I was puzzling over a question this morning that arose from a category theoretical problem and don't think I have the set-theoretic background to answer it myself. The question arose from trying to demonstrate that some weird category was abelian - I constructed a rule for the group structure, but wasn't sure that the Hom collections were sets in the first place.
Given an arbitrary collection $C$ of objects which is not assumed to be a set, if we can construct a binary operation $\cdot$ which satisfies all group axioms, does this imply that $C$ is a set (under ZFC axioms)?
A pedantic answer could be "yes" since then we have a group, and groups are defined over sets, but this does not sit well with me. On the other hand, to me, it seems like a group operation is "local" in that it shouldn't have to consider the size of the universal collection.
Or alternatively, is there anything in ZFC that restricts us from defining groups over arbitrary collections rather than restricting to sets? What goes wrong?