Closure of presentable objects under finite limits












7












$begingroup$


In a locally presentable category $cal E$, there are arbitrarily large regular cardinals $lambda$ such that the $lambda$-presentable (a.k.a. $lambda$-compact) objects are closed under pullbacks. Namely, the pullback functor ${cal E}^{(toleftarrow)}to cal E$ is a right adjoint, hence accesible. Thus it preserves $lambda$-presentable objects for arbitrarily large $lambda$, so it's enough to check that the $lambda$-presentable objects in ${cal E}^{(toleftarrow)}$ are those that are pointwise so in $cal E$. (A version of this argument is given in this answer in the case of finite products.)



Of course "arbitrarily large" means that for any cardinal $mu$ there exists a regular cardinal $lambda>mu$ with this property. A stronger claim would be that this is true for all sufficiently large regular cardinals $lambda$, i.e. to reverse the quantifiers and say there exists a $mu$ such that all regular cardinals $lambda>mu$ have this property (that $lambda$-presentable objects are closed under pullbacks). Is this stronger claim true?



Note that it is certainly not true that all regular cardinals $lambda$ have this property; counterexamples can be found here.






ct.category-theory locally-presentable-categories







share|cite|improve this question









$endgroup$

















    7












    $begingroup$


    In a locally presentable category $cal E$, there are arbitrarily large regular cardinals $lambda$ such that the $lambda$-presentable (a.k.a. $lambda$-compact) objects are closed under pullbacks. Namely, the pullback functor ${cal E}^{(toleftarrow)}to cal E$ is a right adjoint, hence accesible. Thus it preserves $lambda$-presentable objects for arbitrarily large $lambda$, so it's enough to check that the $lambda$-presentable objects in ${cal E}^{(toleftarrow)}$ are those that are pointwise so in $cal E$. (A version of this argument is given in this answer in the case of finite products.)



    Of course "arbitrarily large" means that for any cardinal $mu$ there exists a regular cardinal $lambda>mu$ with this property. A stronger claim would be that this is true for all sufficiently large regular cardinals $lambda$, i.e. to reverse the quantifiers and say there exists a $mu$ such that all regular cardinals $lambda>mu$ have this property (that $lambda$-presentable objects are closed under pullbacks). Is this stronger claim true?



    Note that it is certainly not true that all regular cardinals $lambda$ have this property; counterexamples can be found here.






    ct.category-theory locally-presentable-categories







    share|cite|improve this question









    $endgroup$















      7












      7








      7





      $begingroup$


      In a locally presentable category $cal E$, there are arbitrarily large regular cardinals $lambda$ such that the $lambda$-presentable (a.k.a. $lambda$-compact) objects are closed under pullbacks. Namely, the pullback functor ${cal E}^{(toleftarrow)}to cal E$ is a right adjoint, hence accesible. Thus it preserves $lambda$-presentable objects for arbitrarily large $lambda$, so it's enough to check that the $lambda$-presentable objects in ${cal E}^{(toleftarrow)}$ are those that are pointwise so in $cal E$. (A version of this argument is given in this answer in the case of finite products.)



      Of course "arbitrarily large" means that for any cardinal $mu$ there exists a regular cardinal $lambda>mu$ with this property. A stronger claim would be that this is true for all sufficiently large regular cardinals $lambda$, i.e. to reverse the quantifiers and say there exists a $mu$ such that all regular cardinals $lambda>mu$ have this property (that $lambda$-presentable objects are closed under pullbacks). Is this stronger claim true?



      Note that it is certainly not true that all regular cardinals $lambda$ have this property; counterexamples can be found here.






      ct.category-theory locally-presentable-categories







      share|cite|improve this question









      $endgroup$




      In a locally presentable category $cal E$, there are arbitrarily large regular cardinals $lambda$ such that the $lambda$-presentable (a.k.a. $lambda$-compact) objects are closed under pullbacks. Namely, the pullback functor ${cal E}^{(toleftarrow)}to cal E$ is a right adjoint, hence accesible. Thus it preserves $lambda$-presentable objects for arbitrarily large $lambda$, so it's enough to check that the $lambda$-presentable objects in ${cal E}^{(toleftarrow)}$ are those that are pointwise so in $cal E$. (A version of this argument is given in this answer in the case of finite products.)



