculation of etale Ext groups using the cohomology of simplicial <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane</a> presheaves, and the theory of twisted complexes of Toledo and Tong. <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Stasheff</a> and Wirth <a class="existingWikiWord" href="/nlab/show/gerbe">investigated higher cocycles</a> in the 1960’s although Wirth’s thesis was only recently <a href="http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html">made public</a>. In algebraic topology, Quillen’s theory of <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> was gradually applied to simplicial diagrams. Illusie introduced the notion of <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">weak equivalence of</a> <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaves</a> on a <a class="existingWikiWord" href="/nlab/show/site">Grothendieck site</a>.</p> <p>Grothendieck came out of isolation with the manuscript <a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">La poursuite des champs</a>, which at its start refers to a letter from Joyal to Grothendieck developing the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">model category structure on simplicial sheaves</a>. This led to Jardine’s <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model category structure for simplicial presheaves</a> enhancing Illusie’s weak equivalences. We enter into the modern period in which Jardine’s model structure and its variants have been used and developped with applications in a wide range of mathematics including Thomason’s work in K-theory and then Voevodsky’s theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">A^1</annotation></semantics></math>-homotopy and <a class="existingWikiWord" href="/nlab/show/motive">motive</a>s. <a class="existingWikiWord" href="/nlab/show/algebraic+stack">Algebraic stack</a>s, the first step in the “higher descent!” direction, are now used without restraint in all of <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>.</p> <p>In Grothendieck’s vision as set out in “<a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">La poursuite des champs</a>”, higher descent is just the same as usual descent, but for <a class="existingWikiWord" href="/nlab/show/infinity-stack">n-stacks</a> of <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> over a <a class="existingWikiWord" href="/nlab/show/site">site</a>. The theory of <a class="existingWikiWord" href="/nlab/show/2-category">2-categories</a> was developped early on by Benabou, having occured also in the book of Gabriel and Zisman. The theory of <a class="existingWikiWord" href="/nlab/show/strict+omega-category">strict n-categories</a> was thoroughly investigated by Street and the Australian school, and Brown and Loday introduced other related algebraic objects which could model <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>s. Grothendieck set out the goal of finding an adequate theory of <a class="existingWikiWord" href="/nlab/show/omega-category">weak n-categories</a> where composition would be associative only up to a coherent system of higher equivalences. Similar ideas were being developped by Dwyer and Kan in algebraic topology, and Cordier and <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Porter</a> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. Several definitions of <a class="existingWikiWord" href="/nlab/show/weak+omega-category">weak n-categories</a> have been proposed, by Baez-Dolan, Tamsamani, Batanin and others. It is now well understood that the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherence</a> problems inherent in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categories</a>, are basically the same as those which were studied by <a class="existingWikiWord" href="/nlab/show/topology">topologists</a> for <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping machines</a>. Segal’s simplicial approach and May’s operadic approaches play important roles in all of the current definitions. Maltsiniotis points out that Batanin’s definition is the closest to Grothendieck’s original idea. The topologists, notably Rezk and Bergner, have developped model structures on simplicial categories and simplicial spaces, and Joyal gives a model structure on the restricted Kan complexes originally defined by Boardman and Vogt in the 1960’s. Cisinski and Maltsiniotis have built on a somewhat different direction of “<a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">La poursuite des champs</a>” which aims to characterize the algebraic models for <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>My own work in this area is inspired by the phrase in “<a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">La poursuite des champs</a>” where Grothendieck forsees <a class="existingWikiWord" href="/nlab/show/infinity-stack">n-stacks</a> as the natural coefficients for higher <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>. With Hirschowitz, we have developped the notion of <a class="existingWikiWord" href="/nlab/show/infinity-stack">n-stack</a> based on Tamsamani’s definition of <a class="existingWikiWord" href="/nlab/show/n-category">n-category</a>, and proven that the association <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><mo stretchy="false">{</mo><mi>n</mi><mo>−</mo><mi>stacks</mi><mi>on</mi><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U \mapsto \{n-stacks on U\}</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-stack.</p> <p>The theory of “<a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a>” originated by Kontsevich, Kapranov and Ciocan-Fontanine is now cast by Toen, Vezzosi and <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Lurie</a> in a foundational framework which relies on <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categories</a> and <a class="existingWikiWord" href="/nlab/show/infinity-stack">higher stacks</a> for glueing. In the future derived geometry should be a key ingredient in Hodge theory for higher <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>, to be compared with <a class="existingWikiWord" href="/nlab/show/Katzarkov">Katzarkov</a>, Pantev and <a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Toen</a>‘s <a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a> on the <a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a>. The latter is a higher categorical version of Grothendieck’s reinterpretation of <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a>, forseen in “<a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">La poursuite des champs</a>”, or really its Tannakian counterpart. Grothendieck also mentionned, somewhat cryptically, a potential application to stratified spaces. The respective theses of <a class="existingWikiWord" href="/nlab/show/Treumann">Treumann</a> and Dupont go in this direction by using exit-path n-categories to classify constructible complexes of sheaves.</p> <p>More recent developments include <a href="http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html">Hopkins and Lurie’s proof</a> of a part of the <a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">Baez-Dolan system of conjectures</a> relating higher categories to <a class="existingWikiWord" href="/nlab/show/TQFT">topological</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. And <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> permits us to imagine a local notion of descent as was explained to me by <a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>: using the derived non-transverse intersections, Schlessinger-<a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Stasheff</a>- Deligne-Goldman-Millson theory should be viewed as descent for the inclusion of a <a class="existingWikiWord" href="/nlab/show/point">point</a> into a local formal space, with the neighborhood intersection being the <a class="existingWikiWord" href="/nlab/show/geometric+infinity-function+theory">derived loop space of Ben-Zvi and Nadler</a>. “</p> <h2 id="related_lab_entries">Related <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>Lab entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simpson%27s+conjecture">Simpson's conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent">descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinity-stack">infinity-stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Higgs+bundle">Higgs bundle</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/people">people</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on October 1, 2023 at 12:30:56. See the <a href="/nlab/history/Carlos+Simpson" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Carlos+Simpson" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/17239/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/Carlos+Simpson/14" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Carlos+Simpson" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Carlos+Simpson" accesskey="S" class="navlink" id="history" rel="nofollow">History (14 revisions)</a> <a href="/nlab/show/Carlos+Simpson/cite" style="color: black">Cite</a> <a href="/nlab/print/Carlos+Simpson" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Carlos+Simpson" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>