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More Information: <a href="http://lussumo.com/docs">Documentation</a>, <a href="http://lussumo.com/community">Community Support</a>.</p></div> <div id="Content"><div id="NoticeCollector" class="Notices"><div class="Notice"><strong>Welcome to nForum</strong> <br />If you want to take part in these discussions either <a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F7040%2F">sign in now</a> (if you have an account), <a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">apply for one now</a> (if you don't).</div></div><div class="ContentInfo Top"> <h1><a href="https://nforum.ncatlab.org/5/"><a href="https://nforum.ncatlab.org/18/">nLab</a> > </a> <a href="https://nforum.ncatlab.org/5/">Latest Changes</a>: cylinder object</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 8 of 8</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div> <div id="ContentBody"> <script type="text/javascript"> //<![CDATA[ function toggle_source(id) { var mysrc = document.getElementById("CommentBody_" + id).firstChild; if (mysrc.className == "source") { if (mysrc.style.display == "none") { mysrc.style.display = "block"; } else { mysrc.style.display = "none"; } } } var commentIds = new Array(0); function hide_sources() { for (i = 0; i < commentIds.length; i++) { var myself = document.getElementById("Source" + commentIds[i]); var mycmt = document.getElementById("CommentBody_" + commentIds[i]); if (mycmt.firstChild.className != "source") { myself.style.display = "none"; } } } window.onload = hide_sources; //]]> </script> <ol id="Comments"><li id="Comment_56856"> <a id="Item_1"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>1.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Apr 8th 2016</li><li><em>(edited Apr 8th 2016)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=56856#Comment_56856">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_56856"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I added to _[[cylinder object]]_ a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex). What would be a text that features a _graphics_ which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I'd like to grab such graphics to put it in the entry, too lazy to do it myself. )</code></div><div> <p>I added to <em><a href="https://ncatlab.org/nlab/show/cylinder object">cylinder object</a></em> a pointer to a reference that goes through the trouble of spelling out the precise proof that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a CW-complex, then the standard cyclinder <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \times I</annotation></semantics></math> is again a cell complex (and the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>&sqcup;</mo><mi>X</mi><mo>&rightarrow;</mo><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">X \sqcup X \to X\times I</annotation></semantics></math> a relative cell complex).</p> <p>What would be a text that features a <em>graphics</em> which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math>, then the cells of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math> glued in at top and bottom, then the further <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><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>-cells glued into all the resulting hollow cylinders? (I鈥檇 like to grab such graphics to put it in the entry, too lazy to do it myself. )</p> </div> </div> </li><li id="Comment_90321" class="Alternate"> <a id="Item_2"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>2.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Mar 9th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=90321#Comment_90321">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_90321"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>There is some bicategorical generalization and weakening of a cylinder object, under the same name, which does not factorize identity 1-cell but arbitrary 1-cell and have some other weaker properties in more general context. * M.E. Descotte E.J., Dubuc M. Szyld, _A localization of bicategories via homotopies_, Theory and appl. of categories 35:23, [abs](http://www.tac.mta.ca/tac/volumes/35/23/35-23abs.html), [arxiv/1805.05248](https://arxiv.org/abs/1805.05248) The left homotopies defined using these cylinders do not compose, but one can consider formal finite sequences of left homotopies to define certain localizations. I could write few details about it, but I am not sure if this belongs to this entry, despite the name being the same. Should it be here or having a separate entry ?</code></div><div> <p>There is some bicategorical generalization and weakening of a cylinder object, under the same name, which does not factorize identity 1-cell but arbitrary 1-cell and have some other weaker properties in more general context.</p> <ul> <li>M.E. Descotte E.J., Dubuc M. Szyld, <em>A localization of bicategories via homotopies</em>, Theory and appl. of categories 35:23, <a href="http://www.tac.mta.ca/tac/volumes/35/23/35-23abs.html">abs</a>, <a href="https://arxiv.org/abs/1805.05248">arxiv/1805.05248</a></li> </ul> <p>The left homotopies defined using these cylinders do not compose, but one can consider formal finite sequences of left homotopies to define certain localizations.</p> <p>I could write few details about it, but I am not sure if this belongs to this entry, despite the name being the same. Should it be here or having a separate entry ?</p> </div> </div> </li><li id="Comment_90322"> <a id="Item_3"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>3.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></li> <li><span>CommentTime</span>Mar 9th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=90322#Comment_90322">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_90322"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3/">Mike Shulman</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I'd probably be inclined towards a separate entry, like [[bicategorical cylinder object]] perhaps.