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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='#definitions'>Definitions</a></li> <ul> <li><a href='#plain_definition'>Plain definition</a></li> <ul> <li><a href='#contractible_interval_object'>Contractible interval object</a></li> </ul> <li><a href='#in_homotopical_categories'>In homotopical categories</a></li> <ul> <li><a href='#IntervalForTrimbledOmegaeCategories'>Trimble interval object</a></li> <li><a href='#bergermoerdijk_segment_and_interval_object'>Berger–Moerdijk segment- and interval object</a></li> </ul> <li><a href='#InHomotopyTypeTheory'>Interval type</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#standard_intervals_cubes_and_simplices_in__and_'>Standard intervals, cubes and simplices in <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> and <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></a></li> <li><a href='#homotopy_theory_2'><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</a></li> </ul> <li><a href='#fundamental_categories_induced_from_intervals'>Fundamental <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>-categories induced from intervals</a></li> <li><a href='#homotopy_localization_induced_from_an_interval'>Homotopy localization induced from an interval</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em>interval object</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> in 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">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/object">object</a> that behaves 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> roughly like the standard <a class="existingWikiWord" href="/nlab/show/topological+interval">topological interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>≔</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I \coloneqq [0,1]</annotation></semantics></math> with its two boundary point inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>⨿</mo><mo>*</mo><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mover><mi>I</mi></mrow><annotation encoding="application/x-tex"> {*}\amalg {*} \stackrel{[0, 1]}{\to} I </annotation></semantics></math></div> <p>in the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/copairing">copairing</a> of the <a class="existingWikiWord" href="/nlab/show/global+elements">global elements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">0\colon {*} \to I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">1\colon {*} \to I</annotation></semantics></math>.</p> <p>A bare interval object may be nothing more than such a diagram. 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">C</annotation></semantics></math> admits sufficiently many <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s and <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s, then from this alone a lot of structure derives. The precise definition of further structure and property imposed on an interval object varies with the intended context and applications.</p> <p>Notably in a large class of applications the interval object 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> supposed to be the right structure to ensure</p> <ol> <li> <p>that there is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> 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> such that for every 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>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,X]</annotation></semantics></math> exists and behaves like a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> 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>;</p> </li> <li> <p>that there is a notion of composition on these path objects which induces on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,X]</annotation></semantics></math> a structure of a (higher) category internal to <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>: the <a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a> or <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> of the 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>, or rather its <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">fundamental infinity-groupoid</a>.</p> </li> </ol> <p>For instance the choice <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/Top">Top</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math> should be an instance of a category with interval object, and the fundamental <a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic</a> <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</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><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_n(X)</annotation></semantics></math> obtained for any topological space <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 this data should be the fundamental <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>-groupoid as a <a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble n-category</a>.</p> <p>We give two very similar definitions that differ only in some extra assumptions.</p> <ul> <li> <p>The first one was used by Berger and Moerdijk to generalize the Boardman–Vogt resolution of <a class="existingWikiWord" href="/nlab/show/topological+operad">topological operad</a>s to more general <a class="existingWikiWord" href="/nlab/show/operad">operad</a>s.</p> </li> <li> <p>The second is motivated from constructions appearing in the definitions of <a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble n-category</a> and of <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a>. It includes the possibility that the interval is <em>not</em> weakly equivalent to the point, in which case it may be used nontrivially to test for <a class="existingWikiWord" href="/nlab/show/undirected+object">undirected object</a>s and probe <a class="existingWikiWord" href="/nlab/show/directed+object">directed object</a>s.