CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> isofibration in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> isofibration </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/7640/#Item_4" 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></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="2category_theory">2-category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#section_Idea'>Idea</a></li> <li><a href='#section_Definition'>Definition</a></li> <li><a href='#section_Equivalent_definition'>Equivalent definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#AsFibrationsInCanonicalModelStructures'>As fibrations in canonical model structures</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <p><h2 id='section_Idea'>Idea</h2></p> <p>An <em>isofibration</em> is, roughly speaking, a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">F: E \rightarrow B</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/category">categories</a> such that every <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in <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> can be ‘lifted’ to an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Here ‘lifted’ means that there is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">F(g) = f</annotation></semantics></math>.</p> <p>Alternatively, an isofibration is the analogue of a <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a> for the <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Cat">category of categories</a>, given by the <a class="existingWikiWord" href="/nlab/show/free-standing+isomorphism">free-standing isomorphism</a>. That this definition is equivalent to the former one will be established below.</p> <p>It is the second of these definitions that often generalises better to higher categories.</p> <p><h2 id='section_Definition'>Definition</h2></p> <p> <div class='num_defn'> <h6>Definition</h6> <p>An <strong>isofibration</strong> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p:E\to B</annotation></semantics></math> such that for any <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and any <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>p</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\phi:p(e) \cong b</annotation></semantics></math>, there exists an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>e</mi><mo>≅</mo><mi>e</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi:e \cong e'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">p(\psi)=\phi</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Equivalently: for any commutative diagram</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="103.202" height="90.825" viewBox="0 0 103.202 90.825"> <defs> <g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-0-0"> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-0-1"> <path d="M 4.296875 -9.578125 C 4.296875 -9.921875 4.296875 -9.9375 4 -9.9375 C 3.640625 -9.53125 2.890625 -8.984375 1.359375 -8.984375 L 1.359375 -8.546875 C 1.703125 -8.546875 2.453125 -8.546875 3.265625 -8.9375 L 3.265625 -1.15625 C 3.265625 -0.609375 3.21875 -0.4375 1.90625 -0.4375 L 1.453125 -0.4375 L 1.453125 0 C 1.859375 -0.03125 3.296875 -0.03125 3.796875 -0.03125 C 4.28125 -0.03125 5.71875 -0.03125 6.125 0 L 6.125 -0.4375 L 5.65625 -0.4375 C 4.34375 -0.4375 4.296875 -0.609375 4.296875 -1.15625 Z M 4.296875 -9.578125 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-0"> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-1"> <path d="M 10.375 -3.46875 C 10.390625 -3.515625 10.4375 -3.609375 10.4375 -3.671875 C 10.4375 -3.75 10.375 -3.828125 10.296875 -3.828125 C 10.234375 -3.828125 10.203125 -3.8125 10.15625 -3.765625 C 10.125 -3.75 10.125 -3.71875 10 -3.421875 C 9.109375 -1.328125 8.46875 -0.4375 6.078125 -0.4375 L 3.90625 -0.4375 C 3.6875 -0.4375 3.65625 -0.4375 3.5625 -0.453125 C 3.40625 -0.46875 3.390625 -0.5 3.390625 -0.609375 C 3.390625 -0.71875 3.421875 -0.8125 3.453125 -0.9375 L 4.484375 -5.0625 L 5.953125 -5.0625 C 7.125 -5.0625 7.21875 -4.8125 7.21875 -4.359375 C 7.21875 -4.21875 7.21875 -4.078125 7.109375 -3.625 C 7.078125 -3.5625 7.0625 -3.515625 7.0625 -3.46875 C 7.0625 -3.359375 7.140625 -3.3125 7.234375 -3.3125 C 7.359375 -3.3125 7.375 -3.421875 7.4375 -3.625 L 8.296875 -7.09375 C 8.296875 -7.171875 8.234375 -7.25 8.140625 -7.25 C 8 -7.25 7.984375 -7.1875 7.9375 -6.96875 C 7.640625 -5.828125 7.328125 -5.5 6 -5.5 L 4.578125 -5.5 L 5.515625 -9.171875 C 5.640625 -9.6875 5.671875 -9.734375 6.28125 -9.734375 L 8.421875 -9.734375 C 10.265625 -9.734375 10.640625 -9.25 10.640625 -8.109375 C 10.640625 -8.09375 10.640625 -7.671875 10.578125 -7.1875 C 10.5625 -7.125 10.546875 -7.03125 10.546875 -7 C 10.546875 -6.890625 10.625 -6.84375 10.703125 -6.84375 C 10.8125 -6.84375 10.875 -6.90625 10.90625 -7.171875 L 11.21875 -9.78125 C 11.21875 -9.828125 11.25 -9.984375 11.25 -10.015625 C 11.25 -10.171875 11.109375 -10.171875 10.84375 -10.171875 L 3.5625 -10.171875 C 3.265625 -10.171875 3.125 -10.171875 3.125 -9.90625 C 3.125 -9.734375 3.21875 -9.734375 3.484375 -9.734375 C 4.40625 -9.734375 4.40625 -9.640625 4.40625 -9.46875 C 4.40625 -9.390625 4.390625 -9.328125 4.34375 -9.171875 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.125 -0.4375 C 0.84375 -0.4375 0.6875 -0.4375 0.6875 -0.171875 C 0.6875 0 0.78125 0 1.078125 0 L 8.578125 0 C 8.90625 0 8.921875 -0.015625 9.015625 -0.25 Z M 10.375 -3.46875 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-2"> <path d="M 5.5 -9.09375 C 5.625 -9.625 5.65625 -9.765625 6.75 -9.765625 C 7.078125 -9.765625 7.203125 -9.765625 7.203125 -10.046875 C 7.203125 -10.203125 7.03125 -10.203125 6.984375 -10.203125 C 6.71875 -10.203125 6.390625 -10.171875 6.125 -10.171875 L 4.28125 -10.171875 C 3.984375 -10.171875 3.640625 -10.203125 3.34375 -10.203125 C 3.21875 -10.203125 3.0625 -10.203125 3.0625 -9.921875 C 3.0625 -9.765625 3.1875 -9.765625 3.484375 -9.765625 C 4.40625 -9.765625 4.40625 -9.65625 4.40625 -9.484375 C 4.40625 -9.375 4.375 -9.296875 4.34375 -9.15625 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.078125 -0.4375 C 0.75 -0.4375 0.609375 -0.4375 0.609375 -0.15625 C 0.609375 0 0.765625 0 0.84375 0 C 1.109375 0 1.4375 -0.03125 1.703125 -0.03125 L 3.546875 -0.03125 C 3.84375 -0.03125 4.171875 0 4.46875 0 C 4.578125 0 4.765625 0 4.765625 -0.265625 C 4.765625 -0.4375 4.671875 -0.4375 4.34375 -0.4375 C 3.421875 -0.4375 3.421875 -0.546875 3.421875 -0.734375 C 3.421875 -0.765625 3.421875 -0.84375 3.484375 -1.078125 Z M 5.5 -9.09375 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-3"> <path d="M 5.46875 -9.1875 C 5.59375 -9.734375 5.65625 -9.765625 6.25 -9.765625 L 8.1875 -9.765625 C 9.875 -9.765625 9.875 -8.328125 9.875 -8.203125 C 9.875 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.546875 -5.453125 Z M 7.984375 -5.328125 C 9.625 -5.625 11.09375 -6.765625 11.09375 -8.140625 C 11.09375 -9.3125 10.0625 -10.203125 8.375 -10.203125 L 3.578125 -10.203125 C 3.296875 -10.203125 3.171875 -10.203125 3.171875 -9.921875 C 3.171875 -9.765625 3.296875 -9.765625 3.53125 -9.765625 C 4.4375 -9.765625 4.4375 -9.65625 4.4375 -9.484375 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.140625 L 2.359375 -1.109375 C 2.21875 -0.578125 2.203125 -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 6.234375 0 C 8.515625 0 10.28125 -1.734375 10.28125 -3.234375 C 10.28125 -4.46875 9.203125 -5.21875 7.984375 -5.328125 Z M 5.875 -0.4375 L 3.859375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.53125 -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 -5.15625 L 7.265625 -5.15625 C 9.015625 -5.15625 9.015625 -3.515625 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.875 -0.4375 Z M 5.875 -0.4375 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-0"> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-1"> <path d="M 2 -2.265625 C 2.21875 -2.265625 2.96875 -2.28125 3.5 -2.46875 C 4.359375 -2.765625 4.390625 -3.375 4.390625 -3.515625 C 4.390625 -4.078125 3.875 -4.390625 3.21875 -4.390625 C 2.09375 -4.390625 0.484375 -3.515625 0.484375 -1.75 C 0.484375 -0.734375 1.109375 0.09375 2.203125 0.09375 C 3.765625 0.09375 4.59375 -0.890625 4.59375 -1.03125 C 4.59375 -1.125 4.5 -1.203125 4.4375 -1.203125 C 4.375 -1.203125 4.34375 -1.171875 4.296875 -1.109375 C 3.515625 -0.171875 2.390625 -0.171875 2.21875 -0.171875 C 1.5 -0.171875 1.28125 -0.796875 1.28125 -1.359375 C 1.28125 -1.65625 1.359375 -2.109375 1.40625 -2.265625 Z M 1.484375 -2.546875 C 1.796875 -3.765625 2.71875 -4.109375 3.21875 -4.109375 C 3.625 -4.109375 4 -3.921875 4 -3.515625 C 4 -2.546875 2.359375 -2.546875 1.9375 -2.546875 Z M 1.484375 -2.546875 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-2"> <path d="M 0.515625 1.203125 C 0.4375 1.53125 0.421875 1.609375 0.015625 1.609375 C -0.125 1.609375 -0.234375 1.609375 -0.234375 1.796875 C -0.234375 1.890625 -0.15625 1.9375 -0.09375 1.9375 C 0 1.9375 0.046875 1.90625 0.78125 1.90625 C 1.5 1.90625 1.703125 1.9375 1.78125 1.9375 C 1.8125 1.9375 1.96875 1.9375 1.96875 1.75 C 1.96875 1.609375 1.828125 1.609375 1.703125 1.609375 C 1.21875 1.609375 1.21875 1.546875 1.21875 1.453125 C 1.21875 1.390625 1.40625 0.671875 1.703125 -0.484375 C 1.828125 -0.265625 2.140625 0.09375 2.6875 0.09375 C 3.90625 0.09375 5.1875 -1.3125 5.1875 -2.765625 C 5.1875 -3.75 4.546875 -4.390625 3.75 -4.390625 C 3.15625 -4.390625 2.671875 -3.984375 2.375 -3.6875 C 2.171875 -4.390625 1.5 -4.390625 1.40625 -4.390625 C 1.046875 -4.390625 0.796875 -4.171875 0.640625 -3.859375 C 0.40625 -3.40625 0.296875 -2.90625 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.78125 0.65625 -3.046875 C 0.78125 -3.546875 0.96875 -4.109375 1.375 -4.109375 C 1.625 -4.109375 1.6875 -3.890625 1.6875 -3.65625 C 1.6875 -3.546875 1.65625 -3.3125 1.640625 -3.234375 Z M 2.359375 -3.078125 C 2.40625 -3.234375 2.40625 -3.265625 2.546875 -3.4375 C 2.9375 -3.890625 3.359375 -4.109375 3.71875 -4.109375 C 4.21875 -4.109375 4.40625 -3.625 4.40625 -3.1875 C 4.40625 -2.8125 4.1875 -1.75 3.890625 -1.15625 C 3.625 -0.625 3.15625 -0.171875 2.6875 -0.171875 C 2 -0.171875 1.84375 -0.953125 1.84375 -1.03125 C 1.84375 -1.046875 1.859375 -1.15625 1.875 -1.1875 Z M 2.359375 -3.078125 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-3"> <path d="M 4.609375 -6.78125 C 4.609375 -6.90625 4.5 -6.921875 4.453125 -6.921875 C 4.3125 -6.921875 4.3125 -6.859375 4.265625 -6.703125 L 3.703125 -4.421875 C 1.9375 -4.34375 0.515625 -3.015625 0.515625 -1.71875 C 0.515625 -0.71875 1.375 0.0625 2.5625 0.109375 C 2.484375 0.421875 2.40625 0.734375 2.328125 1.046875 C 2.21875 1.515625 2.125 1.890625 2.125 1.90625 C 2.125 1.921875 2.125 2.03125 2.265625 2.03125 C 2.40625 2.03125 2.40625 1.984375 2.46875 1.796875 L 2.875 0.125 C 4.640625 0.03125 6.0625 -1.28125 6.0625 -2.578125 C 6.0625 -3.71875 5.046875 -4.359375 4.015625 -4.40625 Z M 3.953125 -4.140625 C 4.6875 -4.078125 5.359375 -3.671875 5.359375 -2.75 C 5.359375 -1.796875 4.671875 -0.34375 2.953125 -0.171875 Z M 2.625 -0.15625 C 2.3125 -0.171875 1.21875 -0.34375 1.21875 -1.546875 C 1.21875 -2.609375 2.03125 -3.984375 3.625 -4.125 Z M 2.625 -0.15625 "></path> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-3-0"> </g> <g id="wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-3-1"> <path d="M 4.875 -3.1875 C 4.875 -4.25 4.765625 -4.890625 4.4375 -5.53125 C 4 -6.40625 3.1875 -6.625 2.640625 -6.625 C 1.390625 -6.625 0.921875 -5.6875 0.78125 -5.40625 C 0.421875 -4.6875 0.40625 -3.703125 0.40625 -3.1875 C 0.40625 -2.515625 0.4375 -1.515625 0.921875 -0.71875 C 1.375 0.015625 2.109375 0.203125 2.640625 0.203125 C 3.125 0.203125 3.984375 0.0625 4.46875 -0.921875 C 4.84375 -1.640625 4.875 -2.53125 4.875 -3.1875 Z M 2.640625 -0.0625 C 2.296875 -0.0625 1.609375 -0.234375 1.40625 -1.28125 C 1.296875 -1.84375 1.296875 -2.78125 1.296875 -3.296875 C 1.296875 -3.984375 1.296875 -4.6875 1.40625 -5.234375 C 1.609375 -6.25 2.390625 -6.34375 2.640625 -6.34375 C 2.984375 -6.34375 3.671875 -6.1875 3.875 -5.28125 C 3.984375 -4.71875 3.984375 -3.984375 3.984375 -3.296875 C 3.984375 -2.71875 3.984375 -1.8125 3.875 -1.25 C 3.65625 -0.203125 2.96875 -0.0625 2.640625 -0.0625 Z M 2.640625 -0.0625 "></path> </g> </g> </defs> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-0-1" x="8.368" y="15.681"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-1" x="84.978" y="15.681"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-2" x="8.211" y="78.972"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-1-3" x="84.897" y="78.972"></use> </g> <path 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 -30.121812 30.275687 L 25.327406 30.275687 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.485597 2.870521 C -2.032472 1.147865 -1.020754 0.335365 -0.0012225 -0.0005725 C -1.020754 -0.33651 -2.032472 -1.14901 -2.485597 -2.86776 " transform="matrix(1, 0, 0, -1, 78.05591, 11.94474)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-1" x="47.876" y="8.429"></use> </g> <path 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 -40.465562 20.822562 L -40.465562 -20.345406 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.487514 2.869761 C -2.030482 1.147105 -1.018764 0.334605 0.0007675 -0.0013325 C -1.018764 -0.333364 -2.030482 -1.145864 -2.487514 -2.86852 " transform="matrix(0, 1, 1, 0, 12.02477, 62.80392)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-3-1" x="3.217" y="45.431"></use> </g> <path 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 38.343031 20.822562 L 38.343031 -20.345406 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.487514 2.870665 C -2.030482 1.148009 -1.018764 0.335509 0.0007675 -0.00042875 C -1.018764 -0.336366 -2.030482 -1.148866 -2.487514 -2.867616 " transform="matrix(0, 1, 1, 0, 90.83246, 62.80392)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-2" x="94.348" y="43.397"></use> </g> <path 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 -29.965562 -33.017281 L 25.245375 -33.017281 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.486329 2.869812 C -2.033204 1.147156 -1.021485 0.334656 0.0019525 -0.00128125 C -1.021485 -0.333313 -2.033204 -1.145813 -2.486329 -2.868469 " transform="matrix(1, 0, 0, -1, 77.97461, 75.237)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wcuiY0_430Ry0w2H-Efy1o7xKXo=-glyph-2-3" x="47.222" y="85.671"></use> </g> </svg> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free-standing+isomorphism">free-standing isomorphism</a>, there is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">l: I \rightarrow E</annotation></semantics></math> such that the following diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math> commutes.