Accessible functors not preserving lots of presentable objectsClosure of presentable objects under finite limitsWhat's an example of a locally presentable category “in nature” that's not $aleph_0$-locally presentable?Example of a continous functor between locally presentable categories which has no left adjointColimits of algebras of an endofunctorSharply less regular cardinals in set theoryCan I check the accessibility of a functor on directed colimits of presentable objects?A formal condition for a functor to preserve compact objectsIs every accessible category well-powered?Raising the index of accessibilityClosure of presentable objects under finite limits$mu$-presentable object as $mu$-small colimit of $lambda$-presentable objects

Accessible functors not preserving lots of presentable objects


Closure of presentable objects under finite limitsWhat's an example of a locally presentable category “in nature” that's not $aleph_0$-locally presentable?Example of a continous functor between locally presentable categories which has no left adjointColimits of algebras of an endofunctorSharply less regular cardinals in set theoryCan I check the accessibility of a functor on directed colimits of presentable objects?A formal condition for a functor to preserve compact objectsIs every accessible category well-powered?Raising the index of accessibilityClosure of presentable objects under finite limits$mu$-presentable object as $mu$-small colimit of $lambda$-presentable objects













11












$begingroup$


Let $F:cal Cto D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large regular cardinals $lambda$ such that $F$ preserves $lambda$-presentable objects. It is tempting to expect that $F$ should preserve $lambda$-presentable objects for all sufficiently large $lambda$, but that is not what the theorem says. However, I do not know a counterexample showing that the stronger claim fails. (For instance, this question asks about this property when $F$ is the pullback functor, and has no answer yet in the general case.)



What is an example of an accessible functor $F$ between locally presentable categories for which there exist arbitrarily large regular cardinals $mu$ such that $F$ does not preserve $mu$-presentable objects?










share|cite|improve this question









$endgroup$











  • $begingroup$
    For fixed $alpha$, does $mu^alpha = mu$ hold for all sufficiently large regular $mu$?
    $endgroup$
    – Reid Barton
    Mar 13 at 5:17






  • 1




    $begingroup$
    We may assume that $F$ preserves small $lambda$-filtered colimits. Isn’t it true that, for $mu$ large enough, an object is $mu$-presentable if and only if it is a $mu$-small $lambda$-filtered colimit of $lambda$-presentable objects? Another way to put it, is that for $mu$ large enough (e.g. larger than $lambda$ and than the set of maps between any two $lambda$-presentable objects), the property of $mu$-presentability of an object $X$ is simply the fact that the set of maps from a $lambda$-presentable object to $X$ is of cardinal $leqmu$.
    $endgroup$
    – Denis-Charles Cisinski
    Mar 13 at 7:26







  • 3




    $begingroup$
    @Denis-CharlesCisinski As far as I know that is only true if you either remove the $lambda$-filteredness condition on the colimits (see Remark 1.30 in AR) or add the assumption that $lambdalhdmu$ (which changes it from "for sufficiently large $mu$" to "for arbitrarily large $mu$" -- see Remark 2.15 in AR).
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:03















11












$begingroup$


Let $F:cal Cto D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large regular cardinals $lambda$ such that $F$ preserves $lambda$-presentable objects. It is tempting to expect that $F$ should preserve $lambda$-presentable objects for all sufficiently large $lambda$, but that is not what the theorem says. However, I do not know a counterexample showing that the stronger claim fails. (For instance, this question asks about this property when $F$ is the pullback functor, and has no answer yet in the general case.)



