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'?



Subquotients in ZF
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 $\sf ZF$.


