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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topos_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</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/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <p>This is a sub-entry of <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy groups in an (∞,1)-topos</a>. It discusses the general notions of <strong><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy">étale homotopy</a></strong> in the context of <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-toposes">locally ∞-connected (∞,1)-toposes</a>.</p> <p>For the other notion of homotopy groups in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos see <a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a>.</p> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#GeomIdea'>Idea</a></li> <li><a href='#GeomDef'>Definition</a></li> <li><a href='#LocalContraction'>In terms of local contractions</a></li> <ul> <li><a href='#LocalContractionRef'>References</a></li> </ul> <li><a href='#Monodromy'>In terms of monodromy and Galois theory</a></li> <ul> <li><a href='#InTermsOfMonodromyReferences'>References</a></li> </ul> <li><a href='#Paths'>In terms of concrete paths</a></li> <ul> <li><a href='#references_3'>References</a></li> </ul> <li><a href='#GeomExamples'>Examples</a></li> <ul> <li><a href='#Pi0Ofsheafontopspace'>Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of a sheaf on a locally connected topological space</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#Pi0InLocConTop'>Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of a general object in a locally connected topos</a></li> <li><a href='#geometric__of_objects_in_a_1topos'>Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> of objects in a 1-topos</a></li> <li><a href='#geometric__of_a_topological_space'>Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_2</annotation></semantics></math> of a topological space</a></li> <li><a href='#ooStackOnTopSpace'>Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math> of a topological space</a></li> <ul> <li><a href='#claim'>Claim</a></li> </ul> <li><a href='#GeomPiOfTermObj'>Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math> of the terminal object in a locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</a></li> </ul> <li><a href='#GeneralExamples'>Examples</a></li> </ul> </div> <h2 id="GeomIdea">Idea</h2> <p>An ordinary <a class="existingWikiWord" href="/nlab/show/topos">topos</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> is a <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> if the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a>s <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>E</mi><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex">(LConst \dashv \Gamma) : E \stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set</annotation></semantics></math> is in fact an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> has also a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Pi_0 \dashv LConst)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo>⊣</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>E</mi><mover><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow></mover></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,. </annotation></semantics></math></div> <p>This left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> sends each 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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of connected components. In other words this left adjoint produces the degree 0-part of the homotopy groups of objects 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>.</p> <p>This has an obvious generalization of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es.</p> <h2 id="GeomDef">Definition</h2> <p>The obvious generalization of the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> is to say that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28%E2%88%9E%2C1%29-topos">locally n-connected (n,1)-topos</a> if again the terminal geometric morphism is an essential geometric morphism in that the <a class="existingWikiWord" href="/nlab/show/constant+stack">constant n-stack</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Pi_n</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mi>n</mi></msub><mo>⊣</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mi>n</mi></msub></mrow></mover></mover><mi>n</mi><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi_n \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi_n}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} n Grpd \,. </annotation></semantics></math></div> <p>Here we may take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n = \infty</annotation></semantics></math> and say that an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is <a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28%E2%88%9E%2C1%29-topos">locally contractible</a> if we have an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mover><mo>→</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} \infty Grpd </annotation></semantics></math></div> <p>to <span class="newWikiWord">? Grpd<a href="/nlab/new/%3F+Grpd">?</a></span>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> the left <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> to the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> any object, the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> deserves to be called the <strong><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></strong> 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> Its ordinary <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s are the homotopy groups 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>.</p> <p>While an obvious slight generalization or refinement of what is considered in previous literature, it seems that the simple picture of a left <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> to the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> functor has not been made explicit in the existing literature (though possibly in the thesis by <a class="existingWikiWord" href="/nlab/show/Richard+Williamson">Richard Williamson</a>).</p> <p>However, up to some straightforward translations of concepts and notation, it turns out that essentially all aspects of this picture are present and well known – if somewhat implicitly – in existing literature. A detailed commented account of what is in the literature is in the following subsection and in particular in the section <a href="#GeomExamples">Examples</a> below.</p> <p>There are essentially three different methods concretely constructing the abstractly defined <span class="newWikiWord">homotopy ∞-groupoid<a href="/schreiber/new/homotopy+%E2%88%9E-groupoid">?</a></span>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(-)</annotation></semantics></math>.