</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="103.202" height="90.825" viewBox="0 0 103.202 90.825"> <defs> <g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-0-0"> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-0-1"> <path d="M 4.296875 -9.578125 C 4.296875 -9.921875 4.296875 -9.9375 4 -9.9375 C 3.640625 -9.53125 2.890625 -8.984375 1.359375 -8.984375 L 1.359375 -8.546875 C 1.703125 -8.546875 2.453125 -8.546875 3.265625 -8.9375 L 3.265625 -1.15625 C 3.265625 -0.609375 3.21875 -0.4375 1.90625 -0.4375 L 1.453125 -0.4375 L 1.453125 0 C 1.859375 -0.03125 3.296875 -0.03125 3.796875 -0.03125 C 4.28125 -0.03125 5.71875 -0.03125 6.125 0 L 6.125 -0.4375 L 5.65625 -0.4375 C 4.34375 -0.4375 4.296875 -0.609375 4.296875 -1.15625 Z M 4.296875 -9.578125 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-0"> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-1"> <path d="M 10.375 -3.46875 C 10.390625 -3.515625 10.4375 -3.609375 10.4375 -3.671875 C 10.4375 -3.75 10.375 -3.828125 10.296875 -3.828125 C 10.234375 -3.828125 10.203125 -3.8125 10.15625 -3.765625 C 10.125 -3.75 10.125 -3.71875 10 -3.421875 C 9.109375 -1.328125 8.46875 -0.4375 6.078125 -0.4375 L 3.90625 -0.4375 C 3.6875 -0.4375 3.65625 -0.4375 3.5625 -0.453125 C 3.40625 -0.46875 3.390625 -0.5 3.390625 -0.609375 C 3.390625 -0.71875 3.421875 -0.8125 3.453125 -0.9375 L 4.484375 -5.0625 L 5.953125 -5.0625 C 7.125 -5.0625 7.21875 -4.8125 7.21875 -4.359375 C 7.21875 -4.21875 7.21875 -4.078125 7.109375 -3.625 C 7.078125 -3.5625 7.0625 -3.515625 7.0625 -3.46875 C 7.0625 -3.359375 7.140625 -3.3125 7.234375 -3.3125 C 7.359375 -3.3125 7.375 -3.421875 7.4375 -3.625 L 8.296875 -7.09375 C 8.296875 -7.171875 8.234375 -7.25 8.140625 -7.25 C 8 -7.25 7.984375 -7.1875 7.9375 -6.96875 C 7.640625 -5.828125 7.328125 -5.5 6 -5.5 L 4.578125 -5.5 L 5.515625 -9.171875 C 5.640625 -9.6875 5.671875 -9.734375 6.28125 -9.734375 L 8.421875 -9.734375 C 10.265625 -9.734375 10.640625 -9.25 10.640625 -8.109375 C 10.640625 -8.09375 10.640625 -7.671875 10.578125 -7.1875 C 10.5625 -7.125 10.546875 -7.03125 10.546875 -7 C 10.546875 -6.890625 10.625 -6.84375 10.703125 -6.84375 C 10.8125 -6.84375 10.875 -6.90625 10.90625 -7.171875 L 11.21875 -9.78125 C 11.21875 -9.828125 11.25 -9.984375 11.25 -10.015625 C 11.25 -10.171875 11.109375 -10.171875 10.84375 -10.171875 L 3.5625 -10.171875 C 3.265625 -10.171875 3.125 -10.171875 3.125 -9.90625 C 3.125 -9.734375 3.21875 -9.734375 3.484375 -9.734375 C 4.40625 -9.734375 4.40625 -9.640625 4.40625 -9.46875 C 4.40625 -9.390625 4.390625 -9.328125 4.34375 -9.171875 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.125 -0.4375 C 0.84375 -0.4375 0.6875 -0.4375 0.6875 -0.171875 C 0.6875 0 0.78125 0 1.078125 0 L 8.578125 0 C 8.90625 0 8.921875 -0.015625 9.015625 -0.25 Z M 10.375 -3.46875 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-2"> <path d="M 5.5 -9.09375 C 5.625 -9.625 5.65625 -9.765625 6.75 -9.765625 C 7.078125 -9.765625 7.203125 -9.765625 7.203125 -10.046875 C 7.203125 -10.203125 7.03125 -10.203125 6.984375 -10.203125 C 6.71875 -10.203125 6.390625 -10.171875 6.125 -10.171875 L 4.28125 -10.171875 C 3.984375 -10.171875 3.640625 -10.203125 3.34375 -10.203125 C 3.21875 -10.203125 3.0625 -10.203125 3.0625 -9.921875 C 3.0625 -9.765625 3.1875 -9.765625 3.484375 -9.765625 C 4.40625 -9.765625 4.40625 -9.65625 4.40625 -9.484375 C 4.40625 -9.375 4.375 -9.296875 4.34375 -9.15625 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.078125 -0.4375 C 0.75 -0.4375 0.609375 -0.4375 0.609375 -0.15625 C 0.609375 0 0.765625 0 0.84375 0 C 1.109375 0 1.4375 -0.03125 1.703125 -0.03125 L 3.546875 -0.03125 C 3.84375 -0.03125 4.171875 0 4.46875 0 C 4.578125 0 4.765625 0 4.765625 -0.265625 C 4.765625 -0.4375 4.671875 -0.4375 4.34375 -0.4375 C 3.421875 -0.4375 3.421875 -0.546875 3.421875 -0.734375 C 3.421875 -0.765625 3.421875 -0.84375 3.484375 -1.078125 Z M 5.5 -9.09375 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-3"> <path d="M 5.46875 -9.1875 C 5.59375 -9.734375 5.65625 -9.765625 6.25 -9.765625 L 8.1875 -9.765625 C 9.875 -9.765625 9.875 -8.328125 9.875 -8.203125 C 9.875 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.546875 -5.453125 Z M 7.984375 -5.328125 C 9.625 -5.625 11.09375 -6.765625 11.09375 -8.140625 C 11.09375 -9.3125 10.0625 -10.203125 8.375 -10.203125 L 3.578125 -10.203125 C 3.296875 -10.203125 3.171875 -10.203125 3.171875 -9.921875 C 3.171875 -9.765625 3.296875 -9.765625 3.53125 -9.765625 C 4.4375 -9.765625 4.4375 -9.65625 4.4375 -9.484375 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.140625 L 2.359375 -1.109375 C 2.21875 -0.578125 2.203125 -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 6.234375 0 C 8.515625 0 10.28125 -1.734375 10.28125 -3.234375 C 10.28125 -4.46875 9.203125 -5.21875 7.984375 -5.328125 Z M 5.875 -0.4375 L 3.859375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.53125 -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 -5.15625 L 7.265625 -5.15625 C 9.015625 -5.15625 9.015625 -3.515625 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.875 -0.4375 Z M 5.875 -0.4375 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-0"> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-1"> <path d="M 2 -2.265625 C 2.21875 -2.265625 2.96875 -2.28125 3.5 -2.46875 C 4.359375 -2.765625 4.390625 -3.375 4.390625 -3.515625 C 4.390625 -4.078125 3.875 -4.390625 3.21875 -4.390625 C 2.09375 -4.390625 0.484375 -3.515625 0.484375 -1.75 C 0.484375 -0.734375 1.109375 0.09375 2.203125 0.09375 C 3.765625 0.09375 4.59375 -0.890625 4.59375 -1.03125 C 4.59375 -1.125 4.5 -1.203125 4.4375 -1.203125 C 4.375 -1.203125 4.34375 -1.171875 4.296875 -1.109375 C 3.515625 -0.171875 2.390625 -0.171875 2.21875 -0.171875 C 1.5 -0.171875 1.28125 -0.796875 1.28125 -1.359375 C 1.28125 -1.65625 1.359375 -2.109375 1.40625 -2.265625 Z M 1.484375 -2.546875 C 1.796875 -3.765625 2.71875 -4.109375 3.21875 -4.109375 C 3.625 -4.109375 4 -3.921875 4 -3.515625 C 4 -2.546875 2.359375 -2.546875 1.9375 -2.546875 Z M 1.484375 -2.546875 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-2"> <path d="M 0.515625 1.203125 C 0.4375 1.53125 0.421875 1.609375 0.015625 1.609375 C -0.125 1.609375 -0.234375 1.609375 -0.234375 1.796875 C -0.234375 1.890625 -0.15625 1.9375 -0.09375 1.9375 C 0 1.9375 0.046875 1.90625 0.78125 1.90625 C 1.5 1.90625 1.703125 1.9375 1.78125 1.9375 C 1.8125 1.9375 1.96875 1.9375 1.96875 1.75 C 1.96875 1.609375 1.828125 1.609375 1.703125 1.609375 C 1.21875 1.609375 1.21875 1.546875 1.21875 1.453125 C 1.21875 1.390625 1.40625 0.671875 1.703125 -0.484375 C 1.828125 -0.265625 2.140625 0.09375 2.6875 0.09375 C 3.90625 0.09375 5.1875 -1.3125 5.1875 -2.765625 C 5.1875 -3.75 4.546875 -4.390625 3.75 -4.390625 C 3.15625 -4.390625 2.671875 -3.984375 2.375 -3.6875 C 2.171875 -4.390625 1.5 -4.390625 1.40625 -4.390625 C 1.046875 -4.390625 0.796875 -4.171875 0.640625 -3.859375 C 0.40625 -3.40625 0.296875 -2.90625 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.78125 0.65625 -3.046875 C 0.78125 -3.546875 0.96875 -4.109375 1.375 -4.109375 C 1.625 -4.109375 1.6875 -3.890625 1.6875 -3.65625 C 1.6875 -3.546875 1.65625 -3.3125 1.640625 -3.234375 Z M 2.359375 -3.078125 C 2.40625 -3.234375 2.40625 -3.265625 2.546875 -3.4375 C 2.9375 -3.890625 3.359375 -4.109375 3.71875 -4.109375 C 4.21875 -4.109375 4.40625 -3.625 4.40625 -3.1875 C 4.40625 -2.8125 4.1875 -1.75 3.890625 -1.15625 C 3.625 -0.625 3.15625 -0.171875 2.6875 -0.171875 C 2 -0.171875 1.84375 -0.953125 1.84375 -1.03125 C 1.84375 -1.046875 1.859375 -1.15625 1.875 -1.1875 Z M 2.359375 -3.078125 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-3"> <path d="M 4.609375 -6.78125 C 4.609375 -6.90625 4.5 -6.921875 4.453125 -6.921875 C 4.3125 -6.921875 4.3125 -6.859375 4.265625 -6.703125 L 3.703125 -4.421875 C 1.9375 -4.34375 0.515625 -3.015625 0.515625 -1.71875 C 0.515625 -0.71875 1.375 0.0625 2.5625 0.109375 C 2.484375 0.421875 2.40625 0.734375 2.328125 1.046875 C 2.21875 1.515625 2.125 1.890625 2.125 1.90625 C 2.125 1.921875 2.125 2.03125 2.265625 2.03125 C 2.40625 2.03125 2.40625 1.984375 2.46875 1.796875 L 2.875 0.125 C 4.640625 0.03125 6.0625 -1.28125 6.0625 -2.578125 C 6.0625 -3.71875 5.046875 -4.359375 4.015625 -4.40625 Z M 3.953125 -4.140625 C 4.6875 -4.078125 5.359375 -3.671875 5.359375 -2.75 C 5.359375 -1.796875 4.671875 -0.34375 2.953125 -0.171875 Z M 2.625 -0.15625 C 2.3125 -0.171875 1.21875 -0.34375 1.21875 -1.546875 C 1.21875 -2.609375 2.03125 -3.984375 3.625 -4.125 Z M 2.625 -0.15625 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-4"> <path d="M 2.609375 -6.625 C 2.625 -6.640625 2.65625 -6.765625 2.65625 -6.78125 C 2.65625 -6.828125 2.609375 -6.921875 2.5 -6.921875 L 1.484375 -6.84375 C 1.109375 -6.8125 1.03125 -6.796875 1.03125 -6.625 C 1.03125 -6.484375 1.171875 -6.484375 1.296875 -6.484375 C 1.78125 -6.484375 1.78125 -6.421875 1.78125 -6.328125 C 1.78125 -6.296875 1.78125 -6.28125 1.71875 -6.09375 L 0.484375 -1.15625 C 0.453125 -1 0.453125 -0.84375 0.453125 -0.84375 C 0.453125 -0.21875 0.953125 0.09375 1.453125 0.09375 C 1.890625 0.09375 2.109375 -0.234375 2.21875 -0.453125 C 2.40625 -0.78125 2.546875 -1.375 2.546875 -1.421875 C 2.546875 -1.484375 2.515625 -1.5625 2.390625 -1.5625 C 2.296875 -1.5625 2.265625 -1.5 2.265625 -1.5 C 2.25 -1.46875 2.203125 -1.28125 2.171875 -1.171875 C 2.03125 -0.59375 1.828125 -0.171875 1.46875 -0.171875 C 1.234375 -0.171875 1.171875 -0.40625 1.171875 -0.640625 C 1.171875 -0.84375 1.203125 -0.953125 1.21875 -1.078125 Z M 2.609375 -6.625 "></path> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-3-0"> </g> <g id="v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-3-1"> <path d="M 4.875 -3.1875 C 4.875 -4.25 4.765625 -4.890625 4.4375 -5.53125 C 4 -6.40625 3.1875 -6.625 2.640625 -6.625 C 1.390625 -6.625 0.921875 -5.6875 0.78125 -5.40625 C 0.421875 -4.6875 0.40625 -3.703125 0.40625 -3.1875 C 0.40625 -2.515625 0.4375 -1.515625 0.921875 -0.71875 C 1.375 0.015625 2.109375 0.203125 2.640625 0.203125 C 3.125 0.203125 3.984375 0.0625 4.46875 -0.921875 C 4.84375 -1.640625 4.875 -2.53125 4.875 -3.1875 Z M 2.640625 -0.0625 C 2.296875 -0.0625 1.609375 -0.234375 1.40625 -1.28125 C 1.296875 -1.84375 1.296875 -2.78125 1.296875 -3.296875 C 1.296875 -3.984375 1.296875 -4.6875 1.40625 -5.234375 C 1.609375 -6.25 2.390625 -6.34375 2.640625 -6.34375 C 2.984375 -6.34375 3.671875 -6.1875 3.875 -5.28125 C 3.984375 -4.71875 3.984375 -3.984375 3.984375 -3.296875 C 3.984375 -2.71875 3.984375 -1.8125 3.875 -1.25 C 3.65625 -0.203125 2.96875 -0.0625 2.640625 -0.0625 Z M 2.640625 -0.0625 "></path> </g> </g> </defs> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-0-1" x="8.368" y="15.681"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-1" x="84.978" y="15.681"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-2" x="8.211" y="78.972"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-1-3" x="84.897" y="78.972"></use> </g> <path 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 -30.121812 30.275687 L 25.327406 30.275687 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.485597 2.870521 C -2.032472 1.147865 -1.020754 0.335365 -0.0012225 -0.0005725 C -1.020754 -0.33651 -2.032472 -1.14901 -2.485597 -2.86776 " transform="matrix(1, 0, 0, -1, 78.05591, 11.94474)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-1" x="47.876" y="8.429"></use> </g> <path 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 -40.465562 20.822562 L -40.465562 -20.345406 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.487514 2.869761 C -2.030482 1.147105 -1.018764 0.334605 0.0007675 -0.0013325 C -1.018764 -0.333364 -2.030482 -1.145864 -2.487514 -2.86852 " transform="matrix(0, 1, 1, 0, 12.02477, 62.80392)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-3-1" x="3.217" y="45.431"></use> </g> <path 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 38.343031 20.822562 L 38.343031 -20.345406 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.487514 2.870665 C -2.030482 1.148009 -1.018764 0.335509 0.0007675 -0.00042875 C -1.018764 -0.336366 -2.030482 -1.148866 -2.487514 -2.867616 " transform="matrix(0, 1, 1, 0, 90.83246, 62.80392)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-2" x="94.348" y="43.397"></use> </g> <path 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 -29.965562 -33.017281 L 25.245375 -33.017281 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.486329 2.869812 C -2.033204 1.147156 -1.021485 0.334656 0.0019525 -0.00128125 C -1.021485 -0.333313 -2.033204 -1.145813 -2.486329 -2.868469 " transform="matrix(1, 0, 0, -1, 77.97461, 75.237)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-3" x="47.222" y="85.671"></use> </g> <path 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 -29.965562 -24.587594 L 26.202406 20.521781 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.485854 2.870538 C -2.030938 1.147412 -1.019733 0.33531 0.000306974 0.00210709 C -1.021609 -0.334558 -2.030394 -1.147724 -2.486942 -2.870347 " transform="matrix(0.77965, -0.62614, -0.62614, -0.77965, 78.87608, 21.54871)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#v_dYX5eFkacf8FLAePSy_-w5pvo=-glyph-2-4" x="43.999" y="40.587"></use> </g> </svg> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>We can picture this as follows.</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>e</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo>∃</mo><mi>ψ</mi><mo>∈</mo><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo>∃</mo><mi>e</mi><mo>′</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>b</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ e &\stackrel{\exists \psi \in p^{-1}(\phi)}{\to} & \exists e'&&& E \\ &&&&& \downarrow^p \\ p(e) &\stackrel{\phi}{\to} & b &&& B } \,. </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a>, then being an isofibration says that whatever stuff <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> forgets can be “transported along isomorphisms.”</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Notice that this definition of isofibration violates the 1-categorical <a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a> where it demands that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">p(\psi)=\phi</annotation></semantics></math> (which includes the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>e</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">p(e') = b</annotation></semantics></math>): this condition is not invariant under <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>. If one changed the definition to involve just an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">p(\psi)\cong\phi</annotation></semantics></math>, then of course, any functor would qualify. But the point of isofibrations is rather to help present the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of categories/groupoids in terms of 1-categorical data. For more on this see below at <em><a href="#AsFibrationsInCanonicalModelStructures">As fibrations in canonical model structures</a></em>.</p> </div> </p> <p><h2 id='section_Equivalent_definition'>Equivalent definition</h2></p> <p>The following may at first seem a little surprising. It says that isofibrations have in fact a stronger lifting property, namely the analogue of that of a <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a> with respect to the <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math> given by the <a class="existingWikiWord" href="/nlab/show/interval+groupoid">interval groupoid</a>. This stronger lifting property is more conceptually fundamental with regard to finding the correct generalisation of isofibrations to higher categories, where ‘correct’ refers for instance to defining the fibrations of a <a class="existingWikiWord" href="/nlab/show/model+structure">model structure</a>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p: E \rightarrow B</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between categories. Then <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> is an isofibration if and only if for every commutative diagram</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="136.277" height="91.84" viewBox="0 0 136.277 91.84"> <defs> <g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-0"> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-1"> <path d="M 7.09375 -6.0625 L 5.6875 -9.328125 C 5.890625 -9.6875 6.328125 -9.75 6.515625 -9.765625 C 6.609375 -9.765625 6.765625 -9.78125 6.765625 -10.03125 C 6.765625 -10.203125 6.625 -10.203125 6.546875 -10.203125 C 6.28125 -10.203125 5.984375 -10.171875 5.734375 -10.171875 L 4.875 -10.171875 C 3.953125 -10.171875 3.296875 -10.203125 3.28125 -10.203125 C 3.171875 -10.203125 3.015625 -10.203125 3.015625 -9.921875 C 3.015625 -9.765625 3.15625 -9.765625 3.34375 -9.765625 C 4.21875 -9.765625 4.265625 -9.625 4.421875 -9.265625 L 6.203125 -5.109375 L 2.953125 -1.640625 C 2.421875 -1.0625 1.78125 -0.5 0.671875 -0.4375 C 0.5 -0.421875 0.375 -0.421875 0.375 -0.15625 C 0.375 -0.109375 0.390625 0 0.546875 0 C 0.765625 0 0.984375 -0.03125 1.1875 -0.03125 L 1.890625 -0.03125 C 2.375 -0.03125 2.890625 0 3.359375 0 C 3.46875 0 3.640625 0 3.640625 -0.265625 C 3.640625 -0.421875 3.546875 -0.4375 3.453125 -0.4375 C 3.15625 -0.46875 2.953125 -0.625 2.953125 -0.859375 C 2.953125 -1.125 3.140625 -1.296875 3.5625 -1.75 L 4.90625 -3.203125 C 5.234375 -3.546875 6.015625 -4.40625 6.34375 -4.734375 L 7.921875 -1.0625 C 7.9375 -1.03125 7.984375 -0.875 7.984375 -0.859375 C 7.984375 -0.734375 7.65625 -0.46875 7.1875 -0.4375 C 7.09375 -0.4375 6.9375 -0.421875 6.9375 -0.15625 C 6.9375 0 7.078125 0 7.15625 0 C 7.40625 0 7.703125 -0.03125 7.96875 -0.03125 L 9.609375 -0.03125 C 9.875 -0.03125 10.15625 0 10.40625 0 C 10.515625 0 10.6875 0 10.6875 -0.28125 C 10.6875 -0.4375 10.53125 -0.4375 10.390625 -0.4375 C 9.5 -0.453125 9.46875 -0.515625 9.21875 -1.078125 L 7.25 -5.703125 L 9.140625 -7.734375 C 9.296875 -7.890625 9.640625 -8.265625 9.765625 -8.40625 C 10.40625 -9.078125 11.015625 -9.6875 12.21875 -9.765625 