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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.

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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... $\endgroup$ – green frog 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? $\endgroup$ – User 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... $\endgroup$ – User Dec 6 '18 at 10:09
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    $\begingroup$ 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
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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.

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  • $\begingroup$ Nice! Since I didn't encounter problems of algebra, I didn't think of it right immediately. Thanks a lot! $\endgroup$ – User 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. $\endgroup$ – green frog 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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