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.

languageof category theory is used in e.g. algebraic geometry (when considering moduli spaces it's nice to talk about representable functors), and certainly adjoint functors pop up all over the place, it's not clear to me that any "non-trivial facts" (by which I might mean something like "anything taught in the second half of a first category theory course") have had much of an application in other parts of mathematics. [cue a bunch of people going "but what about the Stone-Cech compactification!" ;-)] $\endgroup$ – Kevin Buzzard Nov 29 '09 at 22:08