C 12.375 -9.78125 12.515625 -9.78125 12.515625 -10.03125 C 12.515625 -10.203125 12.390625 -10.203125 12.328125 -10.203125 C 12.109375 -10.203125 11.890625 -10.171875 11.6875 -10.171875 L 11 -10.171875 C 10.515625 -10.171875 10 -10.203125 9.53125 -10.203125 C 9.421875 -10.203125 9.25 -10.203125 9.25 -9.9375 C 9.25 -9.78125 9.34375 -9.765625 9.4375 -9.765625 C 9.671875 -9.734375 9.9375 -9.625 9.9375 -9.328125 L 9.921875 -9.3125 C 9.90625 -9.203125 9.875 -9.046875 9.703125 -8.875 Z M 7.09375 -6.0625 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-2"> <path d="M 10.375 -3.46875 C 10.390625 -3.515625 10.4375 -3.609375 10.4375 -3.671875 C 10.4375 -3.75 10.375 -3.828125 10.296875 -3.828125 C 10.234375 -3.828125 10.203125 -3.8125 10.15625 -3.765625 C 10.125 -3.75 10.125 -3.71875 10 -3.421875 C 9.109375 -1.328125 8.46875 -0.4375 6.078125 -0.4375 L 3.90625 -0.4375 C 3.6875 -0.4375 3.65625 -0.4375 3.5625 -0.453125 C 3.40625 -0.46875 3.390625 -0.5 3.390625 -0.609375 C 3.390625 -0.71875 3.421875 -0.8125 3.453125 -0.9375 L 4.484375 -5.0625 L 5.953125 -5.0625 C 7.125 -5.0625 7.21875 -4.8125 7.21875 -4.359375 C 7.21875 -4.21875 7.21875 -4.078125 7.109375 -3.625 C 7.078125 -3.5625 7.0625 -3.515625 7.0625 -3.46875 C 7.0625 -3.359375 7.140625 -3.3125 7.234375 -3.3125 C 7.359375 -3.3125 7.375 -3.421875 7.4375 -3.625 L 8.296875 -7.09375 C 8.296875 -7.171875 8.234375 -7.25 8.140625 -7.25 C 8 -7.25 7.984375 -7.1875 7.9375 -6.96875 C 7.640625 -5.828125 7.328125 -5.5 6 -5.5 L 4.578125 -5.5 L 5.515625 -9.171875 C 5.640625 -9.6875 5.671875 -9.734375 6.28125 -9.734375 L 8.421875 -9.734375 C 10.265625 -9.734375 10.640625 -9.25 10.640625 -8.109375 C 10.640625 -8.09375 10.640625 -7.671875 10.578125 -7.1875 C 10.5625 -7.125 10.546875 -7.03125 10.546875 -7 C 10.546875 -6.890625 10.625 -6.84375 10.703125 -6.84375 C 10.8125 -6.84375 10.875 -6.90625 10.90625 -7.171875 L 11.21875 -9.78125 C 11.21875 -9.828125 11.25 -9.984375 11.25 -10.015625 C 11.25 -10.171875 11.109375 -10.171875 10.84375 -10.171875 L 3.5625 -10.171875 C 3.265625 -10.171875 3.125 -10.171875 3.125 -9.90625 C 3.125 -9.734375 3.21875 -9.734375 3.484375 -9.734375 C 4.40625 -9.734375 4.40625 -9.640625 4.40625 -9.46875 C 4.40625 -9.390625 4.390625 -9.328125 4.34375 -9.171875 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.125 -0.4375 C 0.84375 -0.4375 0.6875 -0.4375 0.6875 -0.171875 C 0.6875 0 0.78125 0 1.078125 0 L 8.578125 0 C 8.90625 0 8.921875 -0.015625 9.015625 -0.25 Z M 10.375 -3.46875 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-3"> <path d="M 5.5 -9.09375 C 5.625 -9.625 5.65625 -9.765625 6.75 -9.765625 C 7.078125 -9.765625 7.203125 -9.765625 7.203125 -10.046875 C 7.203125 -10.203125 7.03125 -10.203125 6.984375 -10.203125 C 6.71875 -10.203125 6.390625 -10.171875 6.125 -10.171875 L 4.28125 -10.171875 C 3.984375 -10.171875 3.640625 -10.203125 3.34375 -10.203125 C 3.21875 -10.203125 3.0625 -10.203125 3.0625 -9.921875 C 3.0625 -9.765625 3.1875 -9.765625 3.484375 -9.765625 C 4.40625 -9.765625 4.40625 -9.65625 4.40625 -9.484375 C 4.40625 -9.375 4.375 -9.296875 4.34375 -9.15625 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.078125 -0.4375 C 0.75 -0.4375 0.609375 -0.4375 0.609375 -0.15625 C 0.609375 0 0.765625 0 0.84375 0 C 1.109375 0 1.4375 -0.03125 1.703125 -0.03125 L 3.546875 -0.03125 C 3.84375 -0.03125 4.171875 0 4.46875 0 C 4.578125 0 4.765625 0 4.765625 -0.265625 C 4.765625 -0.4375 4.671875 -0.4375 4.34375 -0.4375 C 3.421875 -0.4375 3.421875 -0.546875 3.421875 -0.734375 C 3.421875 -0.765625 3.421875 -0.84375 3.484375 -1.078125 Z M 5.5 -9.09375 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-4"> <path d="M 5.46875 -9.1875 C 5.59375 -9.734375 5.65625 -9.765625 6.25 -9.765625 L 8.1875 -9.765625 C 9.875 -9.765625 9.875 -8.328125 9.875 -8.203125 C 9.875 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.546875 -5.453125 Z M 7.984375 -5.328125 C 9.625 -5.625 11.09375 -6.765625 11.09375 -8.140625 C 11.09375 -9.3125 10.0625 -10.203125 8.375 -10.203125 L 3.578125 -10.203125 C 3.296875 -10.203125 3.171875 -10.203125 3.171875 -9.921875 C 3.171875 -9.765625 3.296875 -9.765625 3.53125 -9.765625 C 4.4375 -9.765625 4.4375 -9.65625 4.4375 -9.484375 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.140625 L 2.359375 -1.109375 C 2.21875 -0.578125 2.203125 -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 6.234375 0 C 8.515625 0 10.28125 -1.734375 10.28125 -3.234375 C 10.28125 -4.46875 9.203125 -5.21875 7.984375 -5.328125 Z M 5.875 -0.4375 L 3.859375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.53125 -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 -5.15625 L 7.265625 -5.15625 C 9.015625 -5.15625 9.015625 -3.515625 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.875 -0.4375 Z M 5.875 -0.4375 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-1-0"> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-1-1"> <path d="M 5.8125 -4.15625 L 2.828125 -7.125 C 2.640625 -7.296875 2.609375 -7.328125 2.5 -7.328125 C 2.34375 -7.328125 2.203125 -7.203125 2.203125 -7.03125 C 2.203125 -6.9375 2.21875 -6.90625 2.390625 -6.734375 L 5.375 -3.734375 L 2.390625 -0.734375 C 2.21875 -0.5625 2.203125 -0.53125 2.203125 -0.4375 C 2.203125 -0.265625 2.34375 -0.140625 2.5 -0.140625 C 2.609375 -0.140625 2.640625 -0.171875 2.828125 -0.34375 L 5.796875 -3.3125 L 8.890625 -0.21875 C 8.921875 -0.203125 9.015625 -0.140625 9.109375 -0.140625 C 9.296875 -0.140625 9.40625 -0.265625 9.40625 -0.4375 C 9.40625 -0.46875 9.40625 -0.515625 9.359375 -0.59375 C 9.34375 -0.625 6.96875 -2.96875 6.234375 -3.734375 L 8.96875 -6.46875 C 9.03125 -6.5625 9.265625 -6.75 9.328125 -6.84375 C 9.34375 -6.875 9.40625 -6.9375 9.40625 -7.03125 C 9.40625 -7.203125 9.296875 -7.328125 9.109375 -7.328125 C 9 -7.328125 8.9375 -7.28125 8.765625 -7.109375 Z M 5.8125 -4.15625 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-0"> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-1"> <path d="M 3.8125 -3.96875 L 4.75 -3.96875 C 4.9375 -3.96875 5.0625 -3.96875 5.0625 -4.15625 C 5.0625 -4.296875 4.9375 -4.296875 4.765625 -4.296875 L 3.875 -4.296875 C 4.03125 -5.1875 4.140625 -5.765625 4.234375 -6.203125 C 4.28125 -6.375 4.3125 -6.484375 4.453125 -6.609375 C 4.578125 -6.71875 4.671875 -6.734375 4.78125 -6.734375 C 4.921875 -6.734375 5.078125 -6.703125 5.21875 -6.625 C 5.15625 -6.609375 5.109375 -6.578125 5.046875 -6.546875 C 4.890625 -6.453125 4.765625 -6.28125 4.765625 -6.078125 C 4.765625 -5.859375 4.9375 -5.71875 5.15625 -5.71875 C 5.453125 -5.71875 5.71875 -5.96875 5.71875 -6.3125 C 5.71875 -6.78125 5.25 -7.015625 4.765625 -7.015625 C 4.421875 -7.015625 3.796875 -6.859375 3.484375 -5.9375 C 3.390625 -5.71875 3.390625 -5.6875 3.125 -4.296875 L 2.375 -4.296875 C 2.171875 -4.296875 2.046875 -4.296875 2.046875 -4.109375 C 2.046875 -3.96875 2.1875 -3.96875 2.359375 -3.96875 L 3.0625 -3.96875 L 2.34375 -0.09375 C 2.15625 0.90625 2 1.75 1.46875 1.75 C 1.453125 1.75 1.234375 1.75 1.046875 1.640625 C 1.5 1.53125 1.5 1.109375 1.5 1.09375 C 1.5 0.875 1.328125 0.734375 1.109375 0.734375 C 0.84375 0.734375 0.546875 0.953125 0.546875 1.328125 C 0.546875 1.75 0.984375 2.03125 1.46875 2.03125 C 2.078125 2.03125 2.5 1.390625 2.625 1.140625 C 2.984375 0.484375 3.21875 -0.75 3.234375 -0.859375 Z M 3.8125 -3.96875 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-2"> <path d="M 2.96875 -6.21875 C 2.96875 -6.4375 2.8125 -6.59375 2.578125 -6.59375 C 2.328125 -6.59375 2.03125 -6.359375 2.03125 -6.0625 C 2.03125 -5.84375 2.1875 -5.6875 2.421875 -5.6875 C 2.6875 -5.6875 2.96875 -5.921875 2.96875 -6.21875 Z M 1.515625 -2.5625 L 0.984375 -1.1875 C 0.921875 -1.03125 0.875 -0.921875 0.875 -0.75 C 0.875 -0.265625 1.25 0.09375 1.78125 0.09375 C 2.75 0.09375 3.15625 -1.296875 3.15625 -1.421875 C 3.15625 -1.53125 3.078125 -1.5625 3.015625 -1.5625 C 2.890625 -1.5625 2.875 -1.484375 2.84375 -1.390625 C 2.609375 -0.59375 2.203125 -0.171875 1.796875 -0.171875 C 1.6875 -0.171875 1.5625 -0.234375 1.5625 -0.5 C 1.5625 -0.734375 1.640625 -0.921875 1.765625 -1.21875 C 1.859375 -1.5 1.96875 -1.765625 2.078125 -2.03125 L 2.375 -2.84375 C 2.46875 -3.078125 2.59375 -3.375 2.59375 -3.546875 C 2.59375 -4.046875 2.1875 -4.390625 1.6875 -4.390625 C 0.71875 -4.390625 0.296875 -3 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.796875 0.625 -2.90625 C 0.890625 -3.84375 1.359375 -4.109375 1.65625 -4.109375 C 1.796875 -4.109375 1.890625 -4.0625 1.890625 -3.78125 C 1.890625 -3.6875 1.890625 -3.546875 1.78125 -3.25 Z M 1.515625 -2.5625 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-3"> <path d="M 5.359375 -6.625 C 5.375 -6.640625 5.40625 -6.765625 5.40625 -6.78125 C 5.40625 -6.828125 5.359375 -6.921875 5.25 -6.921875 C 5.203125 -6.921875 4.890625 -6.890625 4.671875 -6.875 L 4.109375 -6.828125 C 3.890625 -6.8125 3.78125 -6.796875 3.78125 -6.625 C 3.78125 -6.484375 3.921875 -6.484375 4.046875 -6.484375 C 4.53125 -6.484375 4.53125 -6.421875 4.53125 -6.328125 C 4.53125 -6.265625 4.453125 -5.9375 4.390625 -5.734375 L 3.90625 -3.796875 C 3.8125 -3.96875 3.53125 -4.390625 2.921875 -4.390625 C 1.734375 -4.390625 0.421875 -3.015625 0.421875 -1.53125 C 0.421875 -0.5 1.09375 0.09375 1.859375 0.09375 C 2.5 0.09375 3.046875 -0.40625 3.234375 -0.609375 C 3.40625 0.078125 4.09375 0.09375 4.203125 0.09375 C 4.671875 0.09375 4.890625 -0.28125 4.96875 -0.453125 C 5.171875 -0.8125 5.3125 -1.390625 5.3125 -1.421875 C 5.3125 -1.484375 5.28125 -1.5625 5.15625 -1.5625 C 5.03125 -1.5625 5.015625 -1.5 4.953125 -1.25 C 4.8125 -0.703125 4.625 -0.171875 4.234375 -0.171875 C 4 -0.171875 3.921875 -0.375 3.921875 -0.640625 C 3.921875 -0.84375 3.953125 -0.953125 3.984375 -1.078125 Z M 3.234375 -1.078125 C 2.734375 -0.390625 2.21875 -0.171875 1.890625 -0.171875 C 1.4375 -0.171875 1.203125 -0.59375 1.203125 -1.109375 C 1.203125 -1.578125 1.46875 -2.65625 1.6875 -3.09375 C 1.984375 -3.703125 2.46875 -4.109375 2.9375 -4.109375 C 3.578125 -4.109375 3.765625 -3.390625 3.765625 -3.265625 C 3.765625 -3.234375 3.515625 -2.25 3.453125 -2 C 3.328125 -1.53125 3.328125 -1.5 3.234375 -1.078125 Z M 3.234375 -1.078125 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-4"> <path d="M 0.515625 1.203125 C 0.4375 1.53125 0.421875 1.609375 0.015625 1.609375 C -0.125 1.609375 -0.234375 1.609375 -0.234375 1.796875 C -0.234375 1.890625 -0.15625 1.9375 -0.09375 1.9375 C 0 1.9375 0.046875 1.90625 0.78125 1.90625 C 1.5 1.90625 1.703125 1.9375 1.78125 1.9375 C 1.8125 1.9375 1.96875 1.9375 1.96875 1.75 C 1.96875 1.609375 1.828125 1.609375 1.703125 1.609375 C 1.21875 1.609375 1.21875 1.546875 1.21875 1.453125 C 1.21875 1.390625 1.40625 0.671875 1.703125 -0.484375 C 1.828125 -0.265625 2.140625 0.09375 2.6875 0.09375 C 3.90625 0.09375 5.1875 -1.3125 5.1875 -2.765625 C 5.1875 -3.75 4.546875 -4.390625 3.75 -4.390625 C 3.15625 -4.390625 2.671875 -3.984375 2.375 -3.6875 C 2.171875 -4.390625 1.5 -4.390625 1.40625 -4.390625 C 1.046875 -4.390625 0.796875 -4.171875 0.640625 -3.859375 C 0.40625 -3.40625 0.296875 -2.90625 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.78125 0.65625 -3.046875 C 0.78125 -3.546875 0.96875 -4.109375 1.375 -4.109375 C 1.625 -4.109375 1.6875 -3.890625 1.6875 -3.65625 C 1.6875 -3.546875 1.65625 -3.3125 1.640625 -3.234375 Z M 2.359375 -3.078125 C 2.40625 -3.234375 2.40625 -3.265625 2.546875 -3.4375 C 2.9375 -3.890625 3.359375 -4.109375 3.71875 -4.109375 C 4.21875 -4.109375 4.40625 -3.625 4.40625 -3.1875 C 4.40625 -2.8125 4.1875 -1.75 3.890625 -1.15625 C 3.625 -0.625 3.15625 -0.171875 2.6875 -0.171875 C 2 -0.171875 1.84375 -0.953125 1.84375 -1.03125 C 1.84375 -1.046875 1.859375 -1.15625 1.875 -1.1875 Z M 2.359375 -3.078125 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-5"> <path d="M 4.9375 -3.671875 C 4.984375 -3.828125 4.984375 -3.890625 4.984375 -3.890625 C 4.984375 -4.125 4.796875 -4.203125 4.671875 -4.203125 C 4.4375 -4.203125 4.25 -4.03125 4.21875 -3.8125 C 4.140625 -3.96875 3.84375 -4.390625 3.234375 -4.390625 C 2.046875 -4.390625 0.75 -3.078125 0.75 -1.609375 C 0.75 -0.53125 1.46875 0 2.203125 0 C 2.65625 0 3.078125 -0.25 3.40625 -0.53125 L 3.1875 0.359375 C 3.078125 0.78125 3 1.0625 2.609375 1.390625 C 2.1875 1.75 1.8125 1.75 1.5625 1.75 C 1.3125 1.75 1.078125 1.75 0.84375 1.6875 C 1.0625 1.578125 1.15625 1.359375 1.15625 1.203125 C 1.15625 0.953125 0.984375 0.828125 0.765625 0.828125 C 0.515625 0.828125 0.203125 1.03125 0.203125 1.421875 C 0.203125 2 0.984375 2.03125 1.578125 2.03125 C 3 2.03125 3.703125 1.28125 3.859375 0.671875 Z M 3.59375 -1.3125 C 3.53125 -1.03125 3.3125 -0.84375 3.09375 -0.640625 C 3.015625 -0.578125 2.625 -0.28125 2.21875 -0.28125 C 1.828125 -0.28125 1.53125 -0.609375 1.53125 -1.203125 C 1.53125 -1.625 1.78125 -2.71875 2.046875 -3.21875 C 2.375 -3.78125 2.84375 -4.109375 3.234375 -4.109375 C 3.90625 -4.109375 4.09375 -3.375 4.09375 -3.296875 L 4.0625 -3.15625 Z M 3.59375 -1.3125 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-3-0"> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-3-1"> <path d="M 6.53125 -4.578125 C 6.65625 -4.703125 6.671875 -4.75 6.671875 -4.828125 C 6.671875 -4.9375 6.5625 -5.046875 6.4375 -5.046875 C 6.34375 -5.046875 6.3125 -5.015625 6.203125 -4.90625 L 4.109375 -2.828125 L 2.03125 -4.921875 C 1.90625 -5.03125 1.859375 -5.046875 1.78125 -5.046875 C 1.671875 -5.046875 1.5625 -4.953125 1.5625 -4.828125 C 1.5625 -4.734375 1.59375 -4.6875 1.6875 -4.59375 L 3.78125 -2.5 L 1.703125 -0.40625 C 1.578125 -0.28125 1.5625 -0.234375 1.5625 -0.15625 C 1.5625 -0.03125 1.671875 0.0625 1.78125 0.0625 C 1.875 0.0625 1.90625 0.046875 2.015625 -0.0625 L 4.109375 -2.15625 L 6.28125 0.015625 C 6.328125 0.046875 6.390625 0.0625 6.4375 0.0625 C 6.5625 0.0625 6.671875 -0.046875 6.671875 -0.15625 C 6.671875 -0.234375 6.625 -0.28125 6.625 -0.296875 C 6.578125 -0.34375 4.984375 -1.9375 4.4375 -2.5 Z M 6.53125 -4.578125 "></path> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-4-0"> </g> <g id="FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-4-1"> <path d="M 4.875 -3.1875 C 4.875 -4.25 4.765625 -4.890625 4.4375 -5.53125 C 4 -6.40625 3.1875 -6.625 2.640625 -6.625 C 1.390625 -6.625 0.921875 -5.6875 0.78125 -5.40625 C 0.421875 -4.6875 0.40625 -3.703125 0.40625 -3.1875 C 0.40625 -2.515625 0.4375 -1.515625 0.921875 -0.71875 C 1.375 0.015625 2.109375 0.203125 2.640625 0.203125 C 3.125 0.203125 3.984375 0.0625 4.46875 -0.921875 C 4.84375 -1.640625 4.875 -2.53125 4.875 -3.1875 Z M 2.640625 -0.0625 C 2.296875 -0.0625 1.609375 -0.234375 1.40625 -1.28125 C 1.296875 -1.84375 1.296875 -2.78125 1.296875 -3.296875 C 1.296875 -3.984375 1.296875 -4.6875 1.40625 -5.234375 C 1.609375 -6.25 2.390625 -6.34375 2.640625 -6.34375 C 2.984375 -6.34375 3.671875 -6.1875 3.875 -5.28125 C 3.984375 -4.71875 3.984375 -3.984375 3.984375 -3.296875 C 3.984375 -2.71875 3.984375 -1.8125 3.875 -1.25 C 3.65625 -0.203125 2.96875 -0.0625 2.640625 -0.0625 Z M 2.640625 -0.0625 "></path> </g> </g> </defs> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-1" x="22.65" y="19.325"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-2" x="118.053" y="19.325"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-1" x="9.703" y="82.616"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-1-1" x="26.343" y="82.616"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-3" x="41.28675" y="82.616"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-0-4" x="117.972" y="82.616"></use> </g> <path 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 -27.120656 30.897156 L 41.121531 30.897156 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.487389 2.86931 C -2.030357 1.146654 -1.018639 0.334154 0.0008925 -0.00178375 C -1.018639 -0.333815 -2.030357 -1.146315 -2.487389 -2.868971 " transform="matrix(1, 0, 0, -1, 111.13192, 15.58806)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-1" x="73.919" y="10.136"></use> </g> <path 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 -40.464406 21.444031 L -40.464406 -19.720031 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.486295 2.870315 C -2.033195 1.147615 -1.021472 0.335078 -0.00192756 -0.000886229 C -1.021486 -0.332904 -2.033241 -1.149306 -2.48641 -2.868081 " transform="matrix(0.00002, 0.99998, 0.99998, -0.00002, 29.30948, 66.44724)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-2" x="3.217" y="48.805"></use> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-3" x="6.820481" y="48.805"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-3-1" x="12.26825" y="48.805"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-4-1" x="20.50075" y="48.805"></use> </g> <path 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 54.137156 21.444031 L 54.137156 -19.720031 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.486303 2.868874 C -2.033178 1.146218 -1.021459 0.333718 -0.0019275 0.00168625 C -1.021459 -0.334251 -2.033178 -1.146751 -2.486303 -2.869407 " transform="matrix(0, 1, 1, 0, 123.90847, 66.44724)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-4" x="127.423" y="47.041"></use> </g> <path 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 -14.175344 -32.391906 L 41.0395 -32.391906 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.48812 2.868621 C -2.031089 1.145965 -1.01937 0.333465 0.00016125 0.00143375 C -1.01937 -0.334504 -2.031089 -1.147004 -2.48812 -2.86966 " transform="matrix(1, 0, 0, -1, 111.05062, 78.88034)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#FMhO8emZspfOFUyMB4OdZZU84ZI=-glyph-2-5" x="80.746" y="86.686"></use> </g> </svg> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free-standing+isomorphism">free-standing isomorphism</a>, there is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">l: X \times I \rightarrow E</annotation></semantics></math> such that the following diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math> commutes.