Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$ determined by $\mathbf{Set}_M$ as a category up to equivalence?
For example, suppose $M$ and $N$ are models of ZF. Then are $\mathbf{Set}_M$ and $\mathbf{Set}_N$ equivalent as categories if and only if $M$ and $N$ are isomorphic?
I expect the answer will depend on exactly what we assume about $M$.
For instance, let $M$ be a model of ZFA and let $M'$ be the universe of pure sets in $M$. Then $M \cong M'$ if and only if $M$ has no atoms; but the inclusion $\mathbf{Set}_{M'} \hookrightarrow \mathbf{Set}_M$ is an equivalence as soon as $M$ satisfies the axiom "each set is in bijection with some pure set", which happens if e.g. $M$ satisfies the axiom of choice.
On the other hand, suppose $M$ is a transitive model of ZF. By transitive closure / Mostowski collapse, every set in $M$ is obtained from a "ZF-tree" in $M$, i.e. a set $T$ (in $M$) equipped with a well-founded extensional binary relation $E$ and a unique $E$-maximal element. The notion of ZF-tree is one that can be formulated in the internal language of a topos, so the collection of ZF-trees is recoverable from $\mathbf{Set}_M$ up to equivalence, and hence, $M$ is (exactly!) recoverable from $\mathbf{Set}_M$ up to equivalence.
Following Benedikt Löwe, a somewhat more sophisticated version of the above should work to recover well-founded models $M$ of ZFA with ($M$-)countably many atoms from $\mathbf{Set}_M$.
But what about, say:
- Non-well-founded models of ZF(A)?
- Weaker fragments of ZF, e.g. Mac Lane set theory?
- Set theories where the category of sets is not a topos, e.g. NBG or NF(U)?
To keep the question from being too open-ended, let me say that I would be happy to know the answer just for (possibly non-well-founded) models of ZF.