Short version: what can we say about subsets of $\omega_2$ which - in a generic extension where $\omega_2$ is the new $\omega_1$ - contain a club?

*We could of course generalize beyond $\omega_2$, but already the questions seem hard.*

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**Motivating example**:

Even in much weaker theories than ZFC (although ZF itself is not enough to my knowledge - the obstacle being that $\omega_1$ might be singular!), the club filter on $\omega_2$ is not an ultrafilter. An easy proof of this is to consider the sets $$A=\{x\in\omega_2: cf(x)=\omega\},\quad B=\{y\in\omega_2: cf(y)=\omega_1\};$$ it is easy to see that, while the union of $A$ and $B$ is club, neither $A$ nor $B$ contains a club.

However, consider the forcing notion $\mathbb{P}$ consisting of partial maps $p: \omega_1\rightarrow\omega_2$ such that

 - $dom(p)=\lambda+1$ for some limit $\lambda<\omega_1$ (in particular, $dom(p)$ is countable and has a greatest element), 

 - $p$ is increasing: $\alpha<\beta\in dom(p)\implies p(\alpha)<p(\beta)$,

 - $p$ is *continuous*: if $\lambda$ is a limit ordinal in $dom(p)$, then $p(\lambda)=\sup\{p(\beta): \beta<\lambda\}$, and

 - $cf(p(\alpha))=\omega$ for all $\alpha\in dom(p)$.

Let $C$ be the range of the union of the conditions in some generic filter $G$. Then in $V[G]$, $C$ is a club subset of $\omega_2^V$, and consists entirely of ordinals of countable $V$-cofinality.

From the perspective of $\mathbb{P}$, then, $A$ is "more clubby" than $B$.

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This is the kind of situation I'm interested in: *When does a subset of $\omega_2$ contain a club in some forcing extension, and what can we say about the induced filter on $\omega_2$?*

Specifically, for $\mathbb{P}$ a set forcing, let $\mathcal{F}_\mathbb{P}$ be the filter on $\omega_2$ of subsets of $\omega_2$ which *in $V^\mathbb{P}$* contain a club. (Note that this makes sense even if $\omega_2^V$ is not a cardinal in $V^\mathbb{P}$ - however, it does trivialize, yielding the cofinite filter, if $cf(\omega_2^V)^{V^\mathbb{P}}=\omega$.)

Of particular interest are the forcings which preserve $\omega_2$ but collapse $\omega_1$, like $Col(\omega,\omega_1)$ (call such forcings *relevant*). Unfortunately, the associated filters are never ultrafilters, at least in ZFC: this is because they are always $\omega_2$-closed (since $\omega_1$ is made countable, and $V^\mathbb{P}\models$ the intersection of countably many clubs is countable).

That said, if we drop choice, then such filters *can* be ultrafilters! Let $V$ be a model of $ZF+V=L(\mathbb{R})$ + "The theory of $L(\mathbb{R})$ is absolute" - formally, this last condition is a scheme asserting that for every sentence $\varphi$ in the language of set theory with real parameters and every set forcing $\mathbb{P}$, $V\models\varphi\iff L(\mathbb{R})^{V[G]}\models\varphi$. (Informally, this should suggest that $V$ is the $L(\mathbb{R})$ of some choice model with a proper class of Woodins.)

Now consider the "labelled Levy collapse" $\mathbb{Q}=Col_{\mathbb{R}}(\omega,\omega_1)$ - a condition in this forcing is a finite sequence of reals coding well-orderings, and conditions are ordered by extension. Like the Levy collapse, this forcing makes $\omega_1$ countable, but - importantly - does so by adding a single real, $G$; in particular, the generic extension $V[G]$ satisfies "$V=L(\mathbb{R})$." 

This gives us lots of absoluteness between $V$ and $V[G]$. In particular, the sentence "The club filter on $\omega_1$ is an ultrafilter" is true in $V$, and hence in $V[G]$; and in particular, this means that in $V$ the filter $\mathcal{F}_{Col_\mathbb{R}(\omega, \omega_1)}$ is an ultrafilter on $\omega_2^V$. *Indeed, this filter witnesses that $\omega_2^V$ is measurable in $V$.*
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Hopefully the examples above motivate the following questions.

First, we can consider *specific forcings* in the ZFC context: 

> What is $\mathcal{F}_{Col(\omega,\omega_1)}$?

Second, we can consider *individual sets* in the ZFC context:

> Is there a forcing $\mathbb{P}$ such that $\mathcal{F}_\mathbb{P}$ contains $B$ (the set of $x\in\omega_2$ of uncountable $V$-cofinality)?

I suspect the answer is no, but I don't immediately see how to prove it.

Third, we may consider the natural ideal which this notion induces:

> Let $\mathcal{I}$ denote the set of subsets of $\omega_2$ which are not in any $\mathcal{F}_\mathbb{P}$, and $\mathcal{I}_{rel}$ the set of subsets of $\omega_2$ which are not in any $\mathcal{F}_\mathbb{P}$ with $\mathbb{P}$ relevant. What can we say about $\mathcal{I}$ and $\mathcal{I}_{rel}$?

I suspect $\mathcal{I}=\mathcal{I}_{rel}$ - that is, if we can force $x\subseteq\omega_2$ to contain a club, we can do so with a forcing which collapses $\omega_1$ and preserves $\omega_2$. In terms of understanding the ideals, one thing I'm interested in is the associated forcing notions: force with "positive" subsets of $\omega_2$ modulo "null" sets. 

Finally, we can ask determinacy-flavored questions. The one which seems most interesting to me is:

> Supposing ZF+AD+ whatever else, what are the measures on $\omega_2$ of the form $\mathcal{F}_{\mathbb{P}}$ for some (or some *relevant*) $\mathbb{P}$?

I suspect the answer is "all of them," but I don't see how to prove it.

EDIT: The first three questions were answered by Monroe Eskew below; the fourth, however, seems like a fundamentally different question. I've accepted Monroe's answer, and have asked the fourth question separately [here]().

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And, of course, the standard: what are some good sources on this sort of thing? My problem here is that I don't know how to google this effectively - too many unrelated hits (e.g. about $\omega_2$ satisfying generic versions of large cardinal properties) keep coming up. So I'm probably missing some well-known material.