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width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7040/#Item_8" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #16 to #17: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homotopy_theory'>Homotopy Theory</h4> <div class='hide'> <p><span class='newWikiWord'>!include homotopy - contents<a href='/nlab/new/%21include+homotopy+-+contents'>?</a></span></p> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#examples'>Examples</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>The concept of a <strong>cylinder object</strong> in a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> is an abstraction of the construction in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> which associates to any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> the <em>cylinder</em> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</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'>X \times [0,1]</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_4' xmlns='http://www.w3.org/1998/Math/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 standard <a class='existingWikiWord' href='/nlab/show/diff/topological+interval'>topological interval</a>. It is notably used to define the concept of <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, say in a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a>.</p> <p>The standard topological cylinder <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</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'>X \times [0,1]</annotation></semantics></math> naturally comes equipped with a continuous map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>→</mo><mi>X</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'> X \coprod X \to X \times [0,1] </annotation></semantics></math></div> <p>that identifies <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> as the two ends <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \times \{0\}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \times \{1\}</annotation></semantics></math> of the cylinder, and with a map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> X \times [0,1] \to X </annotation></semantics></math></div> <p>that collapses the cylinder back onto <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>The composite of these two maps is the <a class='existingWikiWord' href='/nlab/show/diff/codiagonal'>codiagonal</a> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Id</mi><mo>,</mo><mi>Id</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>(Id,Id) : X \coprod X \to X</annotation></semantics></math>. Moreover, the cylinder <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</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'>X \times [0,1]</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalent</a> to <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>These properties are the characterizing properties of the cylinder that can be abstracted and realized in other categories.</p> <p>The notion dual to <em>cylinder object</em> is <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path space object</a>, which is thus sometimes alternatively called a cocylinder. Cylinder objects and path space objects are used to define <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopies</a> and <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopies</a>, respectively.</p> <p>There are several views on the role of cylinders / cocylinders in homotopy theory. If there is a natural notion of weak equivalence or <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a> then the cylinder is used to encode a notion of homotopy equivalence compatible with the weak equivalences. In some other situations, a ‘cylinder’ , often functorially given and well structured in some way, may be the <em>primitive</em> notion that allows a notion of ‘homotopy equivalence’ to be put forward. Below we give a definition optimised for the former situation. Some indication of the second context is given in the entry <a class='existingWikiWord' href='/nlab/show/diff/cylinder+functor'>cylinder functor</a>.</p> <h2 id='definition'>Definition</h2> <p>In a <a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> that has <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproducts</a> a <strong>cylinder object</strong> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cyl(X)</annotation></semantics></math> for an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a factorization</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>→</mo><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'> X \coprod X \to Cyl(X) \stackrel{\simeq}{\to} X </annotation></semantics></math></div> <p>of the codiagonal <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \coprod X \to X</annotation></semantics></math> out of the <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> of <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with itself, such that <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Cyl(X) \to X</annotation></semantics></math> is a weak equivalence and such that the morphism <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>→</mo><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \coprod X \to Cyl(X)</annotation></semantics></math> is “nice” in some way.</p> <p>In some situations the assignment of cylinder objects may exist functorially, in which case one speaks of a <a class='existingWikiWord' href='/nlab/show/diff/cylinder+functor'>cylinder functor</a>.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has the structure of a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> then “nice” means that <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>↪</mo><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \coprod X \hookrightarrow Cyl(X)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/cofibration'>cofibration</a>. The factorization axiom of a model category ensures that for each object there is a cylinder object with this property; in fact, one with the additional property that <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Cyl(X) \to X</annotation></semantics></math> is an acyclic fibration. Cylinder objects such that <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>X</mi><mo>↪</mo><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \coprod X \hookrightarrow Cyl(X)</annotation></semantics></math> is a cofibration are sometimes called <em>good</em>, and those for which moreover <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cyl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Cyl(X) \to X</annotation></semantics></math> is an acyclic fibration are then called <em>very good</em>.</p> <h2 id='examples'>Examples</h2> <ul> <li> <p>In <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a> equipped with the standard <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure on simplicial sets</a> the standard cylinder object for any <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>S \in sSet</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>S \times \Delta[1]</annotation></semantics></math>.</p> </li> <li> <p>In <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, the standard cylinder <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</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'>X\times [0,1]</annotation></semantics></math> is a cylinder object for both the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{Quillen}</annotation></semantics></math> (the one with <a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibrations</a>) as well as for the <a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model structure</a> <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{Strom}</annotation></semantics></math> (the one with <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+fibration'>Hurewicz fibrations</a>).</p> <p>This standard cylinder is generally a “good cylinder” in the above sense only for <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>Stron</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{Stron}</annotation></semantics></math> (in which case it is in fact a “very good cylinder”).</p> <p>In <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{Quillen}</annotation></semantics></math> a sufficient condition for the standard cylinder <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X\times I</annotation></semantics></math> to be good is that <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a>.</p> </li><ins class='diffins'> </ins><ins class='diffins'><li> <p>In any <a class='existingWikiWord' href='/nlab/show/diff/topos'>elementary topos</a>, the <a class='existingWikiWord' href='/nlab/show/diff/Lawvere+interval'>Lawvere cylinder</a> is a sort of minimal <a class='existingWikiWord' href='/nlab/show/diff/cylinder+functor'>cylinder functor</a></p> </li></ins> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+functor'>cylinder functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path space object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+spectrum'>cylinder spectrum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/h-cobordism'>h-cobordism</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The general definition in <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> is due to:</p> <ul> <li id='Quillen67'><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Quillen'>Daniel Quillen</a>, Section I.1, Def. 4, p. 9 (15 of 165) in: <em>Axiomatic homotopy theory</em> in: <em><a class='existingWikiWord' href='/nlab/show/diff/Homotopical+Algebra'>Homotopical Algebra</a></em>, Lecture Notes in Mathematics 43, Springer 1967 (<a href='https://doi.org/10.1007/BFb0097438'>doi:10.1007/BFb0097438</a>)</li> </ul> <p>Lecture notes:</p> <ul> <li><em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Homotopy Theory</a></em>, around <a href='Introduction+to+Homotopy+Theory#PathAndCylinderObjectsInAModelCategory'>this Def.</a></li> </ul> <p>The precise argument that for <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a> then also the standard cyclinder <math class='maruku-mathml' display='inline' id='mathml_e8de9d4f7811e00a2885280750f37615102aea78_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X\times I</annotation></semantics></math> is a cell complex is spelled out in</p> <ul> <li id='Ottina14'>Ottina, Prop. 2.9 in: <em>An A-based cofibrantly generated model category</em> (<a href='http://arxiv.org/abs/1405.2086'>arXi:1405.2086</a>)</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on July 9, 2024 at 20:39:00. 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