CINXE.COM

bimodule in nLab

<!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> bimodule 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[/*><!--*/ .toc ul {margin: 0; padding: 0;} .toc ul ul {margin: 0; padding: 0 0 0 10px;} .toc li > p {margin: 0} .toc ul li {list-style-type: none; position: relative;} .toc div {border-top:1px dotted #ccc;} .rightHandSide h2 {font-size: 1.5em;color:#008B26} table.plaintable { border-collapse:collapse; margin-left:30px; border:0; } .plaintable td {border:1px solid #000; padding: 3px;} .plaintable th {padding: 3px;} .plaintable caption { font-weight: bold; font-size:1.1em; text-align:center; margin-left:30px; } /* Query boxes for questioning and answering mechanism */ div.query{ background: #f6fff3; border: solid #ce9; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; max-height: 20em; overflow: auto; } /* Standout boxes for putting important text */ div.standout{ background: #fff1f1; border: solid black; border-width: 2px 1px; padding: 0 1em; margin: 0 1em; overflow: auto; } /* Icon for links to n-category arXiv documents (commented out for now i.e. disabled) a[href*="http://arxiv.org/"] { background-image: url(../files/arXiv_icon.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 22px; } */ /* Icon for links to n-category cafe posts (disabled) a[href*="http://golem.ph.utexas.edu/category"] { background-image: url(../files/n-cafe_5.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pdf files (disabled) a[href$=".pdf"] { background-image: url(../files/pdficon_small.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ /* Icon for links to pages, etc. -inside- pdf files (disabled) a[href*=".pdf#"] { background-image: url(../files/pdf_entry.gif); background-repeat: no-repeat; background-position: right bottom; padding-right: 25px; } */ a.existingWikiWord { color: #226622; } a.existingWikiWord:visited { color: #226622; } a.existingWikiWord[title] { border: 0px; color: #aa0505; text-decoration: none; } a.existingWikiWord[title]:visited { border: 0px; color: #551111; text-decoration: none; } a[href^="http://"] { border: 0px; color: #003399; } a[href^="http://"]:visited { border: 0px; color: #330066; } a[href^="https://"] { border: 0px; color: #003399; } a[href^="https://"]:visited { border: 0px; color: #330066; } div.dropDown .hide { display: none; } div.dropDown:hover .hide { display:block; } div.clickDown .hide { display: none; } div.clickDown:focus { outline:none; } div.clickDown:focus .hide, div.clickDown:hover .hide { display: block; } div.clickDown .clickToReveal, div.clickDown:focus .clickToHide { display:block; } div.clickDown:focus .clickToReveal, div.clickDown .clickToHide { display:none; } div.clickDown .clickToReveal:after { content: "A(Hover to reveal, click to "hold")"; font-size: 60%; } div.clickDown .clickToHide:after { content: "A(Click to hide)"; font-size: 60%; } div.clickDown .clickToHide, div.clickDown .clickToReveal { white-space: pre-wrap; } .un_theorem, .num_theorem, .un_lemma, .num_lemma, .un_prop, .num_prop, .un_cor, .num_cor, .un_defn, .num_defn, .un_example, .num_example, .un_note, .num_note, .un_remark, .num_remark { margin-left: 1em; } span.theorem_label { margin-left: -1em; } .proof span.theorem_label { margin-left: 0em; } :target { background-color: #BBBBBB; border-radius: 5pt; } /*]]>*/--></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[//><!-- MathJax.Ajax.config.path["Contrib"] = "/MathJax"; MathJax.Hub.Config({ MathML: { useMathMLspacing: true }, "HTML-CSS": { scale: 90, extensions: ["handle-floats.js"] } }); MathJax.Hub.Queue( function () { var fos = document.getElementsByTagName('foreignObject'); for (var i = 0; i < fos.length; i++) { MathJax.Hub.Typeset(fos[i]); } }); //--><!]]> </script> <script type="text/javascript"> <!--//--><![CDATA[//><!-- window.addEventListener("DOMContentLoaded", function () { var div = document.createElement('div'); var math = document.createElementNS('http://www.w3.org/1998/Math/MathML', 'math'); document.body.appendChild(div); div.appendChild(math); // Test for MathML support comparable to WebKit version https://trac.webkit.org/changeset/203640 or higher. div.setAttribute('style', 'font-style: italic'); var mathml_unsupported = !(window.getComputedStyle(div.firstChild).getPropertyValue('font-style') === 'normal'); div.parentNode.removeChild(div); if (mathml_unsupported) { // MathML does not seem to be supported... var s = document.createElement('script'); s.src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=MML_HTMLorMML-full"; document.querySelector('head').appendChild(s); } else { document.head.insertAdjacentHTML("beforeend", '<style>svg[viewBox] {max-width: 100%}</style>'); } }); //--><!]]> </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> bimodule </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/4728/#Item_17" 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="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+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='#definition'>Definition</a></li> <ul> <li><a href='#over_a_ring'>Over a ring</a></li> <ul> <li><a href='#with_a_left_action_and_a_right_action'>With a left action and a right action</a></li> <li><a href='#with_a_biaction'>With a biaction</a></li> </ul> <li><a href='#over_a_monoid_in_a_monoidal_category'>Over a monoid in a monoidal category</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#biactions_left_actions_and_right_actions'>Biactions, left actions, and right actions</a></li> <li><a href='#linear_maps'>Linear maps</a></li> <li><a href='#tensor_product_of_bimodules'>Tensor product of bimodules</a></li> <li><a href='#twosided_ideals_of_a_ring'>Two-sided ideals of a ring</a></li> <li><a href='#rings_over_a_ring'>Rings over a ring</a></li> </ul> <li><a href='#categories_of_bimodules'>Categories of bimodules</a></li> <ul> <li><a href='#the_1category_of_bimodules_and_intertwiners'>The 1-category of bimodules and intertwiners</a></li> <li><a href='#AsMorphismsInA2Category'>The 2-category of rings, bimodules, and intertwiners</a></li> <li><a href='#Infinity2CategoryOfInfinityAlgebrasAndBimodules'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category of <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>-algebras and <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>-bimodules</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>bimodule</em> is a <a class="existingWikiWord" href="/nlab/show/module">module</a> in two compatible ways over two <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> <h2 id="definition">Definition</h2> <h3 id="over_a_ring">Over a ring</h3> <h4 id="with_a_left_action_and_a_right_action">With a left action and a right action</h4> <p>Given two <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/bilinear+function">bilinear</a> <a class="existingWikiWord" href="/nlab/show/left+action">left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo>:</mo><mi>R</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha_R:R \times B \to B</annotation></semantics></math> and a bilinear <a class="existingWikiWord" href="/nlab/show/right+action">right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math>-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo>:</mo><mi>B</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha_S:B \times S \to B</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)</annotation></semantics></math>.</p> <h4 id="with_a_biaction">With a biaction</h4> <p>Equivalently, given two <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/multilinear+function">trilinear</a> <a class="existingWikiWord" href="/nlab/show/biaction"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math>-biaction</a>, a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>B</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">(-)(-)(-):R \times B \times S \to B</annotation></semantics></math> such that</p> <ul> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">1_R b 1_S = b</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_1 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>2</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_2 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_1 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_2 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mi>b</mi><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_1 (r_2 b s_1) s_2 = (r_1 \cdot_R r_2) b (s_1 \cdot_S s_2)</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_1 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mn>2</mn></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r_2 \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><mi>s</mi><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><mi>b</mi><mi>s</mi><mo>+</mo><msub><mi>r</mi> <mn>2</mn></msub><mi>b</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">(r_1 + r_2) b s = r_1 b s + r_2 b s</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b_1 \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>2</mn></msub><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b_2 \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>s</mi><mo>=</mo><mi>r</mi><msub><mi>b</mi> <mn>1</mn></msub><mi>s</mi><mo>+</mo><mi>r</mi><msub><mi>b</mi> <mn>2</mn></msub><mi>s</mi></mrow><annotation encoding="application/x-tex">r (b_1 + b_2) s = r b_1 s + r b_2 s</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_1 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_2 \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>b</mi><msub><mi>s</mi> <mn>1</mn></msub><mo>+</mo><mi>r</mi><mi>b</mi><msub><mi>s</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">r b (s_1 + s_2) = r b s_1 + r b s_2</annotation></semantics></math></p> </li> </ul> <p>representing simultaneous left multiplication by scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and right multiplication by scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>.</p> <h3 id="over_a_monoid_in_a_monoidal_category">Over a monoid in a monoidal category</h3> <p>We can define in more generality what is 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><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B)</annotation></semantics></math>-bimodule in a monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes,I)</annotation></semantics></math> where <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><msup><mo>∇</mo> <mi>A</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>A</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\nabla^{A},\eta^{A})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msup><mo>∇</mo> <mi>B</mi></msup><mo>,</mo><msup><mi>η</mi> <mi>B</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,\nabla^{B},\eta^{B})</annotation></semantics></math> are two monoids. It is given by:</p> <ul> <li>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math></li> <li>A left-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">l:A \otimes X \rightarrow X</annotation></semantics></math></li> <li>A right-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><mi>X</mi><mo>⊗</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">r:X \otimes B \rightarrow X</annotation></semantics></math></li> </ul> <p>such that:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,l)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/module+over+a+monoid"> left module</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,r)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/module+over+a+monoid"> right module</a></li> </ul> <p>and moreover this diagram commutes: <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="253.355pt" height="90.138pt" viewBox="0 0 253.355 90.138" version="1.2"> <defs> <g> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-1"> <path style="stroke:none;" d="M 2.515625 -1.640625 C 2 -0.765625 1.5 -0.46875 0.78125 -0.421875 C 0.625 -0.421875 0.5 -0.421875 0.5 -0.140625 C 0.5 -0.0625 0.578125 0 0.6875 0 C 0.953125 0 1.609375 -0.03125 1.875 -0.03125 C 2.3125 -0.03125 2.78125 0 3.203125 0 C 3.28125 0 3.46875 0 3.46875 -0.28125 C 3.46875 -0.421875 3.34375 -0.421875 3.25 -0.421875 C 2.921875 -0.453125 2.640625 -0.578125 2.640625 -0.9375 C 2.640625 -1.140625 2.71875 -1.296875 2.921875 -1.625 L 4.046875 -3.484375 L 7.8125 -3.484375 C 7.828125 -3.359375 7.828125 -3.234375 7.84375 -3.109375 C 7.890625 -2.71875 8.0625 -1.1875 8.0625 -0.90625 C 8.0625 -0.453125 7.3125 -0.421875 7.078125 -0.421875 C 6.90625 -0.421875 6.75 -0.421875 6.75 -0.15625 C 6.75 0 6.875 0 6.96875 0 C 7.21875 0 7.515625 -0.03125 7.765625 -0.03125 L 8.609375 -0.03125 C 9.515625 -0.03125 10.171875 0 10.1875 0 C 10.28125 0 10.453125 0 10.453125 -0.28125 C 10.453125 -0.421875 10.3125 -0.421875 10.09375 -0.421875 C 9.28125 -0.421875 9.265625 -0.5625 9.21875 -1 L 8.3125 -10.234375 C 8.28125 -10.53125 8.234375 -10.5625 8.0625 -10.5625 C 7.921875 -10.5625 7.828125 -10.53125 7.6875 -10.3125 Z M 4.296875 -3.921875 L 7.265625 -8.890625 L 7.765625 -3.921875 Z M 4.296875 -3.921875 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-2"> <path style="stroke:none;" d="M 7.03125 -6 L 5.640625 -9.25 C 5.828125 -9.609375 6.28125 -9.65625 6.453125 -9.671875 C 6.546875 -9.671875 6.703125 -9.6875 6.703125 -9.9375 C 6.703125 -10.109375 6.5625 -10.109375 6.484375 -10.109375 C 6.234375 -10.109375 5.9375 -10.078125 5.6875 -10.078125 L 4.828125 -10.078125 C 3.921875 -10.078125 3.265625 -10.109375 3.25 -10.109375 C 3.140625 -10.109375 2.984375 -10.109375 2.984375 -9.828125 C 2.984375 -9.671875 3.125 -9.671875 3.3125 -9.671875 C 4.171875 -9.671875 4.234375 -9.53125 4.375 -9.171875 L 6.140625 -5.0625 L 2.9375 -1.625 C 2.390625 -1.046875 1.765625 -0.484375 0.671875 -0.421875 C 0.484375 -0.421875 0.375 -0.421875 0.375 -0.140625 C 0.375 -0.109375 0.390625 0 0.546875 0 C 0.75 0 0.984375 -0.03125 1.1875 -0.03125 L 1.875 -0.03125 C 2.359375 -0.03125 2.875 0 3.328125 0 C 3.4375 0 3.609375 0 3.609375 -0.265625 C 3.609375 -0.421875 3.5 -0.421875 3.421875 -0.421875 C 3.125 -0.453125 2.9375 -0.625 2.9375 -0.859375 C 2.9375 -1.109375 3.109375 -1.28125 3.53125 -1.734375 L 4.859375 -3.171875 C 5.171875 -3.5 5.96875 -4.359375 6.28125 -4.6875 L 7.84375 -1.046875 C 7.859375 -1.015625 7.921875 -0.875 7.921875 -0.859375 C 7.921875 -0.71875 7.59375 -0.453125 7.125 -0.421875 C 7.03125 -0.421875 6.859375 -0.421875 6.859375 -0.140625 C 6.859375 0 7.015625 0 7.09375 0 C 7.34375 0 7.640625 -0.03125 7.890625 -0.03125 L 9.515625 -0.03125 C 9.78125 -0.03125 10.0625 0 10.3125 0 C 10.421875 0 10.578125 0 10.578125 -0.28125 C 10.578125 -0.421875 10.4375 -0.421875 10.296875 -0.421875 C 9.40625 -0.4375 9.375 -0.515625 9.125 -1.0625 L 7.171875 -5.65625 L 9.0625 -7.671875 C 9.203125 -7.8125 9.546875 -8.1875 9.671875 -8.328125 C 10.3125 -9 10.90625 -9.609375 12.109375 -9.671875 C 12.25 -9.6875 12.40625 -9.6875 12.40625 -9.9375 C 12.40625 -10.109375 12.265625 -10.109375 12.203125 -10.109375 C 12 -10.109375 11.78125 -10.078125 11.578125 -10.078125 L 10.890625 -10.078125 C 10.421875 -10.078125 9.90625 -10.109375 9.4375 -10.109375 C 9.34375 -10.109375 9.15625 -10.109375 9.15625 -9.84375 C 9.15625 -9.6875 9.265625 -9.671875 9.359375 -9.671875 C 9.59375 -9.640625 9.84375 -9.53125 9.84375 -9.25 L 9.828125 -9.21875 C 9.8125 -9.109375 9.78125 -8.96875 9.625 -8.796875 Z M 7.03125 -6 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-3"> <path style="stroke:none;" d="M 5.421875 -9.09375 C 5.546875 -9.640625 5.609375 -9.671875 6.1875 -9.671875 L 8.109375 -9.671875 C 9.78125 -9.671875 9.78125 -8.25 9.78125 -8.125 C 9.78125 -6.921875 8.578125 -5.40625 6.625 -5.40625 L 4.5 -5.40625 Z M 7.921875 -5.28125 C 9.53125 -5.578125 11 -6.703125 11 -8.0625 C 11 -9.21875 9.96875 -10.109375 8.296875 -10.109375 L 3.546875 -10.109375 C 3.265625 -10.109375 3.140625 -10.109375 3.140625 -9.828125 C 3.140625 -9.671875 3.265625 -9.671875 3.484375 -9.671875 C 4.390625 -9.671875 4.390625 -9.5625 4.390625 -9.390625 C 4.390625 -9.359375 4.390625 -9.28125 4.328125 -9.0625 L 2.34375 -1.09375 C 2.203125 -0.578125 2.171875 -0.421875 1.140625 -0.421875 C 0.859375 -0.421875 0.703125 -0.421875 0.703125 -0.15625 C 0.703125 0 0.796875 0 1.09375 0 L 6.171875 0 C 8.4375 0 10.1875 -1.71875 10.1875 -3.21875 C 10.1875 -4.421875 9.109375 -5.171875 7.921875 -5.28125 Z M 5.8125 -0.421875 L 3.8125 -0.421875 C 3.609375 -0.421875 3.578125 -0.421875 3.484375 -0.4375 C 3.328125 -0.453125 3.3125 -0.484375 3.3125 -0.609375 C 3.3125 -0.703125 3.34375 -0.796875 3.375 -0.9375 L 4.40625 -5.109375 L 7.1875 -5.109375 C 8.9375 -5.109375 8.9375 -3.484375 8.9375 -3.359375 C 8.9375 -1.9375 7.65625 -0.421875 5.8125 -0.421875 Z M 5.8125 -0.421875 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph1-1"> <path style="stroke:none;" d="M 10.6875 -3.703125 C 10.6875 -6.421875 8.46875 -8.625 5.75 -8.625 C 3 -8.625 0.8125 -6.390625 0.8125 -3.703125 C 0.8125 -0.984375 3.03125 1.234375 5.734375 1.234375 C 8.5 1.234375 10.6875 -1 10.6875 -3.703125 Z M 2.8125 -6.875 C 2.78125 -6.90625 2.671875 -7.015625 2.671875 -7.046875 C 2.671875 -7.109375 3.875 -8.25 5.734375 -8.25 C 6.265625 -8.25 7.625 -8.1875 8.828125 -7.046875 L 5.75 -3.953125 Z M 2.375 -0.609375 C 1.484375 -1.609375 1.1875 -2.75 1.1875 -3.703125 C 1.1875 -4.84375 1.609375 -5.921875 2.375 -6.796875 L 5.46875 -3.703125 Z M 9.09375 -6.796875 C 9.8125 -6.03125 10.3125 -4.90625 10.3125 -3.703125 C 10.3125 -2.5625 9.890625 -1.484375 9.109375 -0.609375 L 6.015625 -3.703125 Z M 8.6875 -0.515625 C 8.71875 -0.484375 8.8125 -0.390625 8.8125 -0.359375 C 8.8125 -0.296875 7.625 0.859375 5.75 0.859375 C 5.234375 0.859375 3.875 0.78125 2.65625 -0.359375 L 5.734375 -3.453125 Z M 8.6875 -0.515625 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-1"> <path style="stroke:none;" d="M 2.59375 -6.5625 C 2.59375 -6.578125 2.625 -6.703125 2.625 -6.71875 C 2.625 -6.765625 2.59375 -6.859375 2.46875 -6.859375 L 1.46875 -6.78125 C 1.109375 -6.75 1.03125 -6.734375 1.03125 -6.5625 C 1.03125 -6.421875 1.171875 -6.421875 1.28125 -6.421875 C 1.75 -6.421875 1.75 -6.359375 1.75 -6.265625 C 1.75 -6.234375 1.75 -6.21875 1.703125 -6.046875 L 0.484375 -1.140625 C 0.4375 -0.984375 0.4375 -0.84375 0.4375 -0.828125 C 0.4375 -0.21875 0.953125 0.09375 1.4375 0.09375 C 1.859375 0.09375 2.09375 -0.234375 2.203125 -0.453125 C 2.375 -0.78125 2.53125 -1.359375 2.53125 -1.40625 C 2.53125 -1.46875 2.5 -1.546875 2.375 -1.546875 C 2.28125 -1.546875 2.25 -1.484375 2.25 -1.484375 C 2.234375 -1.453125 2.1875 -1.28125 2.15625 -1.171875 C 2 -0.59375 1.8125 -0.171875 1.46875 -0.171875 C 1.21875 -0.171875 1.15625 -0.40625 1.15625 -0.640625 C 1.15625 -0.828125 1.1875 -0.9375 1.21875 -1.0625 Z M 2.59375 -6.5625 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-2"> <path style="stroke:none;" d="M 1.671875 -0.78125 C 1.578125 -0.40625 1.5625 -0.328125 0.828125 -0.328125 C 0.640625 -0.328125 0.515625 -0.328125 0.515625 -0.140625 C 0.515625 0 0.65625 0 0.8125 0 L 4.484375 0 C 6.109375 0 7.328125 -1.15625 7.328125 -2.109375 C 7.328125 -2.828125 6.703125 -3.421875 5.71875 -3.53125 C 6.859375 -3.75 7.84375 -4.5 7.84375 -5.359375 C 7.84375 -6.109375 7.125 -6.75 5.890625 -6.75 L 2.4375 -6.75 C 2.265625 -6.75 2.140625 -6.75 2.140625 -6.5625 C 2.140625 -6.421875 2.25 -6.421875 2.421875 -6.421875 C 2.75 -6.421875 3.03125 -6.421875 3.03125 -6.265625 C 3.03125 -6.21875 3.015625 -6.21875 2.984375 -6.078125 Z M 3.203125 -3.640625 L 3.8125 -6.046875 C 3.90625 -6.390625 3.90625 -6.421875 4.3125 -6.421875 L 5.734375 -6.421875 C 6.703125 -6.421875 6.921875 -5.78125 6.921875 -5.375 C 6.921875 -4.546875 6.03125 -3.640625 4.765625 -3.640625 Z M 2.53125 -0.328125 C 2.4375 -0.34375 2.40625 -0.34375 2.40625 -0.421875 C 2.40625 -0.5 2.421875 -0.578125 2.453125 -0.625 L 3.140625 -3.375 L 5.140625 -3.375 C 6.0625 -3.375 6.375 -2.75 6.375 -2.1875 C 6.375 -1.21875 5.421875 -0.328125 4.234375 -0.328125 Z M 2.53125 -0.328125 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-3"> <path style="stroke:none;" d="M 1.828125 -1.171875 C 1.375 -0.453125 0.96875 -0.359375 0.609375 -0.328125 C 0.515625 -0.3125 0.390625 -0.3125 0.390625 -0.140625 C 0.390625 -0.109375 0.421875 0 0.53125 0 C 0.625 0 0.6875 -0.03125 1.375 -0.03125 C 2.046875 -0.03125 2.265625 0 2.328125 0 C 2.375 0 2.515625 0 2.515625 -0.1875 C 2.515625 -0.3125 2.390625 -0.328125 2.34375 -0.328125 C 2.140625 -0.328125 1.921875 -0.421875 1.921875 -0.625 C 1.921875 -0.78125 2 -0.90625 2.1875 -1.1875 L 2.859375 -2.28125 L 5.5625 -2.28125 L 5.78125 -0.609375 C 5.78125 -0.46875 5.59375 -0.328125 5.140625 -0.328125 C 4.984375 -0.328125 4.859375 -0.328125 4.859375 -0.140625 C 4.859375 -0.125 4.875 0 5.03125 0 C 5.125 0 5.5 -0.015625 5.59375 -0.03125 L 6.21875 -0.03125 C 7.09375 -0.03125 7.265625 0 7.34375 0 C 7.390625 0 7.546875 0 7.546875 -0.1875 C 7.546875 -0.328125 7.421875 -0.328125 7.265625 -0.328125 C 6.71875 -0.328125 6.703125 -0.421875 6.671875 -0.65625 L 5.890625 -6.765625 C 5.859375 -6.984375 5.859375 -7.046875 5.6875 -7.046875 C 5.515625 -7.046875 5.453125 -6.953125 5.390625 -6.859375 Z M 3.078125 -2.609375 L 5.109375 -5.859375 L 5.53125 -2.609375 Z M 3.078125 -2.609375 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-4"> <path style="stroke:none;" d="M 1.90625 -1.359375 C 2.015625 -1.78125 2.125 -2.21875 2.21875 -2.640625 C 2.234375 -2.671875 2.296875 -2.953125 2.3125 -3 C 2.34375 -3.09375 2.59375 -3.5 2.84375 -3.75 C 3.15625 -4.03125 3.5 -4.078125 3.671875 -4.078125 C 3.78125 -4.078125 3.953125 -4.0625 4.09375 -3.953125 C 3.671875 -3.859375 3.609375 -3.5 3.609375 -3.40625 C 3.609375 -3.1875 3.78125 -3.046875 4 -3.046875 C 4.265625 -3.046875 4.5625 -3.265625 4.5625 -3.65625 C 4.5625 -4.015625 4.25 -4.359375 3.6875 -4.359375 C 3.015625 -4.359375 2.5625 -3.90625 2.359375 -3.640625 C 2.15625 -4.359375 1.484375 -4.359375 1.390625 -4.359375 C 1.03125 -4.359375 0.796875 -4.125 0.625 -3.828125 C 0.40625 -3.375 0.296875 -2.875 0.296875 -2.84375 C 0.296875 -2.75 0.359375 -2.71875 0.4375 -2.71875 C 0.578125 -2.71875 0.578125 -2.75 0.65625 -3.015625 C 0.765625 -3.5 0.953125 -4.078125 1.359375 -4.078125 C 1.625 -4.078125 1.671875 -3.828125 1.671875 -3.609375 C 1.671875 -3.4375 1.625 -3.25 1.546875 -2.921875 C 1.53125 -2.84375 1.375 -2.265625 1.34375 -2.125 L 0.984375 -0.640625 C 0.9375 -0.5 0.875 -0.25 0.875 -0.203125 C 0.875 0.015625 1.0625 0.09375 1.1875 0.09375 C 1.546875 0.09375 