Is the category of fields small?












0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








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




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


















  • $begingroup$
    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










1 Answer
1






active

oldest

votes


















4












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









4












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















4












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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














4












4








4





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • $begingroup$
    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


















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Puebla de Zaragoza

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Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
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Musa

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Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
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