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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="locality_and_descent">Locality and descent</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/localization">localization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+object">local object</a>, <a class="existingWikiWord" href="/nlab/show/local+morphism">local morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+localization">reflective localization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>, <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></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/descent">descent</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+object">descent object</a>, <a class="existingWikiWord" href="/nlab/show/descent+morphism">descent morphism</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/matching+family">matching family</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a>, <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>,<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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+descent">cohomological descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>, <a class="existingWikiWord" href="/nlab/show/higher+monadic+descent">higher monadic descent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a>, <a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/descent+and+locality+-+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> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>In general, given a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/class">class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>, one may ask for the <em><a class="existingWikiWord" href="/nlab/show/localization">localization</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}[W^{-1}]</annotation></semantics></math>, or more specifically for the <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a> (the <a class="existingWikiWord" href="/nlab/show/reflective+localization">reflective localization</a>). Similarly for variants in <a class="existingWikiWord" href="/nlab/show/higher+category">higher category</a>, such as <em><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></em>.</p> <p>If <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">\mathcal{C}</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a>, then for a given <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathbb{A} \in \mathcal{C}</annotation></semantics></math>, one may take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>≔</mo><msub><mi>W</mi> <mi>𝔸</mi></msub></mrow><annotation encoding="application/x-tex">W \coloneqq W_{\mathbb{A}}</annotation></semantics></math> to be the class of morphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mi>𝔸</mi><mover><mo>→</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mo>*</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>𝔸</mi><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \times (\mathbb{A} \overset{\exists!}{\to} \ast) \;\;\colon\;\; X \times \mathbb{A} \overset{p_1}{\longrightarrow} X \,, </annotation></semantics></math></div> <p>where <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> is any <a class="existingWikiWord" href="/nlab/show/object">object</a>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>×</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">(-) \times (-) \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/reflective+localization">reflective localization</a> at such a class of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>𝔸</mi></msub></mrow><annotation encoding="application/x-tex">W_{\mathbb{A}}</annotation></semantics></math> is often referred to as <em>homotopy localization at the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">\mathbb{A}</annotation></semantics></math></em> or similar.</p> <p>The idea is that if <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">\mathbb{A}</annotation></semantics></math> is, or is regarded as, an <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a>, then “geometric” <a class="existingWikiWord" href="/nlab/show/left+homotopies">left homotopies</a> between morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> are, or would be, given by morphisms out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">X \times \mathbb{A}</annotation></semantics></math>, and hence forcing the projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>𝔸</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times \mathbb{A} \to X</annotation></semantics></math> to be equivalences means forcing all morphisms to be <em><a class="existingWikiWord" href="/nlab/show/homotopy+invariance">homotopy invariant</a></em> with respect to <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">\mathbb{A}</annotation></semantics></math>.</p> <p>Typically this is considered in the case that <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">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> with a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> of <a class="existingWikiWord" href="/nlab/show/generators+and+relations">generating objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">G_i</annotation></semantics></math> such that it becomes sufficient to enforce the localization only on the resulting <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>i</mi></msub><mo>×</mo><mo stretchy="false">(</mo><mi>𝔸</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_i \times (\mathbb{A} \to \ast)</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>The localization of the <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> on the <a class="existingWikiWord" href="/nlab/show/Nisnevich+site">Nisnevich site</a> at the <a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1</annotation></semantics></math> is known as <em><a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory">A1-homotopy theory</a></em>.