</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="136.277" height="91.84" viewBox="0 0 136.277 91.84"> <defs> <g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-0"> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-1"> <path d="M 7.09375 -6.0625 L 5.6875 -9.328125 C 5.890625 -9.6875 6.328125 -9.75 6.515625 -9.765625 C 6.609375 -9.765625 6.765625 -9.78125 6.765625 -10.03125 C 6.765625 -10.203125 6.625 -10.203125 6.546875 -10.203125 C 6.28125 -10.203125 5.984375 -10.171875 5.734375 -10.171875 L 4.875 -10.171875 C 3.953125 -10.171875 3.296875 -10.203125 3.28125 -10.203125 C 3.171875 -10.203125 3.015625 -10.203125 3.015625 -9.921875 C 3.015625 -9.765625 3.15625 -9.765625 3.34375 -9.765625 C 4.21875 -9.765625 4.265625 -9.625 4.421875 -9.265625 L 6.203125 -5.109375 L 2.953125 -1.640625 C 2.421875 -1.0625 1.78125 -0.5 0.671875 -0.4375 C 0.5 -0.421875 0.375 -0.421875 0.375 -0.15625 C 0.375 -0.109375 0.390625 0 0.546875 0 C 0.765625 0 0.984375 -0.03125 1.1875 -0.03125 L 1.890625 -0.03125 C 2.375 -0.03125 2.890625 0 3.359375 0 C 3.46875 0 3.640625 0 3.640625 -0.265625 C 3.640625 -0.421875 3.546875 -0.4375 3.453125 -0.4375 C 3.15625 -0.46875 2.953125 -0.625 2.953125 -0.859375 C 2.953125 -1.125 3.140625 -1.296875 3.5625 -1.75 L 4.90625 -3.203125 C 5.234375 -3.546875 6.015625 -4.40625 6.34375 -4.734375 L 7.921875 -1.0625 C 7.9375 -1.03125 7.984375 -0.875 7.984375 -0.859375 C 7.984375 -0.734375 7.65625 -0.46875 7.1875 -0.4375 C 7.09375 -0.4375 6.9375 -0.421875 6.9375 -0.15625 C 6.9375 0 7.078125 0 7.15625 0 C 7.40625 0 7.703125 -0.03125 7.96875 -0.03125 L 9.609375 -0.03125 C 9.875 -0.03125 10.15625 0 10.40625 0 C 10.515625 0 10.6875 0 10.6875 -0.28125 C 10.6875 -0.4375 10.53125 -0.4375 10.390625 -0.4375 C 9.5 -0.453125 9.46875 -0.515625 9.21875 -1.078125 L 7.25 -5.703125 L 9.140625 -7.734375 C 9.296875 -7.890625 9.640625 -8.265625 9.765625 -8.40625 C 10.40625 -9.078125 11.015625 -9.6875 12.21875 -9.765625 C 12.375 -9.78125 12.515625 -9.78125 12.515625 -10.03125 C 12.515625 -10.203125 12.390625 -10.203125 12.328125 -10.203125 C 12.109375 -10.203125 11.890625 -10.171875 11.6875 -10.171875 L 11 -10.171875 C 10.515625 -10.171875 10 -10.203125 9.53125 -10.203125 C 9.421875 -10.203125 9.25 -10.203125 9.25 -9.9375 C 9.25 -9.78125 9.34375 -9.765625 9.4375 -9.765625 C 9.671875 -9.734375 9.9375 -9.625 9.9375 -9.328125 L 9.921875 -9.3125 C 9.90625 -9.203125 9.875 -9.046875 9.703125 -8.875 Z M 7.09375 -6.0625 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-2"> <path d="M 10.375 -3.46875 C 10.390625 -3.515625 10.4375 -3.609375 10.4375 -3.671875 C 10.4375 -3.75 10.375 -3.828125 10.296875 -3.828125 C 10.234375 -3.828125 10.203125 -3.8125 10.15625 -3.765625 C 10.125 -3.75 10.125 -3.71875 10 -3.421875 C 9.109375 -1.328125 8.46875 -0.4375 6.078125 -0.4375 L 3.90625 -0.4375 C 3.6875 -0.4375 3.65625 -0.4375 3.5625 -0.453125 C 3.40625 -0.46875 3.390625 -0.5 3.390625 -0.609375 C 3.390625 -0.71875 3.421875 -0.8125 3.453125 -0.9375 L 4.484375 -5.0625 L 5.953125 -5.0625 C 7.125 -5.0625 7.21875 -4.8125 7.21875 -4.359375 C 7.21875 -4.21875 7.21875 -4.078125 7.109375 -3.625 C 7.078125 -3.5625 7.0625 -3.515625 7.0625 -3.46875 C 7.0625 -3.359375 7.140625 -3.3125 7.234375 -3.3125 C 7.359375 -3.3125 7.375 -3.421875 7.4375 -3.625 L 8.296875 -7.09375 C 8.296875 -7.171875 8.234375 -7.25 8.140625 -7.25 C 8 -7.25 7.984375 -7.1875 7.9375 -6.96875 C 7.640625 -5.828125 7.328125 -5.5 6 -5.5 L 4.578125 -5.5 L 5.515625 -9.171875 C 5.640625 -9.6875 5.671875 -9.734375 6.28125 -9.734375 L 8.421875 -9.734375 C 10.265625 -9.734375 10.640625 -9.25 10.640625 -8.109375 C 10.640625 -8.09375 10.640625 -7.671875 10.578125 -7.1875 C 10.5625 -7.125 10.546875 -7.03125 10.546875 -7 C 10.546875 -6.890625 10.625 -6.84375 10.703125 -6.84375 C 10.8125 -6.84375 10.875 -6.90625 10.90625 -7.171875 L 11.21875 -9.78125 C 11.21875 -9.828125 11.25 -9.984375 11.25 -10.015625 C 11.25 -10.171875 11.109375 -10.171875 10.84375 -10.171875 L 3.5625 -10.171875 C 3.265625 -10.171875 3.125 -10.171875 3.125 -9.90625 C 3.125 -9.734375 3.21875 -9.734375 3.484375 -9.734375 C 4.40625 -9.734375 4.40625 -9.640625 4.40625 -9.46875 C 4.40625 -9.390625 4.390625 -9.328125 4.34375 -9.171875 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.125 -0.4375 C 0.84375 -0.4375 0.6875 -0.4375 0.6875 -0.171875 C 0.6875 0 0.78125 0 1.078125 0 L 8.578125 0 C 8.90625 0 8.921875 -0.015625 9.015625 -0.25 Z M 10.375 -3.46875 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-3"> <path d="M 5.5 -9.09375 C 5.625 -9.625 5.65625 -9.765625 6.75 -9.765625 C 7.078125 -9.765625 7.203125 -9.765625 7.203125 -10.046875 C 7.203125 -10.203125 7.03125 -10.203125 6.984375 -10.203125 C 6.71875 -10.203125 6.390625 -10.171875 6.125 -10.171875 L 4.28125 -10.171875 C 3.984375 -10.171875 3.640625 -10.203125 3.34375 -10.203125 C 3.21875 -10.203125 3.0625 -10.203125 3.0625 -9.921875 C 3.0625 -9.765625 3.1875 -9.765625 3.484375 -9.765625 C 4.40625 -9.765625 4.40625 -9.65625 4.40625 -9.484375 C 4.40625 -9.375 4.375 -9.296875 4.34375 -9.15625 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.078125 -0.4375 C 0.75 -0.4375 0.609375 -0.4375 0.609375 -0.15625 C 0.609375 0 0.765625 0 0.84375 0 C 1.109375 0 1.4375 -0.03125 1.703125 -0.03125 L 3.546875 -0.03125 C 3.84375 -0.03125 4.171875 0 4.46875 0 C 4.578125 0 4.765625 0 4.765625 -0.265625 C 4.765625 -0.4375 4.671875 -0.4375 4.34375 -0.4375 C 3.421875 -0.4375 3.421875 -0.546875 3.421875 -0.734375 C 3.421875 -0.765625 3.421875 -0.84375 3.484375 -1.078125 Z M 5.5 -9.09375 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-4"> <path d="M 5.46875 -9.1875 C 5.59375 -9.734375 5.65625 -9.765625 6.25 -9.765625 L 8.1875 -9.765625 C 9.875 -9.765625 9.875 -8.328125 9.875 -8.203125 C 9.875 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.546875 -5.453125 Z M 7.984375 -5.328125 C 9.625 -5.625 11.09375 -6.765625 11.09375 -8.140625 C 11.09375 -9.3125 10.0625 -10.203125 8.375 -10.203125 L 3.578125 -10.203125 C 3.296875 -10.203125 3.171875 -10.203125 3.171875 -9.921875 C 3.171875 -9.765625 3.296875 -9.765625 3.53125 -9.765625 C 4.4375 -9.765625 4.4375 -9.65625 4.4375 -9.484375 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.140625 L 2.359375 -1.109375 C 2.21875 -0.578125 2.203125 -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 6.234375 0 C 8.515625 0 10.28125 -1.734375 10.28125 -3.234375 C 10.28125 -4.46875 9.203125 -5.21875 7.984375 -5.328125 Z M 5.875 -0.4375 L 3.859375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.53125 -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 -5.15625 L 7.265625 -5.15625 C 9.015625 -5.15625 9.015625 -3.515625 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.875 -0.4375 Z M 5.875 -0.4375 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-1-0"> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-1-1"> <path d="M 5.8125 -4.15625 L 2.828125 -7.125 C 2.640625 -7.296875 2.609375 -7.328125 2.5 -7.328125 C 2.34375 -7.328125 2.203125 -7.203125 2.203125 -7.03125 C 2.203125 -6.9375 2.21875 -6.90625 2.390625 -6.734375 L 5.375 -3.734375 L 2.390625 -0.734375 C 2.21875 -0.5625 2.203125 -0.53125 2.203125 -0.4375 C 2.203125 -0.265625 2.34375 -0.140625 2.5 -0.140625 C 2.609375 -0.140625 2.640625 -0.171875 2.828125 -0.34375 L 5.796875 -3.3125 L 8.890625 -0.21875 C 8.921875 -0.203125 9.015625 -0.140625 9.109375 -0.140625 C 9.296875 -0.140625 9.40625 -0.265625 9.40625 -0.4375 C 9.40625 -0.46875 9.40625 -0.515625 9.359375 -0.59375 C 9.34375 -0.625 6.96875 -2.96875 6.234375 -3.734375 L 8.96875 -6.46875 C 9.03125 -6.5625 9.265625 -6.75 9.328125 -6.84375 C 9.34375 -6.875 9.40625 -6.9375 9.40625 -7.03125 C 9.40625 -7.203125 9.296875 -7.328125 9.109375 -7.328125 C 9 -7.328125 8.9375 -7.28125 8.765625 -7.109375 Z M 5.8125 -4.15625 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-0"> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-1"> <path d="M 3.8125 -3.96875 L 4.75 -3.96875 C 4.9375 -3.96875 5.0625 -3.96875 5.0625 -4.15625 C 5.0625 -4.296875 4.9375 -4.296875 4.765625 -4.296875 L 3.875 -4.296875 C 4.03125 -5.1875 4.140625 -5.765625 4.234375 -6.203125 C 4.28125 -6.375 4.3125 -6.484375 4.453125 -6.609375 C 4.578125 -6.71875 4.671875 -6.734375 4.78125 -6.734375 C 4.921875 -6.734375 5.078125 -6.703125 5.21875 -6.625 C 5.15625 -6.609375 5.109375 -6.578125 5.046875 -6.546875 C 4.890625 -6.453125 4.765625 -6.28125 4.765625 -6.078125 C 4.765625 -5.859375 4.9375 -5.71875 5.15625 -5.71875 C 5.453125 -5.71875 5.71875 -5.96875 5.71875 -6.3125 C 5.71875 -6.78125 5.25 -7.015625 4.765625 -7.015625 C 4.421875 -7.015625 3.796875 -6.859375 3.484375 -5.9375 C 3.390625 -5.71875 3.390625 -5.6875 3.125 -4.296875 L 2.375 -4.296875 C 2.171875 -4.296875 2.046875 -4.296875 2.046875 -4.109375 C 2.046875 -3.96875 2.1875 -3.96875 2.359375 -3.96875 L 3.0625 -3.96875 L 2.34375 -0.09375 C 2.15625 0.90625 2 1.75 1.46875 1.75 C 1.453125 1.75 1.234375 1.75 1.046875 1.640625 C 1.5 1.53125 1.5 1.109375 1.5 1.09375 C 1.5 0.875 1.328125 0.734375 1.109375 0.734375 C 0.84375 0.734375 0.546875 0.953125 0.546875 1.328125 C 0.546875 1.75 0.984375 2.03125 1.46875 2.03125 C 2.078125 2.03125 2.5 1.390625 2.625 1.140625 C 2.984375 0.484375 3.21875 -0.75 3.234375 -0.859375 Z M 3.8125 -3.96875 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-2"> <path d="M 2.96875 -6.21875 C 2.96875 -6.4375 2.8125 -6.59375 2.578125 -6.59375 C 2.328125 -6.59375 2.03125 -6.359375 2.03125 -6.0625 C 2.03125 -5.84375 2.1875 -5.6875 2.421875 -5.6875 C 2.6875 -5.6875 2.96875 -5.921875 2.96875 -6.21875 Z M 1.515625 -2.5625 L 0.984375 -1.1875 C 0.921875 -1.03125 0.875 -0.921875 0.875 -0.75 C 0.875 -0.265625 1.25 0.09375 1.78125 0.09375 C 2.75 0.09375 3.15625 -1.296875 3.15625 -1.421875 C 3.15625 -1.53125 3.078125 -1.5625 3.015625 -1.5625 C 2.890625 -1.5625 2.875 -1.484375 2.84375 -1.390625 C 2.609375 -0.59375 2.203125 -0.171875 1.796875 -0.171875 C 1.6875 -0.171875 1.5625 -0.234375 1.5625 -0.5 C 1.5625 -0.734375 1.640625 -0.921875 1.765625 -1.21875 C 1.859375 -1.5 1.96875 -1.765625 2.078125 -2.03125 L 2.375 -2.84375 C 2.46875 -3.078125 2.59375 -3.375 2.59375 -3.546875 C 2.59375 -4.046875 2.1875 -4.390625 1.6875 -4.390625 C 0.71875 -4.390625 0.296875 -3 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.796875 0.625 -2.90625 C 0.890625 -3.84375 1.359375 -4.109375 1.65625 -4.109375 C 1.796875 -4.109375 1.890625 -4.0625 1.890625 -3.78125 C 1.890625 -3.6875 1.890625 -3.546875 1.78125 -3.25 Z M 1.515625 -2.5625 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-3"> <path d="M 5.359375 -6.625 C 5.375 -6.640625 5.40625 -6.765625 5.40625 -6.78125 C 5.40625 -6.828125 5.359375 -6.921875 5.25 -6.921875 C 5.203125 -6.921875 4.890625 -6.890625 4.671875 -6.875 L 4.109375 -6.828125 C 3.890625 -6.8125 3.78125 -6.796875 3.78125 -6.625 C 3.78125 -6.484375 3.921875 -6.484375 4.046875 -6.484375 C 4.53125 -6.484375 4.53125 -6.421875 4.53125 -6.328125 C 4.53125 -6.265625 4.453125 -5.9375 4.390625 -5.734375 L 3.90625 -3.796875 C 3.8125 -3.96875 3.53125 -4.390625 2.921875 -4.390625 C 1.734375 -4.390625 0.421875 -3.015625 0.421875 -1.53125 C 0.421875 -0.5 1.09375 0.09375 1.859375 0.09375 C 2.5 0.09375 3.046875 -0.40625 3.234375 -0.609375 C 3.40625 0.078125 4.09375 0.09375 4.203125 0.09375 C 4.671875 0.09375 4.890625 -0.28125 4.96875 -0.453125 C 5.171875 -0.8125 5.3125 -1.390625 5.3125 -1.421875 C 5.3125 -1.484375 5.28125 -1.5625 5.15625 -1.5625 C 5.03125 -1.5625 5.015625 -1.5 4.953125 -1.25 C 4.8125 -0.703125 4.625 -0.171875 4.234375 -0.171875 C 4 -0.171875 3.921875 -0.375 3.921875 -0.640625 C 3.921875 -0.84375 3.953125 -0.953125 3.984375 -1.078125 Z M 3.234375 -1.078125 C 2.734375 -0.390625 2.21875 -0.171875 1.890625 -0.171875 C 1.4375 -0.171875 1.203125 -0.59375 1.203125 -1.109375 C 1.203125 -1.578125 1.46875 -2.65625 1.6875 -3.09375 C 1.984375 -3.703125 2.46875 -4.109375 2.9375 -4.109375 C 3.578125 -4.109375 3.765625 -3.390625 3.765625 -3.265625 C 3.765625 -3.234375 3.515625 -2.25 3.453125 -2 C 3.328125 -1.53125 3.328125 -1.5 3.234375 -1.078125 Z M 3.234375 -1.078125 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-4"> <path d="M 0.515625 1.203125 C 0.4375 1.53125 0.421875 1.609375 0.015625 1.609375 C -0.125 1.609375 -0.234375 1.609375 -0.234375 1.796875 C -0.234375 1.890625 -0.15625 1.9375 -0.09375 1.9375 C 0 1.9375 0.046875 1.90625 0.78125 1.90625 C 1.5 1.90625 1.703125 1.9375 1.78125 1.9375 C 1.8125 1.9375 1.96875 1.9375 1.96875 1.75 C 1.96875 1.609375 1.828125 1.609375 1.703125 1.609375 C 1.21875 1.609375 1.21875 1.546875 1.21875 1.453125 C 1.21875 1.390625 1.40625 0.671875 1.703125 -0.484375 C 1.828125 -0.265625 2.140625 0.09375 2.6875 0.09375 C 3.90625 0.09375 5.1875 -1.3125 5.1875 -2.765625 C 5.1875 -3.75 4.546875 -4.390625 3.75 -4.390625 C 3.15625 -4.390625 2.671875 -3.984375 2.375 -3.6875 C 2.171875 -4.390625 1.5 -4.390625 1.40625 -4.390625 C 1.046875 -4.390625 0.796875 -4.171875 0.640625 -3.859375 C 0.40625 -3.40625 0.296875 -2.90625 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.78125 0.65625 -3.046875 C 0.78125 -3.546875 0.96875 -4.109375 1.375 -4.109375 C 1.625 -4.109375 1.6875 -3.890625 1.6875 -3.65625 C 1.6875 -3.546875 1.65625 -3.3125 1.640625 -3.234375 Z M 2.359375 -3.078125 C 2.40625 -3.234375 2.40625 -3.265625 2.546875 -3.4375 C 2.9375 -3.890625 3.359375 -4.109375 3.71875 -4.109375 C 4.21875 -4.109375 4.40625 -3.625 4.40625 -3.1875 C 4.40625 -2.8125 4.1875 -1.75 3.890625 -1.15625 C 3.625 -0.625 3.15625 -0.171875 2.6875 -0.171875 C 2 -0.171875 1.84375 -0.953125 1.84375 -1.03125 C 1.84375 -1.046875 1.859375 -1.15625 1.875 -1.1875 Z M 2.359375 -3.078125 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-5"> <path d="M 4.9375 -3.671875 C 4.984375 -3.828125 4.984375 -3.890625 4.984375 -3.890625 C 4.984375 -4.125 4.796875 -4.203125 4.671875 -4.203125 C 4.4375 -4.203125 4.25 -4.03125 4.21875 -3.8125 C 4.140625 -3.96875 3.84375 -4.390625 3.234375 -4.390625 C 2.046875 -4.390625 0.75 -3.078125 0.75 -1.609375 C 0.75 -0.53125 1.46875 0 2.203125 0 C 2.65625 0 3.078125 -0.25 3.40625 -0.53125 L 3.1875 0.359375 C 3.078125 0.78125 3 1.0625 2.609375 1.390625 C 2.1875 1.75 1.8125 1.75 1.5625 1.75 C 1.3125 1.75 1.078125 1.75 0.84375 1.6875 C 1.0625 1.578125 1.15625 1.359375 1.15625 1.203125 C 1.15625 0.953125 0.984375 0.828125 0.765625 0.828125 C 0.515625 0.828125 0.203125 1.03125 0.203125 1.421875 C 0.203125 2 0.984375 2.03125 1.578125 2.03125 C 3 2.03125 3.703125 1.28125 3.859375 0.671875 Z M 3.59375 -1.3125 C 3.53125 -1.03125 3.3125 -0.84375 3.09375 -0.640625 C 3.015625 -0.578125 2.625 -0.28125 2.21875 -0.28125 C 1.828125 -0.28125 1.53125 -0.609375 1.53125 -1.203125 C 1.53125 -1.625 1.78125 -2.71875 2.046875 -3.21875 C 2.375 -3.78125 2.84375 -4.109375 3.234375 -4.109375 C 3.90625 -4.109375 4.09375 -3.375 4.09375 -3.296875 L 4.0625 -3.15625 Z M 3.59375 -1.3125 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-6"> <path d="M 2.609375 -6.625 C 2.625 -6.640625 