I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's [algebraic set theory](http://www.andrew.cmu.edu/user/awodey/preprints/astIntroFinal.pdf). Both of these perspectives takes a set-theory universe as an object, and consider collections of set-theory universes. The former lives in the setting of first-order logic, while the later lives in the setting of category theory (specifically monads and algebras).


Has there been an attempt to compare these theories? In particular (I hope the following is a well-founded question), does Hamkins' Multiverse axiomatize (the set-theory part, i.e. ignoring the intuitionistic and topos theoretic part) the class category of ZF-algebras?


I'm not too familiar with both these theories, but I have a great interest in Foundations. I would be thankful for more detailed comparisons beyond the particular question that I ask.



Does the class category of ZF-algebras satisfy the Multiverse axioms?
I think the two theories should be regarderd as existing at different levels. A category of classes, in the sense of algebraic set theory, is not a collection of models of set theory, but (an abstraction of) the collection of all classes relative to *one* model of set theory. In particular, if $V$ is a model of set theory, then the collection of all classes in V is a category of classes.


One can then, if one wants, define a notion of "internal model of set theory" inside a category of classes, and in the "canonical" example this would just be the notion of a class-model of set theory relative to $V$. The model $V$ itself is recoverable as the "initial" such internal model. Joel's excellent answer explains why such a collection of "internal models" will not generally satisfy the multiverse axioms.


