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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'?

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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 $\sf ZF$.

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  • $\begingroup$ I was thinking of your paper, but was lazy and didn't check what you said past that Jech had proved the case for $\leq$. $\endgroup$
    – David Roberts ♦
    Commented May 23, 2013 at 4:51
  • $\begingroup$ David, Jech indeed proved for $\leq$, and Roguski extended this to classes of incomparable sets (from which one could actually embed the ground model into the cardinals of the extension), as you have told me! :-) $\endgroup$
    – Asaf Karagila ♦
    Commented May 23, 2013 at 4:55
  • $\begingroup$ :-O so I did! Been thinking about other ways to prove these things, so I've forgotten some of the material set theory stuff. $\endgroup$
    – David Roberts ♦
    Commented May 23, 2013 at 4:58
  • $\begingroup$ <corny joke>Then perhaps it's time to revise that material...</instant rimshot> $\endgroup$
    – Asaf Karagila ♦
    Commented May 23, 2013 at 5:00
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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.

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  • $\begingroup$ Ah, how silly of me. Of course you can map the complement of the image of $C$ in $B$ to some arbitrary point in $A$. In that case, perhaps I should be asking about the behaviour of $\leq^\ast$. $\endgroup$
    – David Roberts ♦
    Commented May 23, 2013 at 3:30
  • $\begingroup$ (Of course, the boolean assumption is redundant since a well-pointed topos has a two-valued logic.) $\endgroup$
    – François G. Dorais
    Commented May 23, 2013 at 3:31
  • $\begingroup$ Your answer is more general François as regarding what $\leq^\ast$ is, in that it deals with well-pointed toposes, but Asaf got the tick for reminding me of the result about the possible structure of $\leq^\ast$ in models of ZF. $\endgroup$
    – David Roberts ♦
    Commented May 23, 2013 at 7:41
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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?

  1. In the effective topos the real numbers are a subquotient of the natural numbers.
  2. 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.

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  • $\begingroup$ Perhaps it's a symptom of my thinking in the well-pointed and classical case. $\endgroup$
    – David Roberts ♦
    Commented May 23, 2013 at 4:53
  • $\begingroup$ My personal opinion is that any general notion of size is more or less trivial. $\endgroup$
    – Andrej Bauer
    Commented May 23, 2013 at 7:38

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