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> reconstruction theorem 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> reconstruction theorem </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/12499/#Item_2" 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> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#tannakian_reconstruction_theorems'>Tannakian reconstruction theorems</a></li> <li><a href='#reconstruction_theorems_for_schemes'>Reconstruction theorems for schemes</a></li> <li><a href='#gabrielulmer_duality'>Gabriel-Ulmer duality</a></li> <li><a href='#heuristics'>Heuristics</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Sometimes for <a class="existingWikiWord" href="/nlab/show/categories">categories</a> having some fixed <a class="existingWikiWord" href="/nlab/show/extra+property">extra property</a> and/or <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a>, one can produce a recipe which gives (up to suitable equivalence) all the examples (and nothing else).</p> <p>There are several typical classes of such <em>reconstruction theorems</em> (which are all to some extent related).</p> <h2 id="tannakian_reconstruction_theorems">Tannakian reconstruction theorems</h2> <p>These theorems reconstruct an algebraic symmetry object (<a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, <a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a>, Hopf algebroid) from the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s of that object (typically rigid and <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> or <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided</a>). The correspondence between the symmetry object and the corresponding <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> is called <strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></strong>.</p> <p>Examples include the classical Tannaka theorem and Krein theorem, <a class="existingWikiWord" href="/nlab/show/Doplicher-Roberts+reconstruction+theorem">Doplicher-Roberts reconstruction theorem</a> in physics, <a class="existingWikiWord" href="/nlab/show/Deligne%27s+theorem+on+tensor+categories">Deligne's theorem on tensor categories</a>, Woronowicz’s Tannaka duality for compact matrix pseudogroups, Saavedra-Rivano and Deligne reconstruction theorems for neutral and mixed Tannakian categories, Ulrich’s reconstruction theorem, reconstruction theorems of Majid, Nori Tannakian theorem, Grothendieck’s version of Galois group in algebraic geometry and so on. The notion of the <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> (due to Grothendieck) is central to these considerations.</p> <h2 id="reconstruction_theorems_for_schemes">Reconstruction theorems for schemes</h2> <p>These theorems for schemes (or varieties only) reconstruct a <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> (variety) out of the category of quasicoherent or only coherent sheaves (or a <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> version of them). In that class one can find the Gabriel–Rosenberg theorem, the <a class="existingWikiWord" href="/nlab/show/Bondal-Orlov+reconstruction+theorem">Bondal-Orlov reconstruction theorem</a>, the reconstruction theorems of <a href="http://www.math.ucla.edu/~balmer">P. Balmer</a> and of <a href="http://www-maths.swan.ac.uk/staff/gg">G. Garkusha</a>, and so on. There is also a class of reconstructions where for some derived categories a realization as derived categories of representation of quivers can be reconstructed.</p> <p>There is a large class of <strong>abelian reconstruction theorems</strong>, for example the <a class="existingWikiWord" href="/nlab/show/Gabriel-Popescu+theorem">Gabriel-Popescu theorem</a>. In <em>topos theory</em> the <strong><a class="existingWikiWord" href="/nlab/show/Giraud+theorem">Giraud theorem</a></strong> is also a reconstruction theorem (of a <a class="existingWikiWord" href="/nlab/show/site">site</a> out of a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, though a nonuniqueness of the resulting site is involved, not affecting cohomology, hence, according to Grothendieck, nonessential).