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In the field of alg. geo. that I'm studying lately, category theory language is employed but not with the highest level of precision. The lack of precision does not obstruct or obfuscate the theory much, but I am a bit curious. One of the ideas used in this discipline is the notion of isomorphism classes of objects in a category C. I have already deduced that C must be locally small and must have fibered products.

However, there is the further assumption in the text I'm reading that the isomorphism classes described above are in fact sets. Any idea what the necessary, sufficient, or necessary&sufficient conditions on C would be to force isomorphism classes of objects of C to be sets?

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    $\begingroup$ For a locally small category the isomorphism classes for a set if and only if the category is essentially small (that is, equivalent to a small category). I don't think you can say much more than that. $\endgroup$
    – Denis Nardin
    Commented Aug 29, 2016 at 15:23
  • $\begingroup$ Alrighty. That's kinda what I figured but thought I'd put the question out there and see what happened. I'm a noob at category theory, having only gotten it through working on Vakil's book this summer. Thanks! :) $\endgroup$
    – Tanner Strunk
    Commented Aug 29, 2016 at 15:25
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    $\begingroup$ Could you clarify the question? Were you asking whether: a) The class of all isomorphism classes is a set, or b) Each isomorphism class is a set? Denis Nardin's answer is appropriate for (a), but not for (b). The full subcategory of the category of sets on the singletons is an example where (a) is true but not (b). The full subcategory of the category of sets on the cardinals is an example where (b) is true but not (a). $\endgroup$
    – Robert Furber
    Commented Aug 29, 2016 at 15:51

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