Which is the relation between objects in a category $C$ and its collection $ob(C)$?

Question

Given a category $C$, we say that objects are in $ob(C)$ which is presented as a class; so it looks obvious that, if $a$ is an object of the category $C$, the relation between $a$ and $ob(C)$ is (a sort of) membership. Nevertheless, in some sources it seems accepted that categorical foundational enterprises are conceptually autonomous from elementhood.

For example in https://plato.stanford.edu/entries/category-theory/#PhilSign we can read this passage: "It has been recognized that it is possible to present a foundational framework in the language of category theory, be it in the form of the Elementary Theory of the Category of Sets, ETCS, or a category of categories, of Makkai Structuralist foundations for abstract mathematics, SFAM. Thus, it seems that the community no longer question the logical and the conceptual autonomy of these approaches, to use the terminology introduced by Linnebo & Pettigrew 2011".

How's it possible to think about categorical framework to be autonomous from a theory of elementhood (such as a set/class theory in whichever form) if categories are dependent from (a sort of) membership relation? I specify that I'm not talking about a particular class theory or an already existing one; I'm just saying that, so to say, membership comes first if you are working in a categorical framework. Where am I wrong?

Answer 1

I'm not qualified to answer this deep question, but to quote a few sources that are too long to fit in a comment.

In a recent post on FOM, Harvey Friedman wrote,

Mac Lane realized that any attempt to remove set theory from the picture, and have notions like category as primitive, is likely doomed not only in terms of its complexity, but also in terms of its Philosophical Coherence. So Mac Lane didn't even attempt to do such a thing, at least in this book.

Even in the exposition of ETCS, it was written that

A lot of confusion seems to reign around “the categorical approach to foundations” and what it might entail; some seem to think it involves a “doing-away with elements” that we all know and love, or doing away with sets and supplanting them with categories, or something like that. That’s an unfortunate misunderstanding.

I guess the above objections are rather philosophical. According to the nLab article,

This page addresses a frequently voiced but easily corrected misconception about categorial approaches to foundations of mathematics, namely that there is a logical circularity in using category theory to give an axiomatic set theory (such as ETCS), since categories themselves are sets (or collections) with extra structure.

The most straightforward response is a formalist one: category theory and various elaborations like ETCS are first-order theories, just as ZFC is. One doesn’t need a theory of sets prior to a theory of categories, any more than one needs a theory of sets prior to a theory of sets...