What do we need to define a category?

I used some results from the category theory without thinking about its foundations. However, after reading a few topics on MSE, this subject haunts me.

My question is:

What do we need to define a category?

According to some books, a category consists of a class $\text{Obj}$ of objects and a set $\text{Hom}$ of morphisms which satisfy some axioms. For me it means, that to define a category we need some set theory. But there are many different set theories. Do they raise different category theories?

Also, as I understand, when we are talking about specific categories, like $\text{Set}$, $\text{Grp}$,... we mean models (interpretations) of the axioms of a category. Is it correct?

• A category is concrete when there is a faithful functor into $\mathrm{Set}$. Loosely speaking, it means that each object is a set, and $\mathrm{Hom}(A,B)$ is a set of maps from $A$ to $B$. Oct 31 '16 at 23:19
• @user10676 oh, sorry.. by concrete categories I mean "specific categories" as $\text{Set}, \text{Grp}, \text{Vec},...$ where the class of objects is determined Oct 31 '16 at 23:30
• There are several different (but more or less equivalent) ways of defining categories even within a single set theory. You can also look at things like ETCS if you want to avoid set theory. Nov 1 '16 at 3:38
• Most categories which have a proper class of objects also have a proper class of morphisms, for example the category of sets. However, it is common in situations like that for each Hom set to have a set of morphisms. Nov 1 '16 at 15:56
• Around ETCS, in particular, I strongly recommend Leinster's Rethinking Set Theory: arxiv.org/abs/1212.6543 Nov 1 '16 at 21:53