2.65625 -6.765625 2.65625 -6.78125 C 2.65625 -6.828125 2.609375 -6.921875 2.5 -6.921875 L 1.484375 -6.84375 C 1.109375 -6.8125 1.03125 -6.796875 1.03125 -6.625 C 1.03125 -6.484375 1.171875 -6.484375 1.296875 -6.484375 C 1.78125 -6.484375 1.78125 -6.421875 1.78125 -6.328125 C 1.78125 -6.296875 1.78125 -6.28125 1.71875 -6.09375 L 0.484375 -1.15625 C 0.453125 -1 0.453125 -0.84375 0.453125 -0.84375 C 0.453125 -0.21875 0.953125 0.09375 1.453125 0.09375 C 1.890625 0.09375 2.109375 -0.234375 2.21875 -0.453125 C 2.40625 -0.78125 2.546875 -1.375 2.546875 -1.421875 C 2.546875 -1.484375 2.515625 -1.5625 2.390625 -1.5625 C 2.296875 -1.5625 2.265625 -1.5 2.265625 -1.5 C 2.25 -1.46875 2.203125 -1.28125 2.171875 -1.171875 C 2.03125 -0.59375 1.828125 -0.171875 1.46875 -0.171875 C 1.234375 -0.171875 1.171875 -0.40625 1.171875 -0.640625 C 1.171875 -0.84375 1.203125 -0.953125 1.21875 -1.078125 Z M 2.609375 -6.625 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-3-0"> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-3-1"> <path d="M 6.53125 -4.578125 C 6.65625 -4.703125 6.671875 -4.75 6.671875 -4.828125 C 6.671875 -4.9375 6.5625 -5.046875 6.4375 -5.046875 C 6.34375 -5.046875 6.3125 -5.015625 6.203125 -4.90625 L 4.109375 -2.828125 L 2.03125 -4.921875 C 1.90625 -5.03125 1.859375 -5.046875 1.78125 -5.046875 C 1.671875 -5.046875 1.5625 -4.953125 1.5625 -4.828125 C 1.5625 -4.734375 1.59375 -4.6875 1.6875 -4.59375 L 3.78125 -2.5 L 1.703125 -0.40625 C 1.578125 -0.28125 1.5625 -0.234375 1.5625 -0.15625 C 1.5625 -0.03125 1.671875 0.0625 1.78125 0.0625 C 1.875 0.0625 1.90625 0.046875 2.015625 -0.0625 L 4.109375 -2.15625 L 6.28125 0.015625 C 6.328125 0.046875 6.390625 0.0625 6.4375 0.0625 C 6.5625 0.0625 6.671875 -0.046875 6.671875 -0.15625 C 6.671875 -0.234375 6.625 -0.28125 6.625 -0.296875 C 6.578125 -0.34375 4.984375 -1.9375 4.4375 -2.5 Z M 6.53125 -4.578125 "></path> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-4-0"> </g> <g id="mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-4-1"> <path d="M 4.875 -3.1875 C 4.875 -4.25 4.765625 -4.890625 4.4375 -5.53125 C 4 -6.40625 3.1875 -6.625 2.640625 -6.625 C 1.390625 -6.625 0.921875 -5.6875 0.78125 -5.40625 C 0.421875 -4.6875 0.40625 -3.703125 0.40625 -3.1875 C 0.40625 -2.515625 0.4375 -1.515625 0.921875 -0.71875 C 1.375 0.015625 2.109375 0.203125 2.640625 0.203125 C 3.125 0.203125 3.984375 0.0625 4.46875 -0.921875 C 4.84375 -1.640625 4.875 -2.53125 4.875 -3.1875 Z M 2.640625 -0.0625 C 2.296875 -0.0625 1.609375 -0.234375 1.40625 -1.28125 C 1.296875 -1.84375 1.296875 -2.78125 1.296875 -3.296875 C 1.296875 -3.984375 1.296875 -4.6875 1.40625 -5.234375 C 1.609375 -6.25 2.390625 -6.34375 2.640625 -6.34375 C 2.984375 -6.34375 3.671875 -6.1875 3.875 -5.28125 C 3.984375 -4.71875 3.984375 -3.984375 3.984375 -3.296875 C 3.984375 -2.71875 3.984375 -1.8125 3.875 -1.25 C 3.65625 -0.203125 2.96875 -0.0625 2.640625 -0.0625 Z M 2.640625 -0.0625 "></path> </g> </g> </defs> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-1" x="22.65" y="19.325"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-2" x="118.053" y="19.325"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-1" x="9.703" y="82.616"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-1-1" x="26.343" y="82.616"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-3" x="41.28675" y="82.616"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-0-4" x="117.972" y="82.616"></use> </g> <path 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 -27.120656 30.897156 L 41.121531 30.897156 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.487389 2.86931 C -2.030357 1.146654 -1.018639 0.334154 0.0008925 -0.00178375 C -1.018639 -0.333815 -2.030357 -1.146315 -2.487389 -2.868971 " transform="matrix(1, 0, 0, -1, 111.13192, 15.58806)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-1" x="73.919" y="10.136"></use> </g> <path 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 -40.464406 21.444031 L -40.464406 -19.720031 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.486295 2.870315 C -2.033195 1.147615 -1.021472 0.335078 -0.00192756 -0.000886229 C -1.021486 -0.332904 -2.033241 -1.149306 -2.48641 -2.868081 " transform="matrix(0.00002, 0.99998, 0.99998, -0.00002, 29.30948, 66.44724)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-2" x="3.217" y="48.805"></use> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-3" x="6.820481" y="48.805"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-3-1" x="12.26825" y="48.805"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-4-1" x="20.50075" y="48.805"></use> </g> <path 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 54.137156 21.444031 L 54.137156 -19.720031 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.486303 2.868874 C -2.033178 1.146218 -1.021459 0.333718 -0.0019275 0.00168625 C -1.021459 -0.334251 -2.033178 -1.146751 -2.486303 -2.869407 " transform="matrix(0, 1, 1, 0, 123.90847, 66.44724)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-4" x="127.423" y="47.041"></use> </g> <path 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 -14.175344 -32.391906 L 41.0395 -32.391906 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.48812 2.868621 C -2.031089 1.145965 -1.01937 0.333465 0.00016125 0.00143375 C -1.01937 -0.334504 -2.031089 -1.147004 -2.48812 -2.86966 " transform="matrix(1, 0, 0, -1, 111.05062, 78.88034)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-5" x="80.746" y="86.686"></use> </g> <path 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 -22.253469 -20.2005 L 41.199656 22.248719 " transform="matrix(1, 0, 0, -1, 69.773, 46.487)"></path> <path 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 -2.485443 2.868471 C -2.034225 1.147222 -1.022492 0.334097 -0.00132848 -0.00123943 C -1.019228 -0.335485 -2.032473 -1.147542 -2.488108 -2.868426 " transform="matrix(0.83112, -0.556, -0.556, -0.83112, 111.17229, 24.1037)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#mU_bIQfxTPabxRod_f1u6XPRiB8=-glyph-2-6" x="72.652" y="41.813"></use> </g> </svg> <p></p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>The “only if” direction is immediate. Let us demonstrate that the “if” direction holds by constructing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math>. To this end, for any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">l_{x}</annotation></semantics></math> be the unique functor such that the following diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{Cat}</annotation></semantics></math> commutes, which exists since <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> is an isofibration.</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="103.202" height="90.825" viewBox="0 0 103.202 90.825"> <defs> <g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-0-0"> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-0-1"> <path d="M 4.296875 -9.578125 C 4.296875 -9.921875 4.296875 -9.9375 4 -9.9375 C 3.640625 -9.53125 2.890625 -8.984375 1.359375 -8.984375 L 1.359375 -8.546875 C 1.703125 -8.546875 2.453125 -8.546875 3.265625 -8.9375 L 3.265625 -1.15625 C 3.265625 -0.609375 3.21875 -0.4375 1.90625 -0.4375 L 1.453125 -0.4375 L 1.453125 0 C 1.859375 -0.03125 3.296875 -0.03125 3.796875 -0.03125 C 4.28125 -0.03125 5.71875 -0.03125 6.125 0 L 6.125 -0.4375 L 5.65625 -0.4375 C 4.34375 -0.4375 4.296875 -0.609375 4.296875 -1.15625 Z M 4.296875 -9.578125 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-0"> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-1"> <path d="M 10.375 -3.46875 C 10.390625 -3.515625 10.4375 -3.609375 10.4375 -3.671875 C 10.4375 -3.75 10.375 -3.828125 10.296875 -3.828125 C 10.234375 -3.828125 10.203125 -3.8125 10.15625 -3.765625 C 10.125 -3.75 10.125 -3.71875 10 -3.421875 C 9.109375 -1.328125 8.46875 -0.4375 6.078125 -0.4375 L 3.90625 -0.4375 C 3.6875 -0.4375 3.65625 -0.4375 3.5625 -0.453125 C 3.40625 -0.46875 3.390625 -0.5 3.390625 -0.609375 C 3.390625 -0.71875 3.421875 -0.8125 3.453125 -0.9375 L 4.484375 -5.0625 L 5.953125 -5.0625 C 7.125 -5.0625 7.21875 -4.8125 7.21875 -4.359375 C 7.21875 -4.21875 7.21875 -4.078125 7.109375 -3.625 C 7.078125 -3.5625 7.0625 -3.515625 7.0625 -3.46875 C 7.0625 -3.359375 7.140625 -3.3125 7.234375 -3.3125 C 7.359375 -3.3125 7.375 -3.421875 7.4375 -3.625 L 8.296875 -7.09375 C 8.296875 -7.171875 8.234375 -7.25 8.140625 -7.25 C 8 -7.25 7.984375 -7.1875 7.9375 -6.96875 C 7.640625 -5.828125 7.328125 -5.5 6 -5.5 L 4.578125 -5.5 L 5.515625 -9.171875 C 5.640625 -9.6875 5.671875 -9.734375 6.28125 -9.734375 L 8.421875 -9.734375 C 10.265625 -9.734375 10.640625 -9.25 10.640625 -8.109375 C 10.640625 -8.09375 10.640625 -7.671875 10.578125 -7.1875 C 10.5625 -7.125 10.546875 -7.03125 10.546875 -7 C 10.546875 -6.890625 10.625 -6.84375 10.703125 -6.84375 C 10.8125 -6.84375 10.875 -6.90625 10.90625 -7.171875 L 11.21875 -9.78125 C 11.21875 -9.828125 11.25 -9.984375 11.25 -10.015625 C 11.25 -10.171875 11.109375 -10.171875 10.84375 -10.171875 L 3.5625 -10.171875 C 3.265625 -10.171875 3.125 -10.171875 3.125 -9.90625 C 3.125 -9.734375 3.21875 -9.734375 3.484375 -9.734375 C 4.40625 -9.734375 4.40625 -9.640625 4.40625 -9.46875 C 4.40625 -9.390625 4.390625 -9.328125 4.34375 -9.171875 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.125 -0.4375 C 0.84375 -0.4375 0.6875 -0.4375 0.6875 -0.171875 C 0.6875 0 0.78125 0 1.078125 0 L 8.578125 0 C 8.90625 0 8.921875 -0.015625 9.015625 -0.25 Z M 10.375 -3.46875 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-2"> <path d="M 5.5 -9.09375 C 5.625 -9.625 5.65625 -9.765625 6.75 -9.765625 C 7.078125 -9.765625 7.203125 -9.765625 7.203125 -10.046875 C 7.203125 -10.203125 7.03125 -10.203125 6.984375 -10.203125 C 6.71875 -10.203125 6.390625 -10.171875 6.125 -10.171875 L 4.28125 -10.171875 C 3.984375 -10.171875 3.640625 -10.203125 3.34375 -10.203125 C 3.21875 -10.203125 3.0625 -10.203125 3.0625 -9.921875 C 3.0625 -9.765625 3.1875 -9.765625 3.484375 -9.765625 C 4.40625 -9.765625 4.40625 -9.65625 4.40625 -9.484375 C 4.40625 -9.375 4.375 -9.296875 4.34375 -9.15625 L 2.328125 -1.109375 C 2.203125 -0.578125 2.171875 -0.4375 1.078125 -0.4375 C 0.75 -0.4375 0.609375 -0.4375 0.609375 -0.15625 C 0.609375 0 0.765625 0 0.84375 0 C 1.109375 0 1.4375 -0.03125 1.703125 -0.03125 L 3.546875 -0.03125 C 3.84375 -0.03125 4.171875 0 4.46875 0 C 4.578125 0 4.765625 0 4.765625 -0.265625 C 4.765625 -0.4375 4.671875 -0.4375 4.34375 -0.4375 C 3.421875 -0.4375 3.421875 -0.546875 3.421875 -0.734375 C 3.421875 -0.765625 3.421875 -0.84375 3.484375 -1.078125 Z M 5.5 -9.09375 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-3"> <path d="M 5.46875 -9.1875 C 5.59375 -9.734375 5.65625 -9.765625 6.25 -9.765625 L 8.1875 -9.765625 C 9.875 -9.765625 9.875 -8.328125 9.875 -8.203125 C 9.875 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.546875 -5.453125 Z M 7.984375 -5.328125 C 9.625 -5.625 11.09375 -6.765625 11.09375 -8.140625 C 11.09375 -9.3125 10.0625 -10.203125 8.375 -10.203125 L 3.578125 -10.203125 C 3.296875 -10.203125 3.171875 -10.203125 3.171875 -9.921875 C 3.171875 -9.765625 3.296875 -9.765625 3.53125 -9.765625 C 4.4375 -9.765625 4.4375 -9.65625 4.4375 -9.484375 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.140625 L 2.359375 -1.109375 C 2.21875 -0.578125 2.203125 -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 6.234375 0 C 8.515625 0 10.28125 -1.734375 10.28125 -3.234375 C 10.28125 -4.46875 9.203125 -5.21875 7.984375 -5.328125 Z M 5.875 -0.4375 L 3.859375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.53125 -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 -5.15625 L 7.265625 -5.15625 C 9.015625 -5.15625 9.015625 -3.515625 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.875 -0.4375 Z M 5.875 -0.4375 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-0"> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-1"> <path d="M 5 -3.984375 C 4.5625 -3.875 4.53125 -3.484375 4.53125 -3.4375 C 4.53125 -3.21875 4.703125 -3.078125 4.921875 -3.078125 C 5.140625 -3.078125 5.484375 -3.234375 5.484375 -3.671875 C 5.484375 -4.234375 4.859375 -4.390625 4.484375 -4.390625 C 4.015625 -4.390625 3.640625 -4.0625 3.40625 -3.671875 C 3.1875 -4.203125 2.671875 -4.390625 2.265625 -4.390625 C 1.171875 -4.390625 0.5625 -3.15625 0.5625 -2.875 C 0.5625 -2.78125 0.640625 -2.734375 0.71875 -2.734375 C 0.84375 -2.734375 0.859375 -2.796875 0.890625 -2.90625 C 1.109375 -3.640625 1.71875 -4.109375 2.234375 -4.109375 C 2.625 -4.109375 2.8125 -3.84375 2.8125 -3.484375 C 2.8125 -3.28125 2.6875 -2.828125 2.609375 -2.5 C 2.546875 -2.21875 2.328125 -1.328125 2.265625 -1.140625 C 2.140625 -0.59375 1.78125 -0.171875 1.328125 -0.171875 C 1.28125 -0.171875 1.03125 -0.171875 0.8125 -0.3125 C 1.28125 -0.421875 1.28125 -0.84375 1.28125 -0.859375 C 1.28125 -1.09375 1.09375 -1.21875 0.875 -1.21875 C 0.609375 -1.21875 0.3125 -1 0.3125 -0.625 C 0.3125 -0.15625 0.8125 0.09375 1.3125 0.09375 C 1.84375 0.09375 2.21875 -0.296875 2.390625 -0.625 C 2.609375 -0.125 3.078125 0.09375 3.546875 0.09375 C 4.640625 0.09375 5.234375 -1.140625 5.234375 -1.421875 C 5.234375 -1.53125 5.15625 -1.5625 5.078125 -1.5625 C 4.96875 -1.5625 4.9375 -1.484375 4.921875 -1.390625 C 4.71875 -0.71875 4.140625 -0.171875 3.5625 -0.171875 C 3.234375 -0.171875 3 -0.40625 3 -0.8125 C 3 -1.015625 3.0625 -1.25 3.203125 -1.796875 C 3.265625 -2.109375 3.484375 -2.984375 3.53125 -3.171875 C 3.671875 -3.6875 4.03125 -4.109375 4.46875 -4.109375 C 4.53125 -4.109375 4.78125 -4.109375 5 -3.984375 Z M 5 -3.984375 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-2"> <path d="M 0.515625 1.203125 C 0.4375 1.53125 0.421875 1.609375 0.015625 1.609375 C -0.125 1.609375 -0.234375 1.609375 -0.234375 1.796875 C -0.234375 1.890625 -0.15625 1.9375 -0.09375 1.9375 C 0 1.9375 0.046875 1.90625 0.78125 1.90625 C 1.5 1.90625 1.703125 1.9375 1.78125 1.9375 C 1.8125 1.9375 1.96875 1.9375 1.96875 1.75 C 1.96875 1.609375 1.828125 1.609375 1.703125 1.609375 C 1.21875 1.609375 1.21875 1.546875 1.21875 1.453125 C 1.21875 1.390625 1.40625 0.671875 1.703125 -0.484375 C 1.828125 -0.265625 2.140625 0.09375 2.6875 0.09375 C 3.90625 0.09375 5.1875 -1.3125 5.1875 -2.765625 C 5.1875 -3.75 4.546875 -4.390625 3.75 -4.390625 C 3.15625 -4.390625 2.671875 -3.984375 2.375 -3.6875 C 2.171875 -4.390625 1.5 -4.390625 1.40625 -4.390625 C 1.046875 -4.390625 0.796875 -4.171875 0.640625 -3.859375 C 0.40625 -3.40625 0.296875 -2.90625 0.296875 -2.875 C 0.296875 -2.78125 0.375 -2.734375 0.453125 -2.734375 C 0.578125 -2.734375 0.59375 -2.78125 0.65625 -3.046875 C 0.78125 -3.546875 0.96875 -4.109375 1.375 -4.109375 C 1.625 -4.109375 1.6875 -3.890625 1.6875 -3.65625 C 1.6875 -3.546875 1.65625 -3.3125 1.640625 -3.234375 Z M 2.359375 -3.078125 C 2.40625 -3.234375 2.40625 -3.265625 2.546875 -3.4375 C 2.9375 -3.890625 3.359375 -4.109375 3.71875 -4.109375 C 4.21875 -4.109375 4.40625 -3.625 4.40625 -3.1875 C 4.40625 -2.8125 4.1875 -1.75 3.890625 -1.15625 C 3.625 -0.625 3.15625 -0.171875 2.6875 -0.171875 C 2 -0.171875 1.84375 -0.953125 1.84375 -1.03125 C 1.84375 -1.046875 1.859375 -1.15625 1.875 -1.1875 Z M 2.359375 -3.078125 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-3"> <path d="M 4.609375 -6.78125 C 4.609375 -6.90625 4.5 -6.921875 4.453125 -6.921875 C 4.3125 -6.921875 4.3125 -6.859375 4.265625 -6.703125 L 3.703125 -4.421875 C 1.9375 -4.34375 0.515625 -3.015625 0.515625 -1.71875 C 0.515625 -0.71875 1.375 0.0625 2.5625 0.109375 C 2.484375 0.421875 2.40625 0.734375 2.328125 1.046875 C 2.21875 1.515625 2.125 1.890625 2.125 1.90625 C 2.125 1.921875 2.125 2.03125 2.265625 2.03125 C 2.40625 2.03125 2.40625 1.984375 2.46875 1.796875 L 2.875 0.125 C 4.640625 0.03125 6.0625 -1.28125 6.0625 -2.578125 C 6.0625 -3.71875 5.046875 -4.359375 4.015625 -4.40625 Z M 3.953125 -4.140625 C 4.6875 -4.078125 5.359375 -3.671875 5.359375 -2.75 C 5.359375 -1.796875 4.671875 -0.34375 2.953125 -0.171875 Z M 2.625 -0.15625 C 2.3125 -0.171875 1.21875 -0.34375 1.21875 -1.546875 C 1.21875 -2.609375 2.03125 -3.984375 3.625 -4.125 Z M 2.625 -0.15625 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-4"> <path d="M 2.609375 -6.625 C 2.625 -6.640625 2.65625 -6.765625 2.65625 -6.78125 C 2.65625 -6.828125 2.609375 -6.921875 2.5 -6.921875 L 1.484375 -6.84375 C 1.109375 -6.8125 1.03125 -6.796875 1.03125 -6.625 C 1.03125 -6.484375 1.171875 -6.484375 1.296875 -6.484375 