I just started reading Categories and Sheaves and noticed that the book's definition of a category requires the collection of objects of a category to be a set. However, from other experiences, I have heard that the object set of a category can also be a class. So in the former, the category of sets doesn't contain all sets (but instead some collection of sets that live in a universe set) whereas in the latter, the category contains all sets. Presumably by only requiring the object collection to be a class, we can apply category theory to a larger collection of mathematical structures. What is a possible reason for this seeming discrepancy in definitions? By following the book's definition, will I be missing out on anything?
category definition: class vs set?
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set-theory
category-theory
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5Category theory with classes is [logically more messy](http://arxiv.org/abs/0810.1279), while, on the other hand, universe axioms are ontologically more questionable. Pick your poison. – 2012-02-19
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0Pfft, ontology is for philosophers. Mathematicians need not go there. – 2012-10-12
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0@Noldorin What is meant by the comment is that the existence of universes has higher consistency strength than $\mathsf{ZFC}$. This has nothing to do with philosophy. – 2018-01-18