How can category theory help my research in set theory?
I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at conferences (except of course when the issue is to apply set theory to category theory rather than conversely). Why is this, when category theoretic language and thinking has proved so successful in other parts of mathematics?
Since there seems to be a relatively sizable community of category theorists on this site, many of whom appear to know a lot of set theory or at least have opinions about it, I hope that I might gain some insight.
Note that I am not looking for a reason to do category theory instead of set theory. I am already inspired by a collection of topics, questions and results within set theory, which I find compelling and sometimes profound. What I want to know is whether category theory can provide me with techniques to use to attack those problems.
Best Answer
I believe this is a question that has not been adequately explored.
I view topos-like set theory and ZF-like set theory as exposing two faces of the same subject. In ZF-like theory, sets come equipped with a "membership" relation $\in$, while in topos-like theory, they do not. The former, which I call "material set theory," is the standard viewpoint of set theorists, but the second, which I call "structural set theory," is much closer to the way sets are used by most mathematicians.
However, the two viewpoints really contain exactly the same information. Of course, any material set theory gives rise to a category of sets, but conversely, as J Williams pointed out, from the topos of sets one can reconstruct the class of well-founded relations. With suitable "axioms of foundation" and/or "transitive-containment" imposed on either side, these two constructions set up an equivalence between "topoi of sets, up to equivalence of categories" and "models of (material) set theory, up to isomorphism."
Of course, it happens quite frequently in mathematics that we have two different viewpoints on one underlying notion, and in such a case it is often very useful to compare the meaning of particular statements from both viewpoints. Usually both viewpoints have advantages and disadvantages and each can easily solve problems that seem difficult to the other. Thus, I see a tremendous and (mostly) untapped potential here, if the ZF-theorists and topos theorists would talk to each other more. How much of the structure studied by ZF-theorists can be naturally seen in categorical language? Does this language provide new insights? Does it suggest new structure that hasn't yet been noticed?
One example is the construction of new models. Many of the constructions used by set theorists, such as forcing, Boolean-valued models, ultrapowers, etc. can be seen very naturally in a topos-theoretic context, where category theory gives us many powerful techniques. I personally never understood set-theoretic forcing until I was told that it was just the construction of the category of sheaves on a site. From this perspective the "generic" objects in forcing models can be seen to actually have a universal property, so that for instance one "freely adjoins" to a model of set theory a particular sort of set (say, for instance, a set with cardinality strictly between $\aleph_0$ and $2^{\aleph_0}$), with exactly the same universal property as when one "freely adjoins" a variable $x$ to a ring $R$ to produce the polynomial ring $R[x]$.
On the other hand, some constructions seem more natural in the world of material set theory, such as Gödel's constructible universe. I don't know what the category-theoretic interpretation of that is. So both viewpoints are important.
Another example is the study of large cardinals. Many or most large cardinal axioms have a natural expression in structural terms. For example, there exists a measurable cardinal if and only if there exists a nontrivial exact endofunctor of $Set$. And there exists a proper class of measurable cardinals if and only if $Set^{op}$ does not have a small dense subcategory. Some people at least would argue that Vopenka's principle is much more naturally formulated in category-theoretic terms. I have asked where there are nontrivial logical endofunctors of $Set$; this seems to be a sort of large-cardinal axiom, but it's unclear how strong it is. It seems possible to me that categorial thinking may suggest new axioms of this sort and new relationships between old ones.