      Of course "arbitrarily large" means that for any cardinal $mu$ there exists a regular cardinal $lambda>mu$ with this property. A stronger claim would be that this is true for all sufficiently large regular cardinals $lambda$, i.e. to reverse the quantifiers and say there exists a $mu$ such that all regular cardinals $lambda>mu$ have this property (that $lambda$-presentable objects are closed under pullbacks). Is this stronger claim true?



      Note that it is certainly not true that all regular cardinals $lambda$ have this property; counterexamples can be found here.






      ct.category-theory locally-presentable-categories




      ct.category-theory locally-presentable-categories






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked yesterday









      Mike ShulmanMike Shulman

      37.1k485230




      37.1k485230






















          1 Answer
          1






          active

          oldest

          votes


















          9












          $begingroup$

          The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Thanks! Would you be able to supply a few more details?
            $endgroup$
            – Mike Shulman
            22 hours ago






          • 1




            $begingroup$
            One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
            $endgroup$
            – Jiří Rosický
            21 hours ago










          • $begingroup$
            Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
            $endgroup$
            – Mike Shulman
            18 hours ago






          • 1




            $begingroup$
            Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
            $endgroup$
            – Jiří Rosický
            9 hours ago






          • 1




            $begingroup$
            I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
            $endgroup$
            – Mike Shulman
            8 hours ago











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "504"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f324621%2fclosure-of-presentable-objects-under-finite-limits%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          9












          $begingroup$

          The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Thanks! Would you be able to supply a few more details?
            $endgroup$
            – Mike Shulman
            22 hours ago






          • 1




            $begingroup$
            One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
            $endgroup$
            – Jiří Rosický
            21 hours ago










          • $begingroup$
            Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
            $endgroup$
            – Mike Shulman
            18 hours ago






          • 1




            $begingroup$
            Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
            $endgroup$
            – Jiří Rosický
            9 hours ago






          • 1




            $begingroup$
            I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
            $endgroup$
            – Mike Shulman
            8 hours ago
















          9












          $begingroup$

          The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Thanks! Would you be able to supply a few more details?
            $endgroup$
            – Mike Shulman
            22 hours ago






          • 1




            $begingroup$
            One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
            $endgroup$
            – Jiří Rosický
            21 hours ago










          • $begingroup$
            Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
            $endgroup$
            – Mike Shulman
            18 hours ago






          • 1




            $begingroup$
            Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
            $endgroup$
            – Jiří Rosický
            9 hours ago






          • 1




            $begingroup$
            I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
            $endgroup$
            – Mike Shulman
            8 hours ago














          9












          9








          9





          $begingroup$

          The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.






          share|cite|improve this answer











          $endgroup$



          The stronger claim is true at least for locally finitely presentable categories; this follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 8 hours ago









          Mike Shulman

          37.1k485230




          37.1k485230










          answered yesterday









          Jiří RosickýJiří Rosický

          1,011166




          1,011166








          • 2




            $begingroup$
            Thanks! Would you be able to supply a few more details?
            $endgroup$
            – Mike Shulman
            22 hours ago






          • 1




            $begingroup$
            One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
            $endgroup$
            – Jiří Rosický
            21 hours ago










          • $begingroup$
            Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
            $endgroup$
            – Mike Shulman
            18 hours ago






          • 1




            $begingroup$
            Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
            $endgroup$
            – Jiří Rosický
            9 hours ago






          • 1




            $begingroup$
            I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
            $endgroup$
            – Mike Shulman
            8 hours ago














          • 2




            $begingroup$
            Thanks! Would you be able to supply a few more details?
            $endgroup$
            – Mike Shulman
            22 hours ago






          • 1




            $begingroup$
            One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
            $endgroup$
            – Jiří Rosický
            21 hours ago










          • $begingroup$
            Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
            $endgroup$
            – Mike Shulman
            18 hours ago






          • 1




            $begingroup$
            Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
            $endgroup$
            – Jiří Rosický
            9 hours ago






          • 1




            $begingroup$
            I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
            $endgroup$
            – Mike Shulman
            8 hours ago








          2




          2




          $begingroup$
          Thanks! Would you be able to supply a few more details?
          $endgroup$
          – Mike Shulman
          22 hours ago




          $begingroup$
          Thanks! Would you be able to supply a few more details?
          $endgroup$
          – Mike Shulman
          22 hours ago




          1




          1




          $begingroup$
          One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
          $endgroup$
          – Jiří Rosický
          21 hours ago




