TITLE: How can category theory help my research in set theory?
QUESTION [29 upvotes]: 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.

REPLY [7 votes]: This is not a direct answer to the question but more a small note on the difference between set theory as usually done and set theory viewed through the lenses of category theory.
For me, the most striking aspect of topos theory was the unearthing of vast families of categories that appear in "ordinary" mathematics and that behave very much like the category of sets (denoted by Set).
One can reverse the point of view and ask what is special from a categorial point of view about the category of sets? Well, one partial answer is that Set is well-pointed, that is, the terminal is a separator (or generator). This means that every arrow $f:a\to b$ is determined uniquely by the family of points $fx:t\to b$, with $t$ the terminal and $x:t\to a$ a (global) point of $a$. This is a precise categorial version of our view of sets as "discrete, structureless piles of sand".
If we did set theory this way, what would be different? There is a (at least one) major philosophical difference. There is no global $\in$. Since the equality predicate of sets is defined by
$X=Y \iff (\forall x, x\in X \iff x\in Y)$
it follows that there is no global equality predicate for sets. This may be unintuitive from our conception of sets, but it is the Right Thing in category land, because equality between objects is not preserved by equivalences. The Freyd / Scedrov book Categories, allegories contains a formal treatment of a language invariant under equivalences and a considerable part of M. Makkai's work on n-categories is building this insight right into the foundation of higher categories.
Hope it helps a little, regards.