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  • $\begingroup$ Thank you for your answer! It resolves some of my concerns. I have a question. As you said, almost all interesting categories are essentially small in the context of Grothendieck groups. Thus I want to construct K_0 as a functor from the category of essentially U-small abelian categories to the category of U-small abelian groups. However, I think that we cannot consider the set of all U-small abelian groups. Is it natural to fix a universe V containing U and consider the category of U-small abelian groups belonging to V? $\endgroup$ Commented Dec 26, 2022 at 13:02
  • $\begingroup$ I would like to know if you know of any literature that has a good discussion of this kind. $\endgroup$ Commented Dec 26, 2022 at 13:03
  • $\begingroup$ Regarding your first question - from a purely category theoretic point of view it is completely fine to consider the category of U-small sets. In the sort of foundation you are using, one needs an additional level of care: you indeed need to consider a second universe $V$ such that $U \in V$, and consider the category of $V$-sets that are $U$-small. The resulting category technically depends on $V$, but not in an essential way: if you replace $V$ by another universe $W$ you get an equivalent categories (because they are both equivalent to the category of $U$-sets). $\endgroup$ Commented Dec 26, 2022 at 13:20
  • $\begingroup$ Regarding litterature - I don't know a place that focuses on the kind of problems that arise out of size issue. They tend to pop-up here and there in isolated way. Mike shulman has a survey paper (arxiv.org/abs/0810.1279) on the various kind of set theoretic foundation that can be use to deal with issue, but I'm not sure that is what you are after. The fact that there are many different foundation that can be used to deal with these problems mean there is no unified account of this sort of things. $\endgroup$ Commented Dec 26, 2022 at 13:23
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    $\begingroup$ @LOCOAS There are some other MO questions which discuss some of these issues, e.g., When size matters in category theory for the working mathematician and Reflection principle vs universes. $\endgroup$ Commented Dec 26, 2022 at 14:43