Category theory without sets

I have been reading Mac Lane's Categories for the Working Mathematician, and the prospect of developing category theory without any use of set theory is mentioned more than once in the book, but never actually realised. I was wondering whether there are any good references (books or online notes) that give an account of such a theory of categories. Looking at this question, it seems that topos theory has been one of the successful ways in which category theory can be defined without sets.

So my question is: What are some good references for how to develop category theory without set theory (using toposes or otherwise).

Solution 1:

Membership relation free set theory

This is what Lawvere did after his PhD. At first, when he discussed this idea with Eilenberg and Mac Lane the latter two did not believe that this was possible and tried to convince him to drop this idea:

“One day, Sammy told me he had a young student who claimed that he could do set theory without elements. It was hard to understand the idea, and he wondered if I could talk with the student. (...) I listened hard, for over an hour. At the end, I said sadly, ‘Bill, this just won’t work. You can’t do sets without elements, sorry,’ and reported this result to Eilenberg. Lawvere’s graduate fellowship at Columbia was not renewed, and he and his wife left for California.” An Interview with F. William Lawvere, p. 40

However, Lawvere came up with the membership relation free axioms (ETCS):

Lawvere, An elementary theory of the category of sets

Another reference for his membership relation free axiomatization of sets is his book:

Lawvere, Sets for mathematics (availlable via google as pdf)

I think this is answering the original question of set free category theory, since the category of sets Set is thereby internalized into category theory, i.e. there is no need for set theoretical membership relation when using notions of sets in category theory.

It is however not the point to avoid membership relation, but to take function as a more natural basic notion instead and derive membership and other relevant concepts from there.

Comparison of Lawvere's axioms with ZFC

Lawvere's axioms are characterizing a two-valued topos with infinite object and axiom of choice:

In summary then, we can say precisely what we mean by a category of abstract sets and arbitrary mappings. It is a topos that is two-valued with an infinite object and the axiom of choice (and hence is also Boolean).

see Lawvere, Sets for mathematics, p. 113.

For a comparison of Lawvere's axioms with ZFC e.g. see Barry W. Cunningham, Boolean topoi and models of ZFC:

Mitchell in [19] showed that categories satisfying Lawvere's axioms were models for finitely axiomatizable set theory $Z_1$ which is strictly weaker than $ZFC$ (Zermelo-Fraenkel set theory with the Axiom of Choice) in that the full axiom scheme of Replacement does not hold.

Compare:

W. Mitchell, Boolean topoi and the theory of sets, J. Pure Appl. Algebra 2 (1972), 261-. 274

The ten axioms [of ETCS] are weaker than ZFC; but when the eleventh [replacement] is added, the two theories have equal strength and are 'bi-interpretable' (the same theormes hold). Moreover it is known to which fragment of ZFC the ten axioms correspond: 'Zermelo with bounded comprehension and choice'. The details of this relationship were mostly worked out in the 70s ...

Compare:

T. Leinster, Rethinking set theory (see p. 7 par. 4 and the references there)

Most mathematicians will never use more properties of sets than those guaranteed by the ten axioms [ETCS]. For example McLarty [C. McLarty. A finite order arithmetic foundation for cohomology. arXiv:1102.1773, 2011] argues that no more is needed anywhere in canons of Grothendieck school of algebraic geometry, the multi-volume works Éléments de Géometrie Algébrique (EGA) and Séminaire de Géometrie Algébrique (SGA).

Compare:

T. Leinster, Rethinking set theory (see p. 6 'How strong are the axioms?')