</code></div><div> <p>I鈥檇 probably be inclined towards a separate entry, like <a href="https://ncatlab.org/nlab/show/bicategorical cylinder object">bicategorical cylinder object</a> perhaps.</p> </div> </div> </li><li id="Comment_90325" class="Alternate"> <a id="Item_4"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>4.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Mar 9th 2021</li><li><em>(edited Mar 9th 2021)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=90325#Comment_90325">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_90325"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>> formal finite sequences of left homotopies to define certain localizations I mean considering equivalence classes of sequences. OK, I will (gradually) write a separate entry and link from [[cylinder object]]. My feeling is that this notion is better thought of as _relative_ to 1-cell. The classical case is an example of rather very special kind.</code></div><div> <blockquote> <p>formal finite sequences of left homotopies to define certain localizations</p> </blockquote> <p>I mean considering equivalence classes of sequences.</p> <p>OK, I will (gradually) write a separate entry and link from <a href="https://ncatlab.org/nlab/show/cylinder object">cylinder object</a>.</p> <p>My feeling is that this notion is better thought of as <em>relative</em> to 1-cell. The classical case is an example of rather very special kind.</p> </div> </div> </li><li id="Comment_90528"> <a id="Item_5"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>5.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Mar 17th 2021</li><li><em>(edited Mar 17th 2021)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=90528#Comment_90528">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_90528"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I did not write the entry yet but I reviewed the paper where the construction appears (without going into the details about the cylinder object itself). The review is in a draft form (very soon to be submitted), so corrections, improvements or suggestions are very welcome. [[zoranskoda:MR4112764]]</code></div><div> <p>I did not write the entry yet but I reviewed the paper where the construction appears (without going into the details about the cylinder object itself). The review is in a draft form (very soon to be submitted), so corrections, improvements or suggestions are very welcome.</p> <p><a href="https://ncatlab.org/zoranskoda/show/MR4112764">MR4112764 (zoranskoda)</a></p> </div> </div> </li><li id="Comment_90555" class="Alternate"> <a id="Item_6"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>6.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Mar 18th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=90555#Comment_90555">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_90555"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>The review has been improved, [[zoranskoda:MR4112764]] and a variant of it electronically submitted (changes can still be made for few days). Quite an interesting construction.</code></div><div> <p>The review has been improved,</p> <p><a href="https://ncatlab.org/zoranskoda/show/MR4112764">MR4112764 (zoranskoda)</a></p> <p>and a variant of it electronically submitted (changes can still be made for few days). Quite an interesting construction.</p> </div> </div> </li><li id="Comment_93887"> <a id="Item_7"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>7.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Jul 13th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=93887#Comment_93887">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_93887"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>added pointer to: * [[Daniel Quillen]], Section I.1, Def. 4, p. 9 (15 of 165) in:, _Axiomatic homotopy theory_ in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)) <a href="https://ncatlab.org/nlab/revision/diff/cylinder+object/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/cylinder+object/16">v16</a>, <a href="https://ncatlab.org/nlab/show/cylinder+object">current</a></code></div><div> <p>added pointer to:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Daniel Quillen">Daniel Quillen</a>, Section I.1, Def. 4, p. 9 (15 of 165) in:, <em>Axiomatic homotopy theory</em> in: <em><a href="https://ncatlab.org/nlab/show/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><a href="https://ncatlab.org/nlab/revision/diff/cylinder+object/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/cylinder+object/16">v16</a>, <a href="https://ncatlab.org/nlab/show/cylinder+object">current</a></p> </div> </div> </li><li id="Comment_117985" class="Alternate"> <a id="Item_8"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>8.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/1623/">Hurkyl</a></li> <li><span>CommentTime</span>Jul 9th 2024</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/7040/cylinder-object/?Focus=117985#Comment_117985">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_117985"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1623/">Hurkyl</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Added the lawvere cylinder as an example. <a href="https://ncatlab.org/nlab/revision/diff/cylinder+object/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/cylinder+object/17">v17</a>, <a href="https://ncatlab.org/nlab/show/cylinder+object">current</a></code></div><div> <p>Added the lawvere cylinder as an example.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/cylinder+object/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/cylinder+object/17">v17</a>, <a href="https://ncatlab.org/nlab/show/cylinder+object">current</a></p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 8 of 8</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>