Non-small objects in categoriesWhy aren't all small categories accessible?K-good trees and K-compactness of...
Non-small objects in categories
Why aren't all small categories accessible?K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST))What's an example of a locally presentable category “in nature” that's not $aleph_0$-locally presentable?compact objects in model categories and $(infty,1)$-categoriesCompact objects in triangulated and infinity categoriesExistence of Colimits in the Definition of Locally Presentable CategoriesSmall objects vs Compact objectsA model category of abelian categories?What are compact objects in the category of topological spaces?Sufficient sets of colimits in small categories
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An object $c$ in a category is called small, if there exists some regular cardinal $kappa$ such that $Hom(c,-)$ preserves $kappa$-filtered colimits.
Is there an example of a (locally small) category $C$ and an object $c$ of $C$, such that $c$ is not small, i.e. such that $Hom(c,-)$ doesn't preserve all $kappa$-filtered colimits for any $kappa$ whatsoever?
ct.category-theory model-categories
asked 9 hours ago
Peter BonartPeter Bonart
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An object $c$ in a category is called small, if there exists some regular cardinal $kappa$ such that $Hom(c,-)$ preserves $kappa$-filtered colimits.
Is there an example of a (locally small) category $C$ and an object $c$ of $C$, such that $c$ is not small, i.e. such that $Hom(c,-)$ doesn't preserve all $kappa$-filtered colimits for any $kappa$ whatsoever?
ct.category-theory model-categories
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
New contributor
Peter Bonart is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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add a comment |
$begingroup$
An object $c$ in a category is called small, if there exists some regular cardinal $kappa$ such that $Hom(c,-)$ preserves $kappa$-filtered colimits.
Is there an example of a (locally small) category $C$ and an object $c$ of $C$, such that $c$ is not small, i.e. such that $Hom(c,-)$ doesn't preserve all $kappa$-filtered colimits for any $kappa$ whatsoever?
ct.category-theory model-categories
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
New contributor
Peter Bonart is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
An object $c$ in a category is called small, if there exists some regular cardinal $kappa$ such that $Hom(c,-)$ preserves $kappa$-filtered colimits.
Is there an example of a (locally small) category $C$ and an object $c$ of $C$, such that $c$ is not small, i.e. such that $Hom(c,-)$ doesn't preserve all $kappa$-filtered colimits for any $kappa$ whatsoever?
ct.category-theory model-categories
ct.category-theory model-categories
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
New contributor
Peter Bonart is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
New contributor
Peter Bonart is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
New contributor
Peter Bonart is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
asked 9 hours ago
Peter BonartPeter Bonart
261 bronze badge
261 bronze badge
New contributor
Peter Bonart is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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2 Answers
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In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and Ulmer's?) theorem that it is impossible for a category and its opposite both to be locally presentable, unless they are both posets.
Indeed, if $A$ is a set with at least two elements, consider functions $f:{0,1}^kappato A$ where $kappa$ is some infinite cardinal. If $lambda<kappa$ then ${0,1}^kappa$ may be viewed as a $lambda$-cofiltered limit of all products of at most $lambda$ of the copies of ${0,1}$. For $A$ to be $lambda$-small in $mathrm{Set}^{mathrm{op}}$, we would have to be able to guarantee that $f$ depends on at most $lambda$ coordinates in the domain.
Since the opposite of the category of sets is the category of complete atomic Boolean algebras (CABAs), we can also make this argument directly in there, where it amounts to the fact that there are elements in a coproduct of CABAs that do not come from any smaller sub-coproduct, since we can always take a join or a meet of elements from every term in the coproduct.
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
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In the category $mathsf{Top}$ of topological spaces and continuous maps the only $lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible categories by Adamek and Rosicky. The reason is explained in 1.2(10) in the same reference.
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
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1
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This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
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– Reid Barton
7 hours ago
add a comment |
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2 Answers
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2 Answers
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$begingroup$
In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and Ulmer's?) theorem that it is impossible for a category and its opposite both to be locally presentable, unless they are both posets.
Indeed, if $A$ is a set with at least two elements, consider functions $f:{0,1}^kappato A$ where $kappa$ is some infinite cardinal. If $lambda<kappa$ then ${0,1}^kappa$ may be viewed as a $lambda$-cofiltered limit of all products of at most $lambda$ of the copies of ${0,1}$. For $A$ to be $lambda$-small in $mathrm{Set}^{mathrm{op}}$, we would have to be able to guarantee that $f$ depends on at most $lambda$ coordinates in the domain.
Since the opposite of the category of sets is the category of complete atomic Boolean algebras (CABAs), we can also make this argument directly in there, where it amounts to the fact that there are elements in a coproduct of CABAs that do not come from any smaller sub-coproduct, since we can always take a join or a meet of elements from every term in the coproduct.