</p> </li> </ul> <h2 id="definitions">Definitions</h2> <h3 id="plain_definition">Plain definition</h3> <div class="un_defn"> <h6 id="definition_plain_interval_object">Definition (plain interval object)</h6> <p>A <strong>plain interval object</strong> in a category <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 just a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> diagram with equal feet</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mover><mo>→</mo><mn>0</mn></mover><mi>I</mi><mover><mo>←</mo><mn>1</mn></mover><mi>pt</mi></mrow><annotation encoding="application/x-tex"> pt \stackrel{0}{\to} I \stackrel{1}{\leftarrow} pt </annotation></semantics></math></div> <p>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>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math> any two objects and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> any two morphisms.</p> </div> <p>In categories with finite limits it is often required that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">pt=*</annotation></semantics></math> is the terminal object and in this case the interval object is called <em>cartesian interval object</em>.</p> <p>Examples for the use of this notion is at <a class="existingWikiWord" href="/nlab/show/fundamental+%28infinity%2C1%29-category">fundamental (infinity,1)-category</a> in the section “fundamental geometric ∞-categories”.</p> <h4 id="contractible_interval_object">Contractible interval object</h4> <div class="un_defn"> <h6 id="definition_contractible_interval_object">Definition (contractible interval object)</h6> <p>A <em>contractible interval object</em> in a category <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 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></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> with two <a class="existingWikiWord" href="/nlab/show/global+elements">global elements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">0:* \to I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">1:* \to I</annotation></semantics></math> such that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 = 1</annotation></semantics></math></li> <li>for any 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">A</annotation></semantics></math> 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> and global elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mi>A</mi></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">0_A:* \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>A</mi></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">1_A:* \to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mi>A</mi></msub><mo>=</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">0_A = 1_A</annotation></semantics></math>, there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f:I \to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mn>0</mn><mo>=</mo><msub><mn>0</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">f \circ 0 = 0_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mn>1</mn><mo>=</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">f \circ 1 = 1_A</annotation></semantics></math>.</li> </ul> </div> <div class="un_defn"> <h6 id="definition_contractible_interval_object_in_n_1categories">Definition (contractible interval object in (n, 1)-categories)</h6> <p>A <em>contractible interval object</em> in an <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>-category <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 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></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> with two <a class="existingWikiWord" href="/nlab/show/global+elements">global elements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">0:* \to I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">1:* \to I</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mn>0</mn><mo>≅</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">s:0 \cong 1</annotation></semantics></math> such that</p> <ul> <li>for any 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">A</annotation></semantics></math> 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> and global elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>0</mn> <mi>A</mi></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">0_A:* \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>A</mi></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">1_A:* \to A</annotation></semantics></math> with an equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>A</mi></msub><mo>:</mo><msub><mn>0</mn> <mi>A</mi></msub><mo>≅</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">s_A:0_A \cong 1_A</annotation></semantics></math>, there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f:I \to A</annotation></semantics></math> with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>′</mo></msup><mo>:</mo><mo stretchy="false">(</mo><mn>0</mn><mo>≅</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mn>0</mn> <mi>A</mi></msub><mo>≅</mo><msub><mn>1</mn> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{'}:(0 \cong 1) \to (0_A \cong 1_A)</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>f</mi><mo>∘</mo><mn>0</mn><mo>≅</mo><msub><mn>0</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">p:f \circ 0 \cong 0_A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>f</mi><mo>∘</mo><mn>1</mn><mo>≅</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">q:f \circ 1 \cong 1_A</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><msup><mi>f</mi> <mo>′</mo></msup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>s</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">r:f^{'}(s) \cong s_A</annotation></semantics></math> satisfy the <a class="existingWikiWord" href="/nlab/show/coherence+laws">coherence laws</a>.