</p> <ol> <li> <p>by constructing the left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(-)</annotation></semantics></math> as the functor that takes an object to its <strong>local contraction</strong> – this is described in the section <em><a href="#LocalContraction">In terms of local contractions</a>;</em></p> </li> <li> <p>by using <strong>monodromy</strong>/<a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> of <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s to reproduce <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi()</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> – this is described in the section <a href="Monodromy">In terms of monodromy and Galois theory</a>;</p> </li> <li> <p>by constructing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(-)</annotation></semantics></math> explicitly as a path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid in terms of paths modeled on an <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="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> – this is described in the section <em><a href="Paths">In terms of concrete paths</a></em> .</p> </li> </ol> <h2 id="LocalContraction">In terms of local contractions</h2> <p>If the <a class="existingWikiWord" href="/nlab/show/locally+contractible+%28%E2%88%9E%2C1%29-topos">locally contractible (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≃</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} \simeq Sh_{(\infty,1)}(C)</annotation></semantics></math> such that the objects of the site are <strong>geometrically contractible</strong> in that <em>constant</em> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaves</a> already satisfy <a class="existingWikiWord" href="/nlab/show/descent">descent</a> over objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi : \mathbf{H} \to \infty Grpd</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> may be constructed explicitly as follows.</p> <p>Following the discussion at <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a> there is a <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mi>loc</mi></msubsup></mrow><annotation encoding="application/x-tex">sPSh(C)_{proj}^{loc}</annotation></semantics></math> wich <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presents</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p><strong>Proposition</strong> The <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>←</mo></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">(\Pi \dashv LConst) : Sh_{(\infty,1)}(C) \stackrel{\leftarrow}{\to} \infty Grpd</annotation></semantics></math> is presented by an <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched</a> <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo><mo>:</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mi>loc</mi></msubsup><mover><mover><mo>←</mo><mi>LConst</mi></mover><mover><mo>→</mo><mi>Π</mi></mover></mover><msub><mi>sSet</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\Pi \dashv LConst) : sPSh(C)_{proj}^{loc} \stackrel{\overset{\Pi}{\to}}{\overset{LConst}{\leftarrow}} sSet_{Quillen} \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">S \in sSet</annotation></semantics></math> the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LConst</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">LConst_S</annotation></semantics></math> sends all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">U \mapsto S</annotation></semantics></math>, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>;</p> </li> <li> <p>the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> acts by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>X</mi><mo>=</mo><msub><mi>lim</mi> <mo>→</mo></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Pi(X) = \int^{U \in C} X = \lim_\to X</annotation></semantics></math>.</p> </li> </ul> <p>The total left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> first takes an 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> to a <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> that is degreewise a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a> and then <em>contracts</em> all these representables to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, regarding the resulting constant simplicial presheaf as a simplicial set:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>Π</mi><mo>:</mo><mi>X</mi><mo>↦</mo><mi>Q</mi><mi>X</mi><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>↦</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></munder><mo>*</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{L} \Pi : X \mapsto Q X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_i \right) \mapsto \Pi(Q X) = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} * \right) \,. </annotation></semantics></math></div> <p><strong>Proof</strong></p> <p>This is discussed at <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a> and <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>.</p> <h3 id="LocalContractionRef">References</h3> <p>Essentially the construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>Π</mi></mrow><annotation encoding="application/x-tex">\mathbb{L} \Pi</annotation></semantics></math> as above is an old construction in terms of – somewhat implicitly – the structure of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>:</p> <p>the discussion on page 18 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Classifying Spaces and Classifying Topoi</em> Lecture Notes in Mathematics 1616, Springer (1995) .</li> </ul> <p>which goes back to</p> <ul> <li>Artin, Mazur, <em>Etale Homotopy</em> Springer Lecture Notes in Mathematics 100, Berlin (1969)</li> </ul> <p>goes as follows:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E = Sh(C)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>←</mo></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex"> (\Pi_0 \dashv LConst) : Sh(C) \stackrel{\leftarrow}{\to} Set </annotation></semantics></math></div> <p>that here we think of as a <a class="existingWikiWord" href="/nlab/show/petit+topos">petit</a> over-topos over a given 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> in some ambient <a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a>. Accordingly we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X = *</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C)</annotation></semantics></math>.</p> <p>Assume that <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> has <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough point</a>. Then <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan-fibrant</a> <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s 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>, i.e. <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>-wise Kan-fibrant simplicial sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>. In particular a fibrant simplicial object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y \in [\Delta^{op}, Sh(C)]</annotation></semantics></math> equipped with an acyclic fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X = *</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a> 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>.