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> Picard groupoid 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> Picard groupoid </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/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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> <blockquote> <p>This page considers Picard groupoids in themselves. For the concept of <a class="existingWikiWord" href="/nlab/show/Picard+groupoid+of+a+monoidal+category">Picard groupoid of a monoidal category</a>, see there.</p> </blockquote> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#introduction'>Introduction</a></li> <li><a href='#2category_of_picard_groupoids'>2-category of Picard groupoids</a></li> <li><a href='#additivity_of_the_homotopy_category_of_picard_groupoids'>Additivity of the homotopy category of Picard groupoids</a></li> <li><a href='#model_for_stable_homotopy_1types'>Model for stable homotopy 1-types</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="introduction">Introduction</h2> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>A <em>Picard groupoid</em> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal">strict symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒜</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{A}, \otimes, 1)</annotation></semantics></math> in which every <a class="existingWikiWord" href="/nlab/show/object">object</a> and every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> is strictly <a class="existingWikiWord" href="/nlab/show/inverse">invertible</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>; that is to say: there is, for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">a^{-1}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊗</mo><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a \otimes a^{-1} = 1</annotation></semantics></math>, and similarly for morphisms.</p> </div> <div class="num_defn"> <h6 id="remark">Remark</h6> <p>The notion of a Picard groupoid <a class="existingWikiWord" href="/nlab/show/categorification">categorifies</a> that of an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>. Note in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> equip the set of objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> with the structure of an abelian group.</p> </div> <div class="num_defn"> <h6 id="remark_2">Remark</h6> <p>The notion of a Picard groupoid can be weakened in two (or three) directions: the monoidal structure can be weak instead of strict (or just the symmetry part), and the invertibility criterion can be asked to hold only up to isomorphism. On this page, we shall, however, work with the fully strict notion.</p> </div> <h2 id="2category_of_picard_groupoids">2-category of Picard groupoids</h2> <p>Picard groupoids assemble into a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a>. The objects are Picard groupoids, the 1-arrows are strict <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functors</a> (these necessarily preserve both the symmetry and the object inverses), and the 2-arrows are <a class="existingWikiWord" href="/nlab/show/monoidal+natural+transformation">monoidal natural transformations</a>.</p> <p>This strict 2-category admits a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+2-category">(fully) strict symmetric monoidal structure</a>, which categorifies the usual closed monoidal structure of the category of abelian groups.</p> <h2 id="additivity_of_the_homotopy_category_of_picard_groupoids">Additivity of the homotopy category of Picard groupoids</h2> <p>The <em>homotopy category</em> of the category of Picard groupoids consists, roughly speaking, of Picard groupoids up to equivalence. Formally, we can obtain it by <a class="existingWikiWord" href="/nlab/show/decategorifying">decategorifying</a> the 2-category of Picard groupoids, namely identifying all 1-arrows which are 2-isomorphic, and throwing away the 2-arrows.</p> <p>The closed monoidal structure of the 2-category of Picard groupoids, together with the fact that the objects of a Picard groupoid define an abelian group under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, provides immediately an <a class="existingWikiWord" href="/nlab/show/enriched+category">enrichment</a> of the homotopy category of Picard groupoids over abelian groups. Moreover, it is obvious that this category has all coproducts. Thus it is an additive category.</p> <h2 id="model_for_stable_homotopy_1types">Model for stable homotopy 1-types</h2> <p>Picard groupoids are well-known to model stable homotopy 1-types, at least if one adopts the weak version of the invertibility condition. This is a stable version of the 1-truncated <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 4, 2023 at 11:06:03. See the <a href="/nlab/history/Picard+groupoid" 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/Picard+groupoid" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/Picard+groupoid/6" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Picard+groupoid" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Picard+groupoid" accesskey="S" class="navlink" id="history" rel="nofollow">History (6 revisions)</a> <a href="/nlab/show/Picard+groupoid/cite" style="color: black">Cite</a> <a href="/nlab/print/Picard+groupoid" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Picard+groupoid" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>