</li> </ul> </div> <p>The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the contractible interval object means that the contractible interval object corresponds to the <a class="existingWikiWord" href="/nlab/show/interval+type">interval type</a> in <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+type+theory">Martin-Löf type theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, and is thus <a class="existingWikiWord" href="/nlab/show/equivalent">equivalent</a> to the terminal object. If a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a> <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>-category has an contractible interval type, the terminal object is a <a class="existingWikiWord" href="/nlab/show/separator">separator</a> (see <a href="https://homotopytypetheory.org/2011/04/04/an-interval-type-implies-function-extensionality/">Mike Shulman’s blogpost</a>).</p> <h3 id="in_homotopical_categories">In homotopical categories</h3> <p>If the ambient category <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 a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>, such as a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, there are natural further conditions to put on an interval object:</p> <h4 id="IntervalForTrimbledOmegaeCategories">Trimble interval object</h4> <p>The following definition is strongly related to the notion of <a class="existingWikiWord" href="/nlab/show/Trimble+omega-category">Trimble omega-category</a> where the interval object gives the internal hom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[I,X]</annotation></semantics></math> the structure of an <a class="existingWikiWord" href="/nlab/show/operad">operad</a> giving (by induction) the model of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/A-infinity-category">category</a> structure on</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>X</mi> <mn>0</mn></msub><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>pt</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><mi>s</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>t</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>τ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>pt</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X_0 := [pt,X]) \stackrel{s := [\sigma,X]}{\leftarrow} [I,X] \stackrel{t := [\tau, X]}{\to} (X_0 := [pt,X]) \,. </annotation></semantics></math></div> <p>This internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category is denoted</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>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Pi_1(X) </annotation></semantics></math></div> <p>This is in (a bit) more detail in <a class="existingWikiWord" href="/nlab/show/Trimble+omega-category">Trimble omega-category</a> and in <a class="existingWikiWord" href="/nlab/show/fundamental+%28infinity%2C1%29-category">fundamental (infinity,1)-category</a> in the section “Fundamental algebraic <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>-categories”.</p> <div class="un_defn"> <h6 id="definition_trimble_interval_object">Definition (Trimble interval object)</h6> <p>A <strong>category with interval object</strong> is</p> <ul> <li> <p>a symmetric <a class="existingWikiWord" href="/nlab/show/closed+monoidal+homotopical+category">closed monoidal homotopical category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> </li> <li> <p>with tensor unit being the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, which we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math>;</p> </li> <li> <p>equipped with a <a class="existingWikiWord" href="/nlab/show/bi-pointed+object">bi-pointed object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>σ</mi></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↖</mo> <mi>τ</mi></msup></mtd></mtr> <mtr><mtd><mi>pt</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && I \\ & {}^\sigma \nearrow && \nwarrow^{\tau} \\ pt &&&& pt } </annotation></semantics></math></div></li> </ul> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> called the <strong>interval object</strong>;</p> <p>such that</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>I</mi> <mrow><mo>∨</mo><mn>2</mn></mrow></msup><mo>:</mo><mo>=</mo><mi>I</mi><msub><mo>⨿</mo> <mi>pt</mi></msub><mi>I</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>I</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mi>τ</mi></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mi>σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && I^{\vee 2} := I \amalg_{pt} I \\ & \nearrow && \nwarrow \\ I &&&& I \\ & {}_{\tau}\nwarrow && \nearrow_{\sigma} \\ && pt } </annotation></semantics></math></div> <p>exists in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, so that all compositions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>I</mi> <mrow><mo>∨</mo><mi>n</mi></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>σ</mi></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↖</mo> <mi>τ</mi></msup></mtd></mtr> <mtr><mtd><mi>pt</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>pt</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && I^{\vee n} \\ & {}^\sigma \nearrow && \nwarrow^{\tau} \\ pt &&&& pt } </annotation></semantics></math></div> <p>of <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> copies of the <a class="existingWikiWord" href="/nlab/show/co-span">co-span</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> with itself by pushout over adjacent legs exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> </li> <li> <p>and for all <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>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-objects of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>pt</mi></msub><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><msup><mi>I</mi> <mrow><mo>∨</mo><mi>n</mi></mrow></msup><msub><mo stretchy="false">]</mo> <mi>pt</mi></msub></mrow><annotation encoding="application/x-tex">{}_{pt}[I, I^{\vee n}]_{pt}</annotation></semantics></math> of cospans (as described at <a class="existingWikiWord" href="/nlab/show/co-span">co-span</a>) are weakly equivalent to the