Closure of presentable objects under finite limitsCartesian product of small objectsThe binary product of two...



Closure of presentable objects under finite limits


Cartesian product of small objectsThe binary product of two presentable objectsWhat's an example of a locally presentable category “in nature” that's not $aleph_0$-locally presentable?Example of a locally presentable $2$-categoryIs the category of small categories locally presentable?About reflective full subcategories and small-orthogonality classesInfinite Fubini rule for co/limitsWhat is known about the large cardinal strength of Shelah's categoricity conjecture?Factorization systems on tensor product of presentable categoriesA formal condition for a functor to preserve compact objectsThe binary product of two presentable objectsIs every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?













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
            yesterday






          • 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
            yesterday










          • $begingroup$
            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
            21 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
            12 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
            11 hours ago











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          1 Answer
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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          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
            yesterday






          • 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
            yesterday










          • $begingroup$
            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
            21 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
            12 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
            11 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
            yesterday






          • 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
            yesterday










          • $begingroup$
            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
            21 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
            12 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
            11 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 11 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
            yesterday






          • 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
            yesterday










          • $begingroup$
            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
            21 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
            12 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
            11 hours ago














          • 2




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






          • 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
            yesterday










          • $begingroup$
            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
            21 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
            12 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
            11 hours ago








          2




          2




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




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




          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
          yesterday




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












          $begingroup$
          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
          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
          21 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
          12 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
          12 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
          11 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
          11 hours ago


















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