</p> </li> <li> <p>The localization of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a> at the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math> is equivalently (<a class="existingWikiWord" href="/nlab/show/geometrically+discrete+%E2%88%9E-groupoids">geometrically discrete</a>) <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>.</p> <p>After realizing <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a> as the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a> over <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Smooth\infty Grpd \simeq Sh_{\infty}(CartSp)</annotation></semantics></math>, this follows from the following Prop. <a class="maruku-ref" href="#HomotopyLocalizationOverSiteOfAns"></a>.</p> </li> </ul> <div class="num_prop" id="HomotopyLocalizationOverSiteOfAns"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^n</annotation></semantics></math>s)</strong></p> <p>Let <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">\mathcal{C}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/site">site</a> (<a href="geometry+of+physics+--+categories+and+toposes#Coverage">this Def.</a>), and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[\mathcal{C}^{op}, sSet_{Qu}]_{proj, loc}</annotation></semantics></math> for its local projective <a class="existingWikiWord" href="/nlab/show/model+category+of+simplicial+presheaves">model category of simplicial presheaves</a> (<a href="geometry+of+physics+--+categories+and+toposes#TopologicalLocalization">this Prop.</a>).</p> <p>Assume that <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">\mathcal{C}</annotation></semantics></math> contains an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathbb{A} \in \mathcal{C}</annotation></semantics></math>, such that every other object is a <a class="existingWikiWord" href="/nlab/show/finite+product">finite product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mi>n</mi></msup><mo>≔</mo><munder><munder><mrow><mi>𝔸</mi><mo>×</mo><mi>⋯</mi><mo>×</mo><mi>𝔸</mi></mrow><mo>⏟</mo></munder><mrow><mi>n</mi><mspace width="thickmathspace"></mspace><mtext>factors</mtext></mrow></munder></mrow><annotation encoding="application/x-tex">\mathbb{A}^n \coloneqq \underset{n \; \text{factors}}{\underbrace{\mathbb{A} \times \cdots \times \mathbb{A}}}</annotation></semantics></math>, for some <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>. (In other words, assume that <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">\mathcal{C}</annotation></semantics></math> is also the <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> of <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>.)</p> <p>Consider the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> (<a href="geometry+of+physics+--+categories+and+toposes#HomotopyLocalizationOfCombinatorialModelCategories">this Def.</a>) of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a> over <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">\mathcal{C}</annotation></semantics></math> (<a href="geometry+of+physics+--+categories+and+toposes#TopologicalLocalization">this Prop.</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> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mi>𝔸</mi></msub><munderover><mo>⊥</mo><munder><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ι</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>L</mi> <mi>𝔸</mi></msub></mrow></mover></munderover><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>CombModCat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat) </annotation></semantics></math></div> <p>hence the <a class="existingWikiWord" href="/nlab/show/left+Bousfield+localization">left Bousfield localization</a> of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</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><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi><mo>,</mo><mi>𝔸</mi></mrow></msub><munderover><mrow><mphantom><mrow><msub><mrow></mrow> <mi>Qu</mi></msub></mrow></mphantom><msub><mo>⊥</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>id</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mi>id</mi></mover></munderover><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>CombModCat</mi></mrow><annotation encoding="application/x-tex"> [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc,\mathbb{A}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc} \;\; \in CombModCat </annotation></semantics></math></div> <p>at the set of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><msup><mi>𝔸</mi> <mi>n</mi></msup><mo>×</mo><mi>𝔸</mi><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><msup><mi>𝔸</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow><annotation encoding="application/x-tex"> S \;\coloneqq\; \big\{ \mathbb{A}^n \times \mathbb{A} \overset{p_1}{\longrightarrow} \mathbb{A}^n \big\} </annotation></semantics></math></div> <p>(according to <a href="geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization">this Prop.</a>).</p> <p>Then this is <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalent</a> (<a href="geometry+of+physics+--+categories+and+toposes#HoCombModCat">this Def.