TITLE: Zermelo-Frankel set theory for algebraists
QUESTION [6 upvotes]: $\DeclareMathOperator\Var{Var}\DeclareMathOperator\CRings{CRings}\DeclareMathOperator\Grp{Grp}\DeclareMathOperator\Sets{Sets}$I'm not a logician/set theorist, and I have some questions on set theory and references that may seem "trivial" for experts. Still I ask the question – if you have references this would be interesting.
In algebraic geometry (See Hartshorne's book, Appendix A) the following theorem is proved:
Let $\Var(k)$ be the "category of non-singular quasi-projective varieties over an algebraically closed field $k$ and morphisms of varieties over $k$. This category is defined in Hartshorne's book.
Theorem 1.1. There is a unique intersection theory $A^*(X)$ for algebraic cycles on $X\in \Var(k)$ modulo rational equivalence satisfying the axioms A1–A7.
The axioms A1–A7 are listed on page 426-427 in the book. For a variety $X\in \Var(k)$ one defines
a commutative unital ring $A^*(X)$ – the Chow ring – and this construction is unique. There is only one way to do this, meaning there is a unique functor
$A^*(-) : \Var(k) \rightarrow \CRings$
such that axioms A1–A7 hold. Here $\CRings$ is the category of commutative unital rings and maps of unital rings.
In algebra one defines a group $(G, \bullet)$ as a set $G$ with an operation $\bullet: G\times G \rightarrow G$ satisfying $3$ axioms: G1 Associativity, G2 existence of identity and G3 existence of inverse. One defines a morphism of groups and the "category of groups" $\Grp$. Clearly the category of groups $\Grp$ contains non-isomorphic groups, hence the axioms G1–G3 does not uniquely determine
one group. There are many different groups satisfying G1–G3.
In ZF set theory set theorists write down 9 axioms ZF1-ZF9, and these axioms determine $\Sets$ – the "category of sets". $\Sets$ is a category with "sets" as objects and "maps between sets" as morphisms. We would like the category $\Sets$ to be uniquely determined by the axioms ZF1–ZF9 similarly to what happens for the Chow ring. Is it? Is there a unique category $\Sets$ fulfilling the axioms ZF1–ZF9? If yes I ask for a reference.
For reference, Wikipedia has the page ZF set theory.

REPLY [3 votes]: You have to specify what you mean by a category 'fulfilling a list of axioms of set theory', because axioms of set theory are usually statements about sets in some logical language.
One reasonable interpretation would be to use the internal logic of a category to form an internal logical language, then formulate the axioms of set theory in the internal language and ask if they're true in the ambient category. Set can trivially internally formulate and satisfy all of the set-theoretic axioms of whatever background set-theoretic universe we're in, so in particular if we're working in $ZF$ then Set will satisfy ZF1-ZF9.
It turns out that other categories do this as well, so Set is not unique in this regard as other answers state. Specifically, Mike Shulman wrote a nice paper exploring how to extend the usual internal logic of a pretopos to a more general interpretation called the stack semantics, which allows us to formulate axioms involving unbounded quantifiers. Using the stack semantics of a topos, we can formulate all the axioms of $ZF$ internally and ask if they're true in the ambient topos.
Collection and replacement are satisfied in the stack semantics of any topos, but the separation axiom requires the introduction of a new topos-theoretic axiom schema -- any topos satisfying this additional axiom schema is called an autological topos, and these are precisely the categories which internally have the full strength of $ZF$ set theory.
So it seems like the answer to your question is:

No, Set is not the only category satisfying ZF1-ZF9 in the above sense, and the (bi)category of all categories satisfying ZF1-ZF9 would be precisely the (bi)category of autological toposes.

If you are looking for a unique characterization of Set, it is the terminal object in the category of Grothendieck toposes and geometric morphisms.
As mentioned in Andrej Bauer's answer it is also the initial 'ZF-algebra'. For a summary on this view with definitions you can take a look at these notes on Algebraic Set Theory by Steve Awodey, or take a look at the classical reference on Algebraic Set Theory by Joyal and Moerdijk for a full explanation.
Linked paper: arXiv:1004.3802 [math.CT]

REPLY [2 votes]: There is such a thing as Algebraic Set Theory where:

models of set theory are simply algebras for a suitably presented algebraic theory and then many familiar set theoretic conditions (such as well-foundedness) are thereby related to familiar algebrauc ones (such as freeness) ... it was developed by Joyal & Moerdjik in 1988 and was first presented in detail in a book published in 1995 by them.