Is the category of fields small?
$begingroup$
I know that the category of groups $textbf{(Group)}$ and rings $textbf{(Ring)}$ are both only locally small, since any non-empty set can be made into a group or a ring.
However, when this comes to fields, I'm not sure if there's also an explicit construction making an infinite set to a field (we know not all finite set can be a field). Therefore I cannot confirm if the category of fields is small.
Thanks for any suggestion in advance.
elementary-set-theory category-theory
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
$endgroup$
|
show 1 more comment
$begingroup$
I know that the category of groups $textbf{(Group)}$ and rings $textbf{(Ring)}$ are both only locally small, since any non-empty set can be made into a group or a ring.
However, when this comes to fields, I'm not sure if there's also an explicit construction making an infinite set to a field (we know not all finite set can be a field). Therefore I cannot confirm if the category of fields is small.
Thanks for any suggestion in advance.
elementary-set-theory category-theory
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
$endgroup$
$begingroup$
Can't an arbitrary countable set be viewed as $mathbb{Q}?$ Or an uncountable one as $mathbb{R}?$ Not at all sure if this helps...
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:04
$begingroup$
Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $textbf{(Field)}$ is not small?
$endgroup$
– tommy xu3
Dec 6 '18 at 10:07
$begingroup$
Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful.
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:07
$begingroup$
Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem...
$endgroup$
– tommy xu3
Dec 6 '18 at 10:09
2
$begingroup$
For $kappa$ large enough, you can biject $2^kappa$ with the transcendence degree $kappa$ extension of $mathbb{F}_2$. So the category (Fields) is not small.
$endgroup$
– user10354138
Dec 6 '18 at 10:10
|
show 1 more comment
$begingroup$
I know that the category of groups $textbf{(Group)}$ and rings $textbf{(Ring)}$ are both only locally small, since any non-empty set can be made into a group or a ring.
However, when this comes to fields, I'm not sure if there's also an explicit construction making an infinite set to a field (we know not all finite set can be a field). Therefore I cannot confirm if the category of fields is small.
Thanks for any suggestion in advance.
elementary-set-theory category-theory
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
$endgroup$
I know that the category of groups $textbf{(Group)}$ and rings $textbf{(Ring)}$ are both only locally small, since any non-empty set can be made into a group or a ring.
However, when this comes to fields, I'm not sure if there's also an explicit construction making an infinite set to a field (we know not all finite set can be a field). Therefore I cannot confirm if the category of fields is small.
Thanks for any suggestion in advance.
elementary-set-theory category-theory
elementary-set-theory category-theory
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
edited Dec 6 '18 at 14:20
Andrés E. Caicedo
65.4k8158248
65.4k8158248
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
asked Dec 6 '18 at 9:56
tommy xu3tommy xu3
9791621
9791621
$begingroup$
Can't an arbitrary countable set be viewed as $mathbb{Q}?$ Or an uncountable one as $mathbb{R}?$ Not at all sure if this helps...
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:04
$begingroup$
Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $textbf{(Field)}$ is not small?
$endgroup$
– tommy xu3
Dec 6 '18 at 10:07
$begingroup$
Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful.
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:07
$begingroup$
Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem...
$endgroup$
– tommy xu3
Dec 6 '18 at 10:09
2
$begingroup$
For $kappa$ large enough, you can biject $2^kappa$ with the transcendence degree $kappa$ extension of $mathbb{F}_2$. So the category (Fields) is not small.
$endgroup$
– user10354138
Dec 6 '18 at 10:10
|
show 1 more comment
$begingroup$
Can't an arbitrary countable set be viewed as $mathbb{Q}?$ Or an uncountable one as $mathbb{R}?$ Not at all sure if this helps...
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:04
$begingroup$
Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $textbf{(Field)}$ is not small?
$endgroup$
– tommy xu3
Dec 6 '18 at 10:07
$begingroup$
Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful.
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:07
$begingroup$
Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem...
$endgroup$
– tommy xu3
Dec 6 '18 at 10:09
2
$begingroup$
For $kappa$ large enough, you can biject $2^kappa$ with the transcendence degree $kappa$ extension of $mathbb{F}_2$. So the category (Fields) is not small.
$endgroup$
– user10354138
Dec 6 '18 at 10:10
$begingroup$
Can't an arbitrary countable set be viewed as $mathbb{Q}?$ Or an uncountable one as $mathbb{R}?$ Not at all sure if this helps...