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
$endgroup$
add a comment |
$begingroup$
In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and Ulmer's?) theorem that it is impossible for a category and its opposite both to be locally presentable, unless they are both posets.
Indeed, if $A$ is a set with at least two elements, consider functions $f:{0,1}^kappato A$ where $kappa$ is some infinite cardinal. If $lambda<kappa$ then ${0,1}^kappa$ may be viewed as a $lambda$-cofiltered limit of all products of at most $lambda$ of the copies of ${0,1}$. For $A$ to be $lambda$-small in $mathrm{Set}^{mathrm{op}}$, we would have to be able to guarantee that $f$ depends on at most $lambda$ coordinates in the domain.
Since the opposite of the category of sets is the category of complete atomic Boolean algebras (CABAs), we can also make this argument directly in there, where it amounts to the fact that there are elements in a coproduct of CABAs that do not come from any smaller sub-coproduct, since we can always take a join or a meet of elements from every term in the coproduct.
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
$endgroup$
add a comment |
$begingroup$
In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and Ulmer's?) theorem that it is impossible for a category and its opposite both to be locally presentable, unless they are both posets.
Indeed, if $A$ is a set with at least two elements, consider functions $f:{0,1}^kappato A$ where $kappa$ is some infinite cardinal. If $lambda<kappa$ then ${0,1}^kappa$ may be viewed as a $lambda$-cofiltered limit of all products of at most $lambda$ of the copies of ${0,1}$. For $A$ to be $lambda$-small in $mathrm{Set}^{mathrm{op}}$, we would have to be able to guarantee that $f$ depends on at most $lambda$ coordinates in the domain.
Since the opposite of the category of sets is the category of complete atomic Boolean algebras (CABAs), we can also make this argument directly in there, where it amounts to the fact that there are elements in a coproduct of CABAs that do not come from any smaller sub-coproduct, since we can always take a join or a meet of elements from every term in the coproduct.
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
$endgroup$
In the opposite category of the category of sets, and of many algebraic categories, the only small objects are the empty set and the singleton. A conceptual reason for this is Freyd's (or Gabriel and Ulmer's?) theorem that it is impossible for a category and its opposite both to be locally presentable, unless they are both posets.
Indeed, if $A$ is a set with at least two elements, consider functions $f:{0,1}^kappato A$ where $kappa$ is some infinite cardinal. If $lambda<kappa$ then ${0,1}^kappa$ may be viewed as a $lambda$-cofiltered limit of all products of at most $lambda$ of the copies of ${0,1}$. For $A$ to be $lambda$-small in $mathrm{Set}^{mathrm{op}}$, we would have to be able to guarantee that $f$ depends on at most $lambda$ coordinates in the domain.
Since the opposite of the category of sets is the category of complete atomic Boolean algebras (CABAs), we can also make this argument directly in there, where it amounts to the fact that there are elements in a coproduct of CABAs that do not come from any smaller sub-coproduct, since we can always take a join or a meet of elements from every term in the coproduct.
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
edited 8 hours ago
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
answered 8 hours ago
Kevin CarlsonKevin Carlson
6444 silver badges10 bronze badges
6444 silver badges10 bronze badges
add a comment |
add a comment |
$begingroup$
In the category $mathsf{Top}$ of topological spaces and continuous maps the only $lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible categories by Adamek and Rosicky. The reason is explained in 1.2(10) in the same reference.
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
$endgroup$
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
7 hours ago
add a comment |
$begingroup$
In the category $mathsf{Top}$ of topological spaces and continuous maps the only $lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible categories by Adamek and Rosicky. The reason is explained in 1.2(10) in the same reference.
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
$endgroup$
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
7 hours ago
add a comment |
$begingroup$
In the category $mathsf{Top}$ of topological spaces and continuous maps the only $lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible categories by Adamek and Rosicky. The reason is explained in 1.2(10) in the same reference.
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
$endgroup$
In the category $mathsf{Top}$ of topological spaces and continuous maps the only $lambda$-presentable objects are discrete spaces. This appears 1.14(6) in Locally presentable and Accessible categories by Adamek and Rosicky. The reason is explained in 1.2(10) in the same reference.
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
answered 9 hours ago
Ivan Di LibertiIvan Di Liberti
1,9121 gold badge7 silver badges21 bronze badges
1,9121 gold badge7 silver badges21 bronze badges
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
7 hours ago
add a comment |
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
7 hours ago
1
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
7 hours ago
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
7 hours ago
add a comment |
Peter Bonart is a new contributor. Be nice, and check out our Code of Conduct.
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