</a>) to <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> (<a href="geometry+of+physics+--+categories+and+toposes#InfinityGroupoid">this Example</a>),</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mi>𝔸</mi></msub><munderover><mo>⊥</mo><munder><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ι</mi><mphantom><mi>AA</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>L</mi> <mi>𝔸</mi></msub></mrow></mover></munderover><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>CombModCat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \infty Grpd \;\simeq\; Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat) </annotation></semantics></math></div> <p>in that the (<a class="existingWikiWord" href="/nlab/show/constant+functor">constant functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/limit">limit</a>)-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> (<a href="geometry+of+physics+--+categories+and+toposes#Limits">this Def.</a>)</p> <div class="maruku-equation" id="eq:QuillenEquivalenceInABousfLocalization"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo stretchy="false">]</mo> <mrow><mi>inj</mi><mo>,</mo><mi>loc</mi><mo>,</mo><mi>𝔸</mi></mrow></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><munder><mi>lim</mi><mo>⟵</mo></munder></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>const</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><msub><mi>sSet</mi> <mi>Qu</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>CombModCat</mi></mrow><annotation encoding="application/x-tex"> [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu} \;\;\;\; \in CombModCat </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> (<a href="geometry+of+physics+--+homotopy+types#QuillenEquivalence">this Def.</a>).</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>First to see that <a class="maruku-eqref" href="#eq:QuillenEquivalenceInABousfLocalization">(1)</a> is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>: Since we have a <a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a> before localization</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><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo stretchy="false">]</mo> <mi>inj</mi></msub><munderover><mo>⊥</mo><munder><mo>⟶</mo><munder><mi>lim</mi><mo>⟵</mo></munder></munder><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>const</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></munderover><msub><mi>sSet</mi> <mi>Qu</mi></msub></mrow><annotation encoding="application/x-tex"> [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu} </annotation></semantics></math></div> <p>(by <a href="geometry+of+physics+--+categories+and+toposes#HomotopyLimitOfSimplicialSets">this Example</a>) and since both <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> here are <a class="existingWikiWord" href="/nlab/show/left+proper+model+category">left proper</a> <a class="existingWikiWord" href="/nlab/show/simplicial+model+categories">simplicial model categories</a> (by <a href="geometry+of+physics+--+categories+and+toposes#SimplicialPresheavesIsProperCombinatorialSimplicial">this Prop.</a> and <a href="geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization">this Prop.</a>), and since <a class="existingWikiWord" href="/nlab/show/left+Bousfield+localization">left Bousfield localization</a> does not change the class of <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> (<a href="geometry+of+physics+--+categories+and+toposes#BousfieldLocalizationOfModelCategories">this Def.</a>) it is sufficient to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>⟵</mo></munder></mrow><annotation encoding="application/x-tex">\underset{\longleftarrow}{\lim}</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a> (by <a href="geometry+of+physics+--+categories+and+toposes#RecognitionOfSimplicialQuillenAdjunction">this Prop.</a>).</p> <p>But by assumption <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">\mathcal{C}</annotation></semantics></math> has a <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><mo>*</mo><mo>=</mo><msup><mi>𝔸</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\ast = \mathbb{A}^0</annotation></semantics></math>, which is hence the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{op}</annotation></semantics></math>, so that the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> operation is given just by evaluation on that object:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>⟵</mo></munder><mstyle mathvariant="bold"><mi>X</mi></mstyle><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\longleftarrow}{\lim} \mathbf{X} \;=\; \mathbf{X}(\mathbb{A}^0) \,. </annotation></semantics></math></div> <p>Hence it is sufficient to see that an injectively fibrant simplicial presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> is objectwise a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>. This is indeed the case, by <a href="geometry+of+physics+--+categories+and+toposes#ModelCategoriesOfSimplicialPresheaves">this Prop.</a>.</p> <p>To check that <a class="maruku-eqref" href="#eq:QuillenEquivalenceInABousfLocalization">(1)</a> is actually a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>, we check that the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> and <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> are <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">X \in sSet</annotation></semantics></math> any simplicial set (necessarily cofibrant), the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>id</mi> <mi>X</mi></msub></mrow></mover><mi>const</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>const</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow></mover><mi>const</mi><mo stretchy="false">(</mo><mi>P</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \overset{id_X}{\longrightarrow} const(X)(\mathbb{A}^0) \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow} const(P X)(\mathbb{A}^0) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mi>X</mi></msub></mrow></mover><mi>P</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">X \overset{j_X}{\longrightarrow} P X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibrant+replacement">fibrant replacement</a> (<a href="geometry+of+physics+--+categories+and+toposes#FibrantCofibrantReplacementFunctorToHomotopyCategory">this Def.