          $begingroup$
          One needs that the pullback functor preserves directed colimits. So, my answer is valid for locally finitely presentable categories only.
          $endgroup$
          – Jiří Rosický
          21 hours ago












          $begingroup$
          Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
          $endgroup$
          – Mike Shulman
          18 hours ago




          $begingroup$
          Ah, that makes more sense. That's interesting, but I would really like to know the answer for arbitrary locally presentable categories.
          $endgroup$
          – Mike Shulman
          18 hours ago




          1




          1




          $begingroup$
          Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
          $endgroup$
          – Jiří Rosický
          9 hours ago




          $begingroup$
          Pullbacks preserve directed colimits also in localizations of locally finitely presentable categories. But, in general, I would expect a negative answer.
          $endgroup$
          – Jiří Rosický
          9 hours ago




          1




          1




          $begingroup$
          I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
          $endgroup$
          – Mike Shulman
          8 hours ago




          $begingroup$
          I would also expect a negative answer in general. But the first paragraph of the proof of Proposition 6.1.6.7 in Higher Topos Theory appears to claim that it is true even for locally presentable $infty$-categories, although I don't follow the proof (it seems to have a gap precisely at the place where I would expect a sharp cardinal inequality to occur). So if it does indeed fail in general, it would be nice to have an explicit counterexample.
          $endgroup$
          – Mike Shulman
          8 hours ago


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to MathOverflow!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f324621%2fclosure-of-presentable-objects-under-finite-limits%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          How to label and detect the document text images

          Image
          1 $begingroup$ This is what I mean as document text image: I want to label the texts in image as separate blocks and my model should detect these labels as classes. NOTE: This is how the end result should be like: The labels like Block 1, Block 2, Block 3,.. should be Logo, Title, Date,.. Others, etc. Work done: First approach : I tried to implement this method via Object Detection, it didn't work. It didn't even detect any text. Second approach : Then I tried it using PixelLink. As this model is build for scene text detection, it detected each and every text in the image. But this method can detect multiple lines of text if the threshold values are increased. But I have no idea how do I add labels to the text blocks. PIXEL_CLS_WEIGHT_all_ones = 'PIXEL_CLS_WEIGHT_all_ones' PIXEL_C...
          Read more

          Vallis Paradisi

          Image
          Coordinata: .mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct{display:none}.mw-parser-output .longitude,.mw-parser-output .latitude{white-space:nowrap} 33°03′S 71°37′W  /  33.05°S 71.617°W  / -33.05; -71.617 Vallis Paradisi Res apud Vicidata repertae: Civitas : Chilia Numerus incolarum : 296 655 Zona horaria : UTC-4 Situs interretialis Nomen officiale : Valparaíso Gubernium Consilium : municipal council of Valparaíso Geographia Superficies : 401 600 000 Metrum quadratum Coniunctiones urbium Urbes gemellae : Ovetum , Badalona , Santa Fe , Ora longa , Sciamhaevum , Quantunia , Callao , Bat Yam , Melaka , Novorossiysk , Busan , Corduba , Odessa , Rosario , Malacca , Barcino , Soteropolis , Franciscopolis (California) , Guanajuato, Silao Tabula aut despectus Despectus in portum sinumque et urbem Vallis Paradisi Vallis Para...
          Read more

          Tabula Rosettana

          Image
          Tabula multilinguis Rosettana in Museo Britannico ostenditur. Tabula Rosettana, [1] etiam titulo OGIS 90 agnita, est stela decreto de rebus sacris in Aegypto anno 196 a.C.n. lato inscripta. Tabula iuxta Rosettam Aegypti, urbem in delta Nili et ad oram maris Mediterranei iacentem, anno 1799 a milite Francico reperta est. Inventio stelae, linguis duabus scripturisque tribus inscriptae, eruditis Instituti Aegypti statim nuntiata est; ibi enim iussu imperatoris Napoleonis eruditi omnium scientiarum (sub aegide Commissionis Scientiarum et Artium) properaverant cum expeditione Francica. Qua a Britannis mox debellata, tabula Rosettana Londinium missa hodie apud Museum Britannicum iacet. Textus Graecus cito lectus interpretationi textuum Aegyptiorum (in formis hieroglyphica et demotica expressorum) gradatim adiuvit. Denique textum plene interpretatus est Ioannes Franciscus Champollion. Ab opere eruditorum cumulativo coepit hodiernus scripturae hieroglyphicae linguaeque Aegyptiae a...
          Read more