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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%2F12682%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>: created limit</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 10 of 10</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_91639"> <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 21st 2021</li><li><em>(edited Apr 21st 2021)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=91639#Comment_91639">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_91639"><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>Where the example of monadic functors creating limits is mentioned, there should be a reference, at least. I have added pointer to MacLane 71, Exercise IV.2.2 (p. 138) Scanning through Borceux II, I don't spot the statement there. (?) <a href="https://ncatlab.org/nlab/revision/diff/created+limit/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/13">v13</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></code></div><div> <p>Where the example of monadic functors creating limits is mentioned, there should be a reference, at least.</p> <p>I have added pointer to MacLane 71, Exercise IV.2.2 (p. 138)</p> <p>Scanning through Borceux II, I don’t spot the statement there. (?)</p> <p><a href="https://ncatlab.org/nlab/revision/diff/created+limit/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/13">v13</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></p> </div> </div> </li><li id="Comment_101858" 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/4/">Urs</a></li> <li><span>CommentTime</span>Aug 11th 2022</li><li><em>(edited Aug 11th 2022)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=101858#Comment_101858">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_101858"><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: * [[Emily Riehl]], §3.3 in: *[[Category Theory in Context]]*, Dover Publications (2017) &lbrack;[pdf](http://www.math.jhu.edu/~eriehl/context.pdf), [book website](http://www.math.jhu.edu/~eriehl/context/)&rbrack; here, and in related entries <a href="https://ncatlab.org/nlab/revision/diff/created+limit/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/16">v16</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></code></div><div> <p>added pointer to:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Emily Riehl">Emily Riehl</a>, §3.3 in: <em><a href="https://ncatlab.org/nlab/show/Category Theory in Context">Category Theory in Context</a></em>, Dover Publications (2017) [<a href="http://www.math.jhu.edu/~eriehl/context.pdf">pdf</a>, <a href="http://www.math.jhu.edu/~eriehl/context/">book website</a>]</li> </ul> <p>here, and in related entries</p> <p><a href="https://ncatlab.org/nlab/revision/diff/created+limit/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/16">v16</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></p> </div> </div> </li><li id="Comment_104267"> <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/1691/">nLab edit announcer</a></li> <li><span>CommentTime</span>Nov 6th 2022</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=104267#Comment_104267">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_104267"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1691/">nLab edit announcer</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Added the remark that for amnestic isofibrations the strict and the non-strict notion of creation of limits are equivalent. Jonas Frey <a href="https://ncatlab.org/nlab/revision/diff/created+limit/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/17">v17</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></code></div><div> <p>Added the remark that for amnestic isofibrations the strict and the non-strict notion of creation of limits are equivalent.</p> <p>Jonas Frey</p> <p><a href="https://ncatlab.org/nlab/revision/diff/created+limit/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/17">v17</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></p> </div> </div> </li><li id="Comment_107080" 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/1573/">jonsterling</a></li> <li><span>CommentTime</span>Feb 12th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=107080#Comment_107080">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_107080"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1573/">jonsterling</a></span><br/><span class="sourceType">Format: Text</span><code>Unless I am confused, my copy of Mac Lane does not in fact require limits that don't exist in the codomain to be created. Mac Lane says: > A functor V : A -> X creates limits for a functor F : J -> A if: > (i) to every limiting cone \tau : x -> VF in X there is exactly one ...... So the limiting cone is assumed to exist in the codomain category. So what is going on in this nlab page? Did I misunderstand the discussion?</code></div><div> Unless I am confused, my copy of Mac Lane does not in fact require limits that don't exist in the codomain to be created. Mac Lane says:<br/><br/>> A functor V : A -> X creates limits for a functor F : J -> A if:<br/>> (i) to every limiting cone \tau : x -> VF in X there is exactly one ......<br/><br/>So the limiting cone is assumed to exist in the codomain category. So what is going on in this nlab page? Did I misunderstand the discussion? </div> </div> </li><li id="Comment_107081"> <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/4/">Urs</a></li> <li><span>CommentTime</span>Feb 12th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=107081#Comment_107081">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_107081"><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>You are probably referring to the subsection [here](https://ncatlab.org/nlab/show/created+limit#creation_of_nonexisting_limits). This was added by Mike in [revision 11](https://ncatlab.org/nlab/revision/diff/created+limit/11#creation_of_nonexisting_limits), Feb 2018. Interesting to compare to the status of [revision 10](https://ncatlab.org/nlab/revision/created+limit/10). Don't know what happened there.</code></div><div> <p>You are probably referring to the subsection <a href="https://ncatlab.org/nlab/show/created+limit#creation_of_nonexisting_limits">here</a>.</p> <p>This was added by Mike in <a href="https://ncatlab.org/nlab/revision/diff/created+limit/11#creation_of_nonexisting_limits">revision 11</a>, Feb 2018.</p> <p>Interesting to compare to the status of <a href="https://ncatlab.org/nlab/revision/created+limit/10">revision 10</a>. Don’t know what happened there.