</p> <p>The definition of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> as defined in the above references (notice that only its homotopy groups are written down explicitly there, but it’s immediate to equivalently write it as we do now) is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mi>lim</mi> <mo>→</mo></munder><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Pi(X) = \lim_\to \Pi_0( Y_\bullet) \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> is taken over the category of acyclic fibrations/hypercovers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \to X</annotation></semantics></math>;</p> </li> <li> <p>the connected components functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Pi_0 : Sh(C) \to Set</annotation></semantics></math> is applied degreewise to the simplicial sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y = (Y_\bullet)</annotation></semantics></math> to produce a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>.</p> </li> </ul> <p>In Artin-Mazur it is discussed that this prescription does produce the right homotopy groups for <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> if one assumes that this space is <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a>.</p> <p>If we therefore interpret this as saying that for the above prescription to yield the correct result we generally ought to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(C)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+contractible+%28%E2%88%9E%2C1%29-topos">locally contractible (∞,1)-topos</a>, then this prescription can be seen to model implicitly the left Quillen functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(-)</annotation></semantics></math> that we described above:</p> <p>In terms of the full <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>proj</mi> <mi>loc</mi></msubsup></mrow><annotation encoding="application/x-tex">sPSh(C)_{proj}^{loc}</annotation></semantics></math> among all these hypercovers is one that is the cofibrant object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>Q</mi><mi>X</mi><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> Y = Q X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_i \right) </annotation></semantics></math></div> <p>mentioned above, consisting degreewise of coproducts of representables with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Pi_0(U_i) = *</annotation></semantics></math>. For instance if <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> admits a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>, we can take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of that good cover. (For more on this see <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a>.) Due to the lifting property of cofibrant objects, any colimit over all hypercovers can be computed by evaluating just at that hypercover.</p> <p>There the Artin-Mazur-Moerdijk-prescription yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Π</mi> <mn>0</mn></msub><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></munder><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>)</mo></mrow><mo>=</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Π</mi> <mn>0</mn></msub><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></munder><mo>*</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(Q X) = \Pi_0((Q X)_\bullet) = \int^{[n] \in \Delta} \Delta[n] \cdot \Pi_0\left( \coprod_{i_n} U_{i_n} \right) = \int^{[n] \in \Delta} \Delta[n] \cdot \Pi_0\left( \coprod_{i_n} * \right) \,. </annotation></semantics></math></div> <p>This is indeed the action of the left Quillen functor from above.</p> <p>It is the <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> that asserts that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>, this simplicial set is homotopy equivalent to the original <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact space</a>.</p> <p>A closely related, implicitly slightly more general statement is in on p. 25 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <em><a class="existingWikiWord" href="/nlab/files/DuggerUniv.pdf" title="Universal homotopy theories">Universal homotopy theories</a></em></li> </ul> <p>which describes this construction for the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = Sh_{(\infty,1)}(Diff)</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks on <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>).</p> <p>With even more general sites allowed, but working only at the level of <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy categories</a> the left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> and its construction is described in <a href="http://math.berkeley.edu/~teleman/math/simpson.pdf#page=5">Proposition 2.18</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>de Rham’s theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em> (<a href="http://math.berkeley.edu/~teleman/math/simpson.pdf">pdf</a>)</li> </ul> <p>See also the discussion at <a class="existingWikiWord" href="/nlab/show/locally+contractible+%28%E2%88%9E%2C1%29-topos">locally contractible (∞,1)-topos</a>.</p> <h2 id="Monodromy">In terms of monodromy and Galois theory</h2> <p>Given an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = Sh_{(\infty,1)}(C)</annotation></semantics></math> we define the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s on an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>LConst</mi> <mrow><mi>Core</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \infty CovBund(X) := \mathbf{H}(X, LConst_{Core(\infty Grpd)}) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LConst</mi> <mrow><mi>Core</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">LConst_{Core(\infty Grpd)}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> on the <a class="existingWikiWord" href="/nlab/show/core">core</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of <span class="newWikiWord">? Grpd<a href="/nlab/new/%3F+Grpd">?</a></span>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+contractible+%28%E2%88%9E%2C1%29-topos">locally contractible (∞,1)-topos</a> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> has the left <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(-)</annotation></semantics></math>, then by definition of adjunction we have the equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \infty CovBund(X) \simeq Func(\Pi(X), \infty Grpd) </annotation></semantics></math></div> <p>with <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a>s on the one hand and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> on the other.