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According to Wikipedia,

The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.

I am aware of Russell's Paradox, which explains why not everything is a set, but how can we show the collection of all groups is a proper class?

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take the free group on each set –  uncookedfalcon Nov 1 '12 at 0:02
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It's too dang big! –  ncmathsadist Nov 1 '12 at 0:08
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When you say "class" you want to say "proper class." Sets are also classes. –  Qiaochu Yuan Nov 1 '12 at 0:08

1 Answer 1

up vote 18 down vote accepted

The collection of singletons is not a set. Therefore the collection of all trivial groups is not a set.

If you wish to consider "up to isomorphism", note that for every infinite cardinal $\kappa$ you can consider the free group, or free abelian group with $\kappa$ generators. These are distinct (up to isomorphism, that is), and since the collection of cardinals is not a set the collection of groups cannot be a set either.

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In more categorical language, the first paragraph shows that $\text{Grp}$ is not a small category, and the second paragraph shows that $\text{Grp}$ is not even an essentially small category (ncatlab.org/nlab/show/small+category). –  Qiaochu Yuan Nov 1 '12 at 0:07
    
@QiaochuYuan: I suppose that they show that $\mathrm{Ab}$ is also neither small nor essentially small. Right? –  Asaf Karagila Nov 1 '12 at 0:16
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Yes, and lots of variations. Much more generally, a theorem of Freyd asserts that if a small category has all small colimits, then it is a poset. –  Qiaochu Yuan Nov 1 '12 at 0:22
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Thanks, I can see how that'd apply to the other types as a well. If anyone else is reading this: Cantor's paradox proves that the collection of cardinals isn't a set –  Casebash Nov 1 '12 at 1:59

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