Subquotients in ZF In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surjection from $B$ to $A$, or $A=\emptyset$. In topos theory people consider the relation '$A$ is a subquotient of $B$' (usually in the non-boolean case): this says that $A$ is equipped with a surjection from a set $C$ which has an injection to $B$, or equivalently that $A$ is equipped with an injection to a set $D$ which has a surjection from $B$. This is a transitive and reflexive relation, and could be seen as a closure of the union of $\leq$ and $\leq^\ast$; indeed $A \leq B$ and $A \leq^* B$ both imply that $A$ is a subquotient of $B$.
It is intuitively clear that if $A$ is a subquotient of $B$ then it is 'smaller' than $B$. My question is whether this relation has been considered in (material) set theory, and even if not, is there much we can say about it? We can have large antichains of cardinals in ZF (by results of Jech and Jech-Sochor) in the relation $\leq$, but what about when we contemplate the relation 'is a subquotient of'?
 A: Assuming that the topos is boolean and well-pointed (which is the case in ZF) if $A$ is a nonempty subquotient of $B$ then $A$ is a quotient of $B$. In other words $A$ is a subquotient of $B$ iff $A = \varnothing$ or $A \leq^\ast B$. So the subquotient relation fixes the "defect" of $\leq^\ast$ that prevents $\varnothing$ from being the $\leq^\ast$-smallest set.
A: In $\sf ZF$ injections can be split, so if $A\leq B$ then we have $A\leq^\ast B$ as well. For this reason I prefer to use a slightly modified (but equivalent) definition for $\leq^\ast$:

$A\leq^\ast B$ if there is some $C\subseteq B$ such that there is a surjection from $C$ onto $A$.

This is dual to the definition of $A\leq B$ if there is a bijection between $A$ and a subset of $B$, and it saves the case of $\varnothing$ as an exception which ruins the nice definition.
As for antichains in $\leq^\ast$, it is not very hard to embed any partial order, and in fact the whole universe, into the cardinals with the $\leq^\ast$ preorder. This way we can prove $\sf WISC$ is independent from $\sf ZF$.
A: I am not so sure that subquotients are about size, at least not in general toposes. How does the "subquotient is smaller" intuition cope with the following examples?


*

*In the effective topos the real numbers are a subquotient of the natural numbers.

*In the realizability topos over infinite-time Turing machines the real numbers are a subobject of the natural numbers.


Because these toposes have number choice Cauchy and Dedekind reals coincide. In the first example we might appeal to the "connectedness of the continuum", i.e., we could say that size measures the connected components of an object. But in the second example the reals fall apart into Borelian dust (not quite Cantorian, though) so connectedness cannot save the intuition.
It may be safest to stick to classical logic, but then others have observed already that we are talking about quotients.