</p> <p>Examples:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/Gabriel-Popescu+embedding+theorem">Gabriel-Popescu embedding theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/Quillen-Gabriel+embedding+theorem">Quillen-Gabriel embedding theorem</a></li> </ul> <h2 id="gabrielulmer_duality">Gabriel-Ulmer duality</h2> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a> is an equivalence of 2-categories LFP of locally finitely presentable categories and Lex of finitely complete categories. It is related to syntax-semantics adjunction and to Tannaka type reconstruction for coalgebra-like objects, with which has a common generalization (enriched Tannaka duality of Day).</p> <p>Gabriel–Ulmer duality has a generalisation to <a class="existingWikiWord" href="/nlab/show/sound+doctrines">sound doctrines</a>. For <a class="existingWikiWord" href="/nlab/show/sifted+colimits">sifted colimits</a> rather than <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>, this gives <a class="existingWikiWord" href="/nlab/show/Lawvere%27s+reconstruction+theorem">Lawvere's reconstruction theorem</a>.</p> <h2 id="heuristics">Heuristics</h2> <p>Typically in the proofs of most reconstruction theorems an implicit use of the Yoneda lemma is involved. Various embedding theorems of classes of categories (as well as theorems on realization as quotient categories) are closely related, e.g. <a class="existingWikiWord" href="/nlab/show/Barr+embedding+theorem">Barr embedding theorem</a> and <a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a>.</p> <p>Many of these examples are corollaries of the theory of <em>lex colimits</em>.</p> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <a href="http://www.maths.mq.edu.au/~street/CT90Como.pdf">An introduction to Tannaka duality and quantum groups</a>, in Part II of <em>Category Theory, Proceedings, Como 1990</em>, eds. A. Carboni, M. C. Pedicchio and G. Rosolini, Lec. Notes in Mathematics <strong>1488</strong>, Springer, Berlin, 1991, pp. 411–492 <a href="http://dx.doi.org/10.1007/BFb0084207">doi:10.1007/BFb0084207</a>.</p> </li> <li> <p>P. Deligne, <a class="existingWikiWord" href="/nlab/show/Cat%C3%A9gories+Tannakiennes">Catégories Tannakiennes</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+Festschrift">Grothendieck Festschrift</a>, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111–195.</p> </li> <li> <p>Alexander L. Rosenberg, <em>The existence of fiber functors</em>, in ‘The Gelfand Mathematical Seminars 1996–1999’, pp. 145–154. Birkhäuser, Boston, MA, 2000.</p> </li> <li> <p>A. L. Rosenberg, <a class="existingWikiWord" href="/nlab/show/Reconstruction+of+groups">Reconstruction of groups</a>, Selecta Math. N.S. 9:1 (2003) <a href="http://dx.doi.org/10.1007/s00029-003-0322-x">doi</a></p> </li> <li> <p>N. Saavedra Rivano, <em>Catégories Tannakiennes</em>, Springer LNM 265, 1972</p> </li> <li> <p>Bodo Pareigis, <a href="http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html">Quantum groups and noncommutative geometry</a>, WS 1999, chapter 3, online notes.</p> </li> <li> <p>S. Majid, <em>Foundations of quantum group theory</em>, chapter 9, Camb. Univ. Press 1995, 2002.</p> </li> <li> <p>S. Majid, <em>Tannaka–Krein theorem for quasiHopf algebras and other results</em>, Contemp. Math. 134 (1992) 219–232.</p> </li> <li> <p>A. L. Rosenberg, <em>Reconstruction of groups</em>, Selecta Math. N.S. <strong>9</strong>:1 (2003)<a href="http://dx.doi.org/10.1007/s00029-003-0322-x">doi:10.1007/s00029-003-0322-x</a> (nlab remark: this paper is on a generalization of Tannaka–Krein and not of the Gabriel–Rosenberg kind of reconstruction)</p> </li> <li> <p>A. Rosenberg, <em>The spectrum of abelian categories and reconstructions of schemes</em>, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. <strong>197</strong>, Marcel Dekker, New York, 257–274, 1998; MR99d:18011; and Max Planck Bonn preprint <em>Reconstruction of Schemes</em>, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=3948">MPIM1996-108</a> (1996).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A.+L.+Rosenberg">A. L. Rosenberg</a>, <em>Spectra of noncommutative spaces</em>, MPIM2003-110 <a href="http://www.mpim-bonn.mpg.de/preblob/1946">ps</a> <a href="http://www.mpim-bonn.mpg.de/preblob/1945">dvi</a>, <em>Underlying spaces of noncommutative schemes</em>, MPIM2003-111, <a href="http://www.mpim-bonn.mpg.de/preblob/1947">dvi</a>, <a href="http://www.mpim-bonn.mpg.de/preblob/1948">ps</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/P.+Gabriel">P. Gabriel</a>, <a class="existingWikiWord" href="/nlab/show/Des+cat%C3%A9gories+ab%C3%A9liennes">Des catégories abéliennes</a>, Bulletin de la Société Mathématique de France <strong>90</strong> (1962), 323–448, <a href="http://www.numdam.org/item?id=BSMF_1962__90__323_0">numdam</a></p> </li> <li> <p>A. I. Bondal, <a class="existingWikiWord" href="/nlab/show/D.