</p> <p>Concrete realizations of this equivalence are discussed in the <a href="GeomExamples">Examples</a>-section below. Here we describe how one may <em><a class="existingWikiWord" href="/nlab/show/reconstruction+theorem">reconstruct</a></em> in terms <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> from just knowing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\infty CovBund(X)</annotation></semantics></math> in terms of the <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> of a fiber functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo>:</mo><mn>∞</mn><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> F_x : \infty CovBund(X) \to \infty Grpd </annotation></semantics></math></div> <p>from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/covering+space">coverin bundle</a>s/<a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s over <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> to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>.</p> <p>– these automorphism are called the <strong>monodromy</strong> 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>.</p> <p>We want to show that these automorphism <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s are the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math>, hence the geometric homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Ω</mi> <mi>x</mi> <mi>geom</mi></msubsup><mi>X</mi><mo>=</mo><mo>:</mo><msub><mi>Ω</mi> <mi>x</mi></msub><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aut (F_x) = \Omega^{geom}_x X =: \Omega_x \Pi(X) \,. </annotation></semantics></math></div> <p>This is the reconstruction of the geometric homotopy <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <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> from its <strong>monodromy</strong> or <strong><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></strong>.</p> <p><strong>Proof</strong></p> <p>The underlying mechanism is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a>, i.e. essentially the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> applied a few times in a row:</p> <p>suppose we knew <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math>, so that by adjunction we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CovBund(X) \simeq \infty Func(\Pi(X), \infty Grpd) \,. </annotation></semantics></math></div> <p>Then for each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in \Pi(X)</annotation></semantics></math> given by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mo>*</mo><mo>→</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i : {*} \to \Pi(X)</annotation></semantics></math> we get a fiber functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo>:</mo><mo>=</mo><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> F_x := \infty Func(i, \infty Grpd) : Func(\Pi(X), \infty Grpd) \to \infty Grpd </annotation></semantics></math></div> <p>which takes a <a class="existingWikiWord" href="/nlab/show/local+system">local system</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\rho : \Pi(X) \to \infty Grpd</annotation></semantics></math> and evaluates it on <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>. By the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">F_x</annotation></semantics></math> is given by homming out of the local system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">Y_{\Pi(X)^{op}} x</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/representable+functor">represented by</a> <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>Y</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><mo stretchy="false">)</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \infty Func(i, \infty Grpd) \simeq Hom_{PSh_{(\infty,1)}(\Pi(X)^{op})}(Y_{\Pi(X)^{op}}) x, -) \,. </annotation></semantics></math></div> <p>But this in turn means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>:</mo><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>∞</mn><mi>Grod</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Func(i,\infty Grpd) : \infty Func(\Pi(X),\infty Grod) \to \infty Grpd</annotation></semantics></math> is itself a representable functor, in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_{(\infty,1)}(PSh_{(\infty,1)}(\Pi(X)^{op})^{op})</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Y</mi> <mrow><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><msub><mi>Y</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><mi>x</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \infty Func(i, \infty Grpd) \simeq Y_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \,. </annotation></semantics></math></div> <p>This way we find, by applying the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> two more times, that the automorphism <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> of the fiber functor is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Aut</mi> <mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow></msub><mn>∞</mn><mi>Func</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mn>∞</mn><mi>Grod</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>Aut</mi> <mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow></msub><msub><mi>Y</mi> <mrow><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><msub><mi>Y</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><mi>x</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Aut</mi> <mrow><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><msub><mi>Y</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><mi>x</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Aut</mi> <mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msub><mi>x</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Ω</mi> <mi>x</mi></msub><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo>:</mo><msubsup><mi>Ω</mi> <mi>x</mi> <mi>geom</mi></msubsup><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Aut_{PSh_{(\infty,1)}((PSh_{(\infty,1)}(\Pi(X)^{op}))^{op})} \infty Func(i, \infty Grod) & = Aut_{PSh_{(\infty,1)}((PSh_{(\infty,1)}(\Pi(X)^{op}))^{op})} Y_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \\ & \simeq Aut_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \\ & \simeq Aut_{\Pi(X)^{op}} x \\ & \simeq \Omega_x \Pi(X) \\ & =: \Omega_x^{geom} X \,. \end{aligned} </annotation></semantics></math></div> <p>Now, the same is of course true even if we don’t have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> in hands yet, but only know the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CovBund(X)</annotation></semantics></math> of covering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles / <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s in <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>: in terms of this we may reconstruct the automorphism <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>CovBund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>F</mi> <mi>x</mi></msub></mrow></mover><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Ω</mi> <mi>x</mi></msub><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><msubsup><mi>Ω</mi> <mi>x</mi> <mi>geom</mi></msubsup><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aut( CovBund(X) \stackrel{F_x}{\to} \infty Grpd ) \simeq \Omega_x \Pi(X) =: \Omega^{geom}_x X \,. </annotation></semantics></math></div> <h3 id="InTermsOfMonodromyReferences">References</h3> <p>The idea that geometric homotopy groups of generalized <a class="existingWikiWord" href="/nlab/show/space">space</a>s given by <a class="existingWikiWord" href="/nlab/show/sheaf">sheaves</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a>s, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>s is detected and definable by the behaviour of locally constant sheaves, stacks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks on these objects goes back to <a class="existingWikiWord" href="/nlab/show/Grothendieck%27s+Galois+theory">Grothendieck's Galois theory</a> and the notion of <a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a>. The state of the art treatment of the Galois theory of coverings in a topos is in</p> <ul> <li>Marta Bunge, <em>Galois groupoids and covering morphisms in topos theory</em>, Galois theory, Hopf algebras, and semiabelian categories, 131–161, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.7071">links</a>.</li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a> <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Grothendieck</a> talked about how this 1-categorical situation generalizes to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>s.</p> <p>After <em>Pursuing Stacks</em>, apparently the first to publish a detailed formalization and proof of how the <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> may be recovered from the behaviour of <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s on <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> was</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a>, <em>Toward a Galoisian interpretation of homotopy theory</em> (<a href="http://arxiv.org/abs/math/0007157">arXiv:0007157</a>)</li> </ul> <p>This has a followup construction in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a> and <a class="existingWikiWord" href="/nlab/show/Gabriele+Vezzosi">Gabriele Vezzosi</a>, <em>Segal topoi and stacks over Segal categories</em> , Proceedings of the <p>Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (<a href="http://arxiv.org/abs/math/0212330">arxiv:math/0212330</a>).</p> </li> </ul> <p>Very similar constructions and statement then appeared in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Polesello">Pietro Polesello</a> and <a class="existingWikiWord" href="/nlab/show/Ingo+Waschkies">Ingo Waschkies</a>, <em>Higher monodromy</em> , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (<a href="http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Parametrized spaces model locally constant homotopy sheaves</em> (<a href="http://arxiv.org/abs/0706.2874">arXiv:0706.2874</a>)</p> </li> </ul> <p>and, building on that, in example 1.8 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Locally constant functors</em> , Math. Proc. Camb. Phil. Soc. , 147 (<a href="http://www.math.univ-toulouse.fr/~dcisinsk/lcmodcat3.pdf">pdf</a></li> </ul> <p>Notably the article by <a class="existingWikiWord" href="/nlab/show/Pietro+Polesello">Pietro Polesello</a> and <a class="existingWikiWord" href="/nlab/show/Ingo+Waschkies">Ingo Waschkies</a> makes fully explicit the observation that <em>locally</em> constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stacks are precisely the sections of the <em>constant</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-stack on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">n Grpd</annotation></semantics></math>. This is a key observation for bringing the full power of the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Pi \dashv LConst)</annotation></semantics></math> into the picture, as we do here.</p> <p>It was pointed out to <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> by <a class="existingWikiWord" href="/nlab/show/Richard+Williamson">Richard Williamson</a> that these constructions should generalize from topological spaces to objects in any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, maybe along the lines outlined above, and that this way suitable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-toposes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> comes canonically equipped with a notion of <span class="newWikiWord">homotopy ∞-groupoid<a href="/schreiber/new/homotopy+%E2%88%9E-groupoid">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> of every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math>.</p> <h2 id="Paths">In terms of concrete paths</h2> <p>…</p> <h3 id="references_3">References</h3> <p>The following references discuss fundamental groupoids of an entire <a class="existingWikiWord" href="/nlab/show/topos">topos</a> constructed from concrete <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a>s. In the context of the above discussion these toposes are to be thought of as <em><a class="existingWikiWord" href="/nlab/show/petit+topos">petit</a></em> over-toposes over a given object in an ambient <a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a>, and as such are concerned with the fundamental groupoid of that object, in our sense.</p> <p>The construction of the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> of a topos from <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a>s is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Gavin+Wraith">Gavin Wraith</a>, <em>Connected locally connected toposes are path-connected</em> , Transactions of the AMS, volume 295, number 2, (1986)</li> </ul> <p>The comparison of this construction with the one by monodromy/Galois theory is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>On the construction of the Grothendieck fundamental group of a topos by paths</em> , J. Pure and Applied Algebra, 116 (1997)</li> </ul> <h2 id="GeomExamples">Examples</h2> <h3 id="Pi0Ofsheafontopspace">Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of a sheaf on a locally connected topological space</h3> <p>Here we discuss the 0-th geometric homotopy group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Pi_0 : Sh(X) \to Set</annotation></semantics></math> of objects in a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">sheaf topos</a> in terms of a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/constant+sheaf">constant sheaf</a> functor. This is a special case of the more general situation discussed in <a href="Pi0InLocConTop">Pi0 of a general object in a locally connected topos</a> below.