</p> </div> </div> </li><li id="Comment_108491" 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/1786/">varkor</a></li> <li><span>CommentTime</span>Apr 11th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=108491#Comment_108491">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_108491"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1786/">varkor</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Mention terminology "closed under limits". <a href="https://ncatlab.org/nlab/revision/diff/created+limit/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/19">v19</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></code></div><div> <p>Mention terminology “closed under limits”.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/created+limit/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/created+limit/19">v19</a>, <a href="https://ncatlab.org/nlab/show/created+limit">current</a></p> </div> </div> </li><li id="Comment_115463"> <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/3308/">ʇɐ</a></li> <li><span>CommentTime</span>Jan 29th 2024</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=115463#Comment_115463">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_115463"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3308/">ʇɐ</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>The stated definition for creation of limits (namely that the functor preserves and reflects limits) doesn’t match that of any of the three references; they all add the requirement that the functor $ F \colon C \to D $ also lifts limit cones, i.e. if every limit cone in the codomain over $ F \circ J $ is the image along $ F $ of some limit cone over $ J $ (following Riehl) or of a unique cone over $ J $ and this cone is limiting (Mac Lane and Adámek–Herrlich–Strecker). Is there any reference for the term as used on this page, or is it a mistake?</code></div><div> <p>The stated definition for creation of limits (namely that the functor preserves and reflects limits) doesn’t match that of any of the three references; they all add the requirement that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo lspace="0.11111em">&colon;</mo><mi>C</mi><mo>&rightarrow;</mo><mi>D</mi></mrow><annotation encoding="application/x-tex"> F \colon C \to D </annotation></semantics></math> also lifts limit cones, i.e. if every limit cone in the codomain over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>&SmallCircle;</mo><mi>J</mi></mrow><annotation encoding="application/x-tex"> F \circ J </annotation></semantics></math> is the image along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex"> F </annotation></semantics></math> of some limit cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex"> J </annotation></semantics></math> (following Riehl) or of a unique cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex"> J </annotation></semantics></math> and this cone is limiting (Mac Lane and Adámek–Herrlich–Strecker). Is there any reference for the term as used on this page, or is it a mistake?</p> </div> </div> </li><li id="Comment_115465" 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/4/">Urs</a></li> <li><span>CommentTime</span>Jan 30th 2024</li><li><em>(edited Jan 30th 2024)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=115465#Comment_115465">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_115465"><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>As mentioned in [#5](#Comment_107081) above, the entry used to say this until [revision 10](https://ncatlab.org/nlab/revision/created+limit/10), it was changed by Mike in [revision 11](https://ncatlab.org/nlab/revision/diff/created+limit/11#creation_of_nonexisting_limits), Feb 2018.</code></div><div> <p>As mentioned in <a href="#Comment_107081">#5</a> above,</p> <p>the entry used to say this until <a href="https://ncatlab.org/nlab/revision/created+limit/10">revision 10</a>,</p> <p>it was changed by Mike in <a href="https://ncatlab.org/nlab/revision/diff/created+limit/11#creation_of_nonexisting_limits">revision 11</a>, Feb 2018.</p> </div> </div> </li><li id="Comment_115473"> <a id="Item_9"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>9.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3308/">ʇɐ</a></li> <li><span>CommentTime</span>Jan 30th 2024</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=115473#Comment_115473">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_115473"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3308/">ʇɐ</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Whoops, missed that completely! &#x1F605; Looking at it again I now see that the existence of a lifting (up to isomorphism this time) is also part of this definition: $ J $ is required to have a limit, and preservation and reflection do the rest. (Maybe the strict variant and the differences should be mentioned earlier up on the page, for undercaffeinated readers…)</code></div><div> <p>Whoops, missed that completely! 😅 Looking at it again I now see that the existence of a lifting (up to isomorphism this time) is also part of this definition: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex"> J </annotation></semantics></math> is required to have a limit, and preservation and reflection do the rest. (Maybe the strict variant and the differences should be mentioned earlier up on the page, for undercaffeinated readers…)</p> </div> </div> </li><li id="Comment_115475" class="Alternate"> <a id="Item_10"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>10.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Jan 30th 2024</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/12682/created-limit/?Focus=115475#Comment_115475">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_115475"><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 think the entry deserves clarification, given that it keeps tripping up people. The traditional definition from [revision 10](https://ncatlab.org/nlab/revision/created+limit/10) should not remain deleted but be re-instantiated, and then referred to for comparison. I don't have the energy now to edit this entry, but if anyone has, it would be welcome.</code></div><div> <p>I think the entry deserves clarification, given that it keeps tripping up people.</p> <p>The traditional definition from <a href="https://ncatlab.org/nlab/revision/created+limit/10">revision 10</a> should not remain deleted but be re-instantiated, and then referred to for comparison.</p> <p>I don’t have the energy now to edit this entry, but if anyone has, it would be welcome.</p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 10 of 10</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>