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:04
$begingroup$
Can't an arbitrary countable set be viewed as $mathbb{Q}?$ Or an uncountable one as $mathbb{R}?$ Not at all sure if this helps...
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:04
$begingroup$
Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $textbf{(Field)}$ is not small?
$endgroup$
– tommy xu3
Dec 6 '18 at 10:07
$begingroup$
Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $textbf{(Field)}$ is not small?
$endgroup$
– tommy xu3
Dec 6 '18 at 10:07
$begingroup$
Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful.
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:07
$begingroup$
Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful.
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:07
$begingroup$
Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem...
$endgroup$
– tommy xu3
Dec 6 '18 at 10:09
$begingroup$
Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem...
$endgroup$
– tommy xu3
Dec 6 '18 at 10:09
2
2
$begingroup$
For $kappa$ large enough, you can biject $2^kappa$ with the transcendence degree $kappa$ extension of $mathbb{F}_2$. So the category (Fields) is not small.
$endgroup$
– user10354138
Dec 6 '18 at 10:10
$begingroup$
For $kappa$ large enough, you can biject $2^kappa$ with the transcendence degree $kappa$ extension of $mathbb{F}_2$. So the category (Fields) is not small.
$endgroup$
– user10354138
Dec 6 '18 at 10:10
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $Bbb Q$, consider the transcendental extension $Bbb Q(A)$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $|A|=|Bbb Q(A)|$ whenever $A$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $Bbb Q$. This can be repeated with any field instead of $Bbb Q$ (e.g. $Bbb F_2$) as well.
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
$endgroup$
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
add a comment |
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1 Answer
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$begingroup$
Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $Bbb Q$, consider the transcendental extension $Bbb Q(A)$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $|A|=|Bbb Q(A)|$ whenever $A$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $Bbb Q$. This can be repeated with any field instead of $Bbb Q$ (e.g. $Bbb F_2$) as well.
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
$endgroup$
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
add a comment |
$begingroup$
Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $Bbb Q$, consider the transcendental extension $Bbb Q(A)$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $|A|=|Bbb Q(A)|$ whenever $A$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $Bbb Q$. This can be repeated with any field instead of $Bbb Q$ (e.g. $Bbb F_2$) as well.
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
$endgroup$
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
add a comment |
$begingroup$
Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $Bbb Q$, consider the transcendental extension $Bbb Q(A)$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $|A|=|Bbb Q(A)|$ whenever $A$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $Bbb Q$. This can be repeated with any field instead of $Bbb Q$ (e.g. $Bbb F_2$) as well.
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
$endgroup$
Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $Bbb Q$, consider the transcendental extension $Bbb Q(A)$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $|A|=|Bbb Q(A)|$ whenever $A$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $Bbb Q$. This can be repeated with any field instead of $Bbb Q$ (e.g. $Bbb F_2$) as well.
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
answered Dec 6 '18 at 10:11
Asaf Karagila♦Asaf Karagila
303k32429762
303k32429762
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
add a comment |
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot!
$endgroup$
– tommy xu3
Dec 6 '18 at 10:14
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
Could you explain under what conditions a category can be considered small? I thought that if it didn't contain a set of every cardinality it can be considered small.
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:16
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 ncatlab.org/nlab/show/small+category
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:17
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, ${X}in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $textsf{Sets}$ with objects $S$ is not small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:26
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
$begingroup$
@伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small.
$endgroup$
– Alex Kruckman
Dec 7 '18 at 19:28
add a comment |
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$begingroup$
Can't an arbitrary countable set be viewed as $mathbb{Q}?$ Or an uncountable one as $mathbb{R}?$ Not at all sure if this helps...
$endgroup$
– 伽罗瓦
Dec 6 '18 at 10:04
$begingroup$
Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $textbf{(Field)}$ is not small?
$endgroup$
– tommy xu3
Dec 6 '18 at 10:07
$begingroup$
Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful.
$endgroup$
– Asaf Karagila♦
Dec 6 '18 at 10:07
$begingroup$
Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem...
$endgroup$
– tommy xu3
Dec 6 '18 at 10:09
2
$begingroup$
For $kappa$ large enough, you can biject $2^kappa$ with the transcendence degree $kappa$ extension of $mathbb{F}_2$. So the category (Fields) is not small.
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– user10354138
Dec 6 '18 at 10:10