I've been reading some of the more philosophical papers by John Lane Bell and Steve Awodey and I'd like some conceptual clarification. I know very little model theory so apologies if this is all too vague. I'd like to get a handle on the significance of cardinal collapse between toposes.
JLB says that topos theory is "pluralistic" because
given an uncountable set $I$ we can produce ... a Boolean extension of the universe of sets in which $I$ is countable. This means that cardinality ... is not an absolute or intrinsic feature of the set but is determined only in relation to the mathematical framework wrt which that cardinality is "measured".
- Geoffrey Hellman and John L. Bell, "Pluralism and the Foundations of Mathematics"
1) It's not stated in the paper but I'm assuming that the extension is of the Von Neumann universe and that this means interpreting it in a Boolean algebra. This strikes me as fulfilling the role of a truth-value object/subobject classifier in category theory - does that sound about right?
2) To make sense of this passage I assume that the cardinality of $I$ is different in two different toposes. What are the toposes here? Is one of them the class of Boolean-valued functions on $V$, i.e. does that form a topos? And the first is just $V$? If these are toposes then I guess they must both be sub-toposes of Set?
3) I'm guessing there has to be something functorial going on here but in Bell's book "Boolean Valued Models and Independence Proofs" the only explicit mention of category theory is in the appendix. Is there some specific kind of functor involved in cardinal collapse?
4) Does this phenomenon of cardinal collapse also take place in exotic places like Homotopy Type Theory? I understand HoTT as taking place in an $(\infty, 1)$-topos. Do we see something analogous, I guess like a logical or geometric $(\infty, 1)$-functor which fails to preserve cardinality?
I recognise that this is a pretty broad question so even some pointers toward relevant literature would be much appreciated.