point</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>pt</mi></msub><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><msup><mi>I</mi> <mrow><mo>∨</mo><mi>n</mi></mrow></msup><msub><mo stretchy="false">]</mo> <mi>pt</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {}_{pt}[I, I^{\vee n}]_{pt} \,. </annotation></semantics></math></div></li> </ul> </div> <h4 id="bergermoerdijk_segment_and_interval_object">Berger–Moerdijk segment- and interval object</h4> <p>In <a href="http://arxiv.org/PS_cache/math/pdf/0502/0502155v2.pdf#page=11">section 4</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Clemens+Berger">Clemens Berger</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>The Boardman-Vogt resolution of operads in monoidal model categories</em> (<a href="http://arxiv.org/abs/math.AT/0502155">arXiv</a>)</li> </ul> <p>the following definition is given:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math> for the tensor unit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (not necessarily the terminal object).</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/segment+object">segment (object)</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is</p> <ul> <li> <p>a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>⨿</mo><mi>pt</mi><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mover><mi>I</mi><mover><mo>→</mo><mi>ϵ</mi></mover><mi>pt</mi></mrow><annotation encoding="application/x-tex"> pt \amalg pt \stackrel{[0 , 1]}{\to} I \stackrel{\epsilon}{\to} pt </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/codiagonal+morphism">codiagonal morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>⨿</mo><mi>pt</mi><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy="false">]</mo></mrow></mover><mi>pt</mi></mrow><annotation encoding="application/x-tex"> pt \amalg pt \stackrel{[Id , Id]}{\to} pt </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math> with itself that sends each component identically to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math>.</p> </li> <li> <p>together with an associative morphsim</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∨</mo><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>I</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> \vee : I \otimes I \to I </annotation></semantics></math></div> <p>which has 0 as its <em>neutral</em> and 1 as its <em>absorbing</em> element, and for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is a counit.</p> </li> </ul> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> then a segment object is an <strong>interval</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> if</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><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><mi>pt</mi><mo>⨿</mo><mi>pt</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> [0, 1]\colon pt \amalg pt \to I </annotation></semantics></math></div> <p>is a cofibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">\epsilon : I \to pt</annotation></semantics></math> a weak equivalence.</p> <h3 id="InHomotopyTypeTheory">Interval type</h3> <p>In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> the cellular interval can be axiomatized as a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a>. See <em><a class="existingWikiWord" href="/nlab/show/interval+type">interval type</a></em> for more.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>In <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> the standard interval object is the 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1]</annotation></semantics></math>.</p> </li> <li> <p>In a <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> the standard interval is the <a class="existingWikiWord" href="/nlab/show/chain+on+a+simplicial+set">simplicial chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(\Delta[1])</annotation></semantics></math> on the 1-simplex, see at <a class="existingWikiWord" href="/nlab/show/interval+object+in+chain+complexes">interval object in chain complexes</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/cube+category">cube category</a> is generated from a single interval object.</p> </li> <li> <p>The <strong>standard interval object</strong> in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is the 1st <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0\to 1\}</annotation></semantics></math> (see <a class="existingWikiWord" href="/nlab/show/co-span+co-trace">co-span co-trace</a>)</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">V = C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equipped with with the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>, the topological <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I \colon [0,1]</annotation></semantics></math> (with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>) with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mover><mo>→</mo><mrow><mi>σ</mi><mo>,</mo><mi>τ</mi></mrow></mover><mi>I</mi></mrow><annotation encoding="application/x-tex">pt \stackrel{\sigma, \tau}{\to}I</annotation></semantics></math> the maps to 0 and 1, respectively. This is the standard <em><a class="existingWikiWord" href="/nlab/show/topological+interval">topological interval</a></em>. This is the case described in detail at <em><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble n-category</a></em>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>ω</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V = \omega Cat</annotation></semantics></math> the category of <a class="existingWikiWord" href="/nlab/show/strict+omega-category">strict omega-categories</a> the first <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a>, the 1-<a class="existingWikiWord" href="/nlab/show/globe">globe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I = \{a \to b\}</annotation></semantics></math> is an interval object. In this strict case in fact all hom objects are already equal to