Are elements of categories sets? (Subtlety in Hungerford's Algebra.)

Question

Hungerford's book Algebra defines class, set, and category informally. Class and $\in$ are left as primitive notions. The definitions are, mostly verbatim:

A class $A$ is defined to be a set if and only if there exists some class $B$ such that $A\in B$.

And:

A category is a class $\mathcal{C}$ of objects (denoted $A,B,C,\ldots$), together with (morphisms, composition, associativity, etc.)

I want to write $A\in\mathcal{C}$, which by the definitions in Hungerford imply that $A$ is a set. However, every document I see uses the terminology "Let $A$ be an object of $\mathcal{C}$" rather than "$A\in\mathcal{C}$". Is this just convention, or is $A\in \mathcal{C}$ incorrect notation?

Secondly, does this cause any problem with the notion of a concretizable category? I can imagine that even though all objects of $\mathcal{C}$ are sets, the morphisms may only be abstract and may not be able to be written as actual functions between the sets. That way, there would be no contradiction between the existence of non-concretizable categories.

Answer 1

Hungerford says that a category is a class $\mathcal C$ of objects, together with a set of morphisms for every pair of objects. ...So what's the category? You're tempted to say the category is a pair $(\mathcal C,\mathcal M)$ of objects and morphisms. But as has been discussed, $\mathcal C$ is generally not a set, so there exists no such class in Hungerford's foundations.

In a sufficiently strong foundation, a category is indeed an ordered pair $\mathcal C=(\mathrm{ob}\mathcal C,\mathrm{mor}\mathcal C)$ as suggested above. Then your notation becomes confusing: if you say $c\in \mathcal{C}$, do you mean $c$ is an object or a morphism? So, as a matter of practice, many authors avoid your usage, in favor of $c\in\mathrm{ob}\mathcal C$.

However, many authors are happy to say $c\in \mathcal C$ when $c$ is an object, by an abuse of notation familiar from group theory etc. In Hungerford's rough foundations, this isn't even an abuse! And finally we return to the question of whether an object of a category is a set. The answer is that it depends on the founda. The most flexible foundation is usually yes, depending on foundations: in particular this is true in ZFC with inaccessible cardinals, probably the best foundation for category theory, and in NBG set theory, which is what Hungerford is approximating.

Answer 2

A category is definitely not a set, since it also contains the information of $Hom(A,B)$ whenever $A$ and $B$ are objects in $\mathcal{C}$. Now, the objects in $\mathcal{C}$ (sometimes denoted $\mathrm{ob}\mathcal{C}$) may form a set, and may not. When they do, $\mathcal{C}$ is called a small category.

I don't understand this stuff very well, but my understanding is that the notion of category is supposed to avoid Russell's paradox. That is, you shouldn't be able to formulate the category of all categories that don't contain themselves.

Answer 3

Categories are classes rather then sets. This is because you can have the category $\mathbb{set}$ which is the category where the objects are sets and the morphisms are functions between these sets.

Since you can't have the set of all sets (at least in ZFC) you have to deal with classes. The operator $\in$ is a relation between sets so saying $A \in \mathcal{C} $ is incorrect as $ \mathcal{C} $ is not a set.

A class is collection of sets for which a certain predicate is true. While a set is a collection of sets created via the axioms of set theory.