C 1.78125 -6.484375 1.78125 -6.421875 1.78125 -6.328125 C 1.78125 -6.296875 1.78125 -6.28125 1.71875 -6.09375 L 0.484375 -1.15625 C 0.453125 -1 0.453125 -0.84375 0.453125 -0.84375 C 0.453125 -0.21875 0.953125 0.09375 1.453125 0.09375 C 1.890625 0.09375 2.109375 -0.234375 2.21875 -0.453125 C 2.40625 -0.78125 2.546875 -1.375 2.546875 -1.421875 C 2.546875 -1.484375 2.515625 -1.5625 2.390625 -1.5625 C 2.296875 -1.5625 2.265625 -1.5 2.265625 -1.5 C 2.25 -1.46875 2.203125 -1.28125 2.171875 -1.171875 C 2.03125 -0.59375 1.828125 -0.171875 1.46875 -0.171875 C 1.234375 -0.171875 1.171875 -0.40625 1.171875 -0.640625 C 1.171875 -0.84375 1.203125 -0.953125 1.21875 -1.078125 Z M 2.609375 -6.625 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-3-0"> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-3-1"> <path d="M 4.875 -3.1875 C 4.875 -4.25 4.765625 -4.890625 4.4375 -5.53125 C 4 -6.40625 3.1875 -6.625 2.640625 -6.625 C 1.390625 -6.625 0.921875 -5.6875 0.78125 -5.40625 C 0.421875 -4.6875 0.40625 -3.703125 0.40625 -3.1875 C 0.40625 -2.515625 0.4375 -1.515625 0.921875 -0.71875 C 1.375 0.015625 2.109375 0.203125 2.640625 0.203125 C 3.125 0.203125 3.984375 0.0625 4.46875 -0.921875 C 4.84375 -1.640625 4.875 -2.53125 4.875 -3.1875 Z M 2.640625 -0.0625 C 2.296875 -0.0625 1.609375 -0.234375 1.40625 -1.28125 C 1.296875 -1.84375 1.296875 -2.78125 1.296875 -3.296875 C 1.296875 -3.984375 1.296875 -4.6875 1.40625 -5.234375 C 1.609375 -6.25 2.390625 -6.34375 2.640625 -6.34375 C 2.984375 -6.34375 3.671875 -6.1875 3.875 -5.28125 C 3.984375 -4.71875 3.984375 -3.984375 3.984375 -3.296875 C 3.984375 -2.71875 3.984375 -1.8125 3.875 -1.25 C 3.65625 -0.203125 2.96875 -0.0625 2.640625 -0.0625 Z M 2.640625 -0.0625 "></path> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-4-0"> </g> <g id="ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-4-1"> <path d="M 4.203125 -2.9375 C 3.9375 -2.859375 3.890625 -2.640625 3.890625 -2.53125 C 3.890625 -2.296875 4.078125 -2.234375 4.1875 -2.234375 C 4.40625 -2.234375 4.609375 -2.421875 4.609375 -2.71875 C 4.609375 -3.109375 4.171875 -3.296875 3.796875 -3.296875 C 3.296875 -3.296875 3 -2.90625 2.921875 -2.765625 C 2.828125 -2.953125 2.546875 -3.296875 1.984375 -3.296875 C 1.125 -3.296875 0.609375 -2.40625 0.609375 -2.140625 C 0.609375 -2.109375 0.640625 -2.03125 0.75 -2.03125 C 0.84375 -2.03125 0.875 -2.078125 0.890625 -2.15625 C 1.078125 -2.75 1.59375 -3.046875 1.953125 -3.046875 C 2.328125 -3.046875 2.4375 -2.8125 2.4375 -2.5625 C 2.4375 -2.46875 2.4375 -2.40625 2.375 -2.171875 C 2.203125 -1.484375 2.03125 -0.796875 2 -0.703125 C 1.890625 -0.421875 1.625 -0.171875 1.3125 -0.171875 C 1.265625 -0.171875 1.046875 -0.171875 0.875 -0.28125 C 1.171875 -0.375 1.203125 -0.625 1.203125 -0.6875 C 1.203125 -0.875 1.046875 -0.984375 0.90625 -0.984375 C 0.6875 -0.984375 0.46875 -0.8125 0.46875 -0.515625 C 0.46875 -0.078125 0.9375 0.078125 1.296875 0.078125 C 1.71875 0.078125 2.03125 -0.21875 2.171875 -0.453125 C 2.3125 -0.140625 2.671875 0.078125 3.09375 0.078125 C 3.984375 0.078125 4.46875 -0.828125 4.46875 -1.078125 C 4.46875 -1.09375 4.46875 -1.1875 4.328125 -1.1875 C 4.234375 -1.1875 4.21875 -1.125 4.1875 -1.0625 C 3.96875 -0.40625 3.4375 -0.171875 3.125 -0.171875 C 2.84375 -0.171875 2.640625 -0.34375 2.640625 -0.65625 C 2.640625 -0.796875 2.6875 -0.953125 2.75 -1.21875 L 2.984375 -2.1875 C 3.0625 -2.484375 3.09375 -2.609375 3.25 -2.796875 C 3.359375 -2.90625 3.546875 -3.046875 3.78125 -3.046875 C 3.8125 -3.046875 4.046875 -3.046875 4.203125 -2.9375 Z M 4.203125 -2.9375 "></path> </g> </g> </defs> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-0-1" x="8.368" y="15.681"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-1" x="84.978" y="15.681"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-2" x="8.211" y="78.972"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-1-3" x="84.897" y="78.972"></use> </g> <path 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 -30.121812 30.275687 L 25.327406 30.275687 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.485597 2.870521 C -2.032472 1.147865 -1.020754 0.335365 -0.0012225 -0.0005725 C -1.020754 -0.33651 -2.032472 -1.14901 -2.485597 -2.86776 " transform="matrix(1, 0, 0, -1, 78.05591, 11.94474)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-1" x="47.352" y="8.429"></use> </g> <path 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 -40.465562 20.822562 L -40.465562 -20.345406 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.487514 2.869761 C -2.030482 1.147105 -1.018764 0.334605 0.0007675 -0.0013325 C -1.018764 -0.333364 -2.030482 -1.145864 -2.487514 -2.86852 " transform="matrix(0, 1, 1, 0, 12.02477, 62.80392)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-3-1" x="3.217" y="45.431"></use> </g> <path 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 38.343031 20.822562 L 38.343031 -20.345406 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.487514 2.870665 C -2.030482 1.148009 -1.018764 0.335509 0.0007675 -0.00042875 C -1.018764 -0.336366 -2.030482 -1.148866 -2.487514 -2.867616 " transform="matrix(0, 1, 1, 0, 90.83246, 62.80392)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-2" x="94.348" y="43.397"></use> </g> <path 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 -29.965562 -33.017281 L 25.245375 -33.017281 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.486329 2.869812 C -2.033204 1.147156 -1.021485 0.334656 0.0019525 -0.00128125 C -1.021485 -0.333313 -2.033204 -1.145813 -2.486329 -2.868469 " transform="matrix(1, 0, 0, -1, 77.97461, 75.237)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-3" x="44.303" y="85.671"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-4-1" x="50.5955" y="86.916"></use> </g> <path 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 -29.965562 -24.587594 L 26.202406 20.521781 " transform="matrix(1, 0, 0, -1, 52.489, 42.221)"></path> <path 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 -2.485854 2.870538 C -2.030938 1.147412 -1.019733 0.33531 0.000306974 0.00210709 C -1.021609 -0.334558 -2.030394 -1.147724 -2.486942 -2.870347 " transform="matrix(0.77965, -0.62614, -0.62614, -0.77965, 78.87608, 21.54871)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-2-4" x="38.28" y="39.342"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ALuxkUaIVwz2mZ1iUqfIFVeEqmQ=-glyph-4-1" x="41.44" y="40.587"></use> </g> </svg> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{x}</annotation></semantics></math> is the functor representing the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x, i)</annotation></semantics></math> of <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>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is the arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \rightarrow 1</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>.</p> <ul> <li>For any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l(0)</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math>, define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l(x, i)</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l_{x}(i)</annotation></semantics></math>, define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">l\left(x,i^{-1}\right)</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>x</mi></msub><mrow><mo>(</mo><msup><mi>x</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">l_{x}\left(x^{-1}\right)</annotation></semantics></math>, and define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l(1)</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l_{x}(1)</annotation></semantics></math>.</li> <li>For any arrow <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>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">r: x \rightarrow x'</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l(r,0)</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(r)</annotation></semantics></math>, and define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l(r, 1)</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mrow><mi>x</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>l</mi> <mi>x</mi></msub><mrow><mo>(</mo><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">l_{x'}(i) \circ f(r) \circ l_{x}\left(i^{-1}\right)</annotation></semantics></math>.</li> </ul> <p>It is immediately checked that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math> is well-defined, is indeed a functor, and fits into the required commutative diagram.</p> <p></p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <p>Isofibrations have a number of good properties. For example, any <a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict pullback</a> of an isofibration is also a <a class="existingWikiWord" href="/nlab/show/2-limit">weak pullback</a>. (This is also explained by the role of isofibrations as the fibrations in the <a class="existingWikiWord" href="/nlab/show/canonical+model+structures">canonical model structures</a>, see <a href="#AsFibrationsInCanonicalModelStructures">below</a>.)</p> <p> <div class='num_remark' id='GrothendieckFibrationsAreIsofibrations'> <h6>Example</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a> or opfibration is an isofibration (take the lifts to be the Cartesian lifts), but not in general conversely, unless <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 <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>.</p> </div> </p> <p> <div class='num_remark' id='GrothIsIsoFibBetweenGroupoids'> <h6>Example</h6> <p>For <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>,</mo><mi>ℬ</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\mathcal{E}, \mathcal{B} \,\in\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℰ</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">P \,\colon\, \mathcal{E} \to \mathcal{B}</annotation></semantics></math> the following are equivalent:</p> <ol> <li> <p><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> is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>,</p> </li> <li> <p><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> is an <a class="existingWikiWord" href="/nlab/show/isofibration">isofibration</a>.</p> </li> </ol> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>The implication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>IsoFib</mi><mo>⇒</mo><mi>GrothFib</mi></mrow><annotation encoding="application/x-tex">IsoFib \Rightarrow GrothFib</annotation></semantics></math> is Ex. <a class="maruku-ref" href="#GrothendieckFibrationsAreIsofibrations"></a>.</p> <p>Conversely, if <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> is an isofibration, it is sufficient to show that any of the lifts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> it provides is a <a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a>. But by assumption the lift is again an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, and all isomorphisms are Cartesian.</p> <p>More explicitly (in the notation <a href="Grothendieck+fibration#CartesianMorphism">there</a>) <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="202.682pt" height="134.33pt" viewBox="0 0 202.682 134.33" version="1.2"> <defs> <g> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-1"> <path style="stroke:none;" d="M 1.515625 -0.96875 C 2.015625 -1.546875 2.4375 -1.921875 3.03125 -2.453125 C 3.75 -3.0625 4.0625 -3.359375 4.21875 -3.546875 C 5.046875 -4.359375 5.46875 -5.046875 5.46875 -5.15625 C 5.46875 -5.25 5.375 -5.25 5.34375 -5.25 C 5.265625 -5.25 5.25 -5.203125 5.1875 -5.109375 C 4.890625 -4.609375 4.609375 -4.359375 4.296875 -4.359375 C 4.046875 -4.359375 3.90625 -4.453125 3.6875 -4.75 C 3.4375 -5.046875 3.234375 -5.25 2.890625 -5.25 C 2.015625 -5.25 1.5 -4.15625 1.5 -3.90625 C 1.5 -3.875 1.515625 -3.8125 1.609375 -3.8125 C 1.71875 -3.8125 1.71875 -3.859375 1.765625 -3.9375 C 1.96875 -4.40625 2.53125 -4.5 2.765625 -4.5 C 3.015625 -4.5 3.25 -4.40625 3.5 -4.296875 C 3.953125 -4.109375 4.140625 -4.109375 4.25 -4.109375 C 4.34375 -4.109375 4.390625 -4.109375 4.453125 -4.125 C 4.0625 -3.65625 3.40625 -3.09375 2.875 -2.609375 L 1.671875 -1.5 C 0.953125 -0.765625 0.515625 -0.0625 0.515625 0.03125 C 0.515625 0.09375 0.578125 0.125 0.640625 0.125 C 0.71875 0.125 0.71875 0.109375 0.8125 -0.03125 C 1 -0.328125 1.375 -0.765625 1.8125 -0.765625 C 2.0625 -0.765625 2.1875 -0.6875 2.421875 -0.390625 C 2.65625 -0.125 2.859375 0.125 3.234375 0.125 C 4.40625 0.125 5.0625 -1.390625 5.0625 -1.671875 C 5.0625 -1.71875 5.046875 -1.78125 4.9375 -1.78125 C 4.84375 -1.78125 4.828125 -1.734375 4.796875 -1.609375 C 4.53125 -0.921875 3.828125 -0.625 3.359375 -0.625 C 3.109375 -0.625 2.875 -0.71875 2.625 -0.828125 C 2.15625 -1.015625 2.015625 -1.015625 1.859375 -1.015625 C 1.75 -1.015625 1.609375 -1.015625 1.515625 -0.96875 Z M 1.515625 -0.96875 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-2"> <path style="stroke:none;" d="M 5.640625 -4.84375 C 5.25 -4.78125 5.109375 -4.5 5.109375 -4.265625 C 5.109375 -3.984375 5.34375 -3.890625 5.5 -3.890625 C 5.859375 -3.890625 6.109375 -4.203125 6.109375 -4.515625 C 6.109375 -5.015625 5.546875 -5.25 5.046875 -5.25 C 4.3125 -5.25 3.90625 -4.53125 3.8125 -4.296875 C 3.53125 -5.203125 2.796875 -5.25 2.578125 -5.25 C 1.375 -5.25 0.71875 -3.6875 0.71875 -3.421875 C 0.71875 -3.375 0.765625 -3.3125 0.859375 -3.3125 C 0.953125 -3.3125 0.96875 -3.390625 1 -3.4375 C 1.40625 -4.75 2.203125 -5 2.546875 -5 C 3.078125 -5 3.1875 -4.5 3.1875 -4.21875 C 3.1875 -3.953125 3.109375 -3.6875 2.96875 -3.109375 L 2.5625 -1.484375 C 2.390625 -0.765625 2.046875 -0.125 1.421875 -0.125 C 1.359375 -0.125 1.0625 -0.125 0.8125 -0.28125 C 1.234375 -0.359375 1.328125 -0.71875 1.328125 -0.859375 C 1.328125 -1.09375 1.15625 -1.234375 0.921875 -1.234375 C 0.640625 -1.234375 0.328125 -0.984375 0.328125 -0.609375 C 0.328125 -0.109375 0.890625 0.125 1.40625 0.125 C 1.96875 0.125 2.375 -0.328125 2.625 -0.828125 C 2.8125 -0.125 3.40625 0.125 3.859375 0.125 C 5.0625 0.125 5.703125 -1.4375 5.703125 -1.703125 C 5.703125 -1.765625 5.65625 -1.8125 5.59375 -1.8125 C 5.484375 -1.8125 5.46875 -1.75 5.4375 -1.65625 C 5.109375 -0.609375 4.421875 -0.125 3.890625 -0.125 C 3.46875 -0.125 3.25 -0.421875 3.25 -0.921875 C 3.25 -1.171875 3.296875 -1.375 3.484375 -2.15625 L 3.90625 -3.765625 C 4.078125 -4.484375 4.484375 -5 5.03125 -5 C 5.046875 -5 5.390625 -5 5.640625 -4.84375 Z M 5.640625 -4.84375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-3"> <path style="stroke:none;" d="M 3.125 1.328125 C 2.8125 1.78125 2.34375 2.1875 1.765625 2.1875 C 1.609375 2.1875 1.046875 2.171875 0.875 1.609375 C 0.90625 1.625 0.96875 1.625 0.984375 1.625 C 1.34375 1.625 1.578125 1.3125 1.578125 1.046875 C 1.578125 0.765625 1.359375 0.671875 1.171875 0.671875 C 0.984375 0.671875 0.578125 0.828125 0.578125 1.40625 C 0.578125 2.015625 1.078125 2.421875 1.765625 2.421875 C 2.953125 2.421875 4.15625 1.328125 4.484375 0.015625 L 5.640625 -4.625 C 5.65625 -4.6875 5.6875 -4.75 5.6875 -4.828125 C 5.6875 -5 5.546875 -5.125 5.359375 -5.125 C 5.25 -5.125 5 -5.078125 4.90625 -4.71875 L 4.03125 -1.21875 C 3.96875 -1.015625 3.96875 -0.984375 3.875 -0.859375 C 3.640625 -0.515625 3.25 -0.125 2.671875 -0.125 C 2.015625 -0.125 1.953125 -0.765625 1.953125 -1.09375 C 1.953125 -1.765625 2.265625 -2.6875 2.59375 -3.546875 C 2.71875 -3.890625 2.796875 -4.0625 2.796875 -4.296875 C 2.796875 -4.796875 2.4375 -5.25 1.859375 -5.25 C 0.765625 -5.25 0.328125 -3.515625 0.328125 -3.421875 C 0.328125 -3.375 0.375 -3.3125 0.453125 -3.3125 C 0.5625 -3.3125 0.578125 -3.359375 0.625 -3.53125 C 0.90625 -4.53125 1.359375 -5 1.8125 -5 C 1.921875 -5 2.125 -5 2.125 -4.609375 C 2.125 -4.296875 2 -3.953125 1.8125 -3.5 C 1.234375 -1.953125 1.234375 -1.5625 1.234375 -1.265625 C 1.234375 -0.140625 2.046875 0.125 2.640625 0.125 C 2.984375 0.125 3.40625 0.015625 3.828125 -0.421875 L 3.84375 -0.421875 C 3.65625 0.28125 3.546875 0.75 3.125 1.328125 Z M 3.125 1.328125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-4"> <path style="stroke:none;" d="M 3.515625 -3.78125 L 5.515625 -3.78125 C 7.15625 -3.78125 8.796875 -5 8.796875 -6.34375 C 8.796875 -7.28125 8.015625 -8.125 6.515625 -8.125 L 2.84375 -8.125 C 2.609375 -8.125 2.515625 -8.125 2.515625 -7.890625 C 2.515625 -7.78125 2.609375 -7.78125 2.796875 -7.78125 C 3.515625 -7.78125 3.515625 -7.6875 