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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="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='#idea'>Idea</a></li> <li><a href='#statement'>Statement</a></li> <ul> <li><a href='#ForTopSpaces'>For topological spaces</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#strict_version'>Strict version</a></li> <ul> <li><a href='#strict_higher_van_kampen_theorem'>Strict Higher van Kampen Theorem</a></li> </ul> </ul> <li><a href='#for_objects_in_a_cohesive_topos'>For objects in a cohesive <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='#examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>One form of a <em>higher homotopy van Kampen theorem</em> is a theorem that asserts that the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> can be computed by a suitable <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> or <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> over homotopy types of its pieces. Another form which allows specific computation deals with spaces with certain kinds of structure, for example filtered spaces or <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>-cubes of spaces.</p> <p>This generalizes the <a class="existingWikiWord" href="/nlab/show/van+Kampen+theorem">van Kampen theorem</a>, which only deals with the underlying 1-type (the fundamental groupoid).</p> <h2 id="statement">Statement</h2> <h3 id="ForTopSpaces">For topological spaces</h3> <h4 id="general">General</h4> <div class="un_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>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \chi : C \to Op(X) </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> out of a <a class="existingWikiWord" href="/nlab/show/small+category">small 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">C</annotation></semantics></math> such that</p> <ul> <li>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>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">C_x</annotation></semantics></math> of objects <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(x)</annotation></semantics></math> contains <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> has a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weakly</a> <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a> <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>.</li> </ul> <p>Then:</p> <p>the canonical morphism in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> out of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow><mi>Sing</mi><mo>∘</mo><mi>χ</mi><mo>→</mo><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> {\lim_\to} Sing \circ \chi \to Sing(X) </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</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> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(X)</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hocolim</mi><mi>Sing</mi><mo>∘</mo><mi>χ</mi></mrow><annotation encoding="application/x-tex">hocolim Sing \circ \chi</annotation></semantics></math>.</p> </div> <p>This is theorem A.1.1 in (<a href="#Lurie">Lurie</a>).</p> <h4 id="strict_version">Strict version</h4> <p>The following is a version of the above general statement restricted to a <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a>-version of the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> and applicable for <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> that are equipped with the extra structure of a <a class="existingWikiWord" href="/nlab/show/filtered+topological+space">filtered topological space</a>.</p> <p>Notice that these strict <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 are equivalent to <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a>es.</p> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">X_*</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filtered+space">filtered space</a> and <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 the union of the interiors of sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">U^i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>U</mi> <mo>*</mo> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">U^i_*</annotation></semantics></math> be the filtered space given by the intersections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mi>i</mi></msup><mo>∩</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">U^i \cap X_n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \geq 0</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>I</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">d=(i,j) \in I^2</annotation></semantics></math> we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">U^d</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mi>i</mi></msup><mo>∩</mo><msup><mi>U</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">U^i \cap U^j</annotation></semantics></math>. We then have a coequaliser diagram of filtered spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨆</mo> <mrow><mi>d</mi><mo>∈</mo><msup><mi>I</mi> <mn>2</mn></msup></mrow></munder><msubsup><mi>U</mi> <mo>*</mo> <mi>d</mi></msubsup><msubsup><mo>⇉</mo> <mi>b</mi> <mi>a</mi></msubsup><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨆</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msubsup><mi>U</mi> <mo>*</mo> <mi>i</mi></msubsup><msup><mo>→</mo> <mi>c</mi></msup><msub><mi>X</mi> <mo>*</mo></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">\bigsqcup_{d \in I^2} U^d_* \rightrightarrows ^a_b \bigsqcup _{i \in I} U^i_* \to ^c X_*.</annotation></semantics></math></div> <div class="un_theorem"> <h6 id="strict_higher_van_kampen_theorem">Strict Higher van Kampen Theorem</h6> <p>If the filtered spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>U</mi> <mo>*</mo> <mi>f</mi></msubsup></mrow><annotation encoding="application/x-tex">U^f_*</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/connected+filtered+space">connected filtered space</a>s for all finite intersections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>U</mi> <mo>*</mo> <mi>f</mi></msubsup></mrow><annotation encoding="application/x-tex">U^f_*</annotation></semantics></math> of the filtered spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>U</mi> <mo>*</mo> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">U^i_*</annotation></semantics></math>, then</p> <ol> <li> <p>(Conn) The filtered space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">X_*</annotation></semantics></math> is connected; and</p> </li> <li> <p>(Iso) The fundamental <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a> 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> takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.