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.
One can do this using less technology, too...
Let $\Sigma$ be an alphabet, $N$ a set of non-terminals, and $\Sigma^\*$ and $(\Sigma\cup N)^\*$ the full languages on $\Sigma$ and $\Sigma\cup N$, respectively. A context-free grammar is a finite subset $G\subset N\times(\Sigma\cup N)^\*$. Given one such grammar $G$ there is a relation $\mathord\rightarrow_G\subseteq(\Sigma\cup N)^\*\times(\Sigma\cup N)^\*$ which is the least transitive reflexive relation which contains $G$ (notice that $N\times(\Sigma\cup N)^\*\subseteq (\Sigma\cup N)^\*\times(\Sigma\cup N)^\*$, so this makes sense) and such that
$$a\rightarrow_Gb \wedge a'\rightarrow_Gb'\implies ab\rightarrow_Ga'b'.$$ The language generated by $G$ from a non-terminal $n\in N$ is just $L(G, n)=\{w\in \Sigma^\*:n\rightarrow_Gw\}$. This is, in fact, the standard way to do this...
Best Answer
Although it may seem on the face of it that this proposal is just a question of terminology — yes, a model of set theory is a certain kind of acyclic digraph — nevertheless, my opinion is that one can indeed get some insight by thinking this way.
In particular, the main results of my paper on the embedding phenomenon arose out of an explicitly graph-theoretic perspective on the models of set theory, viewing the models of set theory as certain special acyclic digraphs.
Joel David Hamkins, Every countable model of set theory embeds into its own constructible universe, J. Math. Log. 13 (2013), no. 2, 1350006, 27. blog post
For example, I proved that the countable models of set theory are linearly pre-ordered by embeddability: for any two such models, one of them is isomorphic to an induced subgraph of the other. Furthermore, embeddability is determined by the heights of the models, and from this it follows that there are precisely $\omega_1+1$ many bi-embeddability classes. So actually, the countable models of set theory are pre-well-ordered by embeddability! Every nonstandard model of set theory is universal for all countable acyclic digraphs.
The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading eventually to what I call the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, which is closely connected with the surreal numbers.
It happens that I am just now at the JMM in Seattle, where I will speak on The hypnagogic digraph, with applications to embeddings of the set-theoretic universe on Friday afternoon at the special session on the surreal numbers.
Finally, let me mention that it is an open question whether one can prove in ZFC that there is no graph-embedding of the set-theoretic universe $V$ to the constructible universe $L$, when $V\neq L$. In a joint project currently underway with many authors, however, we have made some significant progress, without yet settling the full question.