This question fits the Generalised Baire space area. I am interested in the meagre ideal on ${}^\kappa \kappa$, with the bounded topology (or box topology), when, say, $\kappa$ is inaccessible.

To be more precise, basic open sets have the form $[s]=\{t\in {}^\kappa \kappa \mid t\supseteq s\}$, where $s\in {}^{<\kappa}\kappa$. A set $X$ is nowhere dense if every open set has an open subset that does not meet $X$. Finally, $X$ is meagre if it is the $\kappa$-union of meagre sets.

Are there examples or any criterion for a forcing $\mathbb{P}$ to satisfy the following property: "every meagre set in the generic extension is a subset of a meagre set in the sense of the ground model".