the concept of class (category theory)

Question

I have started to give a look at the definition of category, even though I have not started yet to go deep into the study of category theory. The aspect which gives me trouble is the fact that the building blocks of the definition are so called classes. Since I do not have a strong set-theoretical background, I am lead to think about, e.g., the class of objects as a collection of sets (may them be topological spaces, groups,...), but from what I read on the net a class is not really that. Do you know how to get around it? Could you explain it or suggest me a reference which leads to the understanding of the concept of class in set theory (even not a too deep one, since I just want to use it to study topics other than set theory)? Thank you for any help!

Answer 1

Take the category of groups. Its objects are groups, its arrows morphisms between groups. Ok, so how big is this category? Well, how many groups are there? Set-many. As many as there are sets. So, on standard assumptions, too many to form a set.

So we can't say that the category of groups comprises a set of objects (along with a set of the morphisms but let's not worry about them right now).

OK what shall we say? First option: just say that the category of groups comprises groups, plural (and the morphisms).

Second option: say that the category of groups comprises a (proper) class of groups (along with the collection of morphisms) -- where a proper class is a collection too big to be a set.

If you take the second option then further down the road we can wonder about how to handle this notion of classes-which-aren't-sets. But that's really for later. At least at the outset, it is fine to go the first way. In other words just talk of the data of a non-empty category as being the objects (plural, or at least one) and morphisms (plural, or at least one), and forget class talk. Steve Awodey's terrific Category Theory book, for one, goes this way.

Answer 2

There are many paradoxes is set theory if you allow any collection of objects to be a set. Look up Russells paradox on wikipedia there are plenty of elemental examples. The concept of a class is essentially a collection that we cannot call a set as it would lead to paradoxes. Fundamentally classes just contain too many elements. As a reference I would recomend Enderton ELements of set theory a pdf is available here. https://drive.google.com/file/d/0B84Z51KYZrFHYzQ0ODVlODMtNmQ5NC00MDJkLTgxMjEtNTlmMDdkOGU0NTFj/view

It's a book but you'll get a decent idea to be going on with from the first chapter or maybe 2

Answer 3

This is definitely not a very important subject to look if you are in your first course of category theory. Most of these "size issues" are irrelevant (although not all of them are). You need to understand the rudiments of set theory, as in ZFC, and understand that there are constructions that are not sets. For example the category of sets or the category of abelian groups doesn't have a set of objects, informally it is a collection, a very big bag of things.

The formalism of classes allows you to talk about these very big collections which cannot be defined with plain ZFC (you can not construct them as sets, that's why it is safe). There are other formalisms like introducing large cardinal axioms, see https://ncatlab.org/nlab/show/Grothendieck+universe. But my advise is to skip these details until later and first study category theory as in Mac Lane's Categories for the Working Mathematician.