3.515625 -7.546875 C 3.515625 -7.53125 3.515625 -7.453125 3.46875 -7.28125 L 1.859375 -0.875 C 1.765625 -0.46875 1.734375 -0.34375 0.90625 -0.34375 C 0.671875 -0.34375 0.5625 -0.34375 0.5625 -0.125 C 0.5625 0 0.671875 0 0.734375 0 C 0.96875 0 1.203125 -0.03125 1.421875 -0.03125 L 2.8125 -0.03125 C 3.046875 -0.03125 3.296875 0 3.515625 0 C 3.609375 0 3.75 0 3.75 -0.21875 C 3.75 -0.34375 3.640625 -0.34375 3.453125 -0.34375 C 2.75 -0.34375 2.734375 -0.421875 2.734375 -0.546875 C 2.734375 -0.609375 2.75 -0.6875 2.765625 -0.75 Z M 4.375 -7.3125 C 4.484375 -7.75 4.53125 -7.78125 5 -7.78125 L 6.171875 -7.78125 C 7.0625 -7.78125 7.796875 -7.484375 7.796875 -6.59375 C 7.796875 -6.296875 7.640625 -5.28125 7.09375 -4.734375 C 6.890625 -4.515625 6.328125 -4.0625 5.25 -4.0625 L 3.5625 -4.0625 Z M 4.375 -7.3125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph1-1"> <path style="stroke:none;" d="M 2.84375 -4.3125 C 0.46875 -3.03125 0.328125 -1.5 0.328125 -1.21875 C 0.328125 -0.3125 1.21875 0.265625 2.34375 0.265625 C 4.328125 0.265625 5.921875 -1.609375 5.921875 -1.859375 C 5.921875 -1.9375 5.859375 -1.953125 5.796875 -1.953125 C 5.65625 -1.953125 5.203125 -1.796875 4.9375 -1.421875 C 4.6875 -1.078125 4.1875 -0.390625 3.125 -0.390625 C 2.28125 -0.390625 1.34375 -0.828125 1.34375 -1.71875 C 1.34375 -2.3125 2.015625 -4.0625 3.890625 -4.140625 C 4.453125 -4.15625 4.859375 -4.578125 4.859375 -4.703125 C 4.859375 -4.78125 4.796875 -4.796875 4.75 -4.796875 C 3.0625 -4.859375 2.734375 -5.6875 2.734375 -6.15625 C 2.734375 -6.4375 2.90625 -7.734375 4.578125 -7.734375 C 4.796875 -7.734375 5.703125 -7.6875 5.703125 -7.078125 C 5.703125 -6.890625 5.609375 -6.71875 5.5625 -6.640625 C 5.546875 -6.609375 5.5 -6.546875 5.5 -6.515625 C 5.5 -6.4375 5.59375 -6.4375 5.625 -6.4375 C 5.984375 -6.4375 6.71875 -6.890625 6.71875 -7.578125 C 6.71875 -8.28125 5.890625 -8.390625 5.359375 -8.390625 C 3.75 -8.390625 1.71875 -7.09375 1.71875 -5.640625 C 1.71875 -4.953125 2.265625 -4.515625 2.84375 -4.3125 Z M 2.84375 -4.3125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph1-2"> <path style="stroke:none;" d="M 3.328125 -8.078125 C 3.359375 -8.203125 3.359375 -8.234375 3.359375 -8.265625 C 3.359375 -8.34375 3.328125 -8.390625 3.21875 -8.390625 C 3.09375 -8.390625 2.734375 -8.203125 2.5 -8.078125 C 1.953125 -7.8125 1.5 -7.59375 1.5 -7.375 C 1.5 -7.296875 1.578125 -7.296875 1.625 -7.296875 C 1.765625 -7.296875 2.078125 -7.46875 2.3125 -7.578125 C 2.171875 -6.40625 1.875 -4.890625 1.671875 -3.859375 C 1.21875 -1.84375 1 -1.125 0.421875 0.03125 C 0.359375 0.140625 0.359375 0.171875 0.359375 0.171875 C 0.359375 0.265625 0.453125 0.265625 0.46875 0.265625 C 0.671875 0.265625 1.171875 0.046875 1.359375 -0.28125 C 1.515625 -0.5625 1.96875 -1.421875 2.3125 -2.8125 C 2.578125 -3.828125 3.03125 -5.390625 3.875 -6.609375 C 4.421875 -7.390625 4.921875 -7.734375 5.671875 -7.734375 C 6.3125 -7.734375 6.890625 -7.328125 6.890625 -6.6875 C 6.890625 -5.640625 5.765625 -5.265625 4.53125 -4.84375 C 4.390625 -4.796875 3.5625 -4.515625 3.5625 -4.21875 C 3.5625 -4.15625 3.65625 -4.140625 3.6875 -4.140625 C 3.734375 -4.140625 4 -4.1875 4.25 -4.1875 C 5.453125 -4.1875 6.4375 -3.46875 6.4375 -2.34375 C 6.4375 -0.875 5.171875 -0.390625 4.09375 -0.390625 C 3.171875 -0.390625 2.734375 -0.890625 2.578125 -1.0625 C 2.53125 -1.125 2.515625 -1.140625 2.421875 -1.140625 C 2.171875 -1.140625 1.5625 -0.765625 1.5625 -0.5625 C 1.5625 -0.515625 2.015625 0.265625 3.3125 0.265625 C 4.984375 0.265625 7.4375 -0.921875 7.4375 -2.84375 C 7.4375 -3.84375 6.78125 -4.59375 5.5625 -4.8125 C 6.515625 -5.21875 7.90625 -6.078125 7.90625 -7.1875 C 7.90625 -7.890625 7.34375 -8.390625 6.453125 -8.390625 C 6.046875 -8.390625 4.53125 -8.28125 3.09375 -6.5 Z M 3.328125 -8.078125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-1"> <path style="stroke:none;" d="M 3.859375 2.890625 C 3.859375 2.859375 3.859375 2.828125 3.65625 2.625 C 2.46875 1.421875 1.8125 -0.53125 1.8125 -2.953125 C 1.8125 -5.265625 2.359375 -7.25 3.75 -8.65625 C 3.859375 -8.765625 3.859375 -8.78125 3.859375 -8.828125 C 3.859375 -8.890625 3.8125 -8.921875 3.75 -8.921875 C 3.609375 -8.921875 2.625 -8.0625 2.046875 -6.890625 C 1.4375 -5.703125 1.171875 -4.421875 1.171875 -2.953125 C 1.171875 -1.90625 1.328125 -0.484375 1.953125 0.78125 C 2.65625 2.21875 3.625 2.984375 3.75 2.984375 C 3.8125 2.984375 3.859375 2.953125 3.859375 2.890625 Z M 3.859375 2.890625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-2"> <path style="stroke:none;" d="M 3.359375 -2.953125 C 3.359375 -3.859375 3.234375 -5.34375 2.5625 -6.71875 C 1.859375 -8.140625 0.890625 -8.921875 0.765625 -8.921875 C 0.71875 -8.921875 0.65625 -8.890625 0.65625 -8.828125 C 0.65625 -8.78125 0.65625 -8.765625 0.859375 -8.5625 C 2.046875 -7.359375 2.71875 -5.390625 2.71875 -2.96875 C 2.71875 -0.671875 2.15625 1.3125 0.765625 2.71875 C 0.65625 2.828125 0.65625 2.859375 0.65625 2.890625 C 0.65625 2.953125 0.71875 2.984375 0.765625 2.984375 C 0.921875 2.984375 1.890625 2.125 2.46875 0.96875 C 3.078125 -0.25 3.359375 -1.53125 3.359375 -2.953125 Z M 3.359375 -2.953125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph3-1"> <path style="stroke:none;" d="M 4.640625 -5.171875 C 4.6875 -5.28125 4.6875 -5.296875 4.6875 -5.328125 C 4.6875 -5.421875 4.609375 -5.515625 4.5 -5.515625 C 4.375 -5.515625 4.34375 -5.421875 4.3125 -5.328125 L 3.609375 -3.609375 L 1.078125 -3.609375 L 0.375 -5.328125 C 0.34375 -5.40625 0.3125 -5.515625 0.1875 -5.515625 C 0.09375 -5.515625 0 -5.421875 0 -5.328125 C 0 -5.296875 0 -5.28125 0.0625 -5.171875 L 2.140625 -0.015625 C 2.171875 0.0625 2.21875 0.171875 2.34375 0.171875 C 2.453125 0.171875 2.5 0.078125 2.53125 -0.015625 Z M 1.234375 -3.234375 L 3.453125 -3.234375 L 2.34375 -0.515625 Z M 1.234375 -3.234375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph3-2"> <path style="stroke:none;" d="M 4.203125 -5.203125 C 4.203125 -5.46875 4.171875 -5.515625 3.90625 -5.515625 L 0.765625 -5.515625 C 0.640625 -5.515625 0.46875 -5.515625 0.46875 -5.328125 C 0.46875 -5.140625 0.640625 -5.140625 0.765625 -5.140625 L 3.84375 -5.140625 L 3.84375 -2.9375 L 0.890625 -2.9375 C 0.765625 -2.9375 0.59375 -2.9375 0.59375 -2.75 C 0.59375 -2.578125 0.765625 -2.578125 0.890625 -2.578125 L 3.84375 -2.578125 L 3.84375 -0.359375 L 0.765625 -0.359375 C 0.640625 -0.359375 0.46875 -0.359375 0.46875 -0.1875 C 0.46875 0 0.640625 0 0.765625 0 L 3.90625 0 C 4.171875 0 4.203125 -0.03125 4.203125 -0.296875 Z M 4.203125 -5.203125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-1"> <path style="stroke:none;" d="M 3.9375 -2.921875 C 3.96875 -3.046875 3.96875 -3.09375 3.96875 -3.109375 C 3.96875 -3.28125 3.8125 -3.34375 3.71875 -3.34375 C 3.53125 -3.34375 3.390625 -3.203125 3.359375 -3.046875 C 3.296875 -3.15625 3.0625 -3.5 2.578125 -3.5 C 1.640625 -3.5 0.609375 -2.4375 0.609375 -1.28125 C 0.609375 -0.421875 1.171875 0 1.75 0 C 2.125 0 2.4375 -0.203125 2.703125 -0.421875 L 2.53125 0.28125 C 2.4375 0.625 2.390625 0.84375 2.078125 1.109375 C 1.75 1.390625 1.4375 1.390625 1.25 1.390625 C 1.046875 1.390625 0.859375 1.390625 0.671875 1.34375 C 0.84375 1.25 0.921875 1.09375 0.921875 0.953125 C 0.921875 0.765625 0.78125 0.65625 0.609375 0.65625 C 0.40625 0.65625 0.171875 0.8125 0.171875 1.140625 C 0.171875 1.59375 0.78125 1.625 1.265625 1.625 C 2.390625 1.625 2.953125 1.015625 3.078125 0.53125 Z M 2.859375 -1.046875 C 2.8125 -0.828125 2.640625 -0.671875 2.46875 -0.515625 C 2.390625 -0.453125 2.09375 -0.21875 1.765625 -0.21875 C 1.453125 -0.21875 1.21875 -0.484375 1.21875 -0.953125 C 1.21875 -1.296875 1.421875 -2.15625 1.640625 -2.5625 C 1.890625 -3.015625 2.265625 -3.28125 2.578125 -3.28125 C 3.109375 -3.28125 3.25 -2.6875 3.25 -2.625 L 3.234375 -2.515625 Z M 2.859375 -1.046875 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-2"> <path style="stroke:none;" d="M 3.890625 -2.59375 C 3.9375 -2.78125 4.03125 -3.125 4.03125 -3.171875 C 4.03125 -3.375 3.859375 -3.421875 3.765625 -3.421875 C 3.5 -3.421875 3.4375 -3.21875 3.34375 -2.859375 C 3.25 -2.4375 3.21875 -2.296875 3.09375 -1.84375 C 3.03125 -1.546875 2.921875 -1.171875 2.921875 -0.9375 C 2.921875 -0.890625 2.9375 -0.828125 2.9375 -0.796875 C 2.9375 -0.78125 2.71875 -0.140625 2.203125 -0.140625 C 1.875 -0.140625 1.53125 -0.265625 1.53125 -0.859375 C 1.53125 -1.25 1.703125 -1.765625 1.953125 -2.40625 C 2.046875 -2.609375 2.0625 -2.6875 2.0625 -2.828125 C 2.0625 -3.265625 1.71875 -3.5 1.34375 -3.5 C 0.5625 -3.5 0.234375 -2.375 0.234375 -2.28125 C 0.234375 -2.21875 0.296875 -2.1875 0.359375 -2.1875 C 0.453125 -2.1875 0.46875 -2.234375 0.484375 -2.3125 C 0.703125 -3 1.046875 -3.28125 1.328125 -3.28125 C 1.4375 -3.28125 1.515625 -3.203125 1.515625 -3.015625 C 1.515625 -2.84375 1.453125 -2.671875 1.390625 -2.53125 C 1.09375 -1.734375 0.9375 -1.3125 0.9375 -0.953125 C 0.9375 -0.15625 1.609375 0.078125 2.171875 0.078125 C 2.296875 0.078125 2.703125 0.078125 3.03125 -0.4375 C 3.25 -0.015625 3.75 0.078125 4.09375 0.078125 C 4.8125 0.078125 5.140625 -0.59375 5.28125 -0.859375 C 5.546875 -1.375 5.8125 -2.421875 5.8125 -2.890625 C 5.8125 -3.515625 5.46875 -3.515625 5.421875 -3.515625 C 5.234375 -3.515625 5.015625 -3.296875 5.015625 -3.09375 C 5.015625 -2.984375 5.0625 -2.921875 5.125 -2.875 C 5.203125 -2.796875 5.4375 -2.609375 5.4375 -2.21875 C 5.4375 -1.984375 5.21875 -1.234375 5 -0.8125 C 4.78125 -0.421875 4.515625 -0.140625 4.125 -0.140625 C 3.765625 -0.140625 3.5 -0.328125 3.5 -0.828125 C 3.5 -1.046875 3.5625 -1.265625 3.671875 -1.703125 Z M 3.890625 -2.59375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-3"> <path style="stroke:none;" d="M 2.5 -2.46875 L 3.9375 -2.46875 C 5.09375 -2.46875 6.28125 -3.265625 6.28125 -4.203125 C 6.28125 -4.875 5.640625 -5.421875 4.625 -5.421875 L 1.9375 -5.421875 C 1.796875 -5.421875 1.703125 -5.421875 1.703125 -5.265625 C 1.703125 -5.15625 1.796875 -5.15625 1.921875 -5.15625 C 2.1875 -5.15625 2.421875 -5.15625 2.421875 -5.03125 C 2.421875 -5 2.40625 -5 2.390625 -4.890625 L 1.328125 -0.625 C 1.25 -0.328125 1.234375 -0.265625 0.671875 -0.265625 C 0.484375 -0.265625 0.40625 -0.265625 0.40625 -0.109375 C 0.40625 -0.078125 0.421875 0 0.53125 0 C 0.6875 0 0.875 -0.015625 1.03125 -0.03125 L 1.53125 -0.03125 C 2.296875 -0.03125 2.5 0 2.5625 0 C 2.609375 0 2.71875 0 2.71875 -0.15625 C 2.71875 -0.265625 2.609375 -0.265625 2.46875 -0.265625 C 2.453125 -0.265625 2.3125 -0.265625 2.171875 -0.28125 C 2.015625 -0.296875 2 -0.3125 2 -0.390625 C 2 -0.421875 2.015625 -0.46875 2.015625 -0.515625 Z M 3.078125 -4.859375 C 3.15625 -5.140625 3.15625 -5.15625 3.484375 -5.15625 L 4.359375 -5.15625 C 5.015625 -5.15625 5.515625 -4.96875 5.515625 -4.375 C 5.515625 -4.28125 5.46875 -3.578125 5.03125 -3.140625 C 4.90625 -3.015625 4.53125 -2.71875 3.75 -2.71875 L 2.546875 -2.71875 Z M 3.078125 -4.859375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-4"> <path style="stroke:none;" d="M 3.046875 -3.15625 L 3.78125 -3.15625 C 3.9375 -3.15625 4.03125 -3.15625 4.03125 -3.3125 C 4.03125 -3.421875 3.921875 -3.421875 3.796875 -3.421875 L 3.09375 -3.421875 C 3.21875 -4.140625 3.296875 -4.59375 3.375 -4.9375 C 3.40625 -5.078125 3.421875 -5.171875 3.546875 -5.265625 C 3.65625 -5.34375 3.71875 -5.359375 3.796875 -5.359375 C 3.921875 -5.359375 4.046875 -5.34375 4.15625 -5.28125 C 4.109375 -5.265625 4.0625 -5.234375 4.03125 -5.21875 C 3.890625 -5.140625 3.796875 -5 3.796875 -4.84375 C 3.796875 -4.65625 3.9375 -4.546875 4.109375 -4.546875 C 4.34375 -4.546875 4.5625 -4.75 4.5625 -5.03125 C 4.5625 -5.390625 4.171875 -5.59375 3.796875 -5.59375 C 3.53125 -5.59375 3.03125 -5.46875 2.765625 -4.734375 C 2.703125 -4.546875 2.703125 -4.53125 2.484375 -3.421875 L 1.890625 -3.421875 C 1.734375 -3.421875 1.640625 -3.421875 1.640625 -3.265625 C 1.640625 -3.15625 1.734375 -3.15625 1.875 -3.15625 L 2.4375 -3.15625 L 1.859375 -0.078125 C 1.71875 0.71875 1.59375 1.390625 1.171875 1.390625 C 1.15625 1.390625 0.984375 1.390625 0.828125 1.296875 C 1.203125 1.21875 1.203125 0.875 1.203125 0.875 C 1.203125 0.6875 1.0625 0.578125 0.875 0.578125 C 0.671875 0.578125 0.4375 0.765625 0.4375 1.0625 C 0.4375 1.390625 0.78125 1.625 1.171875 1.625 C 1.65625 1.625 2 1.109375 2.09375 0.90625 C 2.375 0.390625 2.5625 -0.609375 2.578125 -0.6875 Z M 3.046875 -3.15625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-1"> <path style="stroke:none;" d="M 1.609375 -5.25 C 1.609375 -5.53125 1.375 -5.671875 1.171875 -5.671875 C 0.9375 -5.671875 0.734375 -5.515625 0.734375 -5.25 C 0.734375 -5.234375 0.75 -5.09375 0.75 -5.0625 L 1.046875 -1.703125 C 1.0625 -1.5625 1.0625 -1.515625 1.171875 -1.515625 C 1.28125 -1.515625 1.296875 -1.5625 1.296875 -1.6875 Z M 1.609375 -0.4375 C 1.609375 -0.703125 1.390625 -0.875 1.171875 -0.875 C 0.90625 -0.875 0.734375 -0.671875 0.734375 -0.4375 C 0.734375 -0.203125 0.921875 0 1.171875 0 C 1.40625 0 1.609375 -0.1875 1.609375 -0.4375 Z M 1.609375 -0.4375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-2"> <path style="stroke:none;" d="M 2.921875 -3.15625 C 2.703125 -3.078125 2.671875 -2.890625 2.671875 -2.796875 C 2.671875 -2.59375 2.8125 -2.421875 3.046875 -2.421875 C 3.25 -2.421875 3.40625 -2.578125 3.40625 -2.8125 C 3.40625 -3.296875 2.875 -3.546875 2.125 -3.546875 C 1.03125 -3.546875 0.28125 -2.65625 0.28125 -1.71875 C 0.28125 -0.703125 1.109375 0.078125 2.109375 0.078125 C 3.21875 0.078125 3.5 -0.859375 3.5 -0.953125 C 3.5 -1.046875 3.390625 -1.046875 3.375 -1.046875 C 3.3125 -1.046875 3.28125 -1.046875 3.25 -0.953125 C 3.203125 -0.796875 2.96875 -0.171875 2.1875 -0.171875 C 1.6875 -0.171875 0.984375 -0.546875 0.984375 -1.71875 C 0.984375 -2.859375 1.578125 -3.296875 2.15625 -3.296875 C 2.21875 -3.296875 2.640625 -3.296875 2.921875 -3.15625 Z M 2.921875 -3.15625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-3"> <path style="stroke:none;" d="M 3.328125 -2.359375 C 3.328125 -3.140625 2.578125 -3.546875 1.859375 -3.546875 C 1.203125 -3.546875 0.609375 -3.28125 0.609375 -2.765625 C 0.609375 -2.53125 0.78125 -2.375 0.984375 -2.375 C 1.203125 -2.375 1.359375 -2.546875 1.359375 -2.75 C 1.359375 -2.9375 1.234375 -3.078125 1.0625 -3.125 C 1.359375 -3.3125 1.78125 -3.3125 1.84375 -3.3125 C 2.28125 -3.3125 2.734375 -3.015625 2.734375 -2.359375 L 2.734375 -2.109375 C 2.265625 -2.09375 1.734375 -2.0625 1.1875 -1.828125 C 0.484375 -1.53125 0.34375 -1.078125 0.34375 -0.8125 C 0.34375 -0.125 1.15625 0.078125 1.703125 0.078125 C 2.28125 0.078125 2.640625 -0.25 2.8125 -0.5625 C 2.84375 -0.265625 3.0625 0.046875 3.40625 0.046875 C 3.484375 0.046875 4.15625 0.015625 4.15625 -0.71875 L 4.15625 -1.15625 L 3.90625 -1.15625 L 3.90625 -0.71875 C 3.90625 -0.390625 3.796875 -0.265625 3.625 -0.265625 C 3.328125 -0.265625 3.328125 -0.625 3.328125 -0.71875 Z M 2.734375 -1.125 C 2.734375 -0.34375 2.078125 -0.140625 1.765625 -0.140625 C 1.34375 -0.140625 1 -0.421875 1 -0.796875 C 1 -1.328125 1.5 -1.859375 2.734375 -1.90625 Z M 2.734375 -1.125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-4"> <path style="stroke:none;" d="M 1.453125 -1.8125 C 1.453125 -2.40625 1.703125 -3.265625 2.46875 -3.28125 C 2.421875 -3.25 2.34375 -3.1875 2.34375 -3 C 2.34375 -2.75 2.53125 -2.640625 2.6875 -2.640625 C 2.875 -2.640625 3.046875 -2.765625 3.046875 -3 C 3.046875 -3.28125 2.796875 -3.5 2.4375 -3.5 C 1.921875 -3.5 1.578125 -3.109375 1.40625 -2.671875 L 1.40625 -3.5 L 0.28125 -3.40625 L 0.28125 -3.15625 C 0.8125 -3.15625 0.875 -3.09375 0.875 -2.703125 L 0.875 -0.625 C 0.875 -0.265625 0.78125 -0.265625 0.28125 -0.265625 L 0.28125 0 C 0.59375 -0.03125 1.03125 -0.03125 1.21875 -0.03125 C 1.6875 -0.03125 1.703125 -0.03125 2.21875 0 L 2.21875 -0.265625 L 2.0625 -0.265625 C 1.46875 -0.265625 1.453125 -0.34375 1.453125 -0.640625 Z M 1.453125 -1.8125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-5"> <path style="stroke:none;" d="M 1.46875 -3.15625 L 2.65625 -3.15625 L 2.65625 -3.421875 L 1.46875 -3.421875 L 1.46875 -4.875 L 1.234375 -4.875 C 1.21875 -4.15625 0.890625 -3.40625 0.15625 -3.375 L 0.15625 -3.15625 L 0.875 -3.15625 L 0.875 -0.984375 C 0.875 -0.0625 1.59375 0.078125 1.953125 0.078125 C 2.484375 0.078125 2.796875 -0.390625 2.796875 -0.984375 L 2.796875 -1.4375 L 2.5625 -1.4375 L 2.5625 -1.015625 C 2.5625 -0.453125 2.3125 -0.171875 2.015625 -0.171875 C 1.46875 -0.171875 1.46875 -0.84375 1.46875 -0.96875 Z M 1.46875 -3.15625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-6"> <path