What is an example of an accessible functor $F$ between locally presentable categories for which there exist arbitrarily large regular cardinals $mu$ such that $F$ does not preserve $mu$-presentable objects?










share|cite|improve this question









$endgroup$











  • $begingroup$
    For fixed $alpha$, does $mu^alpha = mu$ hold for all sufficiently large regular $mu$?
    $endgroup$
    – Reid Barton
    Mar 13 at 5:17






  • 1




    $begingroup$
    We may assume that $F$ preserves small $lambda$-filtered colimits. Isn’t it true that, for $mu$ large enough, an object is $mu$-presentable if and only if it is a $mu$-small $lambda$-filtered colimit of $lambda$-presentable objects? Another way to put it, is that for $mu$ large enough (e.g. larger than $lambda$ and than the set of maps between any two $lambda$-presentable objects), the property of $mu$-presentability of an object $X$ is simply the fact that the set of maps from a $lambda$-presentable object to $X$ is of cardinal $leqmu$.
    $endgroup$
    – Denis-Charles Cisinski
    Mar 13 at 7:26







  • 3




    $begingroup$
    @Denis-CharlesCisinski As far as I know that is only true if you either remove the $lambda$-filteredness condition on the colimits (see Remark 1.30 in AR) or add the assumption that $lambdalhdmu$ (which changes it from "for sufficiently large $mu$" to "for arbitrarily large $mu$" -- see Remark 2.15 in AR).
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:03













11












11








11


3



$begingroup$


Let $F:cal Cto D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large regular cardinals $lambda$ such that $F$ preserves $lambda$-presentable objects. It is tempting to expect that $F$ should preserve $lambda$-presentable objects for all sufficiently large $lambda$, but that is not what the theorem says. However, I do not know a counterexample showing that the stronger claim fails. (For instance, this question asks about this property when $F$ is the pullback functor, and has no answer yet in the general case.)



What is an example of an accessible functor $F$ between locally presentable categories for which there exist arbitrarily large regular cardinals $mu$ such that $F$ does not preserve $mu$-presentable objects?










share|cite|improve this question









$endgroup$




Let $F:cal Cto D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large regular cardinals $lambda$ such that $F$ preserves $lambda$-presentable objects. It is tempting to expect that $F$ should preserve $lambda$-presentable objects for all sufficiently large $lambda$, but that is not what the theorem says. However, I do not know a counterexample showing that the stronger claim fails. (For instance, this question asks about this property when $F$ is the pullback functor, and has no answer yet in the general case.)



What is an example of an accessible functor $F$ between locally presentable categories for which there exist arbitrarily large regular cardinals $mu$ such that $F$ does not preserve $mu$-presentable objects?







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 Mar 13 at 2:25









Mike ShulmanMike Shulman

37.4k485233




37.4k485233











  • $begingroup$
    For fixed $alpha$, does $mu^alpha = mu$ hold for all sufficiently large regular $mu$?
    $endgroup$
    – Reid Barton
    Mar 13 at 5:17






  • 1




    $begingroup$
    We may assume that $F$ preserves small $lambda$-filtered colimits. Isn’t it true that, for $mu$ large enough, an object is $mu$-presentable if and only if it is a $mu$-small $lambda$-filtered colimit of $lambda$-presentable objects? Another way to put it, is that for $mu$ large enough (e.g. larger than $lambda$ and than the set of maps between any two $lambda$-presentable objects), the property of $mu$-presentability of an object $X$ is simply the fact that the set of maps from a $lambda$-presentable object to $X$ is of cardinal $leqmu$.
    $endgroup$
    – Denis-Charles Cisinski
    Mar 13 at 7:26







  • 3




    $begingroup$
    @Denis-CharlesCisinski As far as I know that is only true if you either remove the $lambda$-filteredness condition on the colimits (see Remark 1.30 in AR) or add the assumption that $lambdalhdmu$ (which changes it from "for sufficiently large $mu$" to "for arbitrarily large $mu$" -- see Remark 2.15 in AR).
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:03
















  • $begingroup$
    For fixed $alpha$, does $mu^alpha = mu$ hold for all sufficiently large regular $mu$?
    $endgroup$
    – Reid Barton
    Mar 13 at 5:17