1.609375 -0.171875 1.6875 -0.515625 Z M 1.90625 -1.359375 "></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph3-1"> <path style="stroke:none;" d="M 7.5625 -2.46875 C 7.5625 -4.421875 5.984375 -5.96875 4.078125 -5.96875 C 2.140625 -5.96875 0.578125 -4.390625 0.578125 -2.46875 C 0.578125 -0.515625 2.15625 1.03125 4.0625 1.03125 C 6.015625 1.03125 7.5625 -0.546875 7.5625 -2.46875 Z M 1.9375 -4.828125 C 2.703125 -5.546875 3.59375 -5.65625 4.0625 -5.65625 C 4.671875 -5.65625 5.515625 -5.484375 6.21875 -4.828125 L 4.078125 -2.6875 Z M 1.71875 -0.328125 C 1.171875 -0.90625 0.890625 -1.703125 0.890625 -2.46875 C 0.890625 -3.21875 1.15625 -4.015625 1.71875 -4.609375 L 3.859375 -2.46875 Z M 6.421875 -4.609375 C 6.984375 -4.03125 7.265625 -3.234375 7.265625 -2.46875 C 7.265625 -1.71875 6.984375 -0.921875 6.421875 -0.328125 L 4.28125 -2.46875 Z M 6.21875 -0.109375 C 5.4375 0.609375 4.546875 0.71875 4.078125 0.71875 C 3.46875 0.71875 2.640625 0.546875 1.9375 -0.109375 L 4.0625 -2.25 Z M 6.21875 -0.109375 "></path> </symbol> </g> </defs> <g id="F1qNV8cVsnHhvIH4oLqP87FQtec=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-1" x="7.253565" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph1-1" x="21.408202" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-2" x="36.210397" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph1-1" x="52.692774" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-3" x="67.494969" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-1" x="203.312629" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph1-1" x="217.467267" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-2" x="232.269462" y="18.141511"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-2" x="21.732105" y="82.066192"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph1-1" x="38.214482" y="82.066192"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-3" x="53.016677" y="82.066192"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph0-2" x="217.79117" y="82.066192"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -83.898383 20.821486 L -83.898383 -20.341872 " transform="matrix(0.990527,0,0,-0.990527,126.35756,45.663316)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485066 2.868146 C -2.031508 1.148709 -1.021909 0.33633 -0.000487609 0.0011369 C -1.021895 -0.334097 -2.031462 -1.146516 -2.484952 -2.869915 " transform="matrix(-0.0000198105,0.990508,0.990508,0.0000198105,43.25278,66.051264)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-1" x="19.974909" y="48.575467"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph3-1" x="23.222601" y="48.575467"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-2" x="31.377118" y="48.575467"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -40.873643 31.520489 L 70.529281 31.520489 " transform="matrix(0.990527,0,0,-0.990527,126.35756,45.663316)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487604 2.869834 C -2.03409 1.146478 -1.020583 0.334095 0.000810886 -0.00111107 C -1.020583 -0.336318 -2.03409 -1.1487 -2.487604 -2.868113 " transform="matrix(0.990527,0,0,-0.990527,196.456228,14.440306)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-3" x="130.767388" y="9.930038"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph3-1" x="138.621033" y="9.930038"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-4" x="146.77555" y="9.930038"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 98.970567 20.821486 L 98.966623 -20.341872 " transform="matrix(0.990527,0,0,-0.990527,126.35756,45.663316)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485066 2.869256 C -2.031508 1.145875 -1.021909 0.333496 -0.000487553 -0.00169686 C -1.021895 -0.332987 -2.031462 -1.14935 -2.484952 -2.868805 " transform="matrix(-0.0000198105,0.990508,0.990508,0.0000198105,224.388399,66.051264)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-1" x="227.870777" y="49.089551"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -55.488646 -33.016622 L 85.148228 -33.016622 " transform="matrix(0.990527,0,0,-0.990527,126.35756,45.663316)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485807 2.870323 C -2.032293 1.146967 -1.018786 0.334585 -0.00133559 -0.000621874 C -1.018786 -0.335828 -2.032293 -1.148211 -2.485807 -2.867623 " transform="matrix(0.990527,0,0,-0.990527,210.934917,78.366572)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#F1qNV8cVsnHhvIH4oLqP87FQtec=-glyph2-4" x="138.77184" y="86.097638"></use> </g> </g> </svg></p> <h2 id="properties">Properties</h2> <h3 id="biactions_left_actions_and_right_actions">Biactions, left actions, and right actions</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/ring">ring</a>s, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule.</p> <p>Given a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_R</annotation></semantics></math> and a right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_S</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule, the biaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>B</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">(-)(-)(-):R \times B \times S \to B</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>b</mi><mi>s</mi><mo>≔</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r b s \coloneqq \alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)</annotation></semantics></math></div> <p>The biaction is trilinear because the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action and right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action are bilinear.</p> <p>On the other hand, given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-biaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule, the <a class="existingWikiWord" href="/nlab/show/left+action">left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-action</a> is defined from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-biaction as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>r</mi><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_R(r, b) \coloneqq r b 1_S</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>. It is a left action because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\alpha_R(1_R, b) = 1_R b 1_S = m</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>α</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>2</mn></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>S</mi></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mi>r</mi> <mn>2</mn></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_R(r_1, \alpha_L(r_2, b)) = r_1 (r_2 b 1_S) 1_S = (r_1 \cdot_R r_2) b (1_S \cdot_S 1_S) = (r_1 \cdot_R r_2) b 1_S = \alpha_R(r_1 \cdot_R r_2, b)</annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/right+action">right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>S</mi> </mrow> <annotation encoding="application/x-tex">S</annotation> </semantics> </math>-action</a> is defined from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-biaction as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">\alpha_S(b, s) \coloneqq 1_R b s</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>. It is a right action because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>R</mi></msub><mi>b</mi><msub><mn>1</mn> <mi>S</mi></msub><mo>=</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\alpha_S(b, 1_S) = 1_R b 1_S = m</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>b</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mn>1</mn> <mi>R</mi></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>S</mi></msub><mi>b</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_S(\alpha_S(b, s_1), s_2) = 1_R (1_R, b, s_1) s_2 = (1_R \cdot_R 1_R) b (s_1 \cdot_S s_2) = 1_S b (s_1 \cdot_S s_2) = \alpha_S(b, s_1 \cdot_S s_2)</annotation></semantics></math></div> <p>The left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action and right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action satisfy the following identity:</p> <ul> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)</annotation></semantics></math>.</li> </ul> <p>This is because when expanded out, the identity becomes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>b</mi><mo>,</mo><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha(r, \alpha(1_R, b, s), 1_S) = \alpha(1_R, \alpha(r, b, 1_S), s)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><msub><mo>⋅</mo> <mi>R</mi></msub><msub><mn>1</mn> <mi>R</mi></msub><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><mi>s</mi><msub><mo>⋅</mo> <mi>S</mi></msub><msub><mn>1</mn> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>R</mi></msub><msub><mo>⋅</mo> <mi>R</mi></msub><mi>r</mi><mo stretchy="false">)</mo><mi>b</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>S</mi></msub><msub><mo>⋅</mo> <mi>S</mi></msub><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r \cdot_R 1_R) b (s \cdot_S 1_S) = (1_R \cdot_R r) b (1_S \cdot_S s)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>b</mi><mi>s</mi><mo>=</mo><mi>r</mi><mi>b</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">r b s = r b s</annotation></semantics></math></div> <p>The left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-action and right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-action are bilinear because the original biaction is trilinear.</p> <h3 id="linear_maps">Linear maps</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/rings">rings</a>. A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule homomorphism</strong> between two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodules <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is an abelian group <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mi>a</mi><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">f(r a s) = r f(a) s</annotation></semantics></math></div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/monic">monic</a> or an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule monomorphism</strong> if for every other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h:C \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">k:C \to A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>h</mi><mo>=</mo><mi>f</mi><mo>∘</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f \circ h = f \circ k</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">h = k</annotation></semantics></math>.</p> <p>A <strong>sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule</strong> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <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> with a <a class="existingWikiWord" href="/nlab/show/monic">monic</a> linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i:A \hookrightarrow B</annotation></semantics></math>.</p> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/invertible">invertible</a> or an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule isomorphism</strong> if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">g:B \to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">g \circ f = id_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>g</mi><mo>=</mo><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">f \circ g = id_B</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">id_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">id_B</annotation></semantics></math> are the identity linear maps on <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> respectively.</p> <h3 id="tensor_product_of_bimodules">Tensor product of bimodules</h3> <p>Given <a class="existingWikiWord" href="/nlab/show/rings">rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">R, S, T</annotation></semantics></math> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> bimodule <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> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, the tensor product 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">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is formed as a <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">A \otimes_N B</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\otimes B</annotation></semantics></math>. This is a special case of a more general construction:</p> <p>Given three monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>,</mo><mi>N</mi><mo>,</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">M,N,P</annotation></semantics></math> in a monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes,I)</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-<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>-bimodules <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> and a <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, we denote the monoid actions as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>λ</mi> <mi>A</mi></msup><mo>:</mo><mi>M</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\lambda^{A}:M \otimes A \rightarrow A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mi>A</mi></msup><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>N</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\rho^{A}:A \otimes N \rightarrow N</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>λ</mi> <mi>B</mi></msup><mo>:</mo><mi>N</mi><mo>⊗</mo><mi>B</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\lambda^{B}:N \otimes B \rightarrow N</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mi>B</mi></msup><mo>:</mo><mi>B</mi><mo>⊗</mo><mi>P</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\rho^{B}:B \otimes P \rightarrow P</annotation></semantics></math>. The tensor product, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">A \otimes_{N} B</annotation></semantics></math> is defined as this coequalizer:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="372.113pt" height="39.45pt" viewBox="0 0 372.113 39.45" version="1.2"> <defs> <g> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-1"> <path style="stroke:none;" d="M 2.5 -1.640625 C 1.984375 -0.765625 1.484375 -0.46875 0.78125 -0.421875 C 0.625 -0.40625 0.5 -0.40625 0.5 -0.140625 C 0.5 -0.0625 0.578125 0 0.671875 0 C 0.9375 0 1.609375 -0.03125 1.875 -0.03125 C 2.296875 -0.03125 2.765625 0 3.1875 0 C 3.265625 0 3.453125 0 3.453125 -0.28125 C 3.453125 -0.40625 3.328125 -0.421875 3.234375 -0.421875 C 2.90625 -0.453125 2.625 -0.578125 2.625 -0.921875 C 2.625 -1.140625 2.71875 -1.296875 2.90625 -1.625 L 4.015625 -3.484375 L 7.78125 -3.484375 C 7.796875 -3.34375 7.796875 -3.234375 7.8125 -3.09375 C 7.859375 -2.71875 8.03125 -1.171875 8.03125 -0.90625 C 8.03125 -0.453125 7.28125 -0.421875 7.046875 -0.421875 C 6.875 -0.421875 6.71875 -0.421875 6.71875 -0.15625 C 6.71875 0 6.84375 0 6.9375 0 C 7.1875 0 7.484375 -0.03125 7.734375 -0.03125 L 8.578125 -0.03125 C 9.46875 -0.03125 10.125 0 10.140625 0 C 10.234375 0 10.40625 0 10.40625 -0.28125 C 10.40625 -0.421875 10.265625 -0.421875 10.046875 -0.421875 C 9.234375 -0.421875 9.21875 -0.5625 9.171875 -1 L 8.28125 -10.203125 C 8.25 -10.484375 8.1875 -10.515625 8.03125 -10.515625 C 7.890625 -10.515625 7.796875 -10.484375 7.65625 -10.265625 Z M 4.265625 -3.90625 L 7.234375 -8.859375 L 7.734375 -3.90625 Z M 4.265625 -3.90625 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-2"> <path style="stroke:none;" d="M 10.90625 -8.515625 C 11.0625 -9.15625 11.296875 -9.59375 12.421875 -9.640625 C 12.46875 -9.640625 12.640625 -9.65625 12.640625 -9.90625 C 12.640625 -10.0625 12.515625 -10.0625 12.453125 -10.0625 C 12.15625 -10.0625 11.40625 -10.03125 11.109375 -10.03125 L 10.40625 -10.03125 C 10.203125 -10.03125 9.9375 -10.0625 9.71875 -10.0625 C 9.640625 -10.0625 9.453125 -10.0625 9.453125 -9.78125 C 9.453125 -9.640625 9.578125 -9.640625 9.6875 -9.640625 C 10.5625 -9.609375 10.625 -9.265625 10.625 -9 C 10.625 -8.875 10.609375 -8.828125 10.5625 -8.625 L 8.90625 -1.96875 L 5.75 -9.8125 C 5.640625 -10.046875 5.625 -10.0625 5.296875 -10.0625 L 3.5 -10.0625 C 3.21875 -10.0625 3.078125 -10.0625 3.078125 -9.78125 C 3.078125 -9.640625 3.1875 -9.640625 3.46875 -9.640625 C 3.53125 -9.640625 4.40625 -9.640625 4.40625 -9.5 C 4.40625 -9.46875 4.375 -9.359375 4.359375 -9.3125 L 2.40625 -1.5 C 2.21875 -0.78125 1.875 -0.46875 0.90625 -0.421875 C 0.828125 -0.421875 0.671875 -0.40625 0.671875 -0.140625 C 0.671875 0 0.828125 0 0.875 0 C 1.15625 0 1.921875 -0.03125 2.203125 -0.03125 L 2.921875 -0.03125 C 3.125 -0.03125 3.375 0 3.578125 0 C 3.6875 0 3.84375 0 3.84375 -0.28125 C 3.84375 -0.40625 3.703125 -0.421875 3.640625 -0.421875 C 3.15625 -0.4375 2.6875 -0.53125 2.6875 -1.0625 C 2.6875 -1.171875 2.71875 -1.3125 2.734375 -1.421875 L 4.734375 -9.3125 C 4.8125 -9.171875 4.8125 -9.140625 4.875 -9 L 8.390625 -0.265625 C 8.453125 -0.09375 8.484375 0 8.625 0 C 8.765625 0 8.78125 -0.046875 8.84375 -0.296875 Z M 10.90625 -8.515625 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-3"> <path style="stroke:none;" d="M 5.390625 -9.0625 C 5.53125 -9.609375 5.578125 -9.640625 6.15625 -9.640625 L 8.078125 -9.640625 C 9.734375 -9.640625 9.734375 -8.21875 9.734375 -8.09375 C 9.734375 -6.890625 8.546875 -5.375 6.59375 -5.375 L 4.484375 -5.375 Z M 7.890625 -5.265625 C 9.484375 -5.5625 10.953125 -6.671875 10.953125 -8.03125 C 10.953125 -9.171875 9.9375 -10.0625 8.265625 -10.0625 L 3.53125 -10.0625 C 3.25 -10.0625 3.125 -10.0625 3.125 -9.78125 C 3.125 -9.640625 3.25 -9.640625 3.484375 -9.640625 C 4.375 -9.640625 4.375 -9.515625 4.375 -9.359375 C 4.375 -9.328125 4.375 -9.234375 4.3125 -9.015625 L 2.328125 -1.09375 C 2.203125 -0.578125 2.171875 -0.421875 1.140625 -0.421875 C 0.859375 -0.421875 0.703125 -0.421875 0.703125 -0.15625 C 0.703125 0 0.796875 0 1.09375 0 L 6.140625 0 C 8.40625 0 10.140625 -1.703125 10.140625 -3.203125 C 10.140625 -4.40625 9.078125 -5.140625 7.890625 -5.265625 Z M 5.796875 -0.421875 L 3.796875 -0.421875 C 3.59375 -0.421875 3.5625 -0.421875 3.484375 -0.4375 C 3.3125 -0.453125 3.296875 -0.484375 3.296875 -0.609375 C 3.296875 -0.703125 3.328125 -0.796875 3.359375 -0.921875 L 4.390625 -5.078125 L 7.15625 -5.078125 C 8.90625 -5.078125 8.90625 -3.46875 8.90625 -3.34375 C 8.90625 -1.9375 7.625 -0.421875 5.796875 -0.421875 Z M 5.796875 -0.421875 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph1-1"> <path style="stroke:none;" d="M 10.640625 -3.6875 C 10.640625 -6.390625 8.421875 -8.59375 5.734375 -8.59375 C 2.984375 -8.59375 0.8125 -6.359375 0.8125 -3.6875 C 0.8125 -0.96875 3.015625 1.21875 5.71875 1.21875 C 8.453125 1.21875 10.640625 -1 10.640625 -3.6875 Z M 2.796875 -6.84375 C 2.765625 -6.875 2.671875 -6.984375 2.671875 -7.015625 C 2.671875 -7.078125 3.859375 -8.21875 5.71875 -8.21875 C 6.234375 -8.21875 7.59375 -8.140625 8.796875 -7.015625 L 5.734375 -3.9375 Z M 2.375 -0.609375 C 1.46875 -1.609375 1.171875 -2.734375 1.171875 -3.6875 C 1.171875 -4.8125 1.609375 -5.890625 2.375 -6.765625 L 5.453125 -3.6875 Z M 9.0625 -6.765625 C 9.765625 -6.015625 10.265625 -4.890625 10.265625 -3.6875 C 10.265625 -2.546875 9.84375 -1.46875 9.078125 -0.609375 L 6 -3.6875 Z M 8.65625 -0.515625 C 8.671875 -0.484375 8.78125 -0.390625 8.78125 -0.359375 C 8.78125 -0.296875 7.59375 0.859375 5.734375 0.859375 C 5.21875 0.859375 3.859375 0.78125 2.65625 -0.359375 L 5.71875 -3.4375 Z M 8.65625 -0.515625 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-1"> <path style="stroke:none;" d="M 7.78125 -5.640625 C 7.90625 -6.125 8.125 -6.359375 8.828125 -6.390625 C 8.921875 -6.390625 9 -6.453125 9 -6.578125 C 9 -6.640625 8.953125 -6.71875 8.859375 -6.71875 C 8.78125 -6.71875 8.59375 -6.6875 7.875 -6.6875 C 7.09375 -6.6875 6.953125 -6.71875 6.875 -6.71875 C 6.71875 -6.71875 6.6875 -6.609375 6.6875 -6.53125 C 6.6875 -6.390625 6.8125 -6.390625 6.90625 -6.390625 C 7.5 -6.375 7.5 -6.109375 7.5 -5.96875 C 7.5 -5.921875 7.5 -5.875 7.453125 -5.703125 L 6.375 -1.40625 L 4.015625 -6.53125 C 3.9375 -6.71875 3.90625 -6.71875 3.671875 -6.71875 L 2.390625 -6.71875 C 2.21875 -6.71875 2.09375 -6.71875 2.09375 -6.53125 C 2.09375 -6.390625 2.21875 -6.390625 2.421875 -6.390625 C 2.5 -6.390625 2.796875 -6.390625 3.015625 -6.328125 L 1.703125 -1.046875 C 1.578125 -0.5625 1.328125 -0.34375 0.671875 -0.328125 C 0.609375 -0.328125 0.484375 -0.3125 0.484375 -0.140625 C 0.484375 -0.078125 0.546875 0 0.640625 0 C 0.671875 0 0.90625 -0.03125 1.609375 -0.03125 C 2.390625 -0.03125 2.53125 0 2.625 0 C 2.65625 0 2.8125 0 2.8125 -0.1875 C 2.8125 -0.296875 2.703125 -0.328125 2.640625 -0.328125 C 2.28125 -0.328125 1.984375 -0.390625 1.984375 -0.734375 C 1.984375 -0.78125 2.015625 -0.921875 2.015625 -0.9375 L 3.296875 -6.0625 L 3.3125 -6.0625 L 6.046875 -0.171875 C 6.109375 -0.015625 6.125 0 6.234375 0 C 6.375 0 6.375 -0.046875 6.421875 -0.203125 Z M 7.78125 -5.640625 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-2"> <path style="stroke:none;" d="M 0.359375 1.578125 C 0.328125 1.734375 0.328125 1.78125 0.328125 1.8125 C 0.328125 2.046875 0.515625 2.109375 0.640625 2.109375 C 0.6875 2.109375 0.90625 2.109375 1.046875 1.84375 C 1.09375 1.734375 1.28125 0.828125 1.640625 -0.515625 C 1.765625 -0.296875 2.078125 0.09375 2.671875 0.09375 C 3.875 0.09375 5.203125 -1.21875 5.203125 -2.6875 C 5.203125 -3.796875 4.46875 -4.328125 3.703125 -4.328125 C 2.796875 -4.328125 1.640625 -3.515625 1.28125 -2.109375 Z M 2.65625 -0.171875 C 1.984375 -0.171875 1.765625 -0.890625 1.765625 -1.015625 C 1.765625 -1.0625 2.03125 -2.0625 2.046875 -2.125 C 2.484375 -3.875 3.421875 -4.0625 3.703125 -4.0625 C 4.171875 -4.0625 4.4375 -3.640625 4.4375 -3.09375 C 4.4375 -2.734375 4.234375 -1.765625 3.9375 -1.15625 C 3.65625 -0.59375 3.140625 -0.171875 2.65625 -0.171875 Z M 2.65625 -0.171875 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-3"> <path style="stroke:none;" d="M 1.65625 -0.78125 C 1.578125 -0.40625 1.546875 -0.328125 0.828125 -0.328125 C 0.640625 -0.328125 0.515625 -0.328125 0.515625 -0.140625 C 0.515625 0 0.65625 0 0.8125 0 L 4.46875 0 C 6.078125 0 7.296875 -1.15625 7.296875 -2.109375 C 7.296875 -2.828125 6.671875 -3.40625 5.6875 -3.515625 C 6.828125 -3.71875 7.8125 -4.46875 7.8125 -5.34375 C 7.8125 -6.078125 7.09375 -6.71875 5.859375 -6.71875 L 2.421875 -6.71875 C 2.25 -6.71875 2.125 -6.71875 2.125 -6.53125 C 2.125 -6.390625 2.234375 -6.390625 2.40625 -6.390625 C 2.734375 -6.390625 3.015625 -6.390625 3.015625 -6.234375 C 3.015625 -6.1875 3 -6.1875 2.984375 -6.046875 Z M 3.1875 -3.625 L 3.796875 -6.03125 C 3.875 -6.359375 3.890625 -6.390625 4.296875 -6.390625 L 5.703125 -6.390625 C 6.671875 -6.390625 6.890625 -5.765625 6.890625 -5.359375 C 6.890625 -4.515625 6 -3.625 4.734375 -3.625 Z M 2.515625 -0.328125 C 2.421875 -0.34375 2.390625 -0.34375 2.390625 -0.40625 C 2.390625 -0.484375 2.421875 -0.5625 2.4375 -0.625 L 3.125 -3.34375 L 5.125 -3.34375 C 6.03125 -3.34375 6.34375 -2.734375 6.34375 -2.1875 C 6.34375 -1.21875 5.390625 -0.328125 4.21875 -0.328125 Z M 2.515625 -0.328125 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-4"> <path style="stroke:none;" d="M 1.8125 -1.171875 C 1.359375 -0.453125 0.96875 -0.359375 0.609375 -0.328125 C 0.515625 -0.3125 0.390625 -0.296875 0.390625 -0.140625 C 0.390625 -0.109375 0.40625 0 0.53125 0 C 0.625 0 0.671875 -0.03125 1.359375 -0.03125 C 2.03125 -0.03125 2.265625 0 2.3125 0 C 2.359375 0 2.5 0 2.5 -0.1875 C 2.5 -0.3125 2.375 -0.328125 2.328125 -0.328125 C 2.125 -0.328125 1.921875 -0.40625 1.921875 -0.625 C 1.921875 -0.78125 2 -0.90625 2.1875 -1.1875 L 2.84375 -2.265625 L 5.546875 -2.265625 L 5.75 -0.609375 C 5.75 -0.46875 5.578125 -0.328125 5.109375 -0.328125 C 4.96875 -0.328125 4.828125 -0.328125 4.828125 -0.140625 C 4.828125 -0.125 4.84375 0 5 0 C 5.09375 0 5.46875 -0.015625 5.5625 -0.03125 L 6.1875 -0.03125 C 7.0625 -0.03125 7.234375 0 7.3125 0 C 7.34375 0 7.515625 0 7.515625 -0.1875 C 7.515625 -0.328125 7.375 -0.328125 7.21875 -0.328125 C 6.6875 -0.328125 6.671875 -0.40625 6.640625 -0.65625 L 5.875 -6.734375 C 5.84375 -6.953125 5.828125 -7.015625 5.65625 -7.015625 C 5.484375 -7.015625 5.421875 -6.921875 5.359375 -6.828125 Z M 3.0625 -2.59375 L 5.09375 -5.828125 L 5.5 -2.59375 Z M 3.0625 -2.59375 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-5"> <path style="stroke:none;" d="M 3.703125 -2.828125 C 4.0625 -1.921875 4.65625 -0.328125 4.765625 -0.140625 C 4.9375 0.09375 5.125 0.09375 5.359375 0.09375 C 5.65625 0.09375 5.734375 0.09375 5.734375 -0.015625 C 5.734375 -0.0625 5.703125 -0.09375 5.671875 -0.125 C 5.546875 -0.265625 5.5 -0.359375 5.4375 -0.546875 L 3.21875 -6.15625 C 3.15625 -6.34375 2.96875 -6.828125 1.953125 -6.828125 C 1.859375 -6.828125 1.703125 -6.828125 1.703125 -6.6875 C 1.703125 -6.5625 1.8125 -6.5625 1.859375 -6.546875 C 2.046875 -6.515625 2.234375 -6.5 2.46875 -5.921875 L 3.171875 -4.15625 L 3.5625 -3.171875 L 0.75 -0.5625 C 0.640625 -0.46875 0.546875 -0.34375 0.546875 -0.203125 C 0.546875 -0.015625 0.703125 0.125 0.875 0.125 C 1.015625 0.125 1.15625 0.015625 1.234375 -0.09375 Z M 3.703125 -2.828125 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-6"> <path style="stroke:none;" d="M 2.796875 -3.59375 L 3.921875 -3.59375 C 3.71875 -2.6875 3.53125 -1.96875 3.53125 -1.234375 C 3.53125 -1.171875 3.53125 -0.75 3.640625 -0.390625 C 3.78125 0.015625 3.875 0.09375 4.046875 0.09375 C 4.265625 0.09375 4.484375 -0.09375 4.484375 -0.328125 C 4.484375 -0.390625 4.484375 -0.40625 4.4375 -0.5 C 4.234375 -0.953125 4.09375 -1.421875 4.09375 -2.234375 C 4.09375 -2.453125 4.09375 -2.875 4.234375 -3.59375 L 5.421875 -3.59375 C 5.578125 -3.59375 5.6875 -3.59375 5.78125 -3.671875 C 5.90625 -3.78125 5.9375 -3.90625 5.9375 -3.953125 C 5.9375 -4.234375 5.6875 -4.234375 5.515625 -4.234375 L 1.96875 -4.234375 C 1.765625 -4.234375 1.390625 -4.234375 0.90625 -3.765625 C 0.5625 -3.40625 0.28125 -2.953125 0.28125 -2.890625 C 0.28125 -2.796875 0.359375 -2.765625 0.4375 -2.765625 C 0.53125 -2.765625 0.546875 -2.796875 0.609375 -2.875 C 1.09375 -3.59375 1.671875 -3.59375 1.890625 -3.59375 L 2.46875 -3.59375 C 2.1875 -2.546875 1.65625 -1.359375 1.296875 -0.640625 C 1.234375 -0.484375 1.125 -0.296875 1.125 -0.203125 C 1.125 0 1.296875 0.09375 1.453125 0.09375 C 1.828125 0.09375 1.921875 -0.28125 2.125 -1.078125 Z M 2.796875 -3.59375 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-1"> <path style="stroke:none;" d="M 1.671875 -0.90625 C 1.390625 -0.46875 1.140625 -0.328125 0.71875 -0.296875 C 0.625 -0.28125 0.53125 -0.28125 0.53125 -0.109375 C 0.53125 -0.03125 0.59375 0 0.640625 0 C 0.84375 0 1.09375 -0.03125 1.3125 -0.03125 C 1.484375 -0.03125 1.921875 0 2.09375 0 C 2.171875 0 2.25 -0.03125 2.25 -0.1875 C 2.25 -0.28125 2.15625 -0.296875 2.125 -0.296875 C 1.953125 -0.296875 1.796875 -0.34375 1.796875 -0.515625 C 1.796875 -0.59375 1.875 -0.71875 1.890625 -0.765625 L 2.5 -1.671875 L 4.71875 -1.671875 L 4.90625 -0.484375 C 4.90625 -0.296875 4.484375 -0.296875 4.40625 -0.296875 C 4.3125 -0.296875 4.203125 -0.296875 4.203125 -0.109375 C 4.203125 -0.078125 4.234375 0 4.328125 0 C 4.546875 0 5.078125 -0.03125 5.296875 -0.03125 C 5.453125 -0.03125 5.59375 -0.015625 5.734375 -0.015625 C 5.890625 -0.015625 6.046875 0 6.1875 0 C 6.234375 0 6.34375 0 6.34375 -0.1875 C 6.34375 -0.296875 6.25 -0.296875 6.140625 -0.296875 C 5.6875 -0.296875 5.671875 -0.34375 5.65625 -0.53125 L 4.90625 -5.0625 C 4.890625 -5.203125 4.875 -5.265625 4.71875 -5.265625 C 4.5625 -5.265625 4.515625 -5.203125 4.453125 -5.109375 Z M 2.6875 -1.96875 L 4.28125 -4.359375 L 4.671875 -1.96875 Z M 2.6875 -1.96875 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-2"> <path style="stroke:none;" d="M 1.546875 -0.609375 C 1.484375 -0.34375 1.46875 -0.296875 0.921875 -0.296875 C 0.75 -0.296875 0.734375 -0.296875 0.703125 -0.265625 C 0.671875 -0.25 0.640625 -0.140625 0.640625 -0.109375 C 0.65625 0 0.734375 0 0.875 0 L 3.875 0 C 5.15625 0 6.15625 -0.828125 6.15625 -1.578125 C 6.15625 -2.03125 5.765625 -2.515625 4.84375 -2.640625 C 5.6875 -2.765625 6.53125 -3.3125 6.53125 -3.984375 C 6.53125 -4.515625 5.9375 -5.03125 4.90625 -5.03125 L 2.078125 -5.03125 C 1.9375 -5.03125 1.828125 -5.03125 1.828125 -4.859375 C 1.828125 -4.734375 1.9375 -4.734375 2.078125 -4.734375 C 2.203125 -4.734375 2.390625 -4.734375 2.546875 -4.6875 C 2.546875 -4.609375 2.546875 -4.578125 2.515625 -4.453125 Z M 2.765625 -2.75 L 3.1875 -4.484375 C 3.25 -4.734375 3.265625 -4.734375 3.5625 -4.734375 L 4.78125 -4.734375 C 5.546875 -4.734375 5.78125 -4.28125 5.78125 -3.96875 C 5.78125 -3.359375 5 -2.75 4.09375 -2.75 Z M 2.40625 -0.296875 C 2.265625 -0.296875 2.25 -0.296875 2.15625 -0.3125 L 2.6875 -2.5 L 4.296875 -2.5 C 5.234375 -2.5 5.390625 -1.90625 5.390625 -1.609375 C 5.390625 -0.90625 4.609375 -0.296875 3.6875 -0.296875 Z M 2.40625 -0.296875 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-3"> <path style="stroke:none;" d="M 1.703125 -0.109375 C 1.703125 0.59375 1.328125 1 1.140625 1.171875 C 1.0625 1.234375 1.046875 1.25 1.046875 1.296875 C 1.046875 1.359375 1.109375 1.4375 1.171875 1.4375 C 1.28125 1.4375 1.9375 0.84375 1.9375 -0.0625 C 1.9375 -0.53125 1.75 -0.890625 1.390625 -0.890625 C 1.109375 -0.890625 0.953125 -0.65625 0.953125 -0.453125 C 0.953125 -0.234375 1.109375 0 1.40625 0 C 1.515625 0 1.609375 -0.03125 1.703125 -0.109375 Z M 1.703125 -0.109375 "></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph4-1"> <path style="stroke:none;" d="M 7.53125 -2.453125 C 7.53125 -4.390625 5.953125 -5.9375 4.0625 -5.9375 C 2.125 -5.9375 0.578125 -4.359375 0.578125 -2.453125 C 0.578125 -0.515625 2.15625 1.015625 4.046875 1.015625 C 5.984375 1.015625 7.53125 -0.546875 7.53125 -2.453125 Z M 1.921875 -4.8125 C 2.6875 -5.515625 3.578125 -5.625 4.046875 -5.625 C 4.65625 -5.625 5.484375 -5.453125 6.1875 -4.8125 L 4.0625 -2.671875 Z M 1.703125 -0.328125 C 1.15625 -0.90625 0.890625 -1.703125 0.890625 -2.453125 C 0.890625 -3.203125 1.15625 -4 1.703125 -4.59375 L 3.84375 -2.453125 Z M 6.390625 -4.59375 C 6.953125 -4.015625 7.21875 -3.21875 7.21875 -2.453125 C 7.21875 -1.703125 6.953125 -0.90625 6.390625 -0.328125 L 4.265625 -2.453125 Z M 6.1875 -0.109375 C 5.421875 0.59375 4.53125 0.71875 4.0625 0.71875 C 3.453125 0.71875 2.625 0.546875 1.921875 -0.109375 L 4.046875 -2.234375 Z M 6.1875 -0.109375 "></path> </symbol> </g> </defs> <g id="PpYidnOUOMMo25seLlwKBjGR_Ls=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-1" x="8.466408" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph1-1" x="22.55992" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-2" x="37.298193" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph1-1" x="53.669943" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-3" x="68.408217" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-1" x="203.638379" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph1-1" x="217.731891" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-3" x="232.470165" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-1" x="312.285651" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph1-1" x="326.379164" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-1" x="337.841854" y="25.792657"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph0-3" x="351.063768" y="23.582224"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -100.29453 2.633541 L 11.108354 2.633541 " transform="matrix(0.98625,0,0,-0.98625,185.618605,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486119 2.867625 C -2.030637 1.148677 -1.020656 0.332771 0.00120691 0.0000711914 C -1.020656 -0.336589 -2.030637 -1.148535 -2.486119 -2.867483 " transform="matrix(0.98625,0,0,-0.98625,196.81131,17.054758)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-2" x="127.25233" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-1" x="132.599038" y="8.208312"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph4-1" x="140.023035" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-3" x="148.142338" y="11.676214"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -100.29453 -3.133252 L 11.108354 -3.133252 " transform="matrix(0.98625,0,0,-0.98625,185.618605,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486119 2.869862 C -2.030637 1.146954 -1.020656 0.335008 0.00120691 -0.00165222 C -1.020656 -0.334352 -2.030637 -1.146297 -2.486119 -2.869206 " transform="matrix(0.98625,0,0,-0.98625,196.81131,22.740558)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-4" x="126.964345" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph4-1" x="134.784075" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-5" x="142.904611" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-2" x="148.994704" y="31.243167"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 66.059241 -0.249856 L 121.271535 -0.249856 " transform="matrix(0.98625,0,0,-0.98625,185.618605,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485758 2.870729 C -2.034237 1.14782 -1.020295 0.335875 0.00156815 -0.000785513 C -1.020295 -0.333485 -2.034237 -1.149391 -2.485758 -2.868339 " transform="matrix(0.98625,0,0,-0.98625,305.459391,19.897663)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph2-6" x="266.533514" y="13.345935"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-1" x="272.527448" y="14.996671"></use> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-3" x="279.337381" y="14.996671"></use> <use xlink:href="#PpYidnOUOMMo25seLlwKBjGR_Ls=-glyph3-2" x="282.134752" y="14.996671"></use> </g> </g> </svg> <p>We suppose moreover that this coequalizer is preserved by tensoring on the left by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and tensoring on the right by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>, meaning that these diagrams are coequalizer diagrams: <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="530.247pt" height="39.45pt" viewBox="0 0 530.247 39.45" version="1.2"> <defs> <g> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-1"> <path style="stroke:none;" d="M 13.375 -8.984375 C 13.515625 -9.484375 13.546875 -9.640625 14.59375 -9.640625 C 14.859375 -9.640625 15 -9.640625 15 -9.921875 C 15 -10.0625 14.890625 -10.0625 14.609375 -10.0625 L 12.84375 -10.0625 C 12.484375 -10.0625 12.46875 -10.046875 12.296875 -9.8125 L 6.921875 -1.3125 L 5.8125 -9.734375 C 5.78125 -10.0625 5.765625 -10.0625 5.375 -10.0625 L 3.546875 -10.0625 C 3.265625 -10.0625 3.140625 -10.0625 3.140625 -9.78125 C 3.140625 -9.640625 3.265625 -9.640625 3.484375 -9.640625 C 4.390625 -9.640625 4.390625 -9.515625 4.390625 -9.359375 C 4.390625 -9.328125 4.390625 -9.234375 4.328125 -9.015625 L 2.453125 -1.5 C 2.265625 -0.796875 1.9375 -0.46875 0.9375 -0.421875 C 0.90625 -0.421875 0.71875 -0.40625 0.71875 -0.15625 C 0.71875 0 0.859375 0 0.90625 0 C 1.203125 0 1.953125 -0.03125 2.25 -0.03125 L 2.96875 -0.03125 C 3.171875 -0.03125 3.421875 0 3.625 0 C 3.734375 0 3.890625 0 3.890625 -0.28125 C 3.890625 -0.40625 3.75 -0.421875 3.6875 -0.421875 C 3.203125 -0.4375 2.71875 -0.53125 2.71875 -1.0625 C 2.71875 -1.203125 2.71875 -1.21875 2.78125 -1.421875 L 4.8125 -9.546875 L 4.828125 -9.546875 L 6.0625 -0.390625 C 6.09375 -0.046875 6.109375 0 6.25 0 C 6.40625 0 6.484375 -0.125 6.5625 -0.25 L 12.484375 -9.625 L 12.5 -9.625 L 10.359375 -1.09375 C 10.21875 -0.578125 10.203125 -0.421875 9.171875 -0.421875 C 8.890625 -0.421875 8.734375 -0.421875 8.734375 -0.15625 C 8.734375 0 8.875 0 8.953125 0 C 9.203125 0 9.5 -0.03125 9.75 -0.03125 L 11.5 -0.03125 C 11.75 -0.03125 12.046875 0 12.296875 0 C 12.421875 0 12.578125 0 12.578125 -0.28125 C 12.578125 -0.421875 12.453125 -0.421875 12.234375 -0.421875 C 11.328125 -0.421875 11.328125 -0.546875 11.328125 -0.6875 C 11.328125 -0.703125 11.328125 -0.8125 11.359375 -0.921875 Z M 13.375 -8.984375 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-2"> <path style="stroke:none;" d="M 2.5 -1.640625 C 1.984375 -0.765625 1.484375 -0.46875 0.78125 -0.421875 C 0.625 -0.40625 0.5 -0.40625 0.5 -0.140625 C 0.5 -0.0625 0.578125 0 0.671875 0 C 0.9375 0 1.609375 -0.03125 1.875 -0.03125 C 2.296875 -0.03125 2.765625 0 3.1875 0 C 3.265625 0 3.453125 0 3.453125 -0.28125 C 3.453125 -0.40625 3.328125 -0.421875 3.234375 -0.421875 C 2.90625 -0.453125 2.625 -0.578125 2.625 -0.921875 C 2.625 -1.140625 2.71875 -1.296875 2.90625 -1.625 L 4.015625 -3.484375 L 7.78125 -3.484375 C 7.796875 -3.34375 7.796875 -3.234375 7.8125 -3.09375 C 7.859375 -2.71875 8.03125 -1.171875 8.03125 -0.90625 C 8.03125 -0.453125 7.28125 -0.421875 7.046875 -0.421875 C 6.875 -0.421875 6.71875 -0.421875 6.71875 -0.15625 C 6.71875 0 6.84375 0 6.9375 0 C 7.1875 0 7.484375 -0.03125 7.734375 -0.03125 L 8.578125 -0.03125 C 9.46875 -0.03125 10.125 0 10.140625 0 C 10.234375 0 10.40625 0 10.40625 -0.28125 C 10.40625 -0.421875 10.265625 -0.421875 10.046875 -0.421875 C 9.234375 -0.421875 9.21875 -0.5625 9.171875 -1 L 8.28125 -10.203125 C 8.25 -10.484375 8.1875 -10.515625 8.03125 -10.515625 C 7.890625 -10.515625 7.796875 -10.484375 7.65625 -10.265625 Z M 4.265625 -3.90625 L 7.234375 -8.859375 L 7.734375 -3.90625 Z M 4.265625 -3.90625 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-3"> <path style="stroke:none;" d="M 10.90625 -8.515625 C 11.0625 -9.15625 11.296875 -9.59375 12.421875 -9.640625 C 12.46875 -9.640625 12.640625 -9.65625 12.640625 -9.90625 C 12.640625 -10.0625 12.515625 -10.0625 12.453125 -10.0625 C 12.15625 -10.0625 11.40625 -10.03125 11.109375 -10.03125 L 10.40625 -10.03125 C 10.203125 -10.03125 9.9375 -10.0625 9.71875 -10.0625 C 9.640625 -10.0625 9.453125 -10.0625 9.453125 -9.78125 C 9.453125 -9.640625 9.578125 -9.640625 9.6875 -9.640625 C 10.5625 -9.609375 10.625 -9.265625 10.625 -9 C 10.625 -8.875 10.609375 -8.828125 10.5625 -8.625 L 8.90625 -1.96875 L 5.75 -9.8125 C 5.640625 -10.046875 5.625 -10.0625 5.296875 -10.0625 L 3.5 -10.0625 C 3.21875 -10.0625 3.078125 -10.0625 3.078125 -9.78125 C 3.078125 -9.640625 3.1875 -9.640625 3.46875 -9.640625 C 3.53125 -9.640625 4.40625 -9.640625 4.40625 -9.5 C 4.40625 -9.46875 4.375 -9.359375 4.359375 -9.3125 L 2.40625 -1.5 C 2.21875 -0.78125 1.875 -0.46875 0.90625 -0.421875 C 0.828125 -0.421875 0.671875 -0.40625 0.671875 -0.140625 C 0.671875 0 0.828125 0 0.875 0 C 1.15625 0 1.921875 -0.03125 2.203125 -0.03125 L 2.921875 -0.03125 C 3.125 -0.03125 3.375 0 3.578125 0 C 3.6875 0 3.84375 0 3.84375 -0.28125 C 3.84375 -0.40625 3.703125 -0.421875 3.640625 -0.421875 C 3.15625 -0.4375 2.6875 -0.53125 2.6875 -1.0625 C 2.6875 -1.171875 2.71875 -1.3125 2.734375 -1.421875 L 4.734375 -9.3125 C 4.8125 -9.171875 4.8125 -9.140625 4.875 -9 L 8.390625 -0.265625 C 8.453125 -0.09375 8.484375 0 8.625 0 C 8.765625 0 8.78125 -0.046875 8.84375 -0.296875 Z M 10.90625 -8.515625 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-4"> <path style="stroke:none;" d="M 5.390625 -9.0625 C 5.53125 -9.609375 5.578125 -9.640625 6.15625 -9.640625 L 8.078125 -9.640625 C 9.734375 -9.640625 9.734375 -8.21875 9.734375 -8.09375 C 9.734375 -6.890625 8.546875 -5.375 6.59375 -5.375 L 4.484375 -5.375 Z M 7.890625 -5.265625 C 9.484375 -5.5625 10.953125 -6.671875 10.953125 -8.03125 C 10.953125 -9.171875 9.9375 -10.0625 8.265625 -10.0625 L 3.53125 -10.0625 C 3.25 -10.0625 3.125 -10.0625 3.125 -9.78125 C 3.125 -9.640625 3.25 -9.640625 3.484375 -9.640625 C 4.375 -9.640625 4.375 -9.515625 4.375 -9.359375 C 4.375 -9.328125 4.375 -9.234375 4.3125 -9.015625 L 2.328125 -1.09375 C 2.203125 -0.578125 2.171875 -0.421875 1.140625 -0.421875 C 0.859375 -0.421875 0.703125 -0.421875 0.703125 -0.15625 C 0.703125 0 0.796875 0 1.09375 0 L 6.140625 0 C 8.40625 0 10.140625 -1.703125 10.140625 -3.203125 C 10.140625 -4.40625 9.078125 -5.140625 7.890625 -5.265625 Z M 5.796875 -0.421875 L 3.796875 -0.421875 C 3.59375 -0.421875 3.5625 -0.421875 3.484375 -0.4375 C 3.3125 -0.453125 3.296875 -0.484375 3.296875 -0.609375 C 3.296875 -0.703125 3.328125 -0.796875 3.359375 -0.921875 L 4.390625 -5.078125 L 7.15625 -5.078125 C 8.90625 -5.078125 8.90625 -3.46875 8.90625 -3.34375 C 8.90625 -1.9375 7.625 -0.421875 5.796875 -0.421875 Z M 5.796875 -0.421875 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1"> <path style="stroke:none;" d="M 10.640625 -3.6875 C 10.640625 -6.390625 8.421875 -8.59375 5.734375 -8.59375 C 2.984375 -8.59375 0.8125 -6.359375 0.8125 -3.6875 C 0.8125 -0.96875 3.015625 1.21875 5.71875 1.21875 C 8.453125 1.21875 10.640625 -1 10.640625 -3.6875 Z M 2.796875 -6.84375 C 2.765625 -6.875 2.671875 -6.984375 2.671875 -7.015625 C 2.671875 -7.078125 3.859375 -8.21875 5.71875 -8.21875 C 6.234375 -8.21875 7.59375 -8.140625 8.796875 -7.015625 L 5.734375 -3.9375 Z M 2.375 -0.609375 C 1.46875 -1.609375 1.171875 -2.734375 1.171875 -3.6875 C 1.171875 -4.8125 1.609375 -5.890625 2.375 -6.765625 L 5.453125 -3.6875 Z M 9.0625 -6.765625 C 9.765625 -6.015625 10.265625 -4.890625 10.265625 -3.6875 C 10.265625 -2.546875 9.84375 -1.46875 9.078125 -0.609375 L 6 -3.6875 Z M 8.65625 -0.515625 C 8.671875 -0.484375 8.78125 -0.390625 8.78125 -0.359375 C 8.78125 -0.296875 7.59375 0.859375 5.734375 0.859375 C 5.21875 0.859375 3.859375 0.78125 2.65625 -0.359375 L 5.71875 -3.4375 Z M 8.65625 -0.515625 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-1"> <path style="stroke:none;" d="M 7.78125 -5.640625 C 7.90625 -6.125 8.125 -6.359375 8.828125 -6.390625 C 8.921875 -6.390625 9 -6.453125 9 -6.578125 C 9 -6.640625 8.953125 -6.71875 8.859375 -6.71875 C 8.78125 -6.71875 8.59375 -6.6875 7.875 -6.6875 C 7.09375 -6.6875 6.953125 -6.71875 6.875 -6.71875 C 6.71875 -6.71875 6.6875 -6.609375 6.6875 -6.53125 C 6.6875 -6.390625 6.8125 -6.390625 6.90625 -6.390625 C 7.5 -6.375 7.5 -6.109375 7.5 -5.96875 C 7.5 -5.921875 7.5 -5.875 7.453125 -5.703125 L 6.375 -1.40625 L 4.015625 -6.53125 C 3.9375 -6.71875 3.90625 -6.71875 3.671875 -6.71875 L 2.390625 -6.71875 C 2.21875 -6.71875 2.09375 -6.71875 2.09375 -6.53125 C 2.09375 -6.390625 2.21875 -6.390625 2.421875 -6.390625 C 2.5 -6.390625 2.796875 -6.390625 3.015625 -6.328125 L 1.703125 -1.046875 C 1.578125 -0.5625 1.328125 -0.34375 0.671875 -0.328125 C 0.609375 -0.328125 0.484375 -0.3125 0.484375 -0.140625 C 0.484375 -0.078125 0.546875 0 0.640625 0 C 0.671875 0 0.90625 -0.03125 1.609375 -0.03125 C 2.390625 -0.03125 2.53125 0 2.625 0 C 2.65625 0 2.8125 0 2.8125 -0.1875 C 2.8125 -0.296875 2.703125 -0.328125 2.640625 -0.328125 C 2.28125 -0.328125 1.984375 -0.390625 1.984375 -0.734375 C 1.984375 -0.78125 2.015625 -0.921875 2.015625 -0.9375 L 3.296875 -6.0625 L 3.3125 -6.0625 L 6.046875 -0.171875 C 6.109375 -0.015625 6.125 0 6.234375 0 C 6.375 0 6.375 -0.046875 6.421875 -0.203125 Z M 7.78125 -5.640625 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-2"> <path style="stroke:none;" d="M 9.546875 -5.921875 C 9.640625 -6.296875 9.65625 -6.390625 10.34375 -6.390625 C 10.578125 -6.390625 10.6875 -6.390625 10.6875 -6.578125 C 10.6875 -6.71875 10.5625 -6.71875 10.390625 -6.71875 L 9.125 -6.71875 C 8.859375 -6.71875 8.828125 -6.71875 8.703125 -6.515625 L 5 -0.90625 L 4.125 -6.453125 C 4.078125 -6.6875 4.0625 -6.71875 3.78125 -6.71875 L 2.453125 -6.71875 C 2.265625 -6.71875 2.140625 -6.71875 2.140625 -6.53125 C 2.140625 -6.390625 2.265625 -6.390625 2.421875 -6.390625 C 2.75 -6.390625 3.03125 -6.390625 3.03125 -6.234375 C 3.03125 -6.1875 3.03125 -6.1875 3 -6.046875 L 1.75 -1.046875 C 1.625 -0.5625 1.375 -0.34375 0.71875 -0.328125 C 0.65625 -0.328125 0.546875 -0.3125 0.546875 -0.140625 C 0.546875 -0.078125 0.59375 0 0.6875 0 C 0.734375 0 0.953125 -0.03125 1.65625 -0.03125 C 2.4375 -0.03125 2.578125 0 2.671875 0 C 2.71875 0 2.859375 0 2.859375 -0.1875 C 2.859375 -0.296875 2.75 -0.328125 2.6875 -0.328125 C 2.328125 -0.328125 2.03125 -0.390625 2.03125 -0.734375 C 2.03125 -0.78125 2.0625 -0.921875 2.0625 -0.9375 L 3.40625 -6.265625 L 4.359375 -0.265625 C 4.390625 -0.109375 4.40625 0 4.546875 0 C 4.671875 0 4.75 -0.109375 4.8125 -0.203125 L 8.84375 -6.34375 L 8.859375 -6.34375 L 7.46875 -0.78125 C 7.375 -0.40625 7.359375 -0.328125 6.640625 -0.328125 C 6.453125 -0.328125 6.328125 -0.328125 6.328125 -0.140625 C 6.328125 -0.09375 6.359375 0 6.484375 0 C 6.609375 0 6.90625 -0.015625 7.046875 -0.03125 L 7.671875 -0.03125 C 8.59375 -0.03125 8.828125 0 8.890625 0 C 8.953125 0 9.09375 0 9.09375 -0.1875 C 9.09375 -0.328125 8.96875 -0.328125 8.796875 -0.328125 C 8.765625 -0.328125 8.59375 -0.328125 8.421875 -0.34375 C 8.21875 -0.359375 8.203125 -0.390625 8.203125 -0.484375 C 8.203125 -0.53125 8.21875 -0.59375 8.21875 -0.640625 Z M 9.546875 -5.921875 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-3"> <path style="stroke:none;" d="M 0.359375 1.578125 C 0.328125 1.734375 0.328125 1.78125 0.328125 1.8125 C 0.328125 2.046875 0.515625 2.109375 0.640625 2.109375 C 0.6875 2.109375 0.90625 2.109375 1.046875 1.84375 C 1.09375 1.734375 1.28125 0.828125 1.640625 -0.515625 C 1.765625 -0.296875 2.078125 0.09375 2.671875 0.09375 C 3.875 0.09375 5.203125 -1.21875 5.203125 -2.6875 C 5.203125 -3.796875 4.46875 -4.328125 3.703125 -4.328125 C 2.796875 -4.328125 1.640625 -3.515625 1.28125 -2.109375 Z M 2.65625 -0.171875 C 1.984375 -0.171875 1.765625 -0.890625 1.765625 -1.015625 C 1.765625 -1.0625 2.03125 -2.0625 2.046875 -2.125 C 2.484375 -3.875 3.421875 -4.0625 3.703125 -4.0625 C 4.171875 -4.0625 4.4375 -3.640625 4.4375 -3.09375 C 4.4375 -2.734375 4.234375 -1.765625 3.9375 -1.15625 C 3.65625 -0.59375 3.140625 -0.171875 2.65625 -0.171875 Z M 2.65625 -0.171875 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-4"> <path style="stroke:none;" d="M 1.65625 -0.78125 C 1.578125 -0.40625 1.546875 -0.328125 0.828125 -0.328125 C 0.640625 -0.328125 0.515625 -0.328125 0.515625 -0.140625 C 0.515625 0 0.65625 0 0.8125 0 L 4.46875 0 C 6.078125 0 7.296875 -1.15625 7.296875 -2.109375 C 7.296875 -2.828125 6.671875 -3.40625 5.6875 -3.515625 C 6.828125 -3.71875 7.8125 -4.46875 7.8125 -5.34375 C 7.8125 -6.078125 7.09375 -6.71875 5.859375 -6.71875 L 2.421875 -6.71875 C 2.25 -6.71875 2.125 -6.71875 2.125 -6.53125 C 2.125 -6.390625 2.234375 -6.390625 2.40625 -6.390625 C 2.734375 -6.390625 3.015625 -6.390625 3.015625 -6.234375 C 3.015625 -6.1875 3 -6.1875 2.984375 -6.046875 Z M 3.1875 -3.625 L 3.796875 -6.03125 C 3.875 -6.359375 3.890625 -6.390625 4.296875 -6.390625 L 5.703125 -6.390625 C 6.671875 -6.390625 6.890625 -5.765625 6.890625 -5.359375 C 6.890625 -4.515625 6 -3.625 4.734375 -3.625 Z M 2.515625 -0.328125 C 2.421875 -0.34375 2.390625 -0.34375 2.390625 -0.40625 C 2.390625 -0.484375 2.421875 -0.5625 2.4375 -0.625 L 3.125 -3.34375 L 5.125 -3.34375 C 6.03125 -3.34375 6.34375 -2.734375 6.34375 -2.1875 C 6.34375 -1.21875 5.390625 -0.328125 4.21875 -0.328125 Z M 2.515625 -0.328125 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-5"> <path style="stroke:none;" d="M 1.8125 -1.171875 C 1.359375 -0.453125 0.96875 -0.359375 0.609375 -0.328125 C 0.515625 -0.3125 0.390625 -0.296875 0.390625 -0.140625 C 0.390625 -0.109375 0.40625 0 0.53125 0 C 0.625 0 0.671875 -0.03125 1.359375 -0.03125 C 2.03125 -0.03125 2.265625 0 2.3125 0 C 2.359375 0 2.5 0 2.5 -0.1875 C 2.5 -0.3125 2.375 -0.328125 2.328125 -0.328125 C 2.125 -0.328125 1.921875 -0.40625 1.921875 -0.625 C 1.921875 -0.78125 2 -0.90625 2.1875 -1.1875 L 2.84375 -2.265625 L 5.546875 -2.265625 L 5.75 -0.609375 C 5.75 -0.46875 5.578125 -0.328125 5.109375 -0.328125 C 4.96875 -0.328125 4.828125 -0.328125 4.828125 -0.140625 C 4.828125 -0.125 4.84375 0 5 0 C 5.09375 0 5.46875 -0.015625 5.5625 -0.03125 L 6.1875 -0.03125 C 7.0625 -0.03125 7.234375 0 7.3125 0 C 7.34375 0 7.515625 0 7.515625 -0.1875 C 7.515625 -0.328125 7.375 -0.328125 7.21875 -0.328125 C 6.6875 -0.328125 6.671875 -0.40625 6.640625 -0.65625 L 5.875 -6.734375 C 5.84375 -6.953125 5.828125 -7.015625 5.65625 -7.015625 C 5.484375 -7.015625 5.421875 -6.921875 5.359375 -6.828125 Z M 3.0625 -2.59375 L 5.09375 -5.828125 L 5.5 -2.59375 Z M 3.0625 -2.59375 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-6"> <path style="stroke:none;" d="M 3.703125 -2.828125 C 4.0625 -1.921875 4.65625 -0.328125 4.765625 -0.140625 C 4.9375 0.09375 5.125 0.09375 5.359375 0.09375 C 5.65625 0.09375 5.734375 0.09375 5.734375 -0.015625 C 5.734375 -0.0625 5.703125 -0.09375 5.671875 -0.125 C 5.546875 -0.265625 5.5 -0.359375 5.4375 -0.546875 L 3.21875 -6.15625 C 3.15625 -6.34375 2.96875 -6.828125 1.953125 -6.828125 C 1.859375 -6.828125 1.703125 -6.828125 1.703125 -6.6875 C 1.703125 -6.5625 1.8125 -6.5625 1.859375 -6.546875 C 2.046875 -6.515625 2.234375 -6.5 2.46875 -5.921875 L 3.171875 -4.15625 L 3.5625 -3.171875 L 0.75 -0.5625 C 0.640625 -0.46875 0.546875 -0.34375 0.546875 -0.203125 C 0.546875 -0.015625 0.703125 0.125 0.875 0.125 C 1.015625 0.125 1.15625 0.015625 1.234375 -0.09375 Z M 3.703125 -2.828125 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-7"> <path style="stroke:none;" d="M 2.796875 -3.59375 L 3.921875 -3.59375 C 3.71875 -2.6875 3.53125 -1.96875 3.53125 -1.234375 C 3.53125 -1.171875 3.53125 -0.75 3.640625 -0.390625 C 3.78125 0.015625 3.875 0.09375 4.046875 0.09375 C 4.265625 0.09375 4.484375 -0.09375 4.484375 -0.328125 C 4.484375 -0.390625 4.484375 -0.40625 4.4375 -0.5 C 4.234375 -0.953125 4.09375 -1.421875 4.09375 -2.234375 C 4.09375 -2.453125 4.09375 -2.875 4.234375 -3.59375 L 5.421875 -3.59375 C 5.578125 -3.59375 5.6875 -3.59375 5.78125 -3.671875 C 5.90625 -3.78125 5.9375 -3.90625 5.9375 -3.953125 C 5.9375 -4.234375 5.6875 -4.234375 5.515625 -4.234375 L 1.96875 -4.234375 C 1.765625 -4.234375 1.390625 -4.234375 0.90625 -3.765625 C 0.5625 -3.40625 0.28125 -2.953125 0.28125 -2.890625 C 0.28125 -2.796875 0.359375 -2.765625 0.4375 -2.765625 C 0.53125 -2.765625 0.546875 -2.796875 0.609375 -2.875 C 1.09375 -3.59375 1.671875 -3.59375 1.890625 -3.59375 L 2.46875 -3.59375 C 2.1875 -2.546875 1.65625 -1.359375 1.296875 -0.640625 C 1.234375 -0.484375 1.125 -0.296875 1.125 -0.203125 C 1.125 0 1.296875 0.09375 1.453125 0.09375 C 1.828125 0.09375 1.921875 -0.28125 2.125 -1.078125 Z M 2.796875 -3.59375 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-1"> <path style="stroke:none;" d="M 7.53125 -2.453125 C 7.53125 -4.390625 5.953125 -5.9375 4.0625 -5.9375 C 2.125 -5.9375 0.578125 -4.359375 0.578125 -2.453125 C 0.578125 -0.515625 2.15625 1.015625 4.046875 1.015625 C 5.984375 1.015625 7.53125 -0.546875 7.53125 -2.453125 Z M 1.921875 -4.8125 C 2.6875 -5.515625 3.578125 -5.625 4.046875 -5.625 C 4.65625 -5.625 5.484375 -5.453125 6.1875 -4.8125 L 4.0625 -2.671875 Z M 1.703125 -0.328125 C 1.15625 -0.90625 0.890625 -1.703125 0.890625 -2.453125 C 0.890625 -3.203125 1.15625 -4 1.703125 -4.59375 L 3.84375 -2.453125 Z M 6.390625 -4.59375 C 6.953125 -4.015625 7.21875 -3.21875 7.21875 -2.453125 C 7.21875 -1.703125 6.953125 -0.90625 6.390625 -0.328125 L 4.265625 -2.453125 Z M 6.1875 -0.109375 C 5.421875 0.59375 4.53125 0.71875 4.0625 0.71875 C 3.453125 0.71875 2.625 0.546875 1.921875 -0.109375 L 4.046875 -2.234375 Z M 6.1875 -0.109375 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-1"> <path style="stroke:none;" d="M 1.671875 -0.90625 C 1.390625 -0.46875 1.140625 -0.328125 0.71875 -0.296875 C 0.625 -0.28125 0.53125 -0.28125 0.53125 -0.109375 C 0.53125 -0.03125 0.59375 0 0.640625 0 C 0.84375 0 1.09375 -0.03125 1.3125 -0.03125 C 1.484375 -0.03125 1.921875 0 2.09375 0 C 2.171875 0 2.25 -0.03125 2.25 -0.1875 C 2.25 -0.28125 2.15625 -0.296875 2.125 -0.296875 C 1.953125 -0.296875 1.796875 -0.34375 1.796875 -0.515625 C 1.796875 -0.59375 1.875 -0.71875 1.890625 -0.765625 L 2.5 -1.671875 L 4.71875 -1.671875 L 4.90625 -0.484375 C 4.90625 -0.296875 4.484375 -0.296875 4.40625 -0.296875 C 4.3125 -0.296875 4.203125 -0.296875 4.203125 -0.109375 C 4.203125 -0.078125 4.234375 0 4.328125 0 C 4.546875 0 5.078125 -0.03125 5.296875 -0.03125 C 5.453125 -0.03125 5.59375 -0.015625 5.734375 -0.015625 C 5.890625 -0.015625 6.046875 0 6.1875 0 C 6.234375 0 6.34375 0 6.34375 -0.1875 C 6.34375 -0.296875 6.25 -0.296875 6.140625 -0.296875 C 5.6875 -0.296875 5.671875 -0.34375 5.65625 -0.53125 L 4.90625 -5.0625 C 4.890625 -5.203125 4.875 -5.265625 4.71875 -5.265625 C 4.5625 -5.265625 4.515625 -5.203125 4.453125 -5.109375 Z M 2.6875 -1.96875 L 4.28125 -4.359375 L 4.671875 -1.96875 Z M 2.6875 -1.96875 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-2"> <path style="stroke:none;" d="M 1.546875 -0.609375 C 1.484375 -0.34375 1.46875 -0.296875 0.921875 -0.296875 C 0.75 -0.296875 0.734375 -0.296875 0.703125 -0.265625 C 0.671875 -0.25 0.640625 -0.140625 0.640625 -0.109375 C 0.65625 0 0.734375 0 0.875 0 L 3.875 0 C 5.15625 0 6.15625 -0.828125 6.15625 -1.578125 C 6.15625 -2.03125 5.765625 -2.515625 4.84375 -2.640625 C 5.6875 -2.765625 6.53125 -3.3125 6.53125 -3.984375 C 6.53125 -4.515625 5.9375 -5.03125 4.90625 -5.03125 L 2.078125 -5.03125 C 1.9375 -5.03125 1.828125 -5.03125 1.828125 -4.859375 C 1.828125 -4.734375 1.9375 -4.734375 2.078125 -4.734375 C 2.203125 -4.734375 2.390625 -4.734375 2.546875 -4.6875 C 2.546875 -4.609375 2.546875 -4.578125 2.515625 -4.453125 Z M 2.765625 -2.75 L 3.1875 -4.484375 C 3.25 -4.734375 3.265625 -4.734375 3.5625 -4.734375 L 4.78125 -4.734375 C 5.546875 -4.734375 5.78125 -4.28125 5.78125 -3.96875 C 5.78125 -3.359375 5 -2.75 4.09375 -2.75 Z M 2.40625 -0.296875 C 2.265625 -0.296875 2.25 -0.296875 2.15625 -0.3125 L 2.6875 -2.5 L 4.296875 -2.5 C 5.234375 -2.5 5.390625 -1.90625 5.390625 -1.609375 C 5.390625 -0.90625 4.609375 -0.296875 3.6875 -0.296875 Z M 2.40625 -0.296875 "></path> </symbol> <symbol overflow="visible" id="zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-3"> <path style="stroke:none;" d="M 1.703125 -0.109375 C 1.703125 0.59375 1.328125 1 1.140625 1.171875 C 1.0625 1.234375 1.046875 1.25 1.046875 1.296875 C 1.046875 1.359375 1.109375 1.4375 1.171875 1.4375 C 1.28125 1.4375 1.9375 0.84375 1.9375 -0.0625 C 1.9375 -0.53125 1.75 -0.890625 1.390625 -0.890625 C 1.109375 -0.890625 0.953125 -0.65625 0.953125 -0.453125 C 0.953125 -0.234375 1.109375 0 1.40625 0 C 1.515625 0 1.609375 -0.03125 1.703125 -0.109375 Z M 1.703125 -0.109375 "></path> </symbol> </g> </defs> <g id="zvB7v6tF0ywNHF8WFrliu7kVj54=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-1" x="9.619658" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="28.395392" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-2" x="43.134898" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="57.228411" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-3" x="71.966684" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="88.337201" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-4" x="103.075475" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-1" x="238.306376" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="257.082111" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-2" x="271.821617" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="285.915129" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-4" x="300.653403" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-1" x="435.883811" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="454.659546" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-2" x="469.397819" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph1-1" x="483.491332" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-1" x="494.955255" y="25.792657"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph0-4" x="508.177169" y="23.582224"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -145.379559 2.633541 L -33.976676 2.633541 " transform="matrix(0.98625,0,0,-0.98625,264.751684,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485368 2.867625 C -2.033847 1.148677 -1.019905 0.332771 0.00195776 0.0000711914 C -1.019905 -0.336589 -2.033847 -1.148535 -2.485368 -2.867483 " transform="matrix(0.98625,0,0,-0.98625,231.478538,17.054758)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-2" x="152.324115" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-1" x="163.397237" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-3" x="171.51654" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-1" x="176.863248" y="8.208312"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-1" x="184.287245" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-4" x="192.406548" y="11.676214"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -145.379559 -3.133252 L -33.976676 -3.133252 " transform="matrix(0.98625,0,0,-0.98625,264.751684,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485368 2.869862 C -2.033847 1.146954 -1.019905 0.335008 0.00195776 -0.00165222 C -1.019905 -0.334352 -2.033847 -1.146297 -2.485368 -2.869206 " transform="matrix(0.98625,0,0,-0.98625,231.478538,22.740558)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-2" x="152.03613" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-1" x="163.109252" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-5" x="171.228555" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-1" x="179.048285" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-6" x="187.168821" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-2" x="193.258914" y="31.243167"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.957101 -0.249856 L 166.359985 -0.249856 " transform="matrix(0.98625,0,0,-0.98625,264.751684,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487128 2.870729 C -2.031647 1.14782 -1.021666 0.335875 0.000197376 -0.000785513 C -1.021666 -0.333485 -2.031647 -1.149391 -2.487128 -2.868339 " transform="matrix(0.98625,0,0,-0.98625,429.058399,19.897663)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-2" x="352.826767" y="13.345935"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph3-1" x="363.899889" y="13.345935"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph2-7" x="372.020425" y="13.345935"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-1" x="378.01436" y="14.996671"></use> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-3" x="384.824293" y="14.996671"></use> <use xlink:href="#zvB7v6tF0ywNHF8WFrliu7kVj54=-glyph4-2" x="387.621664" y="14.996671"></use> </g> </g> </svg> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="517.487pt" height="39.45pt" viewBox="0 0 517.487 39.45" version="1.2"> <defs> <g> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph0-1"> <path style="stroke:none;" d="M 2.5 -1.640625 C 1.984375 -0.765625 1.484375 -0.46875 0.78125 -0.421875 C 0.625 -0.40625 0.5 -0.40625 0.5 -0.140625 C 0.5 -0.0625 0.578125 0 0.671875 0 C 0.9375 0 1.609375 -0.03125 1.875 -0.03125 C 2.296875 -0.03125 2.765625 0 3.1875 0 C 3.265625 0 3.453125 0 3.453125 -0.28125 C 3.453125 -0.40625 3.328125 -0.421875 3.234375 -0.421875 C 2.90625 -0.453125 2.625 -0.578125 2.625 -0.921875 C 2.625 -1.140625 2.71875 -1.296875 2.90625 -1.625 L 4.015625 -3.484375 L 7.78125 -3.484375 C 7.796875 -3.34375 7.796875 -3.234375 7.8125 -3.09375 C 7.859375 -2.71875 8.03125 -1.171875 8.03125 -0.90625 C 8.03125 -0.453125 7.28125 -0.421875 7.046875 -0.421875 C 6.875 -0.421875 6.71875 -0.421875 6.71875 -0.15625 C 6.71875 0 6.84375 0 6.9375 0 C 7.1875 0 7.484375 -0.03125 7.734375 -0.03125 L 8.578125 -0.03125 C 9.46875 -0.03125 10.125 0 10.140625 0 C 10.234375 0 10.40625 0 10.40625 -0.28125 C 10.40625 -0.421875 10.265625 -0.421875 10.046875 -0.421875 C 9.234375 -0.421875 9.21875 -0.5625 9.171875 -1 L 8.28125 -10.203125 C 8.25 -10.484375 8.1875 -10.515625 8.03125 -10.515625 C 7.890625 -10.515625 7.796875 -10.484375 7.65625 -10.265625 Z M 4.265625 -3.90625 L 7.234375 -8.859375 L 7.734375 -3.90625 Z M 4.265625 -3.90625 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph0-2"> <path style="stroke:none;" d="M 10.90625 -8.515625 C 11.0625 -9.15625 11.296875 -9.59375 12.421875 -9.640625 C 12.46875 -9.640625 12.640625 -9.65625 12.640625 -9.90625 C 12.640625 -10.0625 12.515625 -10.0625 12.453125 -10.0625 C 12.15625 -10.0625 11.40625 -10.03125 11.109375 -10.03125 L 10.40625 -10.03125 C 10.203125 -10.03125 9.9375 -10.0625 9.71875 -10.0625 C 9.640625 -10.0625 9.453125 -10.0625 9.453125 -9.78125 C 9.453125 -9.640625 9.578125 -9.640625 9.6875 -9.640625 C 10.5625 -9.609375 10.625 -9.265625 10.625 -9 C 10.625 -8.875 10.609375 -8.828125 10.5625 -8.625 L 8.90625 -1.96875 L 5.75 -9.8125 C 5.640625 -10.046875 5.625 -10.0625 5.296875 -10.0625 L 3.5 -10.0625 C 3.21875 -10.0625 3.078125 -10.0625 3.078125 -9.78125 C 3.078125 -9.640625 3.1875 -9.640625 3.46875 -9.640625 C 3.53125 -9.640625 4.40625 -9.640625 4.40625 -9.5 C 4.40625 -9.46875 4.375 -9.359375 4.359375 -9.3125 L 2.40625 -1.5 C 2.21875 -0.78125 1.875 -0.46875 0.90625 -0.421875 C 0.828125 -0.421875 0.671875 -0.40625 0.671875 -0.140625 C 0.671875 0 0.828125 0 0.875 0 C 1.15625 0 1.921875 -0.03125 2.203125 -0.03125 L 2.921875 -0.03125 C 3.125 -0.03125 3.375 0 3.578125 0 C 3.6875 0 3.84375 0 3.84375 -0.28125 C 3.84375 -0.40625 3.703125 -0.421875 3.640625 -0.421875 C 3.15625 -0.4375 2.6875 -0.53125 2.6875 -1.0625 C 2.6875 -1.171875 2.71875 -1.3125 2.734375 -1.421875 L 4.734375 -9.3125 C 4.8125 -9.171875 4.8125 -9.140625 4.875 -9 L 8.390625 -0.265625 C 8.453125 -0.09375 8.484375 0 8.625 0 C 8.765625 0 8.78125 -0.046875 8.84375 -0.296875 Z M 10.90625 -8.515625 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph0-3"> <path style="stroke:none;" d="M 5.390625 -9.0625 C 5.53125 -9.609375 5.578125 -9.640625 6.15625 -9.640625 L 8.078125 -9.640625 C 9.734375 -9.640625 9.734375 -8.21875 9.734375 -8.09375 C 9.734375 -6.890625 8.546875 -5.375 6.59375 -5.375 L 4.484375 -5.375 Z M 7.890625 -5.265625 C 9.484375 -5.5625 10.953125 -6.671875 10.953125 -8.03125 C 10.953125 -9.171875 9.9375 -10.0625 8.265625 -10.0625 L 3.53125 -10.0625 C 3.25 -10.0625 3.125 -10.0625 3.125 -9.78125 C 3.125 -9.640625 3.25 -9.640625 3.484375 -9.640625 C 4.375 -9.640625 4.375 -9.515625 4.375 -9.359375 C 4.375 -9.328125 4.375 -9.234375 4.3125 -9.015625 L 2.328125 -1.09375 C 2.203125 -0.578125 2.171875 -0.421875 1.140625 -0.421875 C 0.859375 -0.421875 0.703125 -0.421875 0.703125 -0.15625 C 0.703125 0 0.796875 0 1.09375 0 L 6.140625 0 C 8.40625 0 10.140625 -1.703125 10.140625 -3.203125 C 10.140625 -4.40625 9.078125 -5.140625 7.890625 -5.265625 Z M 5.796875 -0.421875 L 3.796875 -0.421875 C 3.59375 -0.421875 3.5625 -0.421875 3.484375 -0.4375 C 3.3125 -0.453125 3.296875 -0.484375 3.296875 -0.609375 C 3.296875 -0.703125 3.328125 -0.796875 3.359375 -0.921875 L 4.390625 -5.078125 L 7.15625 -5.078125 C 8.90625 -5.078125 8.90625 -3.46875 8.90625 -3.34375 C 8.90625 -1.9375 7.625 -0.421875 5.796875 -0.421875 Z M 5.796875 -0.421875 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph0-4"> <path style="stroke:none;" d="M 4.359375 -4.6875 L 6.84375 -4.6875 C 8.875 -4.6875 10.90625 -6.1875 10.90625 -7.875 C 10.90625 -9.015625 9.9375 -10.0625 8.078125 -10.0625 L 3.515625 -10.0625 C 3.234375 -10.0625 3.109375 -10.0625 3.109375 -9.78125 C 3.109375 -9.640625 3.234375 -9.640625 3.46875 -9.640625 C 4.359375 -9.640625 4.359375 -9.515625 4.359375 -9.359375 C 4.359375 -9.328125 4.359375 -9.234375 4.296875 -9.015625 L 2.3125 -1.09375 C 2.1875 -0.578125 2.15625 -0.421875 1.125 -0.421875 C 0.84375 -0.421875 0.6875 -0.421875 0.6875 -0.15625 C 0.6875 0 0.828125 0 0.90625 0 C 1.1875 0 1.484375 -0.03125 1.765625 -0.03125 L 3.484375 -0.03125 C 3.765625 -0.03125 4.078125 0 4.359375 0 C 4.484375 0 4.640625 0 4.640625 -0.28125 C 4.640625 -0.421875 4.515625 -0.421875 4.28125 -0.421875 C 3.40625 -0.421875 3.390625 -0.53125 3.390625 -0.671875 C 3.390625 -0.75 3.40625 -0.859375 3.421875 -0.921875 Z M 5.421875 -9.0625 C 5.5625 -9.609375 5.609375 -9.640625 6.1875 -9.640625 L 7.640625 -9.640625 C 8.75 -9.640625 9.671875 -9.28125 9.671875 -8.171875 C 9.671875 -7.796875 9.46875 -6.546875 8.796875 -5.859375 C 8.546875 -5.59375 7.84375 -5.046875 6.5 -5.046875 L 4.421875 -5.046875 Z M 5.421875 -9.0625 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1"> <path style="stroke:none;" d="M 10.640625 -3.6875 C 10.640625 -6.390625 8.421875 -8.59375 5.734375 -8.59375 C 2.984375 -8.59375 0.8125 -6.359375 0.8125 -3.6875 C 0.8125 -0.96875 3.015625 1.21875 5.71875 1.21875 C 8.453125 1.21875 10.640625 -1 10.640625 -3.6875 Z M 2.796875 -6.84375 C 2.765625 -6.875 2.671875 -6.984375 2.671875 -7.015625 C 2.671875 -7.078125 3.859375 -8.21875 5.71875 -8.21875 C 6.234375 -8.21875 7.59375 -8.140625 8.796875 -7.015625 L 5.734375 -3.9375 Z M 2.375 -0.609375 C 1.46875 -1.609375 1.171875 -2.734375 1.171875 -3.6875 C 1.171875 -4.8125 1.609375 -5.890625 2.375 -6.765625 L 5.453125 -3.6875 Z M 9.0625 -6.765625 C 9.765625 -6.015625 10.265625 -4.890625 10.265625 -3.6875 C 10.265625 -2.546875 9.84375 -1.46875 9.078125 -0.609375 L 6 -3.6875 Z M 8.65625 -0.515625 C 8.671875 -0.484375 8.78125 -0.390625 8.78125 -0.359375 C 8.78125 -0.296875 7.59375 0.859375 5.734375 0.859375 C 5.21875 0.859375 3.859375 0.78125 2.65625 -0.359375 L 5.71875 -3.4375 Z M 8.65625 -0.515625 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-1"> <path style="stroke:none;" d="M 7.78125 -5.640625 C 7.90625 -6.125 8.125 -6.359375 8.828125 -6.390625 C 8.921875 -6.390625 9 -6.453125 9 -6.578125 C 9 -6.640625 8.953125 -6.71875 8.859375 -6.71875 C 8.78125 -6.71875 8.59375 -6.6875 7.875 -6.6875 C 7.09375 -6.6875 6.953125 -6.71875 6.875 -6.71875 C 6.71875 -6.71875 6.6875 -6.609375 6.6875 -6.53125 C 6.6875 -6.390625 6.8125 -6.390625 6.90625 -6.390625 C 7.5 -6.375 7.5 -6.109375 7.5 -5.96875 C 7.5 -5.921875 7.5 -5.875 7.453125 -5.703125 L 6.375 -1.40625 L 4.015625 -6.53125 C 3.9375 -6.71875 3.90625 -6.71875 3.671875 -6.71875 L 2.390625 -6.71875 C 2.21875 -6.71875 2.09375 -6.71875 2.09375 -6.53125 C 2.09375 -6.390625 2.21875 -6.390625 2.421875 -6.390625 C 2.5 -6.390625 2.796875 -6.390625 3.015625 -6.328125 L 1.703125 -1.046875 C 1.578125 -0.5625 1.328125 -0.34375 0.671875 -0.328125 C 0.609375 -0.328125 0.484375 -0.3125 0.484375 -0.140625 C 0.484375 -0.078125 0.546875 0 0.640625 0 C 0.671875 0 0.90625 -0.03125 1.609375 -0.03125 C 2.390625 -0.03125 2.53125 0 2.625 0 C 2.65625 0 2.8125 0 2.8125 -0.1875 C 2.8125 -0.296875 2.703125 -0.328125 2.640625 -0.328125 C 2.28125 -0.328125 1.984375 -0.390625 1.984375 -0.734375 C 1.984375 -0.78125 2.015625 -0.921875 2.015625 -0.9375 L 3.296875 -6.0625 L 3.3125 -6.0625 L 6.046875 -0.171875 C 6.109375 -0.015625 6.125 0 6.234375 0 C 6.375 0 6.375 -0.046875 6.421875 -0.203125 Z M 7.78125 -5.640625 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-2"> <path style="stroke:none;" d="M 0.359375 1.578125 C 0.328125 1.734375 0.328125 1.78125 0.328125 1.8125 C 0.328125 2.046875 0.515625 2.109375 0.640625 2.109375 C 0.6875 2.109375 0.90625 2.109375 1.046875 1.84375 C 1.09375 1.734375 1.28125 0.828125 1.640625 -0.515625 C 1.765625 -0.296875 2.078125 0.09375 2.671875 0.09375 C 3.875 0.09375 5.203125 -1.21875 5.203125 -2.6875 C 5.203125 -3.796875 4.46875 -4.328125 3.703125 -4.328125 C 2.796875 -4.328125 1.640625 -3.515625 1.28125 -2.109375 Z M 2.65625 -0.171875 C 1.984375 -0.171875 1.765625 -0.890625 1.765625 -1.015625 C 1.765625 -1.0625 2.03125 -2.0625 2.046875 -2.125 C 2.484375 -3.875 3.421875 -4.0625 3.703125 -4.0625 C 4.171875 -4.0625 4.4375 -3.640625 4.4375 -3.09375 C 4.4375 -2.734375 4.234375 -1.765625 3.9375 -1.15625 C 3.65625 -0.59375 3.140625 -0.171875 2.65625 -0.171875 Z M 2.65625 -0.171875 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-3"> <path style="stroke:none;" d="M 1.65625 -0.78125 C 1.578125 -0.40625 1.546875 -0.328125 0.828125 -0.328125 C 0.640625 -0.328125 0.515625 -0.328125 0.515625 -0.140625 C 0.515625 0 0.65625 0 0.8125 0 L 4.46875 0 C 6.078125 0 7.296875 -1.15625 7.296875 -2.109375 C 7.296875 -2.828125 6.671875 -3.40625 5.6875 -3.515625 C 6.828125 -3.71875 7.8125 -4.46875 7.8125 -5.34375 C 7.8125 -6.078125 7.09375 -6.71875 5.859375 -6.71875 L 2.421875 -6.71875 C 2.25 -6.71875 2.125 -6.71875 2.125 -6.53125 C 2.125 -6.390625 2.234375 -6.390625 2.40625 -6.390625 C 2.734375 -6.390625 3.015625 -6.390625 3.015625 -6.234375 C 3.015625 -6.1875 3 -6.1875 2.984375 -6.046875 Z M 3.1875 -3.625 L 3.796875 -6.03125 C 3.875 -6.359375 3.890625 -6.390625 4.296875 -6.390625 L 5.703125 -6.390625 C 6.671875 -6.390625 6.890625 -5.765625 6.890625 -5.359375 C 6.890625 -4.515625 6 -3.625 4.734375 -3.625 Z M 2.515625 -0.328125 C 2.421875 -0.34375 2.390625 -0.34375 2.390625 -0.40625 C 2.390625 -0.484375 2.421875 -0.5625 2.4375 -0.625 L 3.125 -3.34375 L 5.125 -3.34375 C 6.03125 -3.34375 6.34375 -2.734375 6.34375 -2.1875 C 6.34375 -1.21875 5.390625 -0.328125 4.21875 -0.328125 Z M 2.515625 -0.328125 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-4"> <path style="stroke:none;" d="M 3.09375 -3.0625 L 4.890625 -3.0625 C 6.296875 -3.0625 7.78125 -4.046875 7.78125 -5.203125 C 7.78125 -6.046875 6.984375 -6.71875 5.71875 -6.71875 L 2.40625 -6.71875 C 2.234375 -6.71875 2.109375 -6.71875 2.109375 -6.53125 C 2.109375 -6.390625 2.234375 -6.390625 2.390625 -6.390625 C 2.71875 -6.390625 3 -6.390625 3 -6.234375 C 3 -6.1875 2.984375 -6.1875 2.953125 -6.046875 L 1.640625 -0.78125 C 1.546875 -0.40625 1.53125 -0.328125 0.828125 -0.328125 C 0.609375 -0.328125 0.5 -0.328125 0.5 -0.140625 C 0.5 -0.09375 0.53125 0 0.65625 0 C 0.84375 0 1.078125 -0.015625 1.265625 -0.03125 L 1.890625 -0.03125 C 2.84375 -0.03125 3.09375 0 3.171875 0 C 3.21875 0 3.359375 0 3.359375 -0.1875 C 3.359375 -0.328125 3.234375 -0.328125 3.0625 -0.328125 C 3.03125 -0.328125 2.859375 -0.328125 2.6875 -0.34375 C 2.484375 -0.359375 2.46875 -0.390625 2.46875 -0.484375 C 2.46875 -0.53125 2.484375 -0.59375 2.5 -0.640625 Z M 3.8125 -6.03125 C 3.90625 -6.359375 3.90625 -6.390625 4.3125 -6.390625 L 5.390625 -6.390625 C 6.203125 -6.390625 6.828125 -6.15625 6.828125 -5.421875 C 6.828125 -5.296875 6.765625 -4.421875 6.21875 -3.875 C 6.078125 -3.734375 5.609375 -3.359375 4.640625 -3.359375 L 3.140625 -3.359375 Z M 3.8125 -6.03125 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-5"> <path style="stroke:none;" d="M 1.8125 -1.171875 C 1.359375 -0.453125 0.96875 -0.359375 0.609375 -0.328125 C 0.515625 -0.3125 0.390625 -0.296875 0.390625 -0.140625 C 0.390625 -0.109375 0.40625 0 0.53125 0 C 0.625 0 0.671875 -0.03125 1.359375 -0.03125 C 2.03125 -0.03125 2.265625 0 2.3125 0 C 2.359375 0 2.5 0 2.5 -0.1875 C 2.5 -0.3125 2.375 -0.328125 2.328125 -0.328125 C 2.125 -0.328125 1.921875 -0.40625 1.921875 -0.625 C 1.921875 -0.78125 2 -0.90625 2.1875 -1.1875 L 2.84375 -2.265625 L 5.546875 -2.265625 L 5.75 -0.609375 C 5.75 -0.46875 5.578125 -0.328125 5.109375 -0.328125 C 4.96875 -0.328125 4.828125 -0.328125 4.828125 -0.140625 C 4.828125 -0.125 4.84375 0 5 0 C 5.09375 0 5.46875 -0.015625 5.5625 -0.03125 L 6.1875 -0.03125 C 7.0625 -0.03125 7.234375 0 7.3125 0 C 7.34375 0 7.515625 0 7.515625 -0.1875 C 7.515625 -0.328125 7.375 -0.328125 7.21875 -0.328125 C 6.6875 -0.328125 6.671875 -0.40625 6.640625 -0.65625 L 5.875 -6.734375 C 5.84375 -6.953125 5.828125 -7.015625 5.65625 -7.015625 C 5.484375 -7.015625 5.421875 -6.921875 5.359375 -6.828125 Z M 3.0625 -2.59375 L 5.09375 -5.828125 L 5.5 -2.59375 Z M 3.0625 -2.59375 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-6"> <path style="stroke:none;" d="M 3.703125 -2.828125 C 4.0625 -1.921875 4.65625 -0.328125 4.765625 -0.140625 C 4.9375 0.09375 5.125 0.09375 5.359375 0.09375 C 5.65625 0.09375 5.734375 0.09375 5.734375 -0.015625 C 5.734375 -0.0625 5.703125 -0.09375 5.671875 -0.125 C 5.546875 -0.265625 5.5 -0.359375 5.4375 -0.546875 L 3.21875 -6.15625 C 3.15625 -6.34375 2.96875 -6.828125 1.953125 -6.828125 C 1.859375 -6.828125 1.703125 -6.828125 1.703125 -6.6875 C 1.703125 -6.5625 1.8125 -6.5625 1.859375 -6.546875 C 2.046875 -6.515625 2.234375 -6.5 2.46875 -5.921875 L 3.171875 -4.15625 L 3.5625 -3.171875 L 0.75 -0.5625 C 0.640625 -0.46875 0.546875 -0.34375 0.546875 -0.203125 C 0.546875 -0.015625 0.703125 0.125 0.875 0.125 C 1.015625 0.125 1.15625 0.015625 1.234375 -0.09375 Z M 3.703125 -2.828125 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph2-7"> <path style="stroke:none;" d="M 2.796875 -3.59375 L 3.921875 -3.59375 C 3.71875 -2.6875 3.53125 -1.96875 3.53125 -1.234375 C 3.53125 -1.171875 3.53125 -0.75 3.640625 -0.390625 C 3.78125 0.015625 3.875 0.09375 4.046875 0.09375 C 4.265625 0.09375 4.484375 -0.09375 4.484375 -0.328125 C 4.484375 -0.390625 4.484375 -0.40625 4.4375 -0.5 C 4.234375 -0.953125 4.09375 -1.421875 4.09375 -2.234375 C 4.09375 -2.453125 4.09375 -2.875 4.234375 -3.59375 L 5.421875 -3.59375 C 5.578125 -3.59375 5.6875 -3.59375 5.78125 -3.671875 C 5.90625 -3.78125 5.9375 -3.90625 5.9375 -3.953125 C 5.9375 -4.234375 5.6875 -4.234375 5.515625 -4.234375 L 1.96875 -4.234375 C 1.765625 -4.234375 1.390625 -4.234375 0.90625 -3.765625 C 0.5625 -3.40625 0.28125 -2.953125 0.28125 -2.890625 C 0.28125 -2.796875 0.359375 -2.765625 0.4375 -2.765625 C 0.53125 -2.765625 0.546875 -2.796875 0.609375 -2.875 C 1.09375 -3.59375 1.671875 -3.59375 1.890625 -3.59375 L 2.46875 -3.59375 C 2.1875 -2.546875 1.65625 -1.359375 1.296875 -0.640625 C 1.234375 -0.484375 1.125 -0.296875 1.125 -0.203125 C 1.125 0 1.296875 0.09375 1.453125 0.09375 C 1.828125 0.09375 1.921875 -0.28125 2.125 -1.078125 Z M 2.796875 -3.59375 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph3-1"> <path style="stroke:none;" d="M 1.671875 -0.90625 C 1.390625 -0.46875 1.140625 -0.328125 0.71875 -0.296875 C 0.625 -0.28125 0.53125 -0.28125 0.53125 -0.109375 C 0.53125 -0.03125 0.59375 0 0.640625 0 C 0.84375 0 1.09375 -0.03125 1.3125 -0.03125 C 1.484375 -0.03125 1.921875 0 2.09375 0 C 2.171875 0 2.25 -0.03125 2.25 -0.1875 C 2.25 -0.28125 2.15625 -0.296875 2.125 -0.296875 C 1.953125 -0.296875 1.796875 -0.34375 1.796875 -0.515625 C 1.796875 -0.59375 1.875 -0.71875 1.890625 -0.765625 L 2.5 -1.671875 L 4.71875 -1.671875 L 4.90625 -0.484375 C 4.90625 -0.296875 4.484375 -0.296875 4.40625 -0.296875 C 4.3125 -0.296875 4.203125 -0.296875 4.203125 -0.109375 C 4.203125 -0.078125 4.234375 0 4.328125 0 C 4.546875 0 5.078125 -0.03125 5.296875 -0.03125 C 5.453125 -0.03125 5.59375 -0.015625 5.734375 -0.015625 C 5.890625 -0.015625 6.046875 0 6.1875 0 C 6.234375 0 6.34375 0 6.34375 -0.1875 C 6.34375 -0.296875 6.25 -0.296875 6.140625 -0.296875 C 5.6875 -0.296875 5.671875 -0.34375 5.65625 -0.53125 L 4.90625 -5.0625 C 4.890625 -5.203125 4.875 -5.265625 4.71875 -5.265625 C 4.5625 -5.265625 4.515625 -5.203125 4.453125 -5.109375 Z M 2.6875 -1.96875 L 4.28125 -4.359375 L 4.671875 -1.96875 Z M 2.6875 -1.96875 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph3-2"> <path style="stroke:none;" d="M 1.546875 -0.609375 C 1.484375 -0.34375 1.46875 -0.296875 0.921875 -0.296875 C 0.75 -0.296875 0.734375 -0.296875 0.703125 -0.265625 C 0.671875 -0.25 0.640625 -0.140625 0.640625 -0.109375 C 0.65625 0 0.734375 0 0.875 0 L 3.875 0 C 5.15625 0 6.15625 -0.828125 6.15625 -1.578125 C 6.15625 -2.03125 5.765625 -2.515625 4.84375 -2.640625 C 5.6875 -2.765625 6.53125 -3.3125 6.53125 -3.984375 C 6.53125 -4.515625 5.9375 -5.03125 4.90625 -5.03125 L 2.078125 -5.03125 C 1.9375 -5.03125 1.828125 -5.03125 1.828125 -4.859375 C 1.828125 -4.734375 1.9375 -4.734375 2.078125 -4.734375 C 2.203125 -4.734375 2.390625 -4.734375 2.546875 -4.6875 C 2.546875 -4.609375 2.546875 -4.578125 2.515625 -4.453125 Z M 2.765625 -2.75 L 3.1875 -4.484375 C 3.25 -4.734375 3.265625 -4.734375 3.5625 -4.734375 L 4.78125 -4.734375 C 5.546875 -4.734375 5.78125 -4.28125 5.78125 -3.96875 C 5.78125 -3.359375 5 -2.75 4.09375 -2.75 Z M 2.40625 -0.296875 C 2.265625 -0.296875 2.25 -0.296875 2.15625 -0.3125 L 2.6875 -2.5 L 4.296875 -2.5 C 5.234375 -2.5 5.390625 -1.90625 5.390625 -1.609375 C 5.390625 -0.90625 4.609375 -0.296875 3.6875 -0.296875 Z M 2.40625 -0.296875 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph3-3"> <path style="stroke:none;" d="M 1.703125 -0.109375 C 1.703125 0.59375 1.328125 1 1.140625 1.171875 C 1.0625 1.234375 1.046875 1.25 1.046875 1.296875 C 1.046875 1.359375 1.109375 1.4375 1.171875 1.4375 C 1.28125 1.4375 1.9375 0.84375 1.9375 -0.0625 C 1.9375 -0.53125 1.75 -0.890625 1.390625 -0.890625 C 1.109375 -0.890625 0.953125 -0.65625 0.953125 -0.453125 C 0.953125 -0.234375 1.109375 0 1.40625 0 C 1.515625 0 1.609375 -0.03125 1.703125 -0.109375 Z M 1.703125 -0.109375 "></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="PZpILweU19Qtcou9iIKAPyddh04=-glyph4-1"> <path style="stroke:none;" d="M 7.53125 -2.453125 C 7.53125 -4.390625 5.953125 -5.9375 4.0625 -5.9375 C 2.125 -5.9375 0.578125 -4.359375 0.578125 -2.453125 C 0.578125 -0.515625 2.15625 1.015625 4.046875 1.015625 C 5.984375 1.015625 7.53125 -0.546875 7.53125 -2.453125 Z M 1.921875 -4.8125 C 2.6875 -5.515625 3.578125 -5.625 4.046875 -5.625 C 4.65625 -5.625 5.484375 -5.453125 6.1875 -4.8125 L 4.0625 -2.671875 Z M 1.703125 -0.328125 C 1.15625 -0.90625 0.890625 -1.703125 0.890625 -2.453125 C 0.890625 -3.203125 1.15625 -4 1.703125 -4.59375 L 3.84375 -2.453125 Z M 6.390625 -4.59375 C 6.953125 -4.015625 7.21875 -3.21875 7.21875 -2.453125 C 7.21875 -1.703125 6.953125 -0.90625 6.390625 -0.328125 L 4.265625 -2.453125 Z M 6.1875 -0.109375 C 5.421875 0.59375 4.53125 0.71875 4.0625 0.71875 C 3.453125 0.71875 2.625 0.546875 1.921875 -0.109375 L 4.046875 -2.234375 Z M 6.1875 -0.109375 "></path> </symbol> </g> </defs> <g id="PZpILweU19Qtcou9iIKAPyddh04=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-1" x="9.650283" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="23.743795" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-2" x="38.482068" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="54.853818" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-3" x="69.592092" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="84.574462" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-4" x="99.312736" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-1" x="234.14248" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="248.235993" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-3" x="262.974266" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="277.956636" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-4" x="292.69491" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-1" x="427.525394" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="441.618906" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-1" x="453.081597" y="25.792657"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-3" x="466.303511" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph1-1" x="481.285881" y="23.582224"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph0-4" x="496.024155" y="23.582224"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -143.253728 2.633541 L -31.850844 2.633541 " transform="matrix(0.98625,0,0,-0.98625,258.49102,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486097 2.867625 C -2.030615 1.148677 -1.020634 0.332771 0.00122919 0.0000711914 C -1.020634 -0.336589 -2.030615 -1.148535 -2.486097 -2.867483 " transform="matrix(0.98625,0,0,-0.98625,227.315194,17.054758)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-2" x="149.662278" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph3-1" x="155.007753" y="8.208312"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph4-1" x="162.431749" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-3" x="170.552285" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph4-1" x="178.910754" y="11.676214"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-4" x="187.030057" y="11.676214"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -143.253728 -3.133252 L -31.850844 -3.133252 " transform="matrix(0.98625,0,0,-0.98625,258.49102,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486097 2.869862 C -2.030615 1.146954 -1.020634 0.335008 0.00122919 -0.00165222 C -1.020634 -0.334352 -2.030615 -1.146297 -2.486097 -2.869206 " transform="matrix(0.98625,0,0,-0.98625,227.315194,22.740558)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-5" x="149.374293" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph4-1" x="157.194022" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-6" x="165.313325" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph3-2" x="171.403419" y="31.243167"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph4-1" x="179.198493" y="34.711069"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-4" x="187.319028" y="34.711069"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 52.829131 -0.249856 L 164.232014 -0.249856 " transform="matrix(0.98625,0,0,-0.98625,258.49102,19.652017)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.488499 2.870729 C -2.033018 1.14782 -1.019076 0.335875 -0.00117347 -0.000785513 C -1.019076 -0.333485 -2.033018 -1.149391 -2.488499 -2.868339 " transform="matrix(0.98625,0,0,-0.98625,420.700376,19.897663)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-7" x="345.970409" y="13.345935"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph3-1" x="351.965576" y="14.996671"></use> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph3-3" x="358.775509" y="14.996671"></use> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph3-2" x="361.57288" y="14.996671"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph4-1" x="369.367957" y="13.345935"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#PZpILweU19Qtcou9iIKAPyddh04=-glyph2-4" x="377.48726" y="13.345935"></use> </g> </g> </svg> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \otimes B</annotation></semantics></math> becomes a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>-bimodule with left action defined by the following diagram: <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="530.247pt" height="104.983pt" viewBox="0 0 530.247 104.983" version="1.2"> <defs> <g> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-1"> <path style="stroke:none;" d="M 13.546875 -9.109375 C 13.6875 -9.609375 13.71875 -9.765625 14.765625 -9.765625 C 15.0625 -9.765625 15.1875 -9.765625 15.1875 -10.046875 C 15.1875 -10.1875 15.09375 -10.1875 14.796875 -10.1875 L 13.015625 -10.1875 C 12.640625 -10.1875 12.625 -10.171875 12.453125 -9.9375 L 7.015625 -1.328125 L 5.890625 -9.859375 C 5.84375 -10.1875 5.828125 -10.1875 5.453125 -10.1875 L 3.59375 -10.1875 C 3.3125 -10.1875 3.171875 -10.1875 3.171875 -9.90625 C 3.171875 -9.765625 3.3125 -9.765625 3.53125 -9.765625 C 4.453125 -9.765625 4.453125 -9.640625 4.453125 -9.46875 C 4.453125 -9.453125 4.453125 -9.359375 4.390625 -9.125 L 2.484375 -1.515625 C 2.296875 -0.8125 1.953125 -0.484375 0.953125 -0.4375 C 0.90625 -0.4375 0.734375 -0.421875 0.734375 -0.171875 C 0.734375 0 0.859375 0 0.921875 0 C 1.21875 0 1.984375 -0.03125 2.28125 -0.03125 L 3 -0.03125 C 3.203125 -0.03125 3.46875 0 3.671875 0 C 3.78125 0 3.9375 0 3.9375 -0.28125 C 3.9375 -0.421875 3.796875 -0.4375 3.734375 -0.4375 C 3.234375 -0.453125 2.765625 -0.53125 2.765625 -1.078125 C 2.765625 -1.21875 2.765625 -1.234375 2.8125 -1.453125 L 4.875 -9.671875 L 4.890625 -9.671875 L 6.140625 -0.40625 C 6.171875 -0.046875 6.1875 0 6.328125 0 C 6.484375 0 6.5625 -0.125 6.640625 -0.25 L 12.640625 -9.75 L 12.65625 -9.75 L 10.484375 -1.109375 C 10.359375 -0.578125 10.328125 -0.4375 9.28125 -0.4375 C 9 -0.4375 8.84375 -0.4375 8.84375 -0.171875 C 8.84375 0 8.984375 0 9.078125 0 C 9.328125 0 9.625 -0.03125 9.875 -0.03125 L 11.640625 -0.03125 C 11.890625 -0.03125 12.203125 0 12.453125 0 C 12.578125 0 12.75 0 12.75 -0.28125 C 12.75 -0.4375 12.609375 -0.4375 12.390625 -0.4375 C 11.46875 -0.4375 11.46875 -0.546875 11.46875 -0.703125 C 11.46875 -0.71875 11.46875 -0.828125 11.5 -0.9375 Z M 13.546875 -9.109375 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2"> <path style="stroke:none;" d="M 2.53125 -1.65625 C 2.015625 -0.78125 1.5 -0.484375 0.796875 -0.4375 C 0.625 -0.421875 0.5 -0.421875 0.5 -0.15625 C 0.5 -0.0625 0.578125 0 0.6875 0 C 0.953125 0 1.625 -0.03125 1.890625 -0.03125 C 2.328125 -0.03125 2.8125 0 3.21875 0 C 3.3125 0 3.484375 0 3.484375 -0.28125 C 3.484375 -0.421875 3.375 -0.4375 3.28125 -0.4375 C 2.9375 -0.46875 2.65625 -0.578125 2.65625 -0.9375 C 2.65625 -1.15625 2.75 -1.3125 2.9375 -1.640625 L 4.078125 -3.515625 L 7.875 -3.515625 C 7.890625 -3.390625 7.890625 -3.265625 7.90625 -3.140625 C 7.953125 -2.75 8.125 -1.1875 8.125 -0.90625 C 8.125 -0.46875 7.375 -0.4375 7.125 -0.4375 C 6.96875 -0.4375 6.796875 -0.4375 6.796875 -0.171875 C 6.796875 0 6.9375 0 7.03125 0 C 7.28125 0 7.578125 -0.03125 7.828125 -0.03125 L 8.6875 -0.03125 C 9.59375 -0.03125 10.25 0 10.265625 0 C 10.375 0 10.53125 0 10.53125 -0.28125 C 10.53125 -0.4375 10.40625 -0.4375 10.171875 -0.4375 C 9.359375 -0.4375 9.34375 -0.5625 9.296875 -1.015625 L 8.390625 -10.328125 C 8.359375 -10.625 8.296875 -10.65625 8.125 -10.65625 C 7.984375 -10.65625 7.890625 -10.625 7.765625 -10.40625 Z M 4.328125 -3.953125 L 7.328125 -8.96875 L 7.828125 -3.953125 Z M 4.328125 -3.953125 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-3"> <path style="stroke:none;" d="M 11.046875 -8.625 C 11.203125 -9.265625 11.4375 -9.71875 12.578125 -9.765625 C 12.625 -9.765625 12.796875 -9.78125 12.796875 -10.03125 C 12.796875 -10.1875 12.671875 -10.1875 12.609375 -10.1875 C 12.3125 -10.1875 11.546875 -10.15625 11.25 -10.15625 L 10.53125 -10.15625 C 10.328125 -10.15625 10.0625 -10.1875 9.84375 -10.1875 C 9.765625 -10.1875 9.578125 -10.1875 9.578125 -9.90625 C 9.578125 -9.765625 9.703125 -9.765625 9.796875 -9.765625 C 10.703125 -9.734375 10.765625 -9.390625 10.765625 -9.125 C 10.765625 -8.984375 10.75 -8.9375 10.703125 -8.734375 L 9.015625 -2 L 5.8125 -9.9375 C 5.71875 -10.171875 5.703125 -10.1875 5.375 -10.1875 L 3.546875 -10.1875 C 3.25 -10.1875 3.125 -10.1875 3.125 -9.90625 C 3.125 -9.765625 3.21875 -9.765625 3.5 -9.765625 C 3.578125 -9.765625 4.46875 -9.765625 4.46875 -9.625 C 4.46875 -9.59375 4.4375 -9.46875 4.421875 -9.4375 L 2.4375 -1.515625 C 2.25 -0.796875 1.890625 -0.484375 0.90625 -0.4375 C 0.828125 -0.4375 0.6875 -0.421875 0.6875 -0.15625 C 0.6875 0 0.828125 0 0.875 0 C 1.171875 0 1.9375 -0.03125 2.234375 -0.03125 L 2.953125 -0.03125 C 3.15625 -0.03125 3.421875 0 3.625 0 C 3.734375 0 3.890625 0 3.890625 -0.28125 C 3.890625 -0.421875 3.75 -0.4375 3.6875 -0.4375 C 3.1875 -0.453125 2.71875 -0.53125 2.71875 -1.078125 C 2.71875 -1.1875 2.75 -1.328125 2.78125 -1.453125 L 4.796875 -9.4375 C 4.875 -9.28125 4.875 -9.25 4.9375 -9.125 L 8.484375 -0.265625 C 8.5625 -0.09375 8.59375 0 8.734375 0 C 8.875 0 8.890625 -0.046875 8.953125 -0.296875 Z M 11.046875 -8.625 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4"> <path style="stroke:none;" d="M 5.46875 -9.171875 C 5.59375 -9.734375 5.65625 -9.765625 6.234375 -9.765625 L 8.171875 -9.765625 C 9.859375 -9.765625 9.859375 -8.328125 9.859375 -8.1875 C 9.859375 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.53125 -5.453125 Z M 7.984375 -5.328125 C 9.609375 -5.625 11.09375 -6.765625 11.09375 -8.125 C 11.09375 -9.296875 10.0625 -10.1875 8.375 -10.1875 L 3.578125 -10.1875 C 3.296875 -10.1875 3.15625 -10.1875 3.15625 -9.90625 C 3.15625 -9.765625 3.296875 -9.765625 3.515625 -9.765625 C 4.4375 -9.765625 4.4375 -9.640625 4.4375 -9.46875 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.125 L 2.359375 -1.109375 C 2.21875 -0.578125 2.1875 -0.4375 1.15625 -0.4375 C 0.859375 -0.4375 0.71875 -0.4375 0.71875 -0.171875 C 0.71875 0 0.8125 0 1.109375 0 L 6.21875 0 C 8.5 0 10.265625 -1.734375 10.265625 -3.234375 C 10.265625 -4.46875 9.1875 -5.203125 7.984375 -5.328125 Z M 5.859375 -0.4375 L 3.84375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.515625 -0.453125 C 3.359375 -0.46875 3.34375 -0.5 3.34375 -0.609375 C 3.34375 -0.71875 3.375 -0.8125 3.40625 -0.9375 L 4.453125 -5.140625 L 7.25 -5.140625 C 9.015625 -5.140625 9.015625 -3.5 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.859375 -0.4375 Z M 5.859375 -0.4375 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1"> <path style="stroke:none;" d="M 10.78125 -3.734375 C 10.78125 -6.46875 8.53125 -8.703125 5.796875 -8.703125 C 3.03125 -8.703125 0.828125 -6.453125 0.828125 -3.734375 C 0.828125 -0.984375 3.0625 1.234375 5.796875 1.234375 C 8.5625 1.234375 10.78125 -1.015625 10.78125 -3.734375 Z M 2.828125 -6.9375 C 2.8125 -6.96875 2.703125 -7.078125 2.703125 -7.109375 C 2.703125 -7.15625 3.90625 -8.328125 5.796875 -8.328125 C 6.3125 -8.328125 7.6875 -8.25 8.90625 -7.109375 L 5.796875 -3.984375 Z M 2.40625 -0.609375 C 1.5 -1.625 1.1875 -2.78125 1.1875 -3.734375 C 1.1875 -4.875 1.625 -5.96875 2.40625 -6.84375 L 5.515625 -3.734375 Z M 9.171875 -6.84375 C 9.890625 -6.09375 10.40625 -4.953125 10.40625 -3.734375 C 10.40625 -2.578125 9.96875 -1.5 9.1875 -0.609375 L 6.078125 -3.734375 Z M 8.765625 -0.515625 C 8.78125 -0.5 8.890625 -0.390625 8.890625 -0.359375 C 8.890625 -0.296875 7.6875 0.859375 5.796875 0.859375 C 5.28125 0.859375 3.90625 0.796875 2.6875 -0.359375 L 5.796875 -3.484375 Z M 8.765625 -0.515625 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-1"> <path style="stroke:none;" d="M 7.875 -5.71875 C 8 -6.203125 8.21875 -6.4375 8.9375 -6.46875 C 9.03125 -6.46875 9.109375 -6.53125 9.109375 -6.65625 C 9.109375 -6.71875 9.0625 -6.796875 8.96875 -6.796875 C 8.890625 -6.796875 8.703125 -6.765625 7.96875 -6.765625 C 7.171875 -6.765625 7.046875 -6.796875 6.953125 -6.796875 C 6.796875 -6.796875 6.765625 -6.6875 6.765625 -6.609375 C 6.765625 -6.484375 6.890625 -6.46875 6.984375 -6.46875 C 7.59375 -6.453125 7.59375 -6.1875 7.59375 -6.046875 C 7.59375 -5.984375 7.59375 -5.9375 7.546875 -5.78125 L 6.453125 -1.421875 L 4.0625 -6.625 C 3.984375 -6.796875 3.96875 -6.796875 3.71875 -6.796875 L 2.421875 -6.796875 C 2.25 -6.796875 2.125 -6.796875 2.125 -6.609375 C 2.125 -6.46875 2.234375 -6.46875 2.453125 -6.46875 C 2.53125 -6.46875 2.828125 -6.46875 3.0625 -6.40625 L 1.71875 -1.0625 C 1.609375 -0.5625 1.34375 -0.34375 0.671875 -0.328125 C 0.609375 -0.328125 0.5 -0.3125 0.5 -0.140625 C 0.5 -0.078125 0.546875 0 0.640625 0 C 0.6875 0 0.921875 -0.03125 1.625 -0.03125 C 2.421875 -0.03125 2.5625 0 2.65625 0 C 2.703125 0 2.84375 0 2.84375 -0.1875 C 2.84375 -0.3125 2.734375 -0.328125 2.671875 -0.328125 C 2.3125 -0.34375 2.015625 -0.390625 2.015625 -0.75 C 2.015625 -0.796875 2.046875 -0.9375 2.046875 -0.953125 L 3.34375 -6.140625 L 3.359375 -6.140625 L 6.125 -0.171875 C 6.1875 -0.015625 6.203125 0 6.3125 0 C 6.453125 0 6.453125 -0.046875 6.5 -0.203125 Z M 7.875 -5.71875 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-2"> <path style="stroke:none;" d="M 9.671875 -5.984375 C 9.765625 -6.375 9.78125 -6.46875 10.484375 -6.46875 C 10.703125 -6.46875 10.828125 -6.46875 10.828125 -6.65625 C 10.828125 -6.796875 10.703125 -6.796875 10.515625 -6.796875 L 9.234375 -6.796875 C 8.96875 -6.796875 8.9375 -6.796875 8.8125 -6.59375 L 5.0625 -0.921875 L 4.171875 -6.53125 C 4.125 -6.78125 4.125 -6.796875 3.828125 -6.796875 L 2.484375 -6.796875 C 2.296875 -6.796875 2.171875 -6.796875 2.171875 -6.609375 C 2.171875 -6.46875 2.296875 -6.46875 2.453125 -6.46875 C 2.78125 -6.46875 3.078125 -6.46875 3.078125 -6.3125 C 3.078125 -6.265625 3.0625 -6.265625 3.03125 -6.125 L 1.765625 -1.0625 C 1.65625 -0.5625 1.390625 -0.34375 0.734375 -0.328125 C 0.671875 -0.328125 0.546875 -0.3125 0.546875 -0.140625 C 0.546875 -0.078125 0.59375 0 0.703125 0 C 0.734375 0 0.96875 -0.03125 1.6875 -0.03125 C 2.46875 -0.03125 2.625 0 2.703125 0 C 2.75 0 2.890625 0 2.890625 -0.1875 C 2.890625 -0.3125 2.78125 -0.328125 2.71875 -0.328125 C 2.359375 -0.34375 2.0625 -0.390625 2.0625 -0.75 C 2.0625 -0.796875 2.09375 -0.9375 2.09375 -0.953125 L 3.4375 -6.34375 L 3.453125 -6.34375 L 4.421875 -0.265625 C 4.453125 -0.109375 4.453125 0 4.59375 0 C 4.734375 0 4.8125 -0.109375 4.859375 -0.203125 L 8.953125 -6.421875 L 8.96875 -6.421875 L 7.5625 -0.78125 C 7.46875 -0.40625 7.453125 -0.328125 6.71875 -0.328125 C 6.53125 -0.328125 6.40625 -0.328125 6.40625 -0.140625 C 6.40625 -0.09375 6.4375 0 6.5625 0 C 6.703125 0 7 -0.015625 7.140625 -0.03125 L 7.765625 -0.03125 C 8.703125 -0.03125 8.9375 0 9 0 C 9.0625 0 9.203125 0 9.203125 -0.1875 C 9.203125 -0.328125 9.078125 -0.328125 8.90625 -0.328125 C 8.875 -0.328125 8.703125 -0.328125 8.53125 -0.34375 C 8.328125 -0.375 8.296875 -0.390625 8.296875 -0.484375 C 8.296875 -0.53125 8.328125 -0.59375 8.328125 -0.640625 Z M 9.671875 -5.984375 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-3"> <path style="stroke:none;" d="M 0.375 1.609375 C 0.328125 1.765625 0.328125 1.8125 0.328125 1.828125 C 0.328125 2.078125 0.53125 2.140625 0.640625 2.140625 C 0.703125 2.140625 0.921875 2.125 1.0625 1.875 C 1.109375 1.765625 1.296875 0.84375 1.65625 -0.515625 C 1.78125 -0.3125 2.09375 0.09375 2.703125 0.09375 C 3.921875 0.09375 5.265625 -1.234375 5.265625 -2.734375 C 5.265625 -3.859375 4.515625 -4.390625 3.75 -4.390625 C 2.84375 -4.390625 1.65625 -3.5625 1.296875 -2.125 Z M 2.703125 -0.171875 C 2.015625 -0.171875 1.796875 -0.890625 1.796875 -1.03125 C 1.796875 -1.078125 2.046875 -2.09375 2.0625 -2.15625 C 2.515625 -3.921875 3.46875 -4.109375 3.75 -4.109375 C 4.21875 -4.109375 4.484375 -3.6875 4.484375 -3.140625 C 4.484375 -2.78125 4.296875 -1.78125 3.984375 -1.171875 C 3.703125 -0.609375 3.1875 -0.171875 2.703125 -0.171875 Z M 2.703125 -0.171875 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-4"> <path style="stroke:none;" d="M 1.6875 -0.78125 C 1.59375 -0.40625 1.578125 -0.328125 0.84375 -0.328125 C 0.640625 -0.328125 0.53125 -0.328125 0.53125 -0.140625 C 0.53125 0 0.65625 0 0.828125 0 L 4.515625 0 C 6.15625 0 7.390625 -1.171875 7.390625 -2.125 C 7.390625 -2.859375 6.765625 -3.4375 5.765625 -3.546875 C 6.921875 -3.765625 7.90625 -4.53125 7.90625 -5.40625 C 7.90625 -6.15625 7.1875 -6.796875 5.9375 -6.796875 L 2.453125 -6.796875 C 2.28125 -6.796875 2.15625 -6.796875 2.15625 -6.609375 C 2.15625 -6.46875 2.265625 -6.46875 2.4375 -6.46875 C 2.765625 -6.46875 3.0625 -6.46875 3.0625 -6.3125 C 3.0625 -6.265625 3.046875 -6.265625 3.015625 -6.125 Z M 3.234375 -3.671875 L 3.84375 -6.09375 C 3.9375 -6.4375 3.9375 -6.46875 4.34375 -6.46875 L 5.78125 -6.46875 C 6.765625 -6.46875 6.984375 -5.828125 6.984375 -5.421875 C 6.984375 -4.578125 6.078125 -3.671875 4.796875 -3.671875 Z M 2.546875 -0.328125 C 2.453125 -0.34375 2.421875 -0.34375 2.421875 -0.421875 C 2.421875 -0.5 2.453125 -0.578125 2.46875 -0.640625 L 3.171875 -3.390625 L 5.1875 -3.390625 C 6.109375 -3.390625 6.421875 -2.765625 6.421875 -2.203125 C 6.421875 -1.234375 5.46875 -0.328125 4.265625 -0.328125 Z M 2.546875 -0.328125 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-5"> <path style="stroke:none;" d="M 1.84375 -1.1875 C 1.390625 -0.453125 0.96875 -0.359375 0.609375 -0.328125 C 0.515625 -0.3125 0.390625 -0.3125 0.390625 -0.140625 C 0.390625 -0.109375 0.421875 0 0.53125 0 C 0.625 0 0.6875 -0.03125 1.390625 -0.03125 C 2.0625 -0.03125 2.296875 0 2.34375 0 C 2.390625 0 2.53125 0 2.53125 -0.1875 C 2.53125 -0.3125 2.40625 -0.328125 2.359375 -0.328125 C 2.15625 -0.34375 1.9375 -0.421875 1.9375 -0.640625 C 1.9375 -0.78125 2.015625 -0.921875 2.203125 -1.203125 L 2.890625 -2.296875 L 5.609375 -2.296875 L 5.828125 -0.609375 C 5.828125 -0.484375 5.640625 -0.328125 5.171875 -0.328125 C 5.03125 -0.328125 4.890625 -0.328125 4.890625 -0.140625 C 4.890625 -0.125 4.90625 0 5.0625 0 C 5.171875 0 5.546875 -0.015625 5.640625 -0.03125 L 6.265625 -0.03125 C 7.15625 -0.03125 7.328125 0 7.40625 0 C 7.4375 0 7.609375 0 7.609375 -0.1875 C 7.609375 -0.328125 7.46875 -0.328125 7.3125 -0.328125 C 6.765625 -0.328125 6.765625 -0.421875 6.734375 -0.671875 L 5.9375 -6.8125 C 5.90625 -7.03125 5.90625 -7.109375 5.734375 -7.109375 C 5.546875 -7.109375 5.5 -7 5.4375 -6.90625 Z M 3.09375 -2.625 L 5.15625 -5.90625 L 5.578125 -2.625 Z M 3.09375 -2.625 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-6"> <path style="stroke:none;" d="M 3.75 -2.859375 C 4.109375 -1.953125 4.71875 -0.328125 4.828125 -0.15625 C 5 0.09375 5.203125 0.09375 5.4375 0.09375 C 5.71875 0.09375 5.796875 0.09375 5.796875 -0.015625 C 5.796875 -0.0625 5.78125 -0.09375 5.75 -0.125 C 5.625 -0.265625 5.578125 -0.375 5.5 -0.5625 L 3.265625 -6.234375 C 3.1875 -6.421875 3 -6.90625 1.984375 -6.90625 C 1.875 -6.90625 1.734375 -6.90625 1.734375 -6.765625 C 1.734375 -6.65625 1.828125 -6.640625 1.875 -6.625 C 2.078125 -6.59375 2.265625 -6.578125 2.5 -6 L 3.21875 -4.203125 L 3.609375 -3.203125 L 0.75 -0.5625 C 0.640625 -0.46875 0.546875 -0.34375 0.546875 -0.203125 C 0.546875 -0.015625 0.71875 0.125 0.890625 0.125 C 1.03125 0.125 1.171875 0.015625 1.25 -0.09375 Z M 3.75 -2.859375 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-7"> <path style="stroke:none;" d="M 2.828125 -3.640625 L 3.96875 -3.640625 C 3.765625 -2.734375 3.578125 -1.984375 3.578125 -1.25 C 3.578125 -1.1875 3.578125 -0.75 3.6875 -0.390625 C 3.828125 0.015625 3.921875 0.09375 4.09375 0.09375 C 4.3125 0.09375 4.546875 -0.09375 4.546875 -0.328125 C 4.546875 -0.390625 4.53125 -0.421875 4.5 -0.5 C 4.296875 -0.96875 4.140625 -1.4375 4.140625 -2.265625 C 4.140625 -2.484375 4.140625 -2.90625 4.296875 -3.640625 L 5.5 -3.640625 C 5.65625 -3.640625 5.765625 -3.640625 5.859375 -3.71875 C 5.984375 -3.828125 6.015625 -3.953125 6.015625 -4.015625 C 6.015625 -4.296875 5.765625 -4.296875 5.59375 -4.296875 L 2 -4.296875 C 1.796875 -4.296875 1.40625 -4.296875 0.921875 -3.8125 C 0.5625 -3.453125 0.28125 -3 0.28125 -2.921875 C 0.28125 -2.84375 0.359375 -2.8125 0.4375 -2.8125 C 0.53125 -2.8125 0.5625 -2.84375 0.609375 -2.921875 C 1.109375 -3.640625 1.6875 -3.640625 1.921875 -3.640625 L 2.5 -3.640625 C 2.203125 -2.578125 1.6875 -1.375 1.3125 -0.640625 C 1.25 -0.5 1.140625 -0.296875 1.140625 -0.203125 C 1.140625 0 1.3125 0.09375 1.46875 0.09375 C 1.84375 0.09375 1.953125 -0.28125 2.15625 -1.09375 Z M 2.828125 -3.640625 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1"> <path style="stroke:none;" d="M 7.625 -2.484375 C 7.625 -4.453125 6.03125 -6.015625 4.109375 -6.015625 C 2.15625 -6.015625 0.59375 -4.421875 0.59375 -2.484375 C 0.59375 -0.53125 2.1875 1.03125 4.09375 1.03125 C 6.0625 1.03125 7.625 -0.5625 7.625 -2.484375 Z M 1.953125 -4.859375 C 2.734375 -5.578125 3.625 -5.703125 4.09375 -5.703125 C 4.703125 -5.703125 5.546875 -5.53125 6.265625 -4.859375 L 4.109375 -2.703125 Z M 1.734375 -0.328125 C 1.171875 -0.921875 0.890625 -1.71875 0.890625 -2.484375 C 0.890625 -3.25 1.171875 -4.046875 1.734375 -4.640625 L 3.890625 -2.484375 Z M 6.484375 -4.640625 C 7.03125 -4.0625 7.3125 -3.25 7.3125 -2.484375 C 7.3125 -1.734375 7.046875 -0.921875 6.484375 -0.328125 L 4.3125 -2.484375 Z M 6.265625 -0.109375 C 5.484375 0.609375 4.59375 0.734375 4.109375 0.734375 C 3.5 0.734375 2.65625 0.546875 1.953125 -0.109375 L 4.09375 -2.265625 Z M 6.265625 -0.109375 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1"> <path style="stroke:none;" d="M 1.6875 -0.90625 C 1.40625 -0.484375 1.15625 -0.328125 0.734375 -0.296875 C 0.625 -0.296875 0.53125 -0.28125 0.53125 -0.109375 C 0.53125 -0.03125 0.59375 0 0.65625 0 C 0.859375 0 1.109375 -0.03125 1.328125 -0.03125 C 1.5 -0.03125 1.9375 0 2.125 0 C 2.203125 0 2.28125 -0.03125 2.28125 -0.1875 C 2.28125 -0.296875 2.1875 -0.296875 2.15625 -0.296875 C 1.984375 -0.3125 1.8125 -0.34375 1.8125 -0.515625 C 1.8125 -0.609375 1.890625 -0.734375 1.921875 -0.78125 L 2.53125 -1.703125 L 4.78125 -1.703125 L 4.96875 -0.5 C 4.96875 -0.296875 4.546875 -0.296875 4.46875 -0.296875 C 4.375 -0.296875 4.25 -0.296875 4.25 -0.109375 C 4.25 -0.078125 4.28125 0 4.375 0 C 4.59375 0 5.140625 -0.03125 5.375 -0.03125 C 5.515625 -0.03125 5.65625 -0.015625 5.8125 -0.015625 C 5.953125 -0.015625 6.125 0 6.265625 0 C 6.3125 0 6.4375 0 6.4375 -0.1875 C 6.4375 -0.296875 6.328125 -0.296875 6.21875 -0.296875 C 5.765625 -0.296875 5.75 -0.34375 5.71875 -0.53125 L 4.96875 -5.125 C 4.953125 -5.265625 4.9375 -5.328125 4.78125 -5.328125 C 4.625 -5.328125 4.578125 -5.265625 4.515625 -5.171875 Z M 2.71875 -2 L 4.34375 -4.421875 L 4.734375 -2 Z M 2.71875 -2 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-2"> <path style="stroke:none;" d="M 1.5625 -0.625 C 1.5 -0.34375 1.484375 -0.296875 0.921875 -0.296875 C 0.765625 -0.296875 0.75 -0.296875 0.71875 -0.28125 C 0.671875 -0.25 0.65625 -0.15625 0.65625 -0.109375 C 0.671875 0 0.734375 0 0.890625 0 L 3.921875 0 C 5.21875 0 6.234375 -0.84375 6.234375 -1.59375 C 6.234375 -2.046875 5.828125 -2.546875 4.90625 -2.671875 C 5.765625 -2.8125 6.609375 -3.359375 6.609375 -4.03125 C 6.609375 -4.578125 6.015625 -5.09375 4.96875 -5.09375 L 2.109375 -5.09375 C 1.96875 -5.09375 1.859375 -5.09375 1.859375 -4.921875 C 1.859375 -4.796875 1.96875 -4.796875 2.09375 -4.796875 C 2.234375 -4.796875 2.421875 -4.796875 2.578125 -4.75 C 2.578125 -4.671875 2.578125 -4.625 2.546875 -4.515625 Z M 2.796875 -2.78125 L 3.234375 -4.546875 C 3.296875 -4.78125 3.296875 -4.796875 3.609375 -4.796875 L 4.84375 -4.796875 C 5.625 -4.796875 5.84375 -4.328125 5.84375 -4.015625 C 5.84375 -3.40625 5.078125 -2.78125 4.15625 -2.78125 Z M 2.4375 -0.296875 C 2.296875 -0.296875 2.28125 -0.296875 2.1875 -0.328125 L 2.71875 -2.53125 L 4.359375 -2.53125 C 5.296875 -2.53125 5.453125 -1.921875 5.453125 -1.625 C 5.453125 -0.90625 4.65625 -0.296875 3.734375 -0.296875 Z M 2.4375 -0.296875 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-3"> <path style="stroke:none;" d="M 1.71875 -0.109375 C 1.71875 0.609375 1.34375 1.015625 1.15625 1.1875 C 1.078125 1.25 1.0625 1.265625 1.0625 1.3125 C 1.0625 1.375 1.125 1.453125 1.1875 1.453125 C 1.296875 1.453125 1.96875 0.84375 1.96875 -0.0625 C 1.96875 -0.53125 1.765625 -0.90625 1.40625 -0.90625 C 1.125 -0.90625 0.96875 -0.671875 0.96875 -0.453125 C 0.96875 -0.234375 1.125 0 1.421875 0 C 1.53125 0 1.640625 -0.03125 1.71875 -0.109375 Z M 1.71875 -0.109375 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-4"> <path style="stroke:none;" d="M 6.578125 -4.28125 C 6.65625 -4.59375 6.796875 -4.78125 7.375 -4.796875 C 7.421875 -4.796875 7.53125 -4.8125 7.53125 -4.984375 C 7.53125 -5.0625 7.484375 -5.09375 7.421875 -5.09375 C 7.234375 -5.09375 6.796875 -5.078125 6.625 -5.078125 L 6.203125 -5.078125 C 6.078125 -5.078125 5.921875 -5.09375 5.78125 -5.09375 C 5.65625 -5.09375 5.625 -5 5.625 -4.921875 C 5.625 -4.796875 5.75 -4.796875 5.796875 -4.796875 C 6.15625 -4.796875 6.296875 -4.703125 6.296875 -4.5 C 6.296875 -4.46875 6.28125 -4.421875 6.28125 -4.390625 L 5.453125 -1.109375 L 3.421875 -4.96875 C 3.34375 -5.09375 3.34375 -5.09375 3.15625 -5.09375 L 2.078125 -5.09375 C 1.9375 -5.09375 1.828125 -5.09375 1.828125 -4.921875 C 1.828125 -4.796875 1.9375 -4.796875 2.109375 -4.796875 C 2.15625 -4.796875 2.40625 -4.796875 2.578125 -4.765625 L 1.59375 -0.828125 C 1.53125 -0.546875 1.40625 -0.328125 0.796875 -0.296875 C 0.6875 -0.296875 0.625 -0.25 0.625 -0.109375 C 0.640625 -0.078125 0.640625 0 0.75 0 C 0.9375 0 1.375 -0.03125 1.546875 -0.03125 L 1.96875 -0.015625 C 2.09375 -0.015625 2.25 0 2.375 0 C 2.546875 0 2.546875 -0.15625 2.546875 -0.1875 C 2.53125 -0.296875 2.453125 -0.296875 2.359375 -0.296875 C 2.015625 -0.3125 1.859375 -0.40625 1.859375 -0.59375 C 1.859375 -0.640625 1.875 -0.671875 1.890625 -0.75 L 2.859375 -4.578125 L 5.203125 -0.125 C 5.25 -0.015625 5.265625 0 5.375 0 C 5.5 0 5.515625 -0.03125 5.546875 -0.171875 Z M 6.578125 -4.28125 "></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph5-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3zeE95oATEsba_dbTGyG4gmPBvo=-glyph5-1"> <path style="stroke:none;" d="M 6.484375 -1.859375 C 6.484375 -3.46875 5.1875 -4.765625 3.59375 -4.765625 C 2 -4.765625 0.6875 -3.46875 0.6875 -1.859375 C 0.6875 -0.265625 2 1.03125 3.59375 1.03125 C 5.1875 1.03125 6.484375 -0.265625 6.484375 -1.859375 Z M 1.84375 -3.828125 C 2.234375 -4.1875 2.859375 -4.5 3.59375 -4.5 C 4.3125 -4.5 4.921875 -4.203125 5.34375 -3.828125 L 3.59375 -2.0625 Z M 1.640625 -0.109375 C 1.234375 -0.53125 0.96875 -1.1875 0.96875 -1.859375 C 0.96875 -2.546875 1.234375 -3.1875 1.640625 -3.625 L 3.390625 -1.859375 Z M 5.546875 -3.625 C 5.9375 -3.203125 6.21875 -2.546875 6.21875 -1.859375 C 6.21875 -1.1875 5.953125 -0.546875 5.546875 -0.109375 L 3.78125 -1.859375 Z M 5.34375 0.09375 C 4.9375 0.453125 4.328125 0.765625 3.59375 0.765625 C 2.875 0.765625 2.25 0.46875 1.84375 0.09375 L 3.59375 -1.671875 Z M 5.34375 0.09375 "></path> </symbol> </g> </defs> <g id="3zeE95oATEsba_dbTGyG4gmPBvo=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-1" x="6.424876" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="25.435379" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2" x="40.359186" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="54.628922" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-3" x="69.55148" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="86.126692" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4" x="101.049251" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-1" x="237.971059" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="256.981562" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2" x="271.905369" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="286.175105" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4" x="301.097663" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-1" x="438.018972" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="457.029476" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2" x="471.952034" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="486.22177" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-1" x="497.829037" y="26.181113"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4" x="511.216275" y="23.943041"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2" x="23.391781" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="37.661517" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-3" x="52.584076" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="69.159287" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4" x="84.083094" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2" x="254.937964" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="269.2077" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4" x="284.130259" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-2" x="454.985878" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph1-1" x="469.255613" y="89.38311"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-1" x="480.861632" y="91.621182"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph0-4" x="494.248871" y="89.38311"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -145.378975 35.401398 L -33.974901 35.401398 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487155 2.868326 C -2.033386 1.147135 -1.020231 0.333481 0.000748418 0.000978462 C -1.020231 -0.335436 -2.033386 -1.14909 -2.487155 -2.870281 " transform="matrix(0.998582,0,0,-0.998582,231.057846,17.333008)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-2" x="150.913689" y="11.888161"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="162.125267" y="11.888161"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-3" x="170.346093" y="11.888161"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="175.759655" y="8.376897"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="183.27648" y="11.888161"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-4" x="191.497306" y="11.888161"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -145.378975 29.635409 L -33.974901 29.635409 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487155 2.867427 C -2.033386 1.146236 -1.020231 0.336494 0.000748418 0.0000793333 C -1.020231 -0.336335 -2.033386 -1.146077 -2.487155 -2.867268 " transform="matrix(0.998582,0,0,-0.998582,231.057846,23.089923)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-2" x="150.622103" y="35.21104"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="161.833681" y="35.21104"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-5" x="170.054507" y="35.21104"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="177.972013" y="35.21104"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-6" x="186.194087" y="35.21104"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-2" x="192.36033" y="31.699776"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -205.374209 21.819638 L -205.374209 -20.345624 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486687 2.868966 C -2.032875 1.14775 -1.019683 0.3341 0.0013236 0.00161107 C -1.019669 -0.334831 -2.032829 -1.148521 -2.486572 -2.869755 " transform="matrix(-0.0000199716,0.998562,0.998562,0.0000199716,59.662454,73.23696)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-6" x="8.116474" y="55.972557"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="14.282717" y="52.461293"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="21.799543" y="55.972557"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-1" x="30.021617" y="55.972557"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="39.470698" y="55.972557"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-4" x="47.691524" y="55.972557"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.955897 32.518404 L 166.359971 32.518404 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486743 2.867881 C -2.032974 1.14669 -1.019819 0.333037 0.00116036 0.000533898 C -1.019819 -0.335881 -2.032974 -1.149534 -2.486743 -2.870725 " transform="matrix(0.998582,0,0,-0.998582,431.108216,20.211471)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-2" x="353.923396" y="13.57876"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="365.134974" y="13.57876"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-7" x="373.357048" y="13.57876"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="379.42593" y="15.250136"></use> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-3" x="386.321013" y="15.250136"></use> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-2" x="389.153362" y="15.250136"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 10.729118 21.819638 L 10.729118 -20.345624 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486687 2.870754 C -2.032875 1.149538 -1.019683 0.335889 0.00132364 -0.000512362 C -1.019669 -0.333042 -2.032829 -1.146733 -2.486572 -2.867967 " transform="matrix(-0.0000199716,0.998562,0.998562,0.0000199716,275.461449,73.23696)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-6" x="241.583928" y="55.972557"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="247.750172" y="52.461293"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="255.266997" y="55.972557"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-4" x="263.487823" y="55.972557"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:3.34735,1.91277;stroke-miterlimit:10;" d="M 216.110208 20.82213 L 216.106296 -20.345624 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486687 2.870463 C -2.032875 1.149247 -1.019683 0.335597 0.00132365 -0.000803784 C -1.019669 -0.333334 -2.032829 -1.147024 -2.486572 -2.868258 " transform="matrix(-0.0000199716,0.998562,0.998562,0.0000199716,480.547678,73.23696)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-6" x="484.056595" y="57.032052"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="490.222838" y="53.433412"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph5-1" x="497.118046" y="53.433412"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-4" x="504.302843" y="55.961073"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-2" x="512.94557" y="53.433412"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -162.36791 -30.132941 L -16.985966 -30.132941 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485407 2.867877 C -2.031639 1.146686 -1.018483 0.333032 -0.00141618 0.000529336 C -1.018483 -0.335885 -2.031639 -1.149539 -2.485407 -2.87073 " transform="matrix(0.998582,0,0,-0.998582,248.024852,82.773966)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-3" x="160.629891" y="77.32823"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="166.043453" y="73.816966"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="173.55903" y="77.32823"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-4" x="181.781104" y="77.32823"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -162.36791 -35.89893 L -16.985966 -35.89893 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485407 2.870889 C -2.031639 1.145787 -1.018483 0.336045 -0.00141618 -0.000369793 C -1.018483 -0.332873 -2.031639 -1.146526 -2.485407 -2.867717 " transform="matrix(0.998582,0,0,-0.998582,248.024852,88.530881)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-5" x="160.338305" y="100.651109"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph3-1" x="168.255811" y="100.651109"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-6" x="176.477885" y="100.651109"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-2" x="182.64288" y="97.139845"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 37.96305 -33.015935 L 183.348906 -33.015935 " transform="matrix(0.998582,0,0,-0.998582,264.747035,52.683228)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.484985 2.867412 C -2.031217 1.146221 -1.018062 0.336479 -0.000994233 0.0000647712 C -1.018062 -0.33635 -2.031217 -1.146092 -2.484985 -2.867283 " transform="matrix(0.998582,0,0,-0.998582,448.075212,85.652408)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph2-7" x="363.639598" y="93.44634"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-1" x="369.709728" y="95.118965"></use> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-3" x="376.604811" y="95.118965"></use> <use xlink:href="#3zeE95oATEsba_dbTGyG4gmPBvo=-glyph4-2" x="379.43716" y="95.118965"></use> </g> </g> </svg> and right action defined by the following diagram:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="517.487pt" height="104.983pt" viewBox="0 0 517.487 104.983" version="1.2"> <defs> <g> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1"> <path style="stroke:none;" d="M 2.53125 -1.65625 C 2.015625 -0.78125 1.5 -0.484375 0.796875 -0.4375 C 0.625 -0.421875 0.5 -0.421875 0.5 -0.15625 C 0.5 -0.0625 0.578125 0 0.6875 0 C 0.953125 0 1.625 -0.03125 1.890625 -0.03125 C 2.328125 -0.03125 2.8125 0 3.21875 0 C 3.3125 0 3.484375 0 3.484375 -0.28125 C 3.484375 -0.421875 3.375 -0.4375 3.28125 -0.4375 C 2.9375 -0.46875 2.65625 -0.578125 2.65625 -0.9375 C 2.65625 -1.15625 2.75 -1.3125 2.9375 -1.640625 L 4.078125 -3.515625 L 7.875 -3.515625 C 7.890625 -3.390625 7.890625 -3.265625 7.90625 -3.140625 C 7.953125 -2.75 8.125 -1.1875 8.125 -0.90625 C 8.125 -0.46875 7.375 -0.4375 7.125 -0.4375 C 6.96875 -0.4375 6.796875 -0.4375 6.796875 -0.171875 C 6.796875 0 6.9375 0 7.03125 0 C 7.28125 0 7.578125 -0.03125 7.828125 -0.03125 L 8.6875 -0.03125 C 9.59375 -0.03125 10.25 0 10.265625 0 C 10.375 0 10.53125 0 10.53125 -0.28125 C 10.53125 -0.4375 10.40625 -0.4375 10.171875 -0.4375 C 9.359375 -0.4375 9.34375 -0.5625 9.296875 -1.015625 L 8.390625 -10.328125 C 8.359375 -10.625 8.296875 -10.65625 8.125 -10.65625 C 7.984375 -10.65625 7.890625 -10.625 7.765625 -10.40625 Z M 4.328125 -3.953125 L 7.328125 -8.96875 L 7.828125 -3.953125 Z M 4.328125 -3.953125 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-2"> <path style="stroke:none;" d="M 11.046875 -8.625 C 11.203125 -9.265625 11.4375 -9.71875 12.578125 -9.765625 C 12.625 -9.765625 12.796875 -9.78125 12.796875 -10.03125 C 12.796875 -10.1875 12.671875 -10.1875 12.609375 -10.1875 C 12.3125 -10.1875 11.546875 -10.15625 11.25 -10.15625 L 10.53125 -10.15625 C 10.328125 -10.15625 10.0625 -10.1875 9.84375 -10.1875 C 9.765625 -10.1875 9.578125 -10.1875 9.578125 -9.90625 C 9.578125 -9.765625 9.703125 -9.765625 9.796875 -9.765625 C 10.703125 -9.734375 10.765625 -9.390625 10.765625 -9.125 C 10.765625 -8.984375 10.75 -8.9375 10.703125 -8.734375 L 9.015625 -2 L 5.8125 -9.9375 C 5.71875 -10.171875 5.703125 -10.1875 5.375 -10.1875 L 3.546875 -10.1875 C 3.25 -10.1875 3.125 -10.1875 3.125 -9.90625 C 3.125 -9.765625 3.21875 -9.765625 3.5 -9.765625 C 3.578125 -9.765625 4.46875 -9.765625 4.46875 -9.625 C 4.46875 -9.59375 4.4375 -9.46875 4.421875 -9.4375 L 2.4375 -1.515625 C 2.25 -0.796875 1.890625 -0.484375 0.90625 -0.4375 C 0.828125 -0.4375 0.6875 -0.421875 0.6875 -0.15625 C 0.6875 0 0.828125 0 0.875 0 C 1.171875 0 1.9375 -0.03125 2.234375 -0.03125 L 2.953125 -0.03125 C 3.15625 -0.03125 3.421875 0 3.625 0 C 3.734375 0 3.890625 0 3.890625 -0.28125 C 3.890625 -0.421875 3.75 -0.4375 3.6875 -0.4375 C 3.1875 -0.453125 2.71875 -0.53125 2.71875 -1.078125 C 2.71875 -1.1875 2.75 -1.328125 2.78125 -1.453125 L 4.796875 -9.4375 C 4.875 -9.28125 4.875 -9.25 4.9375 -9.125 L 8.484375 -0.265625 C 8.5625 -0.09375 8.59375 0 8.734375 0 C 8.875 0 8.890625 -0.046875 8.953125 -0.296875 Z M 11.046875 -8.625 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3"> <path style="stroke:none;" d="M 5.46875 -9.171875 C 5.59375 -9.734375 5.65625 -9.765625 6.234375 -9.765625 L 8.171875 -9.765625 C 9.859375 -9.765625 9.859375 -8.328125 9.859375 -8.1875 C 9.859375 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.53125 -5.453125 Z M 7.984375 -5.328125 C 9.609375 -5.625 11.09375 -6.765625 11.09375 -8.125 C 11.09375 -9.296875 10.0625 -10.1875 8.375 -10.1875 L 3.578125 -10.1875 C 3.296875 -10.1875 3.15625 -10.1875 3.15625 -9.90625 C 3.15625 -9.765625 3.296875 -9.765625 3.515625 -9.765625 C 4.4375 -9.765625 4.4375 -9.640625 4.4375 -9.46875 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.125 L 2.359375 -1.109375 C 2.21875 -0.578125 2.1875 -0.4375 1.15625 -0.4375 C 0.859375 -0.4375 0.71875 -0.4375 0.71875 -0.171875 C 0.71875 0 0.8125 0 1.109375 0 L 6.21875 0 C 8.5 0 10.265625 -1.734375 10.265625 -3.234375 C 10.265625 -4.46875 9.1875 -5.203125 7.984375 -5.328125 Z M 5.859375 -0.4375 L 3.84375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.515625 -0.453125 C 3.359375 -0.46875 3.34375 -0.5 3.34375 -0.609375 C 3.34375 -0.71875 3.375 -0.8125 3.40625 -0.9375 L 4.453125 -5.140625 L 7.25 -5.140625 C 9.015625 -5.140625 9.015625 -3.5 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.859375 -0.4375 Z M 5.859375 -0.4375 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-4"> <path style="stroke:none;" d="M 4.421875 -4.75 L 6.921875 -4.75 C 8.984375 -4.75 11.046875 -6.265625 11.046875 -7.96875 C 11.046875 -9.125 10.0625 -10.1875 8.171875 -10.1875 L 3.5625 -10.1875 C 3.28125 -10.1875 3.15625 -10.1875 3.15625 -9.90625 C 3.15625 -9.765625 3.28125 -9.765625 3.5 -9.765625 C 4.421875 -9.765625 4.421875 -9.640625 4.421875 -9.46875 C 4.421875 -9.453125 4.421875 -9.359375 4.359375 -9.125 L 2.34375 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.140625 -0.4375 C 0.84375 -0.4375 0.703125 -0.4375 0.703125 -0.171875 C 0.703125 0 0.828125 0 0.921875 0 C 1.203125 0 1.5 -0.03125 1.796875 -0.03125 L 3.53125 -0.03125 C 3.8125 -0.03125 4.140625 0 4.421875 0 C 4.53125 0 4.703125 0 4.703125 -0.28125 C 4.703125 -0.4375 4.5625 -0.4375 4.34375 -0.4375 C 3.453125 -0.4375 3.4375 -0.53125 3.4375 -0.6875 C 3.4375 -0.765625 3.453125 -0.859375 3.46875 -0.9375 Z M 5.484375 -9.171875 C 5.625 -9.734375 5.6875 -9.765625 6.265625 -9.765625 L 7.75 -9.765625 C 8.859375 -9.765625 9.78125 -9.40625 9.78125 -8.28125 C 9.78125 -7.890625 9.59375 -6.625 8.90625 -5.9375 C 8.65625 -5.671875 7.9375 -5.109375 6.578125 -5.109375 L 4.484375 -5.109375 Z M 5.484375 -9.171875 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1"> <path style="stroke:none;" d="M 10.78125 -3.734375 C 10.78125 -6.46875 8.53125 -8.703125 5.796875 -8.703125 C 3.03125 -8.703125 0.828125 -6.453125 0.828125 -3.734375 C 0.828125 -0.984375 3.0625 1.234375 5.796875 1.234375 C 8.5625 1.234375 10.78125 -1.015625 10.78125 -3.734375 Z M 2.828125 -6.9375 C 2.8125 -6.96875 2.703125 -7.078125 2.703125 -7.109375 C 2.703125 -7.15625 3.90625 -8.328125 5.796875 -8.328125 C 6.3125 -8.328125 7.6875 -8.25 8.90625 -7.109375 L 5.796875 -3.984375 Z M 2.40625 -0.609375 C 1.5 -1.625 1.1875 -2.78125 1.1875 -3.734375 C 1.1875 -4.875 1.625 -5.96875 2.40625 -6.84375 L 5.515625 -3.734375 Z M 9.171875 -6.84375 C 9.890625 -6.09375 10.40625 -4.953125 10.40625 -3.734375 C 10.40625 -2.578125 9.96875 -1.5 9.1875 -0.609375 L 6.078125 -3.734375 Z M 8.765625 -0.515625 C 8.78125 -0.5 8.890625 -0.390625 8.890625 -0.359375 C 8.890625 -0.296875 7.6875 0.859375 5.796875 0.859375 C 5.28125 0.859375 3.90625 0.796875 2.6875 -0.359375 L 5.796875 -3.484375 Z M 8.765625 -0.515625 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-1"> <path style="stroke:none;" d="M 7.875 -5.71875 C 8 -6.203125 8.21875 -6.4375 8.9375 -6.46875 C 9.03125 -6.46875 9.109375 -6.53125 9.109375 -6.65625 C 9.109375 -6.71875 9.0625 -6.796875 8.96875 -6.796875 C 8.890625 -6.796875 8.703125 -6.765625 7.96875 -6.765625 C 7.171875 -6.765625 7.046875 -6.796875 6.953125 -6.796875 C 6.796875 -6.796875 6.765625 -6.6875 6.765625 -6.609375 C 6.765625 -6.484375 6.890625 -6.46875 6.984375 -6.46875 C 7.59375 -6.453125 7.59375 -6.1875 7.59375 -6.046875 C 7.59375 -5.984375 7.59375 -5.9375 7.546875 -5.78125 L 6.453125 -1.421875 L 4.0625 -6.625 C 3.984375 -6.796875 3.96875 -6.796875 3.71875 -6.796875 L 2.421875 -6.796875 C 2.25 -6.796875 2.125 -6.796875 2.125 -6.609375 C 2.125 -6.46875 2.234375 -6.46875 2.453125 -6.46875 C 2.53125 -6.46875 2.828125 -6.46875 3.0625 -6.40625 L 1.71875 -1.0625 C 1.609375 -0.5625 1.34375 -0.34375 0.671875 -0.328125 C 0.609375 -0.328125 0.5 -0.3125 0.5 -0.140625 C 0.5 -0.078125 0.546875 0 0.640625 0 C 0.6875 0 0.921875 -0.03125 1.625 -0.03125 C 2.421875 -0.03125 2.5625 0 2.65625 0 C 2.703125 0 2.84375 0 2.84375 -0.1875 C 2.84375 -0.3125 2.734375 -0.328125 2.671875 -0.328125 C 2.3125 -0.34375 2.015625 -0.390625 2.015625 -0.75 C 2.015625 -0.796875 2.046875 -0.9375 2.046875 -0.953125 L 3.34375 -6.140625 L 3.359375 -6.140625 L 6.125 -0.171875 C 6.1875 -0.015625 6.203125 0 6.3125 0 C 6.453125 0 6.453125 -0.046875 6.5 -0.203125 Z M 7.875 -5.71875 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-2"> <path style="stroke:none;" d="M 0.375 1.609375 C 0.328125 1.765625 0.328125 1.8125 0.328125 1.828125 C 0.328125 2.078125 0.53125 2.140625 0.640625 2.140625 C 0.703125 2.140625 0.921875 2.125 1.0625 1.875 C 1.109375 1.765625 1.296875 0.84375 1.65625 -0.515625 C 1.78125 -0.3125 2.09375 0.09375 2.703125 0.09375 C 3.921875 0.09375 5.265625 -1.234375 5.265625 -2.734375 C 5.265625 -3.859375 4.515625 -4.390625 3.75 -4.390625 C 2.84375 -4.390625 1.65625 -3.5625 1.296875 -2.125 Z M 2.703125 -0.171875 C 2.015625 -0.171875 1.796875 -0.890625 1.796875 -1.03125 C 1.796875 -1.078125 2.046875 -2.09375 2.0625 -2.15625 C 2.515625 -3.921875 3.46875 -4.109375 3.75 -4.109375 C 4.21875 -4.109375 4.484375 -3.6875 4.484375 -3.140625 C 4.484375 -2.78125 4.296875 -1.78125 3.984375 -1.171875 C 3.703125 -0.609375 3.1875 -0.171875 2.703125 -0.171875 Z M 2.703125 -0.171875 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-3"> <path style="stroke:none;" d="M 1.6875 -0.78125 C 1.59375 -0.40625 1.578125 -0.328125 0.84375 -0.328125 C 0.640625 -0.328125 0.53125 -0.328125 0.53125 -0.140625 C 0.53125 0 0.65625 0 0.828125 0 L 4.515625 0 C 6.15625 0 7.390625 -1.171875 7.390625 -2.125 C 7.390625 -2.859375 6.765625 -3.4375 5.765625 -3.546875 C 6.921875 -3.765625 7.90625 -4.53125 7.90625 -5.40625 C 7.90625 -6.15625 7.1875 -6.796875 5.9375 -6.796875 L 2.453125 -6.796875 C 2.28125 -6.796875 2.15625 -6.796875 2.15625 -6.609375 C 2.15625 -6.46875 2.265625 -6.46875 2.4375 -6.46875 C 2.765625 -6.46875 3.0625 -6.46875 3.0625 -6.3125 C 3.0625 -6.265625 3.046875 -6.265625 3.015625 -6.125 Z M 3.234375 -3.671875 L 3.84375 -6.09375 C 3.9375 -6.4375 3.9375 -6.46875 4.34375 -6.46875 L 5.78125 -6.46875 C 6.765625 -6.46875 6.984375 -5.828125 6.984375 -5.421875 C 6.984375 -4.578125 6.078125 -3.671875 4.796875 -3.671875 Z M 2.546875 -0.328125 C 2.453125 -0.34375 2.421875 -0.34375 2.421875 -0.421875 C 2.421875 -0.5 2.453125 -0.578125 2.46875 -0.640625 L 3.171875 -3.390625 L 5.1875 -3.390625 C 6.109375 -3.390625 6.421875 -2.765625 6.421875 -2.203125 C 6.421875 -1.234375 5.46875 -0.328125 4.265625 -0.328125 Z M 2.546875 -0.328125 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-4"> <path style="stroke:none;" d="M 3.140625 -3.109375 L 4.953125 -3.109375 C 6.375 -3.109375 7.875 -4.09375 7.875 -5.28125 C 7.875 -6.125 7.078125 -6.796875 5.796875 -6.796875 L 2.4375 -6.796875 C 2.265625 -6.796875 2.125 -6.796875 2.125 -6.609375 C 2.125 -6.46875 2.265625 -6.46875 2.421875 -6.46875 C 2.75 -6.46875 3.03125 -6.46875 3.03125 -6.3125 C 3.03125 -6.265625 3.03125 -6.265625 3 -6.125 L 1.65625 -0.78125 C 1.578125 -0.421875 1.546875 -0.328125 0.84375 -0.328125 C 0.609375 -0.328125 0.5 -0.328125 0.5 -0.140625 C 0.5 -0.09375 0.53125 0 0.671875 0 C 0.859375 0 1.09375 -0.015625 1.28125 -0.03125 L 1.921875 -0.03125 C 2.875 -0.03125 3.140625 0 3.203125 0 C 3.265625 0 3.40625 0 3.40625 -0.1875 C 3.40625 -0.328125 3.28125 -0.328125 3.109375 -0.328125 C 3.078125 -0.328125 2.890625 -0.328125 2.734375 -0.34375 C 2.515625 -0.375 2.5 -0.390625 2.5 -0.484375 C 2.5 -0.53125 2.515625 -0.59375 2.53125 -0.640625 Z M 3.859375 -6.09375 C 3.953125 -6.4375 3.96875 -6.46875 4.375 -6.46875 L 5.46875 -6.46875 C 6.28125 -6.46875 6.90625 -6.234375 6.90625 -5.5 C 6.90625 -5.359375 6.859375 -4.484375 6.296875 -3.9375 C 6.15625 -3.78125 5.671875 -3.40625 4.703125 -3.40625 L 3.1875 -3.40625 Z M 3.859375 -6.09375 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-5"> <path style="stroke:none;" d="M 1.84375 -1.1875 C 1.390625 -0.453125 0.96875 -0.359375 0.609375 -0.328125 C 0.515625 -0.3125 0.390625 -0.3125 0.390625 -0.140625 C 0.390625 -0.109375 0.421875 0 0.53125 0 C 0.625 0 0.6875 -0.03125 1.390625 -0.03125 C 2.0625 -0.03125 2.296875 0 2.34375 0 C 2.390625 0 2.53125 0 2.53125 -0.1875 C 2.53125 -0.3125 2.40625 -0.328125 2.359375 -0.328125 C 2.15625 -0.34375 1.9375 -0.421875 1.9375 -0.640625 C 1.9375 -0.78125 2.015625 -0.921875 2.203125 -1.203125 L 2.890625 -2.296875 L 5.609375 -2.296875 L 5.828125 -0.609375 C 5.828125 -0.484375 5.640625 -0.328125 5.171875 -0.328125 C 5.03125 -0.328125 4.890625 -0.328125 4.890625 -0.140625 C 4.890625 -0.125 4.90625 0 5.0625 0 C 5.171875 0 5.546875 -0.015625 5.640625 -0.03125 L 6.265625 -0.03125 C 7.15625 -0.03125 7.328125 0 7.40625 0 C 7.4375 0 7.609375 0 7.609375 -0.1875 C 7.609375 -0.328125 7.46875 -0.328125 7.3125 -0.328125 C 6.765625 -0.328125 6.765625 -0.421875 6.734375 -0.671875 L 5.9375 -6.8125 C 5.90625 -7.03125 5.90625 -7.109375 5.734375 -7.109375 C 5.546875 -7.109375 5.5 -7 5.4375 -6.90625 Z M 3.09375 -2.625 L 5.15625 -5.90625 L 5.578125 -2.625 Z M 3.09375 -2.625 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-6"> <path style="stroke:none;" d="M 3.75 -2.859375 C 4.109375 -1.953125 4.71875 -0.328125 4.828125 -0.15625 C 5 0.09375 5.203125 0.09375 5.4375 0.09375 C 5.71875 0.09375 5.796875 0.09375 5.796875 -0.015625 C 5.796875 -0.0625 5.78125 -0.09375 5.75 -0.125 C 5.625 -0.265625 5.578125 -0.375 5.5 -0.5625 L 3.265625 -6.234375 C 3.1875 -6.421875 3 -6.90625 1.984375 -6.90625 C 1.875 -6.90625 1.734375 -6.90625 1.734375 -6.765625 C 1.734375 -6.65625 1.828125 -6.640625 1.875 -6.625 C 2.078125 -6.59375 2.265625 -6.578125 2.5 -6 L 3.21875 -4.203125 L 3.609375 -3.203125 L 0.75 -0.5625 C 0.640625 -0.46875 0.546875 -0.34375 0.546875 -0.203125 C 0.546875 -0.015625 0.71875 0.125 0.890625 0.125 C 1.03125 0.125 1.171875 0.015625 1.25 -0.09375 Z M 3.75 -2.859375 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-7"> <path style="stroke:none;" d="M 2.828125 -3.640625 L 3.96875 -3.640625 C 3.765625 -2.734375 3.578125 -1.984375 3.578125 -1.25 C 3.578125 -1.1875 3.578125 -0.75 3.6875 -0.390625 C 3.828125 0.015625 3.921875 0.09375 4.09375 0.09375 C 4.3125 0.09375 4.546875 -0.09375 4.546875 -0.328125 C 4.546875 -0.390625 4.53125 -0.421875 4.5 -0.5 C 4.296875 -0.96875 4.140625 -1.4375 4.140625 -2.265625 C 4.140625 -2.484375 4.140625 -2.90625 4.296875 -3.640625 L 5.5 -3.640625 C 5.65625 -3.640625 5.765625 -3.640625 5.859375 -3.71875 C 5.984375 -3.828125 6.015625 -3.953125 6.015625 -4.015625 C 6.015625 -4.296875 5.765625 -4.296875 5.59375 -4.296875 L 2 -4.296875 C 1.796875 -4.296875 1.40625 -4.296875 0.921875 -3.8125 C 0.5625 -3.453125 0.28125 -3 0.28125 -2.921875 C 0.28125 -2.84375 0.359375 -2.8125 0.4375 -2.8125 C 0.53125 -2.8125 0.5625 -2.84375 0.609375 -2.921875 C 1.109375 -3.640625 1.6875 -3.640625 1.921875 -3.640625 L 2.5 -3.640625 C 2.203125 -2.578125 1.6875 -1.375 1.3125 -0.640625 C 1.25 -0.5 1.140625 -0.296875 1.140625 -0.203125 C 1.140625 0 1.3125 0.09375 1.46875 0.09375 C 1.84375 0.09375 1.953125 -0.28125 2.15625 -1.09375 Z M 2.828125 -3.640625 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1"> <path style="stroke:none;" d="M 1.6875 -0.90625 C 1.40625 -0.484375 1.15625 -0.328125 0.734375 -0.296875 C 0.625 -0.296875 0.53125 -0.28125 0.53125 -0.109375 C 0.53125 -0.03125 0.59375 0 0.65625 0 C 0.859375 0 1.109375 -0.03125 1.328125 -0.03125 C 1.5 -0.03125 1.9375 0 2.125 0 C 2.203125 0 2.28125 -0.03125 2.28125 -0.1875 C 2.28125 -0.296875 2.1875 -0.296875 2.15625 -0.296875 C 1.984375 -0.3125 1.8125 -0.34375 1.8125 -0.515625 C 1.8125 -0.609375 1.890625 -0.734375 1.921875 -0.78125 L 2.53125 -1.703125 L 4.78125 -1.703125 L 4.96875 -0.5 C 4.96875 -0.296875 4.546875 -0.296875 4.46875 -0.296875 C 4.375 -0.296875 4.25 -0.296875 4.25 -0.109375 C 4.25 -0.078125 4.28125 0 4.375 0 C 4.59375 0 5.140625 -0.03125 5.375 -0.03125 C 5.515625 -0.03125 5.65625 -0.015625 5.8125 -0.015625 C 5.953125 -0.015625 6.125 0 6.265625 0 C 6.3125 0 6.4375 0 6.4375 -0.1875 C 6.4375 -0.296875 6.328125 -0.296875 6.21875 -0.296875 C 5.765625 -0.296875 5.75 -0.34375 5.71875 -0.53125 L 4.96875 -5.125 C 4.953125 -5.265625 4.9375 -5.328125 4.78125 -5.328125 C 4.625 -5.328125 4.578125 -5.265625 4.515625 -5.171875 Z M 2.71875 -2 L 4.34375 -4.421875 L 4.734375 -2 Z M 2.71875 -2 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2"> <path style="stroke:none;" d="M 1.5625 -0.625 C 1.5 -0.34375 1.484375 -0.296875 0.921875 -0.296875 C 0.765625 -0.296875 0.75 -0.296875 0.71875 -0.28125 C 0.671875 -0.25 0.65625 -0.15625 0.65625 -0.109375 C 0.671875 0 0.734375 0 0.890625 0 L 3.921875 0 C 5.21875 0 6.234375 -0.84375 6.234375 -1.59375 C 6.234375 -2.046875 5.828125 -2.546875 4.90625 -2.671875 C 5.765625 -2.8125 6.609375 -3.359375 6.609375 -4.03125 C 6.609375 -4.578125 6.015625 -5.09375 4.96875 -5.09375 L 2.109375 -5.09375 C 1.96875 -5.09375 1.859375 -5.09375 1.859375 -4.921875 C 1.859375 -4.796875 1.96875 -4.796875 2.09375 -4.796875 C 2.234375 -4.796875 2.421875 -4.796875 2.578125 -4.75 C 2.578125 -4.671875 2.578125 -4.625 2.546875 -4.515625 Z M 2.796875 -2.78125 L 3.234375 -4.546875 C 3.296875 -4.78125 3.296875 -4.796875 3.609375 -4.796875 L 4.84375 -4.796875 C 5.625 -4.796875 5.84375 -4.328125 5.84375 -4.015625 C 5.84375 -3.40625 5.078125 -2.78125 4.15625 -2.78125 Z M 2.4375 -0.296875 C 2.296875 -0.296875 2.28125 -0.296875 2.1875 -0.328125 L 2.71875 -2.53125 L 4.359375 -2.53125 C 5.296875 -2.53125 5.453125 -1.921875 5.453125 -1.625 C 5.453125 -0.90625 4.65625 -0.296875 3.734375 -0.296875 Z M 2.4375 -0.296875 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-3"> <path style="stroke:none;" d="M 1.71875 -0.109375 C 1.71875 0.609375 1.34375 1.015625 1.15625 1.1875 C 1.078125 1.25 1.0625 1.265625 1.0625 1.3125 C 1.0625 1.375 1.125 1.453125 1.1875 1.453125 C 1.296875 1.453125 1.96875 0.84375 1.96875 -0.0625 C 1.96875 -0.53125 1.765625 -0.90625 1.40625 -0.90625 C 1.125 -0.90625 0.96875 -0.671875 0.96875 -0.453125 C 0.96875 -0.234375 1.125 0 1.421875 0 C 1.53125 0 1.640625 -0.03125 1.71875 -0.109375 Z M 1.71875 -0.109375 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-4"> <path style="stroke:none;" d="M 6.578125 -4.28125 C 6.65625 -4.59375 6.796875 -4.78125 7.375 -4.796875 C 7.421875 -4.796875 7.53125 -4.8125 7.53125 -4.984375 C 7.53125 -5.0625 7.484375 -5.09375 7.421875 -5.09375 C 7.234375 -5.09375 6.796875 -5.078125 6.625 -5.078125 L 6.203125 -5.078125 C 6.078125 -5.078125 5.921875 -5.09375 5.78125 -5.09375 C 5.65625 -5.09375 5.625 -5 5.625 -4.921875 C 5.625 -4.796875 5.75 -4.796875 5.796875 -4.796875 C 6.15625 -4.796875 6.296875 -4.703125 6.296875 -4.5 C 6.296875 -4.46875 6.28125 -4.421875 6.28125 -4.390625 L 5.453125 -1.109375 L 3.421875 -4.96875 C 3.34375 -5.09375 3.34375 -5.09375 3.15625 -5.09375 L 2.078125 -5.09375 C 1.9375 -5.09375 1.828125 -5.09375 1.828125 -4.921875 C 1.828125 -4.796875 1.9375 -4.796875 2.109375 -4.796875 C 2.15625 -4.796875 2.40625 -4.796875 2.578125 -4.765625 L 1.59375 -0.828125 C 1.53125 -0.546875 1.40625 -0.328125 0.796875 -0.296875 C 0.6875 -0.296875 0.625 -0.25 0.625 -0.109375 C 0.640625 -0.078125 0.640625 0 0.75 0 C 0.9375 0 1.375 -0.03125 1.546875 -0.03125 L 1.96875 -0.015625 C 2.09375 -0.015625 2.25 0 2.375 0 C 2.546875 0 2.546875 -0.15625 2.546875 -0.1875 C 2.53125 -0.296875 2.453125 -0.296875 2.359375 -0.296875 C 2.015625 -0.3125 1.859375 -0.40625 1.859375 -0.59375 C 1.859375 -0.640625 1.875 -0.671875 1.890625 -0.75 L 2.859375 -4.578125 L 5.203125 -0.125 C 5.25 -0.015625 5.265625 0 5.375 0 C 5.5 0 5.515625 -0.03125 5.546875 -0.171875 Z M 6.578125 -4.28125 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1"> <path style="stroke:none;" d="M 7.625 -2.484375 C 7.625 -4.453125 6.03125 -6.015625 4.109375 -6.015625 C 2.15625 -6.015625 0.59375 -4.421875 0.59375 -2.484375 C 0.59375 -0.53125 2.1875 1.03125 4.09375 1.03125 C 6.0625 1.03125 7.625 -0.5625 7.625 -2.484375 Z M 1.953125 -4.859375 C 2.734375 -5.578125 3.625 -5.703125 4.09375 -5.703125 C 4.703125 -5.703125 5.546875 -5.53125 6.265625 -4.859375 L 4.109375 -2.703125 Z M 1.734375 -0.328125 C 1.171875 -0.921875 0.890625 -1.71875 0.890625 -2.484375 C 0.890625 -3.25 1.171875 -4.046875 1.734375 -4.640625 L 3.890625 -2.484375 Z M 6.484375 -4.640625 C 7.03125 -4.0625 7.3125 -3.25 7.3125 -2.484375 C 7.3125 -1.734375 7.046875 -0.921875 6.484375 -0.328125 L 4.3125 -2.484375 Z M 6.265625 -0.109375 C 5.484375 0.609375 4.59375 0.734375 4.109375 0.734375 C 3.5 0.734375 2.65625 0.546875 1.953125 -0.109375 L 4.09375 -2.265625 Z M 6.265625 -0.109375 "></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph5-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph5-1"> <path style="stroke:none;" d="M 6.484375 -1.859375 C 6.484375 -3.46875 5.1875 -4.765625 3.59375 -4.765625 C 2 -4.765625 0.6875 -3.46875 0.6875 -1.859375 C 0.6875 -0.265625 2 1.03125 3.59375 1.03125 C 5.1875 1.03125 6.484375 -0.265625 6.484375 -1.859375 Z M 1.84375 -3.828125 C 2.234375 -4.1875 2.859375 -4.5 3.59375 -4.5 C 4.3125 -4.5 4.921875 -4.203125 5.34375 -3.828125 L 3.59375 -2.0625 Z M 1.640625 -0.109375 C 1.234375 -0.53125 0.96875 -1.1875 0.96875 -1.859375 C 0.96875 -2.546875 1.234375 -3.1875 1.640625 -3.625 L 3.390625 -1.859375 Z M 5.546875 -3.625 C 5.9375 -3.203125 6.21875 -2.546875 6.21875 -1.859375 C 6.21875 -1.1875 5.953125 -0.546875 5.546875 -0.109375 L 3.78125 -1.859375 Z M 5.34375 0.09375 C 4.9375 0.453125 4.328125 0.765625 3.59375 0.765625 C 2.875 0.765625 2.25 0.46875 1.84375 0.09375 L 3.59375 -1.671875 Z M 5.34375 0.09375 "></path> </symbol> </g> </defs> <g id="JQ-hfNoqJORrDWk3nJsG3BgZpGw=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1" x="6.427628" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="20.703476" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-2" x="35.632427" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="52.215987" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3" x="67.144937" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="82.321143" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-4" x="97.250093" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1" x="233.824203" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="248.100051" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3" x="263.029002" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="278.205207" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-4" x="293.134158" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1" x="429.709017" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="443.984865" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-1" x="455.595855" y="26.169843"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3" x="468.988828" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="484.165033" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-4" x="499.093984" y="23.930813"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1" x="21.277907" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="35.553755" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-2" x="50.482705" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="67.065016" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3" x="81.993967" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1" x="248.673483" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="262.949331" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3" x="277.878281" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-1" x="444.558296" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph1-1" x="458.834144" y="89.398913"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-1" x="470.446383" y="91.637943"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph0-3" x="483.839356" y="89.398913"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -143.250715 35.401964 L -31.851329 35.401964 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485269 2.867664 C -2.031695 1.14721 -1.018973 0.333904 0.00156893 0.00154385 C -1.018973 -0.334727 -2.031695 -1.148032 -2.485269 -2.868486 " transform="matrix(0.99901,0,0,-0.99901,226.908589,17.317949)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-2" x="148.251034" y="11.870769"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1" x="153.665667" y="8.358001"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="161.185712" y="11.870769"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-3" x="169.411308" y="11.870769"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="177.877914" y="11.870769"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-4" x="186.102261" y="11.870769"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -143.250715 29.634533 L -31.851329 29.634533 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485269 2.869233 C -2.031695 1.148779 -1.018973 0.335474 0.00156893 -0.000796661 C -1.018973 -0.333157 -2.031695 -1.146463 -2.485269 -2.870826 " transform="matrix(0.99901,0,0,-0.99901,226.908589,23.077329)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-5" x="147.959324" y="35.203638"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="155.880221" y="35.203638"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-6" x="164.104568" y="35.203638"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="170.273453" y="31.69087"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="178.169375" y="35.203638"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-4" x="186.393722" y="35.203638"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -201.120526 21.818199 L -201.124437 -20.344651 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486697 2.868251 C -2.033079 1.147772 -1.020321 0.33447 0.000248427 -0.00178664 C -1.020307 -0.334174 -2.033033 -1.147516 -2.486582 -2.868013 " transform="matrix(-0.0000199802,0.99899,0.99899,0.0000199802,57.564285,73.245846)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-5" x="6.919141" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="14.840039" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-1" x="23.065634" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="32.518763" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-2" x="40.744359" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="46.158991" y="52.012974"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 52.830192 32.516294 L 164.233488 32.516294 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485662 2.868453 C -2.032088 1.148 -1.019366 0.334694 0.00117591 -0.00157647 C -1.019366 -0.333937 -2.032088 -1.147242 -2.485662 -2.867696 " transform="matrix(0.99901,0,0,-0.99901,422.7957,20.197644)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-7" x="347.098911" y="13.562092"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1" x="353.171641" y="15.234184"></use> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-3" x="360.069678" y="15.234184"></use> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="362.90324" y="15.234184"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="370.799166" y="13.562092"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-4" x="379.023513" y="13.562092"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 10.729904 21.818199 L 10.729904 -20.344651 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486697 2.870342 C -2.033079 1.145952 -1.020321 0.336561 0.000248385 0.000304066 C -1.020307 -0.335994 -2.033033 -1.149336 -2.486582 -2.869833 " transform="matrix(-0.0000199802,0.99899,0.99899,0.0000199802,269.206727,73.245846)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-5" x="236.237811" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="244.158708" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-2" x="252.383055" y="55.524493"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="257.798936" y="52.012974"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:3.34735,1.91277;stroke-miterlimit:10;" d="M 211.854869 20.821117 L 211.854869 -20.344651 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486697 2.870996 C -2.033079 1.146607 -1.020321 0.333306 0.000248372 0.000958933 C -1.020307 -0.335339 -2.033033 -1.148681 -2.486582 -2.869178 " transform="matrix(-0.0000199802,0.99899,0.99899,0.0000199802,470.131855,73.245846)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-2" x="473.641465" y="56.066956"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1" x="479.057346" y="52.465526"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph5-1" x="485.955508" y="52.465526"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-4" x="493.143383" y="54.995518"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="501.789811" y="52.465526"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -158.117001 -30.131687 L -16.985043 -30.131687 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487404 2.867902 C -2.03383 1.147449 -1.021108 0.334143 -0.000565909 0.00178276 C -1.021108 -0.334488 -2.03383 -1.147793 -2.487404 -2.868247 " transform="matrix(0.99901,0,0,-0.99901,241.758378,82.786937)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-2" x="156.449907" y="77.338868"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1" x="161.864539" y="73.8261"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="169.384584" y="77.338868"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-3" x="177.61018" y="77.338868"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -158.117001 -35.899118 L -16.985043 -35.899118 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487404 2.869472 C -2.03383 1.149018 -1.021108 0.335713 -0.000565909 -0.000557748 C -1.021108 -0.332918 -2.03383 -1.146224 -2.487404 -2.870588 " transform="matrix(0.99901,0,0,-0.99901,241.758378,88.546318)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-5" x="156.158196" y="100.671738"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph4-1" x="164.079094" y="100.671738"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-6" x="172.303441" y="100.671738"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="178.472325" y="97.15897"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 37.963906 -33.017358 L 179.095864 -33.017358 " transform="matrix(0.99901,0,0,-0.99901,258.487754,52.68331)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487787 2.868672 C -2.034213 1.148218 -1.021491 0.334913 -0.000948925 -0.00135755 C -1.021491 -0.333718 -2.034213 -1.147023 -2.487787 -2.867477 " transform="matrix(0.99901,0,0,-0.99901,437.645479,85.666613)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph2-7" x="355.297783" y="93.463883"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-1" x="361.370513" y="95.137224"></use> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-3" x="368.26855" y="95.137224"></use> <use xlink:href="#JQ-hfNoqJORrDWk3nJsG3BgZpGw=-glyph3-2" x="371.102112" y="95.137224"></use> </g> </g> </svg> <p>Assuming all requisite <a class="existingWikiWord" href="/nlab/show/reflexive+coequalizer">(reflective) coequalizers</a> exist, universal property arguments guarantee associativity isomorphisms of type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mi>B</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>P</mi></msub><mi>C</mi><mo>→</mo><mi>A</mi><msub><mo>⊗</mo> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>B</mi><msub><mo>⊗</mo> <mi>P</mi></msub><mi>C</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">(A \otimes_N B) \otimes_P C \to A \otimes_N (B \otimes_P C).</annotation></semantics></math></div> <p>In fact, this tensor product defines composition in a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> where objects or 0-cells are monoids in a monoidal category, where 1-cells <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> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> bimodules, and where 2-cells from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> are morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> bimodules.</p> <p>This in turn can be seen as a special case of a bicategory of <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with suitably nice cocompleteness properties – see <a class="existingWikiWord" href="/nlab/show/monoidally+cocomplete+category">monoidally cocomplete category</a> and <a class="existingWikiWord" href="/nlab/show/Benabou+cosmos">Benabou cosmos</a>.</p> <h3 id="twosided_ideals_of_a_ring">Two-sided ideals of a ring</h3> <p>Every ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule, with the <a class="existingWikiWord" href="/nlab/show/biaction">biaction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">(-)(-)(-):R \times R \times R \to R</annotation></semantics></math> defined by the ternary product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>b</mi><mi>c</mi><mo>≔</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>⋅</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a b c \coloneqq a \cdot b \cdot c</annotation></semantics></math> for elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">b \in R</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">c \in R</annotation></semantics></math>.</p> <p>Given a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, a <strong><a class="existingWikiWord" href="/nlab/show/two-sided+ideal">two-sided ideal</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a sub-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> <h3 id="rings_over_a_ring">Rings over a ring</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>. An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-ring</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule with a <a class="existingWikiWord" href="/nlab/show/bilinear+function">bilinear function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>S</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">(-)\cdot(-):S \times S \to S</annotation></semantics></math> and an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">1 \in S</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mo>⋅</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, \cdot, 1)</annotation></semantics></math> forms a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>.</p> <h2 id="categories_of_bimodules">Categories of bimodules</h2> <h3 id="the_1category_of_bimodules_and_intertwiners">The 1-category of bimodules and intertwiners</h3> <div class="num_defn" id="1CategoryOfBimodulesAndIntertwiners"> <h6 id="definition_2">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BMod</mi></mrow><annotation encoding="application/x-tex">BMod</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(R,S,B)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/rings">rings</a> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g, \phi)</annotation></semantics></math> consisting of two ring <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><mi>R</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f \colon R \to R'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>S</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \colon S \to S'</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">S'</annotation></semantics></math>-bimodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>⋅</mo><mi>g</mi><mo>→</mo><mi>f</mi><mo>⋅</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi \colon B \cdot g \to f \cdot B'</annotation></semantics></math>. This we may depict as a</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>R</mi></mtd> <mtd><mover><mo>→</mo><mi>B</mi></mover></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇓</mo> <mi>ϕ</mi></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>R</mi><mo>′</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>B</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>S</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ R &amp;\stackrel{B}{\to}&amp; S \\ {}^{\mathllap{f}}\downarrow &amp;\Downarrow_{\phi}&amp; \downarrow^{\mathrlap{g}} \\ R' &amp;\stackrel{B'}{\to}&amp; S' } \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>As this notation suggests, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BMod</mi></mrow><annotation encoding="application/x-tex">BMod</annotation></semantics></math> is naturally the vertical category of a <a class="existingWikiWord" href="/nlab/show/pseudo+double+category">pseudo double category</a> whose horizontal composition is given by tensor product of bimodules.</p> </div> <h3 id="AsMorphismsInA2Category">The 2-category of rings, bimodules, and intertwiners</h3> <p>Consider bimodules over <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> <div class="num_prop" id="AlgebrasAndBimodules"> <h6 id="proposition">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/rings">rings</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are bimodules;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/intertwiners">intertwiners</a>.</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of 1-morphisms is given by the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> over the middle algebra.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> from the above 2-category of rings and bimodules to <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> which</p> <ul> <li> <p>sends an ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>;</p> </li> <li> <p>sends a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mi>B</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Mod</mi> <mi>R</mi></msub><mo>→</mo><msub><mi>Mod</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex"> (-)\otimes_{R} B \;\colon\; Mod_{R} \to Mod_{S} </annotation></semantics></math></div></li> <li> <p>sends an <a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a> to the evident <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of the above functors.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>This construction has as its image precisely the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a>.</p> </div> <p>This is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Watts+theorem">Eilenberg-Watts theorem</a>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>In the context of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> one may interpret this as says that the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of those <a class="existingWikiWord" href="/nlab/show/2-modules">2-modules</a> over the given ring which are equivalent to a <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> is that of rings, bimodules and intertwiners. See also at <em><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>The 2-category of rings and bimodules is an archtypical example for a <a class="existingWikiWord" href="/nlab/show/2-category+with+proarrow+equipment">2-category with proarrow equipment</a>, hence for a <a class="existingWikiWord" href="/nlab/show/pseudo+double+category">pseudo double category</a> with niche-fillers. Or in the language of <a class="existingWikiWord" href="/nlab/show/internal+%28infinity%2C1%29-category">internal (infinity,1)-category</a>-theory: it naturally induces the structure of a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>⋯</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><msub><mi>X</mi> <mn>1</mn></msub><mover><munder><mo>→</mo><mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></munder><mover><mo>→</mo><mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></mover></mover><msub><mi>X</mi> <mn>0</mn></msub><mo>)</mo></mrow><mo>∈</mo><msup><mi>Cat</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 \right) \in Cat^{\Delta^{op}} </annotation></semantics></math></div> <p>which satisfies the <a class="existingWikiWord" href="/nlab/show/Segal+conditions">Segal conditions</a>. Here</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>Ring</mi></mrow><annotation encoding="application/x-tex"> X_0 = Ring </annotation></semantics></math></div> <p>is the category of <a class="existingWikiWord" href="/nlab/show/rings">rings</a> and <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between them, while</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mi>BMod</mi></mrow><annotation encoding="application/x-tex"> X_1 = BMod </annotation></semantics></math></div> <p>is the category of def. <a class="maruku-ref" href="#1CategoryOfBimodulesAndIntertwiners"></a>, whose objects are pairs consisting of two rings <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and an <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> bimodule <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> between them, and whose morphisms are pairs consisting of two ring homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f \colon A \to A'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g \colon B \to B'</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>N</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">N \cdot (g) \to (f) \cdot N'</annotation></semantics></math>.</p> </div> <h3 id="Infinity2CategoryOfInfinityAlgebrasAndBimodules">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,2)</annotation></semantics></math>-category of <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>-algebras and <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>-bimodules</h3> <p>The above has a generalization to <em><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-bimodules">(infinity,1)-bimodules</a></em>. See there for more.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monad">module over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule+object">bimodule object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+bimodule">quotient bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+bimodule">Hilbert bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+bimodule">Noetherian bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Artinian+bimodule">Artinian bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/two-sided+ideal">two-sided ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+two-sided+ideals+in+a+ring">category of two-sided ideals in a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/biaction">biaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/amplimorphism">amplimorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-bimodule">(infinity,1)-bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule+category">bimodule category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a>, <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a></p> </li> </ul> <h2 id="references">References</h2> <p>The 2-category of bimodules in its incarnation as a <a class="existingWikiWord" href="/nlab/show/2-category+with+proarrow+equipment">2-category with proarrow equipment</a> appears as example 2.3 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em>Framed bicategories and monoidal fibrations</em> (<a href="http://arxiv.org/abs/0706.1286">arXiv:0706.1286</a>)</li> </ul> <p>Bimodules in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> are discussed in section 4.3 of</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>For more on that see at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-bimodule">(∞,1)-bimodule</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on August 20, 2024 at 13:10:19. See the <a href="/nlab/history/bimodule" 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/bimodule" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4728/#Item_17">Discuss</a><span class="backintime"><a href="/nlab/revision/bimodule/48" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/bimodule" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/bimodule" accesskey="S" class="navlink" id="history" rel="nofollow">History (48 revisions)</a> <a href="/nlab/show/bimodule/cite" style="color: black">Cite</a> <a href="/nlab/print/bimodule" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/bimodule" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10