Characterization of Cohen reals

Question

The following is a well-know fact:

Theorem The real $r$ is Cohen over $V$ iff if it does not belong to any meager Borel set coded in $V$.

Now suppose that $\kappa$ is an uncountable cardinal and let $(r_i: i<\kappa)$ be a sequence of reals.

Question. Is there a characterization theorem as above for the sequence $(r_i: i<\kappa)$ to be $Add(\omega, \kappa)$-generic over $V$, where $Add(\omega, \kappa)$ is the Cohen forcing for adding $\kappa$-many Cohen reals.

If there is a known characterization, giving a reference is welcome.

Remark. I know there are characterizations of Cohen algebra, for example the one given in the paper Characterizations of Cohen algebras, but I'm interested in a characterization parallel to the one given in the above theorem.

Remark 2. The reason I'm asking the question is the following: In the paper Adding many random reals may add many Cohen reals I showed that forcing with $R(\kappa) \times R(\kappa)$ adds a generic filter for $Add(\omega, \kappa)$ (where $R(\kappa)$ is the usual forcing for adding $\kappa$-many random reals), which generalizes the well-known fact that forcing with $R \times R$ adds a Cohen real (where $R$ is the random forcing). The usual proof of the above known result uses characterization of Cohen reals given in the above theorem, while I presented a direct proof without using any characterization result. If there is a characterization as asked above, there might be a different proof parallel to the known one.

Answer 1

Chapter 20 of the Handbook of Set Theoretic Topology ("Random and Cohen reals" by Ken Kunen, pp 887-911) deals with such questions.

Quoting from Truss's review:

"Quite a proportion of the paper is devoted to a study of the properties of Cohen extensions of a countable transitive model by $2^I/\mathcal{I}$, where I is an arbitrary index set and $\mathcal{I}$ is the natural "lifting'' to $\mathcal{P}(I)$ of one of the ideals under consideration."

More specifically, Lemma 3.8 on page 903 gives a general result akin to what Will Brian suggested in the comments above.

3.8 Lemma

Let $\mathcal{I}$ be a reasonable ideal. Let $M$ be a countable transitive model for ZFC with $I\in M$. Let $F\in 2^I$. Then $F$ is $\mathcal{I}$-generic over $M$ if and only if $F$ is not in any $M$-coded Baire set in $\mathcal{I}$.

Recall that the Baire sets are defined as belonging to the $\sigma$-algebra generated by the clopen sets. The paper is written from an abstract point of view so there's a lot of notation, but the intent is to lay out what Cohen and Random forcing have in common, and why they can be lifted to large index sets.