  • 1




    $begingroup$
    We may assume that $F$ preserves small $lambda$-filtered colimits. Isn’t it true that, for $mu$ large enough, an object is $mu$-presentable if and only if it is a $mu$-small $lambda$-filtered colimit of $lambda$-presentable objects? Another way to put it, is that for $mu$ large enough (e.g. larger than $lambda$ and than the set of maps between any two $lambda$-presentable objects), the property of $mu$-presentability of an object $X$ is simply the fact that the set of maps from a $lambda$-presentable object to $X$ is of cardinal $leqmu$.
    $endgroup$
    – Denis-Charles Cisinski
    Mar 13 at 7:26







  • 3




    $begingroup$
    @Denis-CharlesCisinski As far as I know that is only true if you either remove the $lambda$-filteredness condition on the colimits (see Remark 1.30 in AR) or add the assumption that $lambdalhdmu$ (which changes it from "for sufficiently large $mu$" to "for arbitrarily large $mu$" -- see Remark 2.15 in AR).
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:03















$begingroup$
For fixed $alpha$, does $mu^alpha = mu$ hold for all sufficiently large regular $mu$?
$endgroup$
– Reid Barton
Mar 13 at 5:17




$begingroup$
For fixed $alpha$, does $mu^alpha = mu$ hold for all sufficiently large regular $mu$?
$endgroup$
– Reid Barton
Mar 13 at 5:17




1




1




$begingroup$
We may assume that $F$ preserves small $lambda$-filtered colimits. Isn’t it true that, for $mu$ large enough, an object is $mu$-presentable if and only if it is a $mu$-small $lambda$-filtered colimit of $lambda$-presentable objects? Another way to put it, is that for $mu$ large enough (e.g. larger than $lambda$ and than the set of maps between any two $lambda$-presentable objects), the property of $mu$-presentability of an object $X$ is simply the fact that the set of maps from a $lambda$-presentable object to $X$ is of cardinal $leqmu$.
$endgroup$
– Denis-Charles Cisinski
Mar 13 at 7:26





$begingroup$
We may assume that $F$ preserves small $lambda$-filtered colimits. Isn’t it true that, for $mu$ large enough, an object is $mu$-presentable if and only if it is a $mu$-small $lambda$-filtered colimit of $lambda$-presentable objects? Another way to put it, is that for $mu$ large enough (e.g. larger than $lambda$ and than the set of maps between any two $lambda$-presentable objects), the property of $mu$-presentability of an object $X$ is simply the fact that the set of maps from a $lambda$-presentable object to $X$ is of cardinal $leqmu$.
$endgroup$
– Denis-Charles Cisinski
Mar 13 at 7:26





3




3




$begingroup$
@Denis-CharlesCisinski As far as I know that is only true if you either remove the $lambda$-filteredness condition on the colimits (see Remark 1.30 in AR) or add the assumption that $lambdalhdmu$ (which changes it from "for sufficiently large $mu$" to "for arbitrarily large $mu$" -- see Remark 2.15 in AR).
$endgroup$
– Mike Shulman
Mar 13 at 13:03




$begingroup$
@Denis-CharlesCisinski As far as I know that is only true if you either remove the $lambda$-filteredness condition on the colimits (see Remark 1.30 in AR) or add the assumption that $lambdalhdmu$ (which changes it from "for sufficiently large $mu$" to "for arbitrarily large $mu$" -- see Remark 2.15 in AR).
$endgroup$
– Mike Shulman
Mar 13 at 13:03










1 Answer
1






active

oldest

votes


















12












$begingroup$

An example is given in my paper with Tibor Beke,




Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005.2910.




see Remark 3.2(4). This is what Reid Barton indicated.






share|cite|improve this answer











$endgroup$








  • 1




    $begingroup$
    Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:11






  • 1




    $begingroup$
    The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
    $endgroup$
    – Dap
    Mar 13 at 15:26






  • 2




    $begingroup$
    So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
    $endgroup$
    – Reid Barton
    Mar 13 at 16:21










