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](http://onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1993.tb52525.x/abstract;jsessionid=086B060DF38C2FCD265C4E5B54282196.f02t03), 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](https://arxiv.org/abs/1701.04156) 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.



Characterization of Cohen reals
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. 


