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 projection $\pi: P\to Q$?

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 $\pi[G]$?

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 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 projection $\pi: P\to Q$?