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Consider the category of all groups. Now a group is a set with extra structure, so the collection of all groups must be "at least as large" as the collection of all sets. Since the category of all sets is the prototypical example of a large category. This it seems to me that we are forced to conclude that the category of all groups is large. Going on from here we should also be able to conclude the same thing for the category of rings, of vector spaces, of topological spaces . . . i.e. fo any mathematical object consisting of a set plus some "extra structure".

This seems clear, but I get worried when things are so big.

Dick Johnson
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  • Yes, those categories are large because at least one among the class of objects or the class of morphisms is proper (in those cases, both). –  Sep 25 '21 at 13:43
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    I don't think that the fact that "object $X$ is a set with extra structure" in general implies that the collection of objects $X$ is at least as large as the collection of all sets. – Giorgos Giapitzakis Sep 25 '21 at 13:49
  • For every set $X$ you can have a group structure on the set ${X}$. –  Sep 25 '21 at 14:14

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