Closure of presentable objects under finite limits
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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
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
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add a comment |
$begingroup$
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
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
$endgroup$
add a comment |
$begingroup$
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
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
$endgroup$
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
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 12 hours ago
Mike ShulmanMike Shulman
37.1k485230
37.1k485230
add a comment |
add a comment |
1 Answer
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The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
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2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 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
8 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
6 hours ago
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
$endgroup$
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 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
8 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
6 hours ago
add a comment |
$begingroup$
The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
$endgroup$
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 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
8 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
6 hours ago
add a comment |
$begingroup$
The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
$endgroup$
The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
answered 11 hours ago
Jiří RosickýJiří Rosický
1,001166
1,001166
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 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
8 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
6 hours ago
add a comment |
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 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
8 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
6 hours ago
2
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 hours ago
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
10 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
8 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
8 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
6 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
6 hours ago
add a comment |
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