style="stroke:none;" d="M 3.28125 -1.8125 C 3.453125 -1.8125 3.5 -1.8125 3.5 -2 C 3.5 -2.703125 3.109375 -3.546875 2 -3.546875 C 1.015625 -3.546875 0.234375 -2.71875 0.234375 -1.734375 C 0.234375 -0.71875 1.09375 0.078125 2.09375 0.078125 C 3.109375 0.078125 3.5 -0.765625 3.5 -0.953125 C 3.5 -0.984375 3.484375 -1.0625 3.375 -1.0625 C 3.28125 -1.0625 3.265625 -1.015625 3.25 -0.953125 C 2.96875 -0.1875 2.28125 -0.171875 2.140625 -0.171875 C 1.78125 -0.171875 1.421875 -0.328125 1.1875 -0.703125 C 0.9375 -1.0625 0.9375 -1.578125 0.9375 -1.8125 Z M 0.953125 -2.015625 C 1.03125 -3.125 1.703125 -3.3125 2 -3.3125 C 2.921875 -3.3125 2.953125 -2.203125 2.953125 -2.015625 Z M 0.953125 -2.015625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-7"> <path style="stroke:none;" d="M 2.828125 -3.328125 C 2.828125 -3.453125 2.828125 -3.546875 2.71875 -3.546875 C 2.6875 -3.546875 2.65625 -3.546875 2.53125 -3.40625 C 2.515625 -3.40625 2.4375 -3.328125 2.421875 -3.328125 C 2.40625 -3.328125 2.390625 -3.328125 2.34375 -3.359375 C 2.21875 -3.453125 2 -3.546875 1.640625 -3.546875 C 0.53125 -3.546875 0.28125 -2.9375 0.28125 -2.5625 C 0.28125 -2.15625 0.578125 -1.921875 0.59375 -1.90625 C 0.90625 -1.671875 1.09375 -1.640625 1.625 -1.546875 C 2 -1.46875 2.609375 -1.359375 2.609375 -0.8125 C 2.609375 -0.515625 2.40625 -0.140625 1.671875 -0.140625 C 0.875 -0.140625 0.640625 -0.765625 0.546875 -1.1875 C 0.515625 -1.28125 0.5 -1.328125 0.40625 -1.328125 C 0.28125 -1.328125 0.28125 -1.265625 0.28125 -1.109375 L 0.28125 -0.125 C 0.28125 0 0.28125 0.078125 0.375 0.078125 C 0.421875 0.078125 0.4375 0.078125 0.578125 -0.078125 C 0.625 -0.125 0.703125 -0.21875 0.75 -0.265625 C 1.109375 0.0625 1.46875 0.078125 1.6875 0.078125 C 2.6875 0.078125 3.046875 -0.5 3.046875 -1.03125 C 3.046875 -1.40625 2.8125 -1.953125 1.859375 -2.140625 C 1.796875 -2.15625 1.359375 -2.234375 1.328125 -2.234375 C 1.078125 -2.28125 0.703125 -2.453125 0.703125 -2.765625 C 0.703125 -3.015625 0.875 -3.34375 1.640625 -3.34375 C 2.53125 -3.34375 2.5625 -2.6875 2.578125 -2.46875 C 2.59375 -2.40625 2.640625 -2.375 2.703125 -2.375 C 2.828125 -2.375 2.828125 -2.4375 2.828125 -2.59375 Z M 2.828125 -3.328125 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-8"> <path style="stroke:none;" d="M 1.546875 -4.890625 C 1.546875 -5.125 1.359375 -5.328125 1.109375 -5.328125 C 0.875 -5.328125 0.671875 -5.15625 0.671875 -4.890625 C 0.671875 -4.625 0.890625 -4.453125 1.109375 -4.453125 C 1.375 -4.453125 1.546875 -4.6875 1.546875 -4.890625 Z M 0.359375 -3.40625 L 0.359375 -3.15625 C 0.859375 -3.15625 0.9375 -3.109375 0.9375 -2.71875 L 0.9375 -0.625 C 0.9375 -0.265625 0.84375 -0.265625 0.328125 -0.265625 L 0.328125 0 C 0.640625 -0.03125 1.09375 -0.03125 1.203125 -0.03125 C 1.3125 -0.03125 1.78125 -0.03125 2.0625 0 L 2.0625 -0.265625 C 1.546875 -0.265625 1.515625 -0.296875 1.515625 -0.609375 L 1.515625 -3.5 Z M 0.359375 -3.40625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-9"> <path style="stroke:none;" d="M 3.859375 -2.40625 C 3.859375 -3.078125 3.5625 -3.5 2.71875 -3.5 C 1.9375 -3.5 1.578125 -2.921875 1.484375 -2.734375 L 1.46875 -2.734375 L 1.46875 -3.5 L 0.328125 -3.40625 L 0.328125 -3.15625 C 0.859375 -3.15625 0.921875 -3.09375 0.921875 -2.703125 L 0.921875 -0.625 C 0.921875 -0.265625 0.828125 -0.265625 0.328125 -0.265625 L 0.328125 0 C 0.671875 -0.03125 1.015625 -0.03125 1.234375 -0.03125 C 1.453125 -0.03125 1.796875 -0.03125 2.140625 0 L 2.140625 -0.265625 C 1.625 -0.265625 1.53125 -0.265625 1.53125 -0.625 L 1.53125 -2.0625 C 1.53125 -2.890625 2.171875 -3.28125 2.65625 -3.28125 C 3.140625 -3.28125 3.25 -2.9375 3.25 -2.4375 L 3.25 -0.625 C 3.25 -0.265625 3.15625 -0.265625 2.65625 -0.265625 L 2.65625 0 C 3 -0.03125 3.34375 -0.03125 3.5625 -0.03125 C 3.78125 -0.03125 4.125 -0.03125 4.453125 0 L 4.453125 -0.265625 C 3.953125 -0.265625 3.859375 -0.265625 3.859375 -0.625 Z M 3.859375 -2.40625 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-10"> <path style="stroke:none;" d="M 2.640625 1.984375 C 2.703125 1.984375 2.796875 1.984375 2.796875 1.890625 C 2.796875 1.859375 2.796875 1.84375 2.6875 1.75 C 1.609375 0.71875 1.328125 -0.75 1.328125 -1.984375 C 1.328125 -4.265625 2.28125 -5.34375 2.6875 -5.703125 C 2.796875 -5.8125 2.796875 -5.8125 2.796875 -5.859375 C 2.796875 -5.890625 2.765625 -5.953125 2.6875 -5.953125 C 2.5625 -5.953125 2.171875 -5.546875 2.109375 -5.484375 C 1.046875 -4.359375 0.8125 -2.9375 0.8125 -1.984375 C 0.8125 -0.203125 1.5625 1.21875 2.640625 1.984375 Z M 2.640625 1.984375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-11"> <path style="stroke:none;" d="M 2.453125 -1.984375 C 2.453125 -2.734375 2.328125 -3.640625 1.828125 -4.578125 C 1.4375 -5.3125 0.71875 -5.953125 0.578125 -5.953125 C 0.5 -5.953125 0.46875 -5.890625 0.46875 -5.859375 C 0.46875 -5.828125 0.46875 -5.8125 0.578125 -5.71875 C 1.6875 -4.65625 1.9375 -3.203125 1.9375 -1.984375 C 1.9375 0.296875 0.984375 1.375 0.59375 1.734375 C 0.484375 1.84375 0.46875 1.84375 0.46875 1.890625 C 0.46875 1.921875 0.5 1.984375 0.578125 1.984375 C 0.703125 1.984375 1.109375 1.578125 1.171875 1.515625 C 2.234375 0.390625 2.453125 -1.03125 2.453125 -1.984375 Z M 2.453125 -1.984375 "></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph6-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph6-1"> <path style="stroke:none;" d="M 2.34375 -5.890625 L -0.0625 -4.640625 L 0.0625 -4.421875 L 2.328125 -5.375 L 4.625 -4.421875 L 4.734375 -4.640625 Z M 2.34375 -5.890625 "></path> </symbol> </g> </defs> <g id="t_pf165XCL7aQfxUkuJPz3gsyZA=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-1" x="52.804088" y="9.475738"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph1-1" x="5.883117" y="46.137877"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-2" x="106.999776" y="46.137877"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-3" x="181.701186" y="46.137877"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-4" x="43.71144" y="85.906523"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-1" x="52.836925" y="85.906523"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-1" x="57.367328" y="85.906523"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-2" x="63.307699" y="85.906523"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph1-2" x="5.466196" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-4" x="97.907127" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-1" x="107.033607" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-2" x="111.563015" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-2" x="118.182002" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-4" x="172.608538" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-1" x="181.734022" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph0-3" x="186.263431" y="126.335872"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph2-2" x="192.369973" y="126.335872"></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 -37.301509 61.233217 C 8.354769 64.801709 38.304188 56.479154 75.186452 30.137484 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.487931 2.868658 C -2.031758 1.148984 -1.019614 0.332866 -0.000110917 -0.000407132 C -1.021061 -0.333897 -2.032027 -1.147495 -2.48541 -2.870258 " transform="matrix(0.809731,0.578236,0.578236,-0.809731,176.187825,36.980203)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 115.601562 16.332031 L 131.105469 16.332031 L 131.105469 4.160156 L 115.601562 4.160156 Z M 115.601562 16.332031 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph3-1" x="118.163096" y="12.230995"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-1" x="124.248743" y="12.230995"></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 -37.301509 55.030558 L 0.102878 29.882311 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.485833 2.870524 C -2.031592 1.146466 -1.022329 0.334852 0.00140335 0.00130044 C -1.020802 -0.335874 -2.031356 -1.147743 -2.487406 -2.868817 " transform="matrix(0.825731,0.555151,0.555151,-0.825731,101.482494,37.230763)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 72.96875 30.144531 L 92.777344 30.144531 L 92.777344 19.292969 L 72.96875 19.292969 Z M 72.96875 30.144531 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph3-2" x="75.529739" y="27.583422"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-1" x="80.210393" y="27.583422"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph6-1" x="85.447273" y="27.583422"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-2" x="83.955713" y="27.583422"></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 -92.089042 16.220759 L -92.089042 -46.579196 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.484744 2.867799 C -2.033284 1.148327 -1.020445 0.335701 0.00024546 -0.00191237 C -1.020445 -0.3356 -2.033284 -1.148226 -2.484744 -2.867698 " transform="matrix(0,0.995037,0.995037,0,9.552684,113.417725)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-3" x="12.352848" y="84.881634"></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 17.843266 23.785647 L 75.09616 23.785647 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.487895 2.869228 C -2.03251 1.145831 -1.019671 0.333205 0.00102008 -0.00048262 C -1.019671 -0.33417 -2.03251 -1.146797 -2.487895 -2.870194 " transform="matrix(0.995037,0,0,-0.995037,176.143516,43.163582)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 142.636719 49.25 L 152.683594 49.25 L 152.683594 37.078125 L 142.636719 37.078125 Z M 142.636719 49.25 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-4" x="145.199247" y="45.14682"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-2" x="131.248828" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-3" x="134.993631" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-4" x="139.20713" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-5" x="142.491168" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-6" x="145.768069" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-7" x="149.512873" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-8" x="152.836564" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-3" x="155.176868" y="52.272281"></use> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-9" x="159.390366" y="52.272281"></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 -28.166328 -15.389245 C 12.029255 -13.583408 38.241376 -21.807819 69.812123 -45.970707 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.485285 2.86772 C -2.033614 1.146945 -1.021444 0.334608 0.00102311 -0.000210903 C -1.018102 -0.33516 -2.0334 -1.146705 -2.48631 -2.867598 " transform="matrix(0.790169,0.604664,0.604664,-0.790169,170.839163,112.717965)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 113.925781 94.230469 L 136.410156 94.230469 L 136.410156 81.179688 L 113.925781 81.179688 Z M 113.925781 94.230469 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-3" x="116.487454" y="89.687663"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-10" x="122.999971" y="89.687663"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-1" x="126.276628" y="89.687663"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-11" x="130.571208" y="89.687663"></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 -31.401132 -26.734614 L -5.436332 -45.974633 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.486966 2.870229 C -2.032787 1.149608 -1.019835 0.336611 0.00212883 -0.000606803 C -1.021327 -0.33394 -2.031828 -1.146318 -2.484569 -2.869586 " transform="matrix(0.799403,0.592415,0.592415,-0.799403,95.963501,112.72091)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 74.3125 108.460938 L 91.777344 108.460938 L 91.777344 97.832031 L 74.3125 97.832031 Z M 74.3125 108.460938 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph3-1" x="76.873039" y="105.900797"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-2" x="82.958686" y="105.900797"></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 26.982373 -56.813582 L 65.957053 -56.813582 " transform="matrix(0.995037,0,0,-0.995037,101.182789,66.831663)"></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 -2.485076 2.868859 C -2.033617 1.149387 -1.020778 0.332835 -0.000086969 -0.000852362 C -1.020778 -0.33454 -2.033617 -1.147166 -2.485076 -2.870563 " transform="matrix(0.995037,0,0,-0.995037,167.050868,123.362433)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 136.105469 129.890625 L 159.214844 129.890625 L 159.214844 116.835938 L 136.105469 116.835938 Z M 136.105469 129.890625 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-3" x="138.665834" y="125.344816"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-10" x="145.178352" y="125.344816"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph4-4" x="148.455009" y="125.344816"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#t_pf165XCL7aQfxUkuJPz3gsyZA=-glyph5-11" x="153.377457" y="125.344816"></use> </g> </g> </svg> the <a class="existingWikiWord" href="/nlab/show/invertible+morphism">invertibility</a> of all morphisms implies that both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">w'</annotation></semantics></math> are uniquely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, and then <a class="existingWikiWord" href="/nlab/show/functor">functoriality</a> of <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> implies that indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>w</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">P(\widehat{w}) = w</annotation></semantics></math>.</p> </div> </p> <h3 id="AsFibrationsInCanonicalModelStructures">As fibrations in canonical model structures</h3> <p>The isofibrations are the <em><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a></em> in the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+categories">canonical model structure on categories</a> and the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">canonical model structure on groupoids</a>. More generally, the fibrations in <a class="existingWikiWord" href="/nlab/show/canonical+model+structures">canonical model structures</a> on various types of higher categories are usually some generalization of isofibrations. For example, the fibrations in the Lack model structure on 2-Cat, known as <a class="existingWikiWord" href="/nlab/show/equifibrations">equifibrations</a>, have “equivalence lifting” and “local isomorphism lifting,” and the fibrations in the Joyal model structure for <a class="existingWikiWord" href="/nlab/show/quasi-category">quasicategories</a> have “equivalence lifting” at all levels.</p> <p>Generalizing in another direction, internalized isofibrations are the fibrations in the <a class="existingWikiWord" href="/nlab/show/2-trivial+model+structure">2-trivial model structure</a> on any finitely complete and cocomplete <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/amnestic+isofibration">amnestic isofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isocofibration">isocofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifibration">equifibration</a></p> </li> </ul> <h2 id="references">References</h2> <p>For groupoids the definition appears (called “star surjectivity” there) on p. 105 (3 of 30) in</p> <ul> <li id="Brown70"><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Fibrations of groupoids</em>, Journal of Algebra Volume 15, Issue 1, May 1970, Pages 103-132 doi:<a href="https://doi.org/10.1016/0021-8693%2870%2990089-X">10.1016/0021-8693(70)90089-X</a>, <a href="http://groupoids.org.uk/pdffiles/fibrationsgpds.pdf">author’s pdf</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 4, 2023 at 21:18:13. See the <a href="/nlab/history/isofibration" 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/isofibration" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7640/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/isofibration/18" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/isofibration" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/isofibration" accesskey="S" class="navlink" id="history" rel="nofollow">History (18 revisions)</a> <a href="/nlab/show/isofibration/cite" style="color: black">Cite</a> <a href="/nlab/print/isofibration" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/isofibration" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>