TITLE: A rigid type of structure that can be put on every set?
QUESTION [16 upvotes]: Call a type of structure rigid if any automorphism of such a structure is an identity.  (This is a bit different from some other uses of the word, but hopefully I'll be forgiven.)  For example, well-orderings are rigid.  It follows that assuming the axiom of choice, there is a rigid type of structure (namely, a well-ordering) such that any set can be equipped with that type of structure (in a non-unique way, of course).
Now the axiom of choice isn't necessary for that conclusion.  Extensional well-founded relations are also rigid, and the axiom of foundation implies that any set injects into an extensional well-founded relation (its transitive closure).  Aczel's axiom of anti-foundation also suffices, since strongly extensional relations are also rigid, and anti-foundation implies that any set injects into such a relation.  But with neither foundation nor anti-foundation, the membership relation $\in$ needn't be rigid.  For instance, the set {a,b}, where a={a} and b={b} are unequal ill-founded sets with the same membership tree, has a nonidentity $\in$-automorphism which swaps a and b.
Now my question: If we don't assume choice or any sort of foundation, does there still exist a rigid type of structure with the property that any set can be equipped with that type of structure?
Edit: Of course, as Steven points out in the comments, I haven't said exactly what I mean by "type of structure."  I'm using the word "structure" in the Bourbaki-sense, not in the sense of stuff, structure, property.  Here's one way to make this question precise: does there exist a theory in higher-order logic which is rigid, in the sense that any automorphism of one of its models is the identity, and which admits a definable functor to Set which is essentially surjective?

REPLY [3 votes]: This doesn't completely answer the question, but I think it's relevant.
Theorem: Let $T$ and $W$ be theories in higher-order logic (by which I mean the internal type theory associated to elementary topoi), where $W$ has a specified underlying object $X$ (i.e. we regard a model of $W$ as an object $X$ equipped with some additional stuff).  Then it is not the case that both (1) $T$ proves that $W$ is rigid (as structure on $X$) and (2) in any topos satisfying $T$, every object can be equipped with W-structure.
Proof: Suppose that both the given conditions hold.  Let $C[T,X]$ be the free topos on the theory $T$ and an additional object $X$, and let C[T,W] be the free topos on $T$ and $W$.  (Recall that for any theory $S$ and any topos $E$, the category $\operatorname{Log}(C[S],E)$ of logical functors and natural isomorphisms from $C[S]$ to $E$ is equivalent to the category of models  of $S$ and their isomorphisms in $E$.)  Now the generic model of $W$ in $C[T,W]$ has an underlying object, giving a logical functor $C[T,X]\to C[T,W]$.  And $C[T,X]$ satisfies $T$, so by assumption, its generic object $X$ can be equipped with $W$-structure; thus we also have a logical functor $C[T,W]\to C[T,X]$, and the composite $C[T,X]\to C[T,W]\to C[T,X]$ is isomorphic to the identity.  In other words, $C[T,X]$ is a pseudo-retract of $C[T,W]$.  Now let $E$ be any other topos satisfying $T$ and $Y$ any object of $E$ admitting a nonidentity automorphism (such as $Y=1+1$); then we have a logical functor $C[T,X]\to E$ sending $X$ to $Y$, which in turn has a nonidentity automorphism.  But because $C[T,X]\to C[T,W]\to C[T,X]$ is the identity, this functor $C[T,X]\to E$ must factor through $C[T,W]$, and since $T$ proves that $W$ is rigid, any automorphism of this functor must be the identity, a contradiction. ∎
This means that if you want a rigid type of structure that can be put on every set, you need to use more about sets than the fact that they form an elementary topos, or even a Boolean topos with a NNO, or any additional property of $\mathsf{Set}$ that can be expressed as a theory $T$ in HOL.  Note that AC, although it can be phrased as a "categorical" property not referring directly to $\in$ (every surjection splits), cannot be expressed as a theory in HOL since it involves a quantification over all sets.  Likewise for the "topos-theoretic axiom of foundation" that every set injects into some well-founded extensional relation.  So the question is, what can we do with the remaining non-HOL axioms of ZF, i.e. basically replacement and (its consequence) unbounded separation.