</p> <p>Let <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> be a sufficiently nice <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <div class="un_proposition"> <h6 id="observation">Observation</h6> <p>There is a triple of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo>⊣</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow></mover></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex"> (\Pi_0 \dashv LConst \dashv \Gamma) \;\;\; : \;\;\; Sh(X) \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(LConst \dashv \Gamma)</annotation></semantics></math> is the usual <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LConst</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">LConst_S</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/constant+sheaf">constant sheaf</a> of <a class="existingWikiWord" href="/nlab/show/locally+constant+function">locally constant function</a>s with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S \in Set</annotation></semantics></math> and</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Pi : Sh(X) \to Set</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> and sends each sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to the set of connected components of the corresponding <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>A</mi></msub><mo>:</mo><mi>Et</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p_A : Et(A) \to X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>Et</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi_0(A) = \pi_0 Et(A) \,. </annotation></semantics></math></div></li> </ul> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LConst</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">LConst_S</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>LConst</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">E(LConst_S) = X \times S</annotation></semantics></math>. By the relation of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> on <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> with <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a>s over <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 have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>LConst</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mi>Et</mi><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo>×</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{Sh(X)}(A, LConst_S) \simeq Hom_{Et/X}(E(A), X \times S) </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>E</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma : I \to E(A)</annotation></semantics></math> any continuous path in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(A)</annotation></semantics></math>, and for <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 stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f : E(A) \to X \times S</annotation></semantics></math> a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Et</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Et/X</annotation></semantics></math>, the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> is fixed by, say, the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\gamma(0)) = (p_A(\gamma_0),s)</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>γ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>t</mi><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\gamma) : t \mapsto (p_A(\gamma(t)),s)</annotation></semantics></math>. This means that the value of <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> on any path component of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(A)</annotation></semantics></math> is uniquely fixed by its value on any point in that path component.</p> <p>Choosing a basepoint in each path component therefore induces bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Et</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \simeq Hom_{Set}(\pi_0(Et(A)), S) = Hom_{Set}(\Pi_0(A),S) \,. </annotation></semantics></math></div></div> <h3 id="Pi0InLocConTop">Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> of a general object in a locally connected topos</h3> <p>More generally, if <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> is a <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> then the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a>s <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>E</mi><mover><mo>→</mo><mo>←</mo></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex">(LConst \dashv \Gamma) : E \stackrel{\leftarrow}{\to} Set</annotation></semantics></math> has also a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo>⊣</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>E</mi><mover><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow></mover></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,. </annotation></semantics></math></div> <p>For instance page 17 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Classifying Spaces and Classifying Topoi</em> Lecture Notes in Mathematics 1616, Springer (1995)</li> </ul> <h3 id="geometric__of_objects_in_a_1topos">Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> of objects in a 1-topos</h3> <p>The general idea is that of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck%27s+Galois+theory">Grothendieck's Galois theory</a>.</li> </ul> <p>A discussion of of how this produces first homotopy groups of a 1-<a class="existingWikiWord" href="/nlab/show/topos">topos</a> is at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a>.</li> </ul> <p>The general construction of the first geometric homotopy group of objects in a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> is for instance in section 8.4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Topos theory</em> .</li> </ul> <h3 id="geometric__of_a_topological_space">Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_2</annotation></semantics></math> of a topological space</h3> <p>This case is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pietro+Polesello">Pietro Polesello</a> and <a class="existingWikiWord" href="/nlab/show/Ingo+Waschkies">Ingo Waschkies</a>, <em>Higher monodromy</em> , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (<a href="http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf">pdf</a>)</li> </ul> <p>We indicate briefly how the results stated in this article fit into the general abstract picture as indicated above:</p> <p>The authors consider locally constant 1-stacks and 2-stacks on sites of open subsets of sufficiently nice <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s.</p> <p>Prop. 1.1.9 gives the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>LConst</mi></mover></mover><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (LConst \dashv \Gamma) : Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Grpd </annotation></semantics></math></div> <p>between forming constant stacks and taking global sections.