+O.+Orlov">D. O. Orlov</a>, <em>Reconstruction of a variety from the derived category and groups of autoequivalences</em>, Compos. Math. 125 (2001), 327–344 <a href="http://dx.doi.org/10.1023/A:1002470302976">doi:10.1023/A:1002470302976</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K.+Szlach%C3%A1nyi">K. Szlachányi</a>, <em>Fiber functors, monoidal sites and Tannaka duality for bialgebroids</em> <a href="http://arxiv.org/abs/0907.1578">arxiv:0907.1578</a></p> </li> <li> <p>Phùng Hô Hai, <em>Tannaka–Krein duality for Hopf algebroids</em>, <a href="http://arxiv.org/abs/math/0206113">arxiv:math.QA/0206113</a></p> </li> <li> <p>Hélène Esnault, Phùng Hô Hai, <em>Gauß–Manin connection and Tannaka duality</em>, <a href="http://arxiv.org/abs/math/0509111">math.AG/0509111</a></p> </li> <li> <p>H. Pfeiffer, <em>Tannaka–Krein reconstruction and a characterization of modular tensor categories</em>, <a href="http://arxiv.org/abs/0711.1402">arxiv:math.QA/0711.1402</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S.+L.+Woronowicz">S. L. Woronowicz</a>, <em>Tannaka–Kreĭn duality for compact matrix pseudogroups. Twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)</annotation></semantics></math> groups</em>, Invent. Math. <strong>93</strong> (1988), no. 1, 35–76</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>, <em>Abstract duality for symmetric tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-categories</em> (App. to Hans Halvorson: ‘Algebraic Quantum Field Theory’.) In: J. Butterfield and J. Earman (eds.): “Handbook of the Philosophy of Physics”, p. 865–922. North Holland, 2007; <a href="http://arxiv.org/abs/math-ph/0602036">arxiv:math-ph/0602036</a>; cf. The String Coffee Table, <a href="http://golem.ph.utexas.edu/string/archives/000711.html">Müger on Doplicher–Roberts</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S.+Doplicher">S. Doplicher</a>, J. E. Roberts, <em>A new duality theory for compact groups</em>, Inventiones Math., 98(1):157–218, 1989.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/P.+Balmer">P. Balmer</a>, <em>The spectrum of prime ideals in tensor triangulated categories</em>, J. Reine Angew. Math. <strong>588</strong> (2005), pp. 149–168 <a href="http://www.math.ucla.edu/~balmer/research/Pubfile/Spectrum.dvi">dvi</a> <a href="http://www.math.ucla.edu/~balmer/research/Pubfile/Spectrum.pdf">pdf</a> <a href="http://www.math.ucla.edu/~balmer/research/Pubfile/Spectrum.ps">ps</a>.</p> </li> <li> <p><a href="http://www.maths.man.ac.uk/~mprest">M. Prest</a>, <a class="existingWikiWord" href="/nlab/show/G.+Garkusha">G. Garkusha</a>, <em>Reconstructing projective schemes from Serre subcategories</em>, J. Algebra <strong>319</strong> (3) (2008), 1132–1153 (<a href="http://www-maths.swan.ac.uk/staff/gg/papers/garkpr44.pdf">pdf</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/P.+Deligne">P. Deligne</a>, J. S. Milne, <em>Tannakian categories</em>, Lect. notes in math. 900, 101–228, Springer 1982.</p> </li> <li> <p>A. Bruguières, <em>On a tannakian theorem due to Nori</em>, <a href="http://imag.umontpellier.fr/~bruguieres/docs/ntan.pdf">pdf</a>; <em>Théorie tannakienne non commutative</em>, Comm. Algebra <strong>22</strong>, 5817–5860, 1994</p> </li> </ul> <p>Lex colimits are discussed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a> and <a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>. <em>Lex colimits</em>. Journal of Pure and Applied Algebra 216.6 (2012): 1372-1396.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 15, 2023 at 15:25:55. See the <a href="/nlab/history/reconstruction+theorem" 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/reconstruction+theorem" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12499/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/reconstruction+theorem/28" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/reconstruction+theorem" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/reconstruction+theorem" accesskey="S" class="navlink" id="history" rel="nofollow">History (28 revisions)</a> <a href="/nlab/show/reconstruction+theorem/cite" style="color: black">Cite</a> <a href="/nlab/print/reconstruction+theorem" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/reconstruction+theorem" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>