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My recent paper with Dan Seabold,

has various sections describing the basic algebraic connections between the Boolean ultrapower and the corresponding Boolean forcing extensions and how they relate to subalgebras, ideals, quotients, products limits and so on, and this may be a part of the solution you seek.

Meanwhile, to my way of thinking, the most important thing to say about the Galois connection aspect of forcing extensions is the following.

Theorem. Suppose that $V\subset V[G]$ is a forcing extension arising by forcing with a complete Boolean algebra $\mathbb{B}$, where $G\subset\mathbb{B}$ is $V$-generic. Then every intermediate model $M$ of ZFC with $V\subset M\subset V[G]$, is itself a forcing extension of $V$, and specifically $M=V[G_0]$ where $G=G\cap\mathbb{B}_0$ for some complete subalgebra $\mathbb{B}_0\subset\mathbb{B}$. Furthermore, $V[G]$ is also a forcing extension of $M$ by the quotient forcing $\mathbb{B}/G_0$. Conversely, every complete subalgebra $\mathbb{B}_0\subset\mathbb{B}$ gives rise to such an intermediate forcing extension $V[G_0]$, where $G_0=G\cap\mathbb{B}_0$.

Thus, there is a connection between the intermediate models of ZFC in a forcing extension and the complete subalgebras of the original Boolean algebra. The theorem is definitely not obvious, but is proved in any of the standard accounts of forcing, such as Jech's book Set Theory.

So the theorem provides a correspondence between intermediate models of a given forcing extension $V\subset M\subset V[G]$ and complete subalgebras $\mathbb{B}_0\subset \mathbb{B}$ of the complete Boolean algebra giving rise to the original extension, and the two intermediate extensions $V\subset M$ and $M\subset V[G]$ arising as forcing extensions by $\mathbb{B}_0$ and by the quotient $\mathbb{B}/G_0$, respectively.

But the theorem does not provide a Galois correspondence in terms of automorphism groups and fixed points of automorphisms. In part, this is because transitive sets simply have no nontrivial automorphisms. There are no nontrivial forcing extensions automorphisms of $V[G]$, since it is a transitive class, and so one wouldn't ordinarily want to consider the group of automorphisms of $V[G]$. Two transitive classes are isomorphic if and only if they are identical, and so this that approach also is not fruitful in the context of forcing extensions of $V$. But you ask about automorphisms of the $\mathbb{B}$-valued structure, and here of course there will be a rich automorphism theory.

show/hide this revision's text 3 added 425 characters in body

My recent paper with Dan Seabold,

has various sections describing the basic algebraic connections between the Boolean ultrapower and the corresponding Boolean forcing extensions and how they relate to subalgebras, ideals, quotients, products limits and so on, and this may be a part of the solution you seek.

Meanwhile, to my way of thinking, the most important thing to say about the Galois connection aspect of forcing extensions is the following.

Theorem. Suppose that $V\subset V[G]$ is a forcing extension arising by forcing with a complete Boolean algebra $\mathbb{B}$, where $G\subset\mathbb{B}$ is $V$-generic. Then every intermediate model $M$ of ZFC with $V\subset M\subset V[G]$, is itself a forcing extension of $V$, and specifically $M=V[G_0]$ where $G=G\cap\mathbb{B}_0$ for some complete subalgebra $\mathbb{B}_0\subset\mathbb{B}$. Furthermore, $V[G]$ is also a forcing extension of $M$ by the quotient forcing $\mathbb{B}/G_0$. Conversely, every complete subalgebra $\mathbb{B}_0\subset\mathbb{B}$ gives rise to such an intermediate forcing extension $V[G_0]$, where $G_0=G\cap\mathbb{B}_0$.

Thus, there is a connection between the intermediate models of ZFC in a forcing extension and the complete subalgebras of the original Boolean algebra. The theorem is definitely not obvious, but is proved in any of the standard accounts of forcing, such as Jech's book Set Theory.

So the theorem provides a correspondence between intermediate models of a given forcing extension $V\subset M\subset V[G]$ and complete subalgebras $\mathbb{B}_0\subset \mathbb{B}$ of the complete Boolean algebra giving rise to the original extension, and the two intermediate extensions $V\subset M$ and $M\subset V[G]$ arising as forcing extensions by $\mathbb{B}_0$ and by the quotient $\mathbb{B}/G_0$, respectively.

But the theorem does not provide a Galois correspondence in terms of automorphism groups and fixed points of automorphisms. In part, this is because transitive sets simply have no nontrivial automorphisms. There are no nontrivial forcing extensions of $V[G]$, since it is a transitive class, and so one wouldn't ordinarily want to consider the group of automorphisms of $V[G]$. Two transitive classes are isomorphic if and only if they are identical, so this approach also is not fruitful in the context of forcing extensions of $V$. But you ask about automorphisms of the $\mathbb{B}$-valued structure, and here of course there will be a rich automorphism theory.

show/hide this revision's text 2 added 199 characters in body

My recent paper with Dan Seabold,

has several various sections describing the basic algebraic connections between the Boolean ultrapower and the corresponding Boolean forcing extensions and how they relate to subalgebras, ideals, quotients, products limits and so on, and this may be a part of the solution you seek.

Meanwhile, to my way of thinking, the most important thing to say about the Galois connection aspects aspect of forcing extensions is the following.

Theorem. Suppose that $V\subset V[G]$ is a forcing extension arising by forcing with a complete Boolean algebra $\mathbb{B}$, where $G\subset\mathbb{B}$ is $V$-generic. Then every intermediate model $M$ of ZFC with $V\subset M\subset V[G]$, is itself a forcing extension of $V$, and specifically $M=V[G_0]$ where $G=G\cap\mathbb{B}_0$ for some complete subalgebra $\mathbb{B}_0\subset\mathbb{B}$. Furthermore, $V[G]$ is also a forcing extension of $M$ by the quotient forcing $\mathbb{B}/G_0$. Conversely, every complete subalgebra $\mathbb{B}_0\subset\mathbb{B}$ gives rise to such an intermediate forcing extension $V[G_0]$, where $G_0=G\cap\mathbb{B}_0$.

Thus, there is a connection between the intermediate models of ZFC in a forcing extension and the complete subalgebras of the original Boolean algebra. The theorem is definitely not obvious, but is proved in any of the standard accounts of forcing, such as Jech's book Set Theory.

But the theorem does not provide a Galois correspondence in terms of automorphism groups and fixed points of automorphisms. In part, this is because transitive sets simply have no nontrivial automorphisms. There are no nontrivial forcing extensions of $V[G]$, since it is a transitive class, and so one wouldn't ordinarily want to consider the group of automorphisms of $V[G]$. Two transitive classes are isomorphic if and only if they are identical, so this approach also is not fruitful in the context of forcing extensions of $V$. But you ask about automorphisms of the $\mathbb{B}$-valued structure, and here of course there will be a rich automorphism theory.

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