</p> <p>Then prop 1.2.5, 1.2.6, culminating in <a href="http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf#page=13">theorem 1.2.9, p. 121</a> gives (somewhat implicitly) the other adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munder><mo>←</mo><mi>LConst</mi></munder><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mn>1</mn></msub></mrow></mover></mover><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi_1\dashv LConst) : Op(X) \hookrightarrow Sh_{(2,1)}(X) \stackrel{\overset{\Pi_1}{\to}}{\underset{LConst}{\leftarrow}} Grpd </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> functor on representables. (Where we change a bit the perspective on the results as presented there, to amplify the pattern indicated above. For instance where the authors write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>C</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(X,C_X)</annotation></semantics></math> we think of this here equivalently as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>LConst</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(2,1)}(X)(X,LConst(C))</annotation></semantics></math>, so that the theorem then gives the adjunction equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><mi>Grpd</mi><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cdots \simeq Grpd(\Pi_1(X),C)</annotation></semantics></math>).</p> <p>Then in essentially verbatim analogy, these results are lifted from stacks to 2-stacks in section 2, where now prop 2.2.2, 2.2.3, culminating in <a href="http://www.intlpress.com/HHA/v7/n1/a7/v7n1a7.pdf#page=24">theorem 2.2.5, p. 132</a> gives (somewhat implicitly) the adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>2</mn></msub><mo>⊣</mo><mi>LConst</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munder><mo>←</mo><mi>LConst</mi></munder><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mn>2</mn></msub></mrow></mover></mover><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi_2\dashv LConst) : Op(X) \hookrightarrow Sh_{(3,1)}(X) \stackrel{\overset{\Pi_2}{\to}}{\underset{LConst}{\leftarrow}} Grpd </annotation></semantics></math></div> <p>now with the <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path 2-groupoid</a> operation (locally) left adjoint to forming constant 2-stacks.</p> <h3 id="ooStackOnTopSpace">Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math> of a topological space</h3> <p>Let <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> be a sufficiently nice (I think this should be locally (relatively) contractible. -DR) (<a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>) <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. The canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to {*}</annotation></semantics></math> induces the <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>LConst</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is taking <a class="existingWikiWord" href="/nlab/show/global+section">global section</a>s and the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> is forming the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>Core</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K = Core (\infty Grpd)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LConst</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">LConst_K</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s and we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>LConst</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Γ</mi><msub><mi>LConst</mi> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> LConst(X) := Sh_{(\infty,1)}(X, LConst_{\infty Grpd})= \Gamma LConst_{\infty Grpd} </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of locally constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks on <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>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Pi(X) := Sing X</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> 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>.</p> <div class="un_theorem"> <h6 id="claim">Claim</h6> <p>There is an equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> LConst(X) \simeq \infty Grpd(\Pi(X), \infty Grpd) \,. </annotation></semantics></math></div></div> <blockquote> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: I think this is proven in the literature, if maybe slightly implicitly so. I’ll now go through the available references to discuss this.</p> </blockquote> <p>After old ideas by <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a> from <a class="existingWikiWord" href="/nlab/show/Pursuing+Stacks">Pursuing Stacks</a>, it seems that the first explicit formalization and proof of this statement is given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a>, <em>Toward a Galoisian interpretation of homotopy theory</em> (<a href="http://arxiv.org/abs/math/0007157">arXiv:0007157</a>)</li> </ul> <p>In <a href="http://arxiv.org/PS_cache/math/pdf/0007/0007157v4.pdf#page=25">theorem 2.13, p. 25</a> the author proves an equivalence of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> (modeled there as <a class="existingWikiWord" href="/nlab/show/Segal+category">Segal categories</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Fib</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> LConst(X) \simeq Fib(\Pi(X)) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">locally constant ∞-stack</a>s on <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> and <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>s over the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X) = Sing(X)</annotation></semantics></math>.</p> <p>But Kan fibrations over a Kan complex such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> are equivalently <a class="existingWikiWord" href="/nlab/show/left+fibration">left fibration</a>s (as discussed there) and by by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a> these are equivalent to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi(X) \to \infty Grpd</annotation></semantics></math>. So under the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a> Toën’s result does actually produce an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> LConst(X) \simeq Func(\Pi(X), \infty Grpd) \,. </annotation></semantics></math></div> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Toen">Bertrand Toen</a> and <a class="existingWikiWord" href="/nlab/show/Gabriele+Vezzosi">Gabriele Vezzosi</a>, <em>Segal topoi and stacks over Segal categories</em> , Proceedings of the <p>Program Stacks, Intersection theory and Non-abelian Hodge Theory, MSRI, Berkeley, January-May 2002 (<a href="http://arxiv.org/abs/math/0212330">arxiv:math/0212330</a>).</p> </li> </ul> <p>this is discussed in the context of <a class="existingWikiWord" href="/nlab/show/Segal-topos">Segal-topos</a>es.