Non-small objects in categoriesWhy 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))Equivalence of the two definitions of k-compactness/k-presentabilityWhat's an example of a locally presentable category “in nature” that's not $aleph_0$-locally presentable?Is the category of small categories locally presentable?compact objects in model categories and $(infty,1)$-categoriesSmall objects vs Compact objectsA model category of abelian categories?Locally presentable abelian categories with enough injective objects
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))Equivalence of the two definitions of k-compactness/k-presentabilityWhat's an example of a locally presentable category “in nature” that's not $aleph_0$-locally presentable?Is the category of small categories locally presentable?compact objects in model categories and $(infty,1)$-categoriesSmall objects vs Compact objectsA model category of abelian categories?Locally presentable abelian categories with enough injective objects
$begingroup$
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 accessible-categories locally-presentable-categories
edited 2 days ago
Ivan Di Liberti
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asked Aug 6 at 17:02
Peter BonartPeter Bonart
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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 accessible-categories locally-presentable-categories
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
$endgroup$
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 accessible-categories locally-presentable-categories
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
$endgroup$
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 accessible-categories locally-presentable-categories
ct.category-theory model-categories accessible-categories locally-presentable-categories
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
edited 2 days ago
Ivan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
1,9741 gold badge7 silver badges23 bronze badges
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
asked Aug 6 at 17:02
Peter BonartPeter Bonart
461 bronze badge
461 bronze badge
add a comment |
add a comment |
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 $mathrmSet^mathrmop$, 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 Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 bronze badges
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In the category $mathsfTop$ 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 Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
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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
Aug 6 at 18:58
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 $mathrmSet^mathrmop$, 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 Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 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 $mathrmSet^mathrmop$, 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 Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 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 $mathrmSet^mathrmop$, 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 Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 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 $mathrmSet^mathrmop$, 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 Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 bronze badges
edited Aug 6 at 17:55
answered Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 bronze badges
answered Aug 6 at 17:48
Kevin CarlsonKevin Carlson
7144 silver badges11 bronze badges
answered Aug 6 at 17:48
Kevin CarlsonKevin Carlson
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7144 silver badges11 bronze badges
add a comment |
add a comment |
$begingroup$
In the category $mathsfTop$ 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 Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 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
Aug 6 at 18:58
add a comment |
$begingroup$
In the category $mathsfTop$ 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 Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 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
Aug 6 at 18:58
add a comment |
$begingroup$
In the category $mathsfTop$ 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 Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
$endgroup$
In the category $mathsfTop$ 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 Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
answered Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
answered Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
answered Aug 6 at 17:10
Ivan Di LibertiIvan Di Liberti
1,9741 gold badge7 silver badges23 bronze badges
1,9741 gold badge7 silver badges23 bronze badges
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
Aug 6 at 18:58
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
Aug 6 at 18:58
1
1
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
Aug 6 at 18:58
$begingroup$
This example is also discussed before Lemma 2.4.1 of Mark Hovey's book Model Categories.
$endgroup$
– Reid Barton
Aug 6 at 18:58
add a comment |
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