the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>pt</mi></msub><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><msup><mi>I</mi> <mrow><mo>∨</mo><mi>n</mi></mrow></msup><msub><mo stretchy="false">]</mo> <mi>pt</mi></msub><mo>=</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">{}_{pt}[I, I^{\vee n}]_{pt} = pt</annotation></semantics></math> and</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>X</mi><mo>=</mo><mo stretchy="false">[</mo><mi>pt</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><mi>s</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>t</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>τ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mo stretchy="false">[</mo><mi>pt</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X = [pt,X]) \stackrel{s := [\sigma,X]}{\leftarrow} [I,X] \stackrel{t := [\tau, X]}{\to} (X = [pt,X]) </annotation></semantics></math></div> <p>is a strict co-category internal to <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>Cat. In this case, 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> any <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>-category the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(X)</annotation></semantics></math> is just an ordinary category, namely the 1-category obtained from truncation 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>. Similarly, probably <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>ω</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Pi_\omega(X) = X</annotation></semantics></math> in this case.</p> </li> </ul> <h3 id="standard_intervals_cubes_and_simplices_in__and_">Standard intervals, cubes and simplices in <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> and <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></h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">X = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> or <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> be the category of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s or of <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>s.</p> <p>A standard choice of interval object 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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">I = [0,1] \subset \mathbb{R}</annotation></semantics></math> with the obvious two boundary inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>:</mo><mo>*</mo><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0,1 : {*} \to [0,1]</annotation></semantics></math>.</p> <p>But another possible choice is to let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">I = \mathbb{R}</annotation></semantics></math> be the whole real line, but still equipped with the two maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>:</mo><mo>*</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">0,1 : {*} \to \mathbb{R}</annotation></semantics></math>, that hit the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">1 \in \mathbb{R}</annotation></semantics></math>, respectively.</p> <p>Either of these two examples will do in the following discussion. The second choice is to be thought of as obtained from the first choice by adding “infinitely wide <a class="existingWikiWord" href="/nlab/show/collars">collars</a>” at both boundaries of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math>. While <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mn>0</mn></mover><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mover><mo>←</mo><mn>1</mn></mover><mo>*</mo></mrow><annotation encoding="application/x-tex">{*} \stackrel{0}{\to}[0,1] \stackrel{1}{\leftarrow} {*}</annotation></semantics></math> may seem like a more natural choice for a representative of the idea of the “standard interval”, the choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mn>0</mn></mover><mi>ℝ</mi><mover><mo>←</mo><mn>1</mn></mover><mo>*</mo></mrow><annotation encoding="application/x-tex">{*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}</annotation></semantics></math> is actually more useful for many <a class="existingWikiWord" href="/nlab/show/category+theory">abstract nonsense</a> constructions.</p> <p>But since it is hard to draw the full real line, in the following we depict the situation for the choice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = [0,1]</annotation></semantics></math>.</p> <p>Then for low <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> the above construction yields this</p> <ul> <li> <p><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=0</annotation></semantics></math> – here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mi>I</mi> <mn>0</mn></msubsup><mo>=</mo><msup><mi>I</mi> <mrow><mo>×</mo><mn>0</mn></mrow></msup><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Delta_I^0 = I^{\times 0} = {*}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/point">point</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math> – here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mi>I</mi> <mn>1</mn></msubsup><mo>=</mo><msup><mi>I</mi> <mrow><mo>×</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\Delta_I^1 = I^{\times 1} = I</annotation></semantics></math> is just the interval itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0) \to (1) } </annotation></semantics></math></div> <p>The two face maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>1</mn></msub><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\delta_1 {*} \to I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mn>0</mn></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\delta_0 : {*} \to I</annotation></semantics></math> pick the boundary points in the obvious way. The unique degeneracy map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>0</mn></msub><mo>:</mo><mi>I</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\sigma_0 : I \to {*}</annotation></semantics></math> maps all points of the interval to the single point of the point.