</p> <p>Very similar statements are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Parametrized spaces model locally constant homotopy sheaves</em> (<a href="http://arxiv.org/abs/0706.2874">arXiv:0706.2874</a>)</li> </ul> <p>and, building on that, in example 1.8 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Locally constant functors</em> , Math. Proc. Camb. Phil. Soc. , 147 (<a href="http://www-math.univ-paris13.fr/~cisinski/lcmodcat3.pdf">pdf</a>)</li> </ul> <p>A variant of this statement – more general in one respect, less general in another – appears in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>as <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=545">theorem 7.1.0.1</a>.</p> <p>There it is shown that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">K \in \infty Grpd</annotation></semantics></math> there is a bijection of homotopy sets of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">|</mo><mi>K</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mo>*</mo></msub><msup><mi>p</mi> <mo>*</mo></msup><mi>K</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \pi_0 Top(X, |K|) \simeq \pi_0(p_* p^* K) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>p</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>p</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">(p^* \dashv p_*) : Sh_{(\infty,1)}(X) \to \infty Grpd</annotation></semantics></math> is the geometric morphism we denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(LConst \dashv \Gamma)</annotation></semantics></math> above.</p> <p>If we also rewrite the left using <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">the equivalence</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>, this reads</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Γ</mi><msub><mi>LConst</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>LConst</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \infty Grpd(\Pi(X), K) \simeq \pi_0(\Gamma LConst_K) = \pi_0 Sh_{(\infty,1)}(X,LConst_K) \,, </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>Core</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K = Core(\infty Grpd)</annotation></semantics></math> this is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> of the above statement.</p> <h3 id="GeomPiOfTermObj">Geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math> of the terminal object in a locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connected <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</h3> <p>The geometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_\infty</annotation></semantics></math> of the terminal object in a locally ∞-connected (∞,1)-topos can be called the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+an+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid</a> of the topos. It <a class="existingWikiWord" href="/nlab/show/representable+functor">represents</a> the <a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> of the topos.</p> <p>On page 18-19 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Classifying Spaces and Classifying Topoi</em> Lecture Notes in Mathematics 1616, Springer (1995)</li> </ul> <p>is described the construction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi(X) \in \infty Grpd</annotation></semantics></math> for <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> the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(C)</annotation></semantics></math> on an ordinary <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> as described above in <a href="PiConstruction">Geometric fundamental oo-groupoid</a>.</p> <p>This reviews in particular (slightly implicitly)</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> that has a basis of contractible open subsets. Write <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> also for <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> regarded as the terminal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(X)</annotation></semantics></math>. Then the image 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> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Pi : Sh_{(\infty,1)}(X) \to \infty Grpd</annotation></semantics></math> has the same homtopy groups as <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> regarded as an object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_n \Pi(X) \simeq \pi_n(X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is a slight reformulation of the statement in</p> <p>M. Artin, B. Mazur, <em>Etale homotopy</em> , Springer lecture notes in mathematics 100, Berlin 1969</p> </div> <p>Notice the local contractibility assumption. This is necessary in general for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> to make sense.</p> <h2 id="GeneralExamples">Examples</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> and consider in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(Diff)</annotation></semantics></math> the two objects</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack represented by the standard circle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math> – the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack constant on the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of the additive group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>.</p> </li> </ul> <p>Then</p> <ul> <li> <p>the <em>categorical</em> homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> are all trivial</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mi>n</mi> <mi>cat</mi></msubsup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \pi_n^{cat}(S^1) = {*} </annotation></semantics></math></div></li> <li> <p>the <em>geometric</em> homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> are the usual ones obtained from regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> as an object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mn>0</mn> <mi>geom</mi></msubsup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \pi^{geom}_0(S^1) = * </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mn>1</mn> <mi>geom</mi></msubsup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \pi^{geom}_1(S^1) = \mathbb{Z} </annotation></semantics></math></div> <p>etc.</p> </li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math> it is the other way round:</p> <ul> <li> <p>the <em>categorical</em> homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{Z}</annotation></semantics></math> are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mi>n</mi> <mi>cat</mi></msubsup><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo stretchy="false">|</mo><mi>if</mi><mspace width="thickmathspace"></mspace><mi>n</mi><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo stretchy="false">|</mo><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_n^{cat}(\mathbf{B}\mathbb{Z}) = \left\{ \array{ \mathbb{Z} & | if\; n=1 \\ * & | otherwise } \right. \,. </annotation></semantics></math></div></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 5, 2023 at 09:43:45. 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