TITLE: Is it possible to formulate category theory without set theory?
QUESTION [66 upvotes]: I have never understood why set theory has so many detractors, or what is gained by avoiding its use.
It is well known that the naive concept of a set as a collection of objects leads to logical paradoxes (when dealing with infinite sets) that can only be resolved by defining the concept of a set according to a system of axioms.
With this context, can someone please help me to understand the following passage from Categories for the Working Mathematician, Chapter 1, first two paragraphs (emphasis added):

First we describe categories directly, by  means of axioms, without using any set theory, and call them "metacategories".  Actually we being with a simpler notion, a (meta)graph.
A metagraph consists of objects a,b,c..., arrows f,g,h..., and two operations as follows:

Is it really possible to avoid the use of set theory in the foundations of category theory simply by using the phrase "objects a,b,c ..." instead?
Thanks!

REPLY [10 votes]: You have asked three questions and so far answers have concentrated on the first one since it is the most polemical. 

Why does set theory have so many detractors, and what is gained by avoiding its use?
What does Mac Lane mean when he says he can describe categories without using set theory?
Is it really possible to avoid the use of set theory in the foundations of category theory simply by using the phrase "objects a,b,c ..." instead?

I'll answer the first question last.

For the second question, Mac Lane attempts to weasel his way out of discussing logic. He says a metacategory "consists of" objects, arrows, and operations between them. But what he really means is that a metacategory is a collection of objects, a collection of arrows, and graph-of-function like collections of ordered pairs between them. Formally, a collection, also called a class when working with set theory, is a formula (in first-order logic) with free variables. This is why later on Mac Lane can talk about  "the metacategory of all sets".
In my opinion, what Mac Lane calls metacategories should simply be called categories. My reasons are

Mac Lane's metacategories are elements of the theory, first-order logic, nevermind that they are not objects (e.g. are not sets) that the formal theory is describing. A true metacategory by contrast would consist of all classes (formulas) and "class functions" between them, since such a thing would be part of the metatheoretical analysis of first-order logic.
Mac Lane is not using standard set theory (ZFC). Instead he is assuming that there is a set closed under all the operations of set theory, i.e. a universe (ZFC+U). He refers to elements of the universe as small sets and to all other sets as large sets. He then has a large category of small sets and a metacategory of all sets. But large sets (i.e. ZFC+U) are (for the basic category theory he's doing) merely a way to get at the classes and metacategories of ZFC without having to think about first-order logic.


The answer to the third question is no. This is because Mac Lane's metacategories depend on what you have axiomatized using first-order logic.
Most of category theory really is the study of the metacategories that arise when you axiomatize set theory. In turn, this is because set theory is used as the foundation of modern mathematics and we would like category theory to be applicable to modern mathematics. The use of universes and large vs small sets is primarily a way to make the application of category theory to modern mathematics easier and more convenient.
A very thorough discussion of how category theory interacts with set theory is Mike Shulman's Set Theory for Category Theory, though it does presuppose some familiarity with category theory. 
However, there are other mathematical theories of independent interest, such as Peano arithmetic, which certainly do not require you to axiomatize set theory to do mathematics with. In particular, the metacategories of Peano arithmetic are related to computability theory; sadly I don't know any good references for this -- most everyone does tend to stick to modern mathematics and hence does everything with some set theory in mind.

The answer to the first question is that set theory doesn't actually have many detractors. Rather its detractors are quite vocal but many of the usually given objections are frankly irrelevant.
Currently, my primary objection to set theory is that it's purpose from a purely logical perspective is to mimic higher-order logic inside first-order logic. But why not just directly use higher-order logic (even better: dependent type theory) instead of trying to mimic it in first-order logic? Furthermore, logical systems naturally have the structure of fibred categories, so from a more modern viewpoint set theory ought to be presented as a fibred category satisfying appropriate conditions (this is what a topos is; a mind-blowing reference for some of this stuff is Jacobs' Categorical Logic and Type Theory). 
My secondary objection is that set theory and first-order logic are actively harmful when it comes to learning type theory or more advanced algebraic geometry. One has to actively unlearn some (tacitly) internalized set-theoretic expectations in order to grok those subjects (types are not sets, the "points" of schemes seem weird).