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math> – here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mi>i</mi> <mn>2</mn></msubsup><mo>=</mo><msup><mi>I</mi> <mrow><mo>×</mo><mn>2</mn></mrow></msup><mo>=</mo><mi>I</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\Delta_i^2 = I^{\times 2} = I \times I</annotation></semantics></math> is the <strong>standard square</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0,1) &\to& (1,1) \\ \uparrow && \uparrow \\ (0,0) &\to& (1,0) } </annotation></semantics></math></div> <p>But the three face maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>I</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\delta_i : I \to I\times I</annotation></semantics></math> of the cosimplicial object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_I</annotation></semantics></math> constructed above don’t regard the full square here, but just a triangle sitting inside it, in that pictorially they identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>I</mi> <mn>1</mn></msubsup><mo>=</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Delta_I^1 = I)</annotation></semantics></math>-shaped boundaries in <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 \times I</annotation></semantics></math> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msup><mo></mo><mrow><mo>=</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mrow><mo>=</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>=</mo><msub><mi>δ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0,1) &\to& (1,1) \\ \uparrow &^{= \delta_1(I)}\nearrow& \uparrow^{ = \delta_0(I)} \\ (0,0) &\stackrel{= \delta_2(I)}{\to}& (1,0) } </annotation></semantics></math></div> <p>(here the arrows do not depict morphisms, but the standard topological interval, i don’t know how to typeset just lines without arrow heads in this fashion!)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n=3</annotation></semantics></math> – here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Δ</mi> <mi>i</mi> <mn>3</mn></msubsup><mo>=</mo><msup><mi>I</mi> <mrow><mo>×</mo><mn>3</mn></mrow></msup><mo>=</mo><mi>I</mi><mo>×</mo><mi>I</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\Delta_i^3 = I^{\times 3} = I \times I \times I</annotation></semantics></math> is the <strong>standard cube</strong></p> <div class="un_example"> <h6 id="exercise">Exercise</h6> <p>Insert the analog of the above discussion here and upload a nice graphics that shows the standard cube and how the cosimplicial object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_I</annotation></semantics></math> picks a solid tetrahedron inside it.</p> </div> </li> </ul> <p>As a start, we can illustrate how there are 6 3-simplices sitting inside each 3-cube.</p> <p><img src="http://ncatlab.org/nlab/files/3simplex_in_3cube.jpg" width="550" /></p> <p>Once you see how the 3-simplices sit inside the 3-cube, the facemaps can be illustrated as follows:</p> <p><img src="http://ncatlab.org/nlab/files/3simplex_facemaps.jpg" width="550" /></p> <p>Note that these face maps are to be thought of as maps into 3-simplices sitting inside a 3-cube.</p> <h3 id="homotopy_theory_2"><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</h3> <p>See <a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory">A1-homotopy theory</a>.</p> <h2 id="fundamental_categories_induced_from_intervals">Fundamental <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>-categories induced from intervals</h2> <p>The interest in interval objects is that various further structures of interest may be built up from them. In particular, since picking an interval object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is like picking a notion of <em>path</em>, in a category with interval object there is, under mild assumptions, for 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> an <a class="existingWikiWord" href="/nlab/show/infinity-category">infinity-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>I</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_I(X)</annotation></semantics></math> – the fundamental <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>-category 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> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> – whose <a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>s are <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>-fold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-paths 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>.</p> <p>This is described for two models for <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>-categories at <a class="existingWikiWord" href="/nlab/show/fundamental+%28infinity%2C1%29-category">fundamental (infinity,1)-category</a></p> <h2 id="homotopy_localization_induced_from_an_interval">Homotopy localization induced from an interval</h2> <p>Given a suitable interval obect in 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>, one may ask for <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>s 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> that are invariant under the notion of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. These are obtained by <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> of a full <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 <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>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/interval">interval</a>, <a class="existingWikiWord" href="/nlab/show/interval+type">interval type</a></li> </ul> <h2 id="references">References</h2> <ul> <li>Clemens Berger, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>The Boardman-Vogt resolution of operads in monoidal model categories</em> (<a href="http://arxiv.org/abs/math.AT/0502155">arXiv</a>), section 4, p.11</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 14, 2022 at 04:10:05. 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