My Confusion about the Definition for a Category

Question

I need you to validate me about my confusion in a category's definition.

My reference book is telling that a category is "a class $ \mathscr{C} $ of objects $ (A,B,C,...) $ together with a class of pairwise disjoint sets, $ hom(A,B) $, and a composite relation satisfying some conditions."

I won't make it detailed. I'm confused with the beginning:

1. Is the class of $ hom(A,B) $ a part of the category $ \mathscr{C} $? ie. Is the category $ \mathscr{C} $ a mixed whole containing different kinds of other mathematical "tools" (i'm using "tools" because, here, the word "object" goes specific for $ (A,B,C,...) $)? My comprehension of the subject says that yes they are. otherwise we needn't to use classes to define categories. they would have been these objects themselves with some properties...

2. Can we talk about cardinality of a category? The relation of my second question to the first one is here: I don't get if homomorphisms are contained by the category, along side objects, too?

If I mistook some concept, please forgive me, I'm not well acquainted with categories yet.

Thank you very much.

Answer 1

A few things here.

• Your definition is that of a locally-small category: one in which the arrows from $A$ to $B$ (for any fixed $A, B$ objects in the category) really do form a set rather than a proper class. Just bear that in mind if you're reading other resources.
• There's not just one hom-set $\mathrm{hom}(A, B)$. There is one such set for every pair $A, B \in \mathcal{C}$ (that is, for each pair of objects).
• $\mathcal{C}$ is a collection of stuff, yes. Just as a group is a collection of stuff - a set with an operation on that set such that some conditions hold - so a category is a collection of stuff. Specifically, it is a class of objects, together with a class of hom-sets (one hom-set per pair of objects in that class of objects), and a relation-class that holds with certain properties.
• A category might have a bona-fide set of objects (remember that the definition of a category allows any class of objects), in which case you can talk about its cardinality in the usual way. Otherwise, you've got only the much more limited ways you can talk about the size of a class. Usually one does not talk about the cardinality of a category, but when one does, one is talking about the cardinality of the collection of objects, and not the cardinality of some larger structure (e.g. "objects unioned with arrows" in some way).

Remember the difference between a thing and the encoding of that thing in some theory. Categories are often considered as "simply existing"; there's no need for a category per se to be represented by some particular object-thing in some space. One instead considers the objects of the category to exist, and the arrows in the category to exist, without actually considering the category as having any particular existence unto itself. (Of course, this slightly-sloppy way of thinking about categories can all go out the window when you've familiarised yourself with them; they are useful things to study.)

That is to say, the category of sets is perfectly well allowed to exist, and you don't need to find some collection of things which contains the objects and which contains the arrows and which contains the composition relations if you want to study it. Fear not.