</p> </li> </ol> <p>A full account is given in (<a href="#NAT">Brown-Higgins-Sivera</a> and the methodology is discussed in (<a href="#RBrown">Brown</a>).</p> </div> <p><strong>Remarks</strong></p> <ul> <li> <p>Note that because <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> uses groupoids, it obviously takes disjoint unions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">⨆</mo></mrow><annotation encoding="application/x-tex">\bigsqcup</annotation></semantics></math> of filtered spaces into disjoint unions (= coproducts) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">⨆</mo></mrow><annotation encoding="application/x-tex">\bigsqcup</annotation></semantics></math> of crossed complexes.</p> </li> <li> <p>The proof of the theorem is not direct but goes via the fundamental cubical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-groupoid with connections of the filtered spaces, as that context allows the notions of <em>algebraic inverse to subdivision</em> and of <em>commutative cube</em>. However the proof is a direct generalisation of a proof for the <a class="existingWikiWord" href="/nlab/show/van+Kampen+theorem">van Kampen theorem</a> for the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a>.</p> </li> <li> <p>Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of <em>singular homology theory</em> or of <em>simplicial approximation</em>. Also included is a version of the “small simplex theorem”, see Theorem 10.4.20 of (<a href="#NAT">Brown-Higgins-Sivera</a>).</p> </li> </ul> <h3 id="for_objects_in_a_cohesive_topos">For objects in a cohesive <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>In a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> (already in a <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>) higher van Kampen theorems hold in great generality.</p> <p>See the section <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#vanKampenTheorem">cohesive (∞,1)-topos – van Kampen theorem</a>.</p> <p>In particular for the cohesive <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 class="existingWikiWord" href="/nlab/show/%3FTopGrpd">?TopGrpd</a> of <a class="existingWikiWord" href="/nlab/show/topological+%E2%88%9E-groupoid">topological ∞-groupoid</a>s this reproduces the <a href="#ForTopSpaces">topological higher van Kampen theorem</a> discussed above.</p> <h2 id="examples">Examples</h2> <p>Here is one application in dimension 2 not easily obtainable by traditional <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>P</mi><mo>→</mo><mi>Q</mi><mo>→</mo><mi>R</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to P \to Q \to R \to 0</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>. 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 the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> of the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>Q</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(P,1) \to K(Q,1)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+space">Eilenberg-Mac Lane space</a>s. Then a <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> representing the <a class="existingWikiWord" href="/nlab/show/homotopy+2-type">homotopy 2-type</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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">\mu: C \to Q</annotation></semantics></math> where <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> is abelian and is the direct sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊕</mo> <mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow></msub><msup><mi>P</mi> <mi>r</mi></msup></mrow><annotation encoding="application/x-tex">\oplus_{r \in R} P^r</annotation></semantics></math> of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> one for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> and the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> and permutes the copies by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>r</mi><msup><mo stretchy="false">)</mo> <mi>s</mi></msup><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>r</mi><mo>+</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,r)^s=(p,r+s)</annotation></semantics></math>. Similar examples for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">P,Q,R</annotation></semantics></math> nonabelian are do-able, more complicated, and certainly <strong>not</strong> obtainable by traditional methods.</p> <h2 id="references">References</h2> <p>The version for topological spaces and the <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">fundamental infinity-groupoid</a> functor is discussed in Appendix A of</p> <ul id="Lurie"> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Ek-Algebras">Ek-Algebras</a></em></li> </ul> <p>The version for filtered topological spaces and the strict 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>-groupoid functor is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Higgins">Philip Higgins</a>, <a class="existingWikiWord" href="/nlab/show/Rafael+Sivera">Rafael Sivera</a>, <em><a class="existingWikiWord" href="/nlab/show/Nonabelian+Algebraic+Topology">Nonabelian Algebraic Topology</a></em> (#NAT)</li> </ul> <p>while the general methodology is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>“Modelling and computing homotopy types: I” to appear in 2017 in a special issue of Indagationes Math in honour of L.E.J. Brouwer. (https://arxiv.org/abs/1610.07421) (#RBrown)</li> </ul> <p>One area of application of work of Brown and Loday is to a nonabelian tensor product of groups, see:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a> Bibliography on the nonabelian tensor product. (http://www.groupoids.org.uk/nonabtens.html) (#tens)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 16, 2017 at 12:22:27. 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