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"A category $\mathfrak{C}$ is the data of the collection $\textrm{Obj}(\mathfrak{C})$ of objects of $\mathfrak{C}$ and for each objects $X,Y$ of a set of morphisms $\textrm{Hom}_{\mathfrak{C}} (X,Y)$, such that ..." A perfectly fine classic where one informally speaks about collections of objects.

First example, the category $S$ of sets. Then whatever kind of "construction" $\textrm{Obj}(S)$ may be, it is for sure not a set, as "the set of all sets does not exist". Another perfectly fine (Russell's) classic in naive set theory, "solved" in ZFC.

I never thought about which set theory I was in when I was considering cathegory theory or doing algebraic geometry (even if Bourbaki's appendix in SGA IV 1 and some "non-boundedness" questions around flat topologies puzzled me) but apparently I was in ZFC. But in ZFC, I don't know what a "collection" is.

What are the reactions (common or not) to this "issue" ?

Olórin
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    It might help to read the foundation part of introduction of Abstract and Concrete Categories starting at page 13. – drhab Sep 16 '19 at 09:53
  • In ZFC, a "collection" is a class: https://en.wikipedia.org/wiki/Class_(set_theory) – Qiaochu Yuan Sep 16 '19 at 10:20
  • Maybe you find https://math.stackexchange.com/questions/1164627/is-category-theory-constructive interesting. – Hanno Sep 16 '19 at 10:26
  • And perhaps https://math.stackexchange.com/questions/432172/developing-category-theory-inside-etcs – Hanno Sep 16 '19 at 10:27
  • @QiaochuYuan I thought that a class was something informal in ZFC. – Olórin Sep 16 '19 at 14:19
  • @drhab I don't see a proper definition of a class there, though. – Olórin Sep 16 '19 at 14:22
  • It would not surprise me if classes are actually not defined but primitive. It cannot be avoided that some mathematical concepts are not defined. This because the relation of "being defined" on math is a well-founded relation. – drhab Sep 16 '19 at 14:32
  • Is the only formal (I'm leaving aside intuition about sizing etc) difference between sets and classes the fact that altough both are defined by a formula, the former are parametrized by a set whereas the latter have a "free" parametrization ? Or is there no formal way of defining a class and dinstinguishing it from a set, thereby meaning that classes can only be defined in the metalanguage of the set theory ? – Olórin Sep 16 '19 at 15:27
  • I mean, I wouldn't mind defining class theory (not the one from alg number though :)) and then being able to state formally, inside the theory, what a set is and why a class isn't a set, such that the trace of the class theory on the sets is ZFC (or anything else). I don't know if I make sense. – Olórin Sep 16 '19 at 15:31
  • Remark : I wrote that all classes are defined by a formula because I don't see how "abstract classes", even in category theory, could be useful. (With algebraic geometry in view, though.) – Olórin Sep 16 '19 at 15:33

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You should read Mike Shulman's Set theory for category theory. It gives lots of ways to tackle the issue and compares them (their strenghts and weaknesses). This allows you to 1- make sure that what you do is valid in some/all of these formalizations, 2-pick your favorite solution if you have one and stick to it and maybe 3- forget about it if you only want to know you're on solid ground (no matter what the ground is)

Maxime Ramzi
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