Does it make sense to form a set whose elements are proper classes?

In chapter 4 of Handbook of Categorical Algebra, vol 1, the author defines a "subobject of $A$" as "an equivalence class of monomorphisms with codomain $A$" (for a suitable notion of equivalence). He then defines what it means for a category to be well-powered: "$\mathcal{A}$ is well-powered when the subobjects of every object constitute a set". Thus, for instance, the category of sets is well-powered.

I'm having trouble understanding exactly what it means to have such a set of subobjects. As far as I can tell, each element of a set should also be a set, but an equivalence class of monomorphisms could be a proper class: for instance, the class of singleton sets is not a set. On the other hand, it seems that one can cheat by defining a subobject to be a class containing one representative of each equivalence class of monomorphisms, even though this is not, strictly speaking, what's stated in the book.

How can one solve this problem? Is there a "normal" set theory where such a set of subobjects can contain proper classes? Or does one need to cheat like suggested above?

• Elements of a set must be a set so there is no set which has a proper class as element. You need a another word to denote `family of proper classes'... – Hanul Jeon Nov 5 '14 at 14:59
• That's what I suspected indeed, but how to make sense of the definition of "well-powered", then? – Arthur Azevedo De Amorim Nov 5 '14 at 15:20
• If classes are objects, then it seems to make just as much sense to have a set of them as it does to have a set of cats. We need a slightly different theory of sets, see en.wikipedia.org/wiki/Urelement, though. – GME Nov 5 '14 at 15:22
• This just makes sense with a global choice. Maybe he's assuming locally small categories... – user40276 Nov 13 '14 at 21:10
• @user40276 Yes, he is assuming locally small categories, but even with this assumption it seems a bit strange, because the monomorphisms in the definition need not have the same domain. Each hom set is indeed a set, but when you add up hom sets for lots of different objects, you can get something that is not one. – Arthur Azevedo De Amorim Nov 14 '14 at 14:57

• If $M$ is a model of set theory, then $M$ thinks that $\varphi(y)$ defines a set if and only if $M\models\exists x\forall y(y\in x\leftrightarrow \varphi(y))$. Do you agree with that? – Asaf Karagila Nov 5 '14 at 15:53
• Good. So what does it mean when $\varphi$ defines a proper class? It means that $M$ does not satisfy the statement "There exists $x$ ...", so proper classes are collections of objects in $M$ which do not exist in $M$. But being elements, from the point of view of $M$, requires first to exist, because the $\in$ is only defined on the objects inside $M$. Ergo, proper classes are collections of objects of $M$ which do not exist in $M$. So they cannot be elements of other sets, since in this context we limit "set" to mean "an object inside $M$". – Asaf Karagila Nov 5 '14 at 15:58
• @GME: No, it means exactly that. If categories are not sets, they don't exist in the universe. I really don't know how to make this any clearer than that. As for your second remark, that might depend on your axioms, but in the usual axiomatization of $\sf ZFA+AC$ you can prove that every set is in bijection with a pure set (more specifically, an ordinal). So this remark is certainly false there. – Asaf Karagila Nov 5 '14 at 16:16