TITLE: Characterization of intermediate submodels of generic extensions
QUESTION [6 upvotes]: Question 1: Suppose that $V[G]$ is a set generic extension of $V$ by some forcing notion $P\in V,$ and suppose that $W$ is a model of $ZFC, V \subset W \subset V[G].$ Can we find a forcing notion $Q\in V$ and a map $\pi: P\to Q$ such that:
1) $|Q|\leq |P|,$
2) $\pi$ and $G$ generate a filter $H$ which is $Q-$generic over $V$ (for example if $\pi$ is a projection, then we can take $H$ to be the filter generated by $\pi''[G]$),
3) $W=V[H].$
Remark. Let $B=r.o(P).$ Then for some $\bar{G}, B-$generic over $V$, we have $V[G]=V[\bar{G}].$ Also for some complete subalgebra $C$ of $B, W=V[\bar{G}\cap C].$ If $P$ satisfies the $\kappa-c.c.$, then $|C|\leq |B| \leq |P|^{<\kappa},$ so clearly the answer is yes if $|P|=|P|^{<\kappa}.$ 
Question 2: Suppose that $P, Q\in V$ are two forcing notions such that for any $G$ which is $P-$generic over $V$, there is $H\in V[G]$ which is $Q-$generic over $V$. Is there a map $\pi: P\to Q$ such that for any $G$ as above, we can choose $H$ to be the filter generated by $\pi[G]$?

REPLY [6 votes]: The answer to your first question is yes.  Let $C \subseteq B$ be as in your remark.  Let $\pi : B \to C$ be the standard projection map, $\pi(b) = \inf \{ c \in C : c \geq b \}$.  The restriction of $\pi$ to $P$ maps a dense subset of $B$ onto a dense subset $Q \subseteq C$, it is a projection, and $|Q| \leq |P|$.