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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?

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  • $\begingroup$ 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 $B$. $\endgroup$
    – user10676
    Oct 31 '16 at 23:19
  • $\begingroup$ @user10676 oh, sorry.. by concrete categories I mean "specific categories" as $\text{Set}, \text{Grp}, \text{Vec},...$ where the class of objects is determined $\endgroup$
    – nightked
    Oct 31 '16 at 23:30
  • $\begingroup$ 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. $\endgroup$ Nov 1 '16 at 3:38
  • $\begingroup$ 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. $\endgroup$ Nov 1 '16 at 15:56
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    $\begingroup$ Around ETCS, in particular, I strongly recommend Leinster's Rethinking Set Theory: arxiv.org/abs/1212.6543 $\endgroup$
    – Berci
    Nov 1 '16 at 21:53
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First, one needs to adopt a foundation of mathematics to define a set, category, and other mathematical objects!

Different foundations give different category theories. Some 'categories', which are called big in one foundation, do not exists in another, e.g. the functor category between two large categories and the localisation of a category with respect to a proper class (large set) of its morphisms. The meanings of the terms (small) set and (proper) class, and the operations you can perform on them, depend on the adopted foundation. Shulman's Set theory for category theory and Mac Lane's One universe as a foundation for category theory discuss the effect of the foundation on the resulting category theory, although the latter is more focused on the advantages of a specific foundation.

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