</a>). But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">const(-)(\mathbb{A}^0)</annotation></semantics></math> is clearly the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> and the plain adjunction unit is the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a>, so that this composite is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">j_X</annotation></semantics></math> itself, which is indeed a weak equivalence.</p> <p>For the other case, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>sSet</mi> <mi>Qu</mi></msub><msub><mo stretchy="false">]</mo> <mrow><mi>inj</mi><mo>,</mo><mi>loc</mi><mo>,</mo><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{X} \in [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}^1}</annotation></semantics></math> be fibrant. This means (by <a href="geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization">this Prop.</a>) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> is fibrant in the injective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> as well as in the local model structure, and is a derived-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+object">local object</a> (<a href="geometry+of+physics+--+categories+and+toposes#DerivedLocalObjects">this Def.</a>), in that the <a class="existingWikiWord" href="/nlab/show/derived+hom-functor">derived hom-functor</a> out of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>𝔸</mi> <mn>1</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><msup><mi>𝔸</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^n \times \mathbb{A}^1 \overset{p_1}{\longrightarrow} \mathbb{A}^n</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mi>n</mi></msup><mo>,</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>𝔸</mi> <mn>1</mn></msup><mo>,</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}Hom( p_1 ) \;\colon\; \mathbb{R}Hom( \mathbb{A}^n , \mathbf{X}) \overset{\in W}{\longrightarrow} \mathbb{R}Hom( \mathbb{A}^n \times \mathbb{A}^1 , \mathbf{X}) </annotation></semantics></math></div> <p>But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> is fibrant, this derived hom is equivalent to the ordinary <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> (<a href="geometry+of+physics+--+categories+and+toposes#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory">this Lemma</a>), and hence with the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this Prop.</a>) we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X}(p_1) \;\colon\; \mathbf{X}(\mathbb{A}^n) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^{n+1}) </annotation></semantics></math></div> <p>is a weak equivalence, for all <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>. By <a class="existingWikiWord" href="/nlab/show/induction">induction</a> on <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> this means that in fact</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow></mover><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X}(\mathbb{A}^0) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^n) </annotation></semantics></math></div> <p>is a weak equivalence for all <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>. But these are just the components of the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>ϵ</mi></munderover><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex"> const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X} </annotation></semantics></math></div> <p>which is hence also a weak equivalence. Hence for the <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo stretchy="false">(</mo><mi>Q</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>const</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mstyle mathvariant="bold"><mi>X</mi></mstyle></msub><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mi>const</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msup><mi>𝔸</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>ϵ</mi></munderover><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex"> const (Q \mathbf{X})(\mathbb{A}^0) \overset{const(p_{\mathbf{X}}(\mathbb{A}^0))}{\longrightarrow} const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X} </annotation></semantics></math></div> <p>to be a weak equivalence, it is now sufficient to see that the value of a <a class="existingWikiWord" href="/nlab/show/cofibrant+replacement">cofibrant replacement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mstyle mathvariant="bold"><mi>X</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">p_{\mathbf{X}}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^0</annotation></semantics></math> is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.</p> </div> <h2 id="references">References</h2> <ul> <li id="MorelVoevodsky99"><a class="existingWikiWord" href="/nlab/show/Fabien+Morel">Fabien Morel</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Voevodsky">Vladimir Voevodsky</a>, Def. 3.1 in <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1</annotation></semantics></math>-homotopy theory of schemes</em>, Publications Mathématiques de l’IHÉS, Volume 90 (1999), p. 45-143 (<a href="http://www.numdam.org/item/?id=PMIHES_1999__90__45_0">Numdam:PMIHES_1999__90__45_0</a> <a href="http://www.math.uiuc.edu/K-theory/0305/">K-Theory:0305</a> )</li> </ul> <p>For more references see also at <em><a class="existingWikiWord" href="/nlab/show/motivic+homotopy+theory">motivic homotopy theory</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on January 10, 2019 at 18:10:12. 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