Reverse of a termspace forcing fact

Question

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable partial functions from $\kappa$ to 2, ordered by $\tau \leq \sigma$ iff $1 \Vdash \tau \leq \sigma$. Call this $T(\mathbb P, Add(\kappa))$.

It is well known that the identity map is a projection from $\mathbb P \times T(\mathbb P, Add(\kappa))$ to the iteration $\mathbb P * \dot Add(\kappa)$. Furthermore in this situation it is not hard to show that $T(\mathbb P, Add(\kappa)) \cong Add(\kappa)$. So if $G \times H$ is $\mathbb P \times Add(\kappa)$-generic, then in the extension there is $I$ such that $G * \dot I$ is $\mathbb P * \dot Add(\kappa)$-generic.

My question is what about the opposite. In $V[G* \dot I]$ is there an $Add(\kappa)^V$-generic?

Answer 1

The answer is no. Forcing with $\mathbb{P}*\dot{\text{Add}}(\kappa,1)$ adds no fresh subsets to $\kappa$, that is, a new subset of $\kappa$ all of whose initial segments are in $V$. Thus, it cannot add a $V$-generic filter for $\text{Add}(\kappa,1)^V$.

To see this, note that $\mathbb{P}$ is productively $\kappa$-c.c., meaning that $\mathbb{P}\times\mathbb{P}$ is $\kappa$-c.c., since $\mathbb{P}$ remains $\kappa$-c.c. after forcing with $\mathbb{P}$, since in the extension it amounts to $\text{Add}(\omega,\kappa)$, which is c.c.c. there.

It now follows from Spencer Unger's improved version of my lemma on the approximation and cover properties, which go back to results implicit in Mitchell's dissertation. Specifically, it follows from lemma 1.3 in his paper Spencer Unger, FRAGILITY AND INDESTRUCTIBILITY II that $\mathbb{P}*\dot{\text{Add}}(\kappa,1)$ has the $\kappa$-approximation property, meaning this forcing adds no new sets of ordinals all of whose small approximations are in the ground model. Thus, it can add no $V$-generic subsets of $\kappa$ using $\text{Add}(\kappa,1)^V$.