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This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein:

start from a ZF transitive model $M$, and consider the category $CBA(M)$ of complete boolean algebras of $M$ with the obvious boolean maps.

This category corresponds to the category of boolean valued boolean-valued extensions of $M$, via the Scott-Solovay-Vopenka (SSV) construction. $CBA(M)$ contains the initial algebra $2$ (which corresponds to the trivial extension), and the forcing extensions of $M$ are the maps $B\rightarrow 2$ (ie the ultrafilters; notice that these maps in general do not exist in $M$).

Now the questions:

1) We can also consider maps $B_1\rightarrow B_2$ (the $B_2$ "points"of $B_1$). What do they represent when one goes to the boolean valued models? I would think they stand for "relative extensions", but can one make this precise?

2) Similarly, one can construct, given two such algebras $B_1$, $B_2$ where $B_1$ is a subalgebra of $B_2$, the Galois group $Gal(B_2/B_1)$, of boolean automorphisms of $B_2$ leaving fixed the ground algebra $B_1$. Can one use Gal to classify relative extensions?

3) (change of base model) Let $M_1$ and $M_2$ two transitive models (ie two objects of the ZF-multiverse, here to be thought of as the large category of transitive models of ZF), and a map of the first to the second, it looks like that this induces a functor between $CBA(M_1 )$ and $CBA(M_2 )$ (something like akin to the change of base of bundles)bundles in geometry). Is this (alleged) functor strong enough to capture the isomorphisms of the underlying multiverse?

More generally, what is the relation of the 2-valued multiverse to the extended multiverse of boolean valued models?

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# The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein:

start from a ZF transitive model $M$, and consider the category $CBA(M)$ of complete boolean algebras of $M$ with the obvious boolean maps.

This category corresponds to the category of boolean valued extensions of $M$, via the Scott-Solovay-Vopenka (SSV) construction. $CBA(M)$ contains the initial algebra $2$ (which corresponds to the trivial extension), and the forcing extensions of $M$ are the maps $B\rightarrow 2$ (ie the ultrafilters; notice that these maps in general do not exist in $M$).

Now the questions:

1) We can also consider maps $B_1\rightarrow B_2$ (the $B_2$ "points"of $B_1$). What do they represent when one goes to the boolean valued models? I would think they stand for "relative extensions", but can one make this precise?

2) Similarly, one can construct, given two such algebras $B_1$, $B_2$ where $B_1$ is a subalgebra of $B_2$, the Galois group $Gal(B_2/B_1)$, of boolean automorphisms of $B_2$ leaving fixed the ground algebra $B_1$. Can one use Gal to classify relative extensions?

3) (change of base model) Let $M_1$ and $M_2$ two transitive models (ie two objects of the ZF-multiverse, here to be thought of as the large category of transitive models of ZF), and a map of the first to the second, it looks like that this induces a functor between $CBA(M_1 )$ and $CBA(M_2 )$ (something like the change of base of bundles). Is this (alleged) functor strong enough to capture the isomorphisms of the underlying multiverse? More generally, what is the relation of the 2-valued multiverse to the extended multiverse of boolean valued models?