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1 Answer
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oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









12












$begingroup$

An example is given in my paper with Tibor Beke,




Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005.2910.




see Remark 3.2(4). This is what Reid Barton indicated.






share|cite|improve this answer











$endgroup$








  • 1




    $begingroup$
    Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:11






  • 1




    $begingroup$
    The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
    $endgroup$
    – Dap
    Mar 13 at 15:26






  • 2




    $begingroup$
    So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
    $endgroup$
    – Reid Barton
    Mar 13 at 16:21















12












$begingroup$

An example is given in my paper with Tibor Beke,




Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005.2910.




see Remark 3.2(4). This is what Reid Barton indicated.






share|cite|improve this answer











$endgroup$








  • 1




    $begingroup$
    Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:11






  • 1




    $begingroup$
    The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
    $endgroup$
    – Dap
    Mar 13 at 15:26






  • 2




    $begingroup$
    So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
    $endgroup$
    – Reid Barton
    Mar 13 at 16:21













12












12








12





$begingroup$

An example is given in my paper with Tibor Beke,




Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005.2910.




see Remark 3.2(4). This is what Reid Barton indicated.






share|cite|improve this answer











$endgroup$



An example is given in my paper with Tibor Beke,




Abstract elementary classes and accessible categories, Annals Pure Appl. Logic 163 (2012), 2008-2017, doi:10.1016/j.apal.2012.06.003, arXiv:1005.2910.




see Remark 3.2(4). This is what Reid Barton indicated.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 13 at 10:46









David Roberts

17.5k463177




17.5k463177










answered Mar 13 at 7:50









Jiří RosickýJiří Rosický

1,222178




1,222178







  • 1




    $begingroup$
    Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:11






  • 1




    $begingroup$
    The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
    $endgroup$
    – Dap
    Mar 13 at 15:26






  • 2




    $begingroup$
    So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
    $endgroup$
    – Reid Barton
    Mar 13 at 16:21












  • 1




    $begingroup$
    Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
    $endgroup$
    – Mike Shulman
    Mar 13 at 13:11






  • 1




    $begingroup$
    The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
    $endgroup$
    – Dap
    Mar 13 at 15:26






  • 2




    $begingroup$
    So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
    $endgroup$
    – Reid Barton
    Mar 13 at 16:21







1




1




$begingroup$
Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
$endgroup$
– Mike Shulman
Mar 13 at 13:11




$begingroup$
Ah, of course. For the non-set-theorist readers, can you give a quick reference or sketch of why $mu mapsto mu^alpha$ has arbitrarily large non-fixed-points for $alpha$ infinite?
$endgroup$
– Mike Shulman
Mar 13 at 13:11




1




1




$begingroup$
The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
$endgroup$
– Dap
Mar 13 at 15:26




$begingroup$
The question requires $mu$ to be regular, but that example in Remark 3.2(4) doesn't assume $mu$ is regular, right? It follows from GCH that $mu^alpha=mu$ for all regular $mu>alpha.$ @MikeShulman: By a diagonalization argument, if $mu$ has cofinality $alpha$ then $mu^alpha>mu.$ A reference is Jech's Set Theory, 3rd Ed, Theorem 3.11.
$endgroup$
– Dap
Mar 13 at 15:26




2




2




$begingroup$
So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
$endgroup$
– Reid Barton
Mar 13 at 16:21




$begingroup$
So I was slightly confused when I posted my comment and maybe caused more confusion. Let's switch notation and ask whether $kappa^alpha > kappa$ for arbitrarily large $kappa$. It doesn't matter whether $kappa$ is regular, because if $kappa$ is such that $kappa^alpha > kappa$, then the accessible functor $FX = X^alpha$ fails to preserve $mu$-presentable objects for the regular cardinal $mu = kappa^+$.
$endgroup$
– Reid Barton
Mar 13 at 16:21

















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