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?
$begingroup$
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 15 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 15 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 15 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 15 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 15 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 15 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 15 hours ago
Mike ShulmanMike Shulman
37.1k485230
asked 15 hours ago
Mike ShulmanMike Shulman
37.1k485230
37.1k485230
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 14 hours ago
Jiří RosickýJiří Rosický
1,001166
$endgroup$
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
12 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
11 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
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
28 mins ago
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
$begingroup$
The stronger claim is true and follows from Proposition 4.3 in https://arxiv.org/pdf/1005.2910.pdf.
answered 14 hours ago
Jiří RosickýJiří Rosický
1,001166
$endgroup$
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
12 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
11 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
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
28 mins 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 14 hours ago
Jiří RosickýJiří Rosický
1,001166
$endgroup$
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
12 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
11 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
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
28 mins 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 14 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 14 hours ago
Jiří RosickýJiří Rosický
1,001166
answered 14 hours ago
Jiří RosickýJiří Rosický
1,001166
answered 14 hours ago
Jiří RosickýJiří Rosický
1,001166
answered 14 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
12 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
11 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
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
28 mins ago
add a comment |
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
12 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
11 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
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
28 mins ago
2
2
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
12 hours ago
$begingroup$
Thanks! Would you be able to supply a few more details?
$endgroup$
– Mike Shulman
12 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
11 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
